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THE SYSTEM ORTOCARTAN USER'S MANUAL

1. where the lt n1 gt lt n2 gt lt n3 gt are any numbers Note just like in the program Ortocartan each of these numbers should be put in parentheses If you wish the program to stop working before it has calculated all the coefficients write the following item in the data stop after eguation lt n gt where lt n gt is the number of the last eguation to be printed Note again here the eguation lt n gt must be put in parentheses and the number lt n gt must be in its own paren theses This may be confusing but has a justification in the algorithm of the program Since the justification would be highly technical we shall skip it You can also write 48 stop after maineg and then the program will not print any of the coefficients In practice you will solve the eguations one by one and substitute the solutions in the remaining eguations until all of them are satisfied When this finally happens the program does not print either the maineq or the equations but just prints the following message THE FIRST INTEGRAL IS ALREADY MAXIMALLY SIMPLIFIED AND IS EXPLICITLY CONSTANT The substitutions are carried out when you write more eguations in the substitutions list shown above You can direct the additional substitutions either to maineq or to any equation Directing the additional substitutions to maineq is in most cases not reasonable This is often a long and complicated expression and t
2. calculate The second item of the data is again a title the last line must contain the two right parentheses Section 11 The number of coordinates i e independent variables in differentiation is arbitrary here Coordinates can be omitted If no differentiations are going to be done then all the symbols used can be called constants Section 14 The heart of the data are here the expressions to be simplified They are written as operation lt an arbitrary expression written according to the rules of section 5 gt operation lt another arbitrary expression gt operation lt another arbitrary expression gt 99 The number of operations is arbitrary It is understood that the program will be used to calculate complicated derivatives or to multiply large polynomials or to carry out series of substitutions in given expressions Section 16 Does not apply here at all Section 18 The substitutions should be directed here to result index where the index is the subscript that appears in print If no address is specified then they will be carried out everywhere i e in each result in the present call to calculate All the rules explained in sec 18 apply here without any modification Section 20 Does not apply here at all C The programs ellisevol curvature landlagr eulagr and squint C 1 The program Ellisevol This program calculates all the quantities appearing in the
3. 2 f weyl 1 6 r exp 2 mu 1 6 r exp 2 mu nu 1 6 0202 r 1 2 r exp 2 mu mu 1 6 exp 2 nu mu 1 6 exp 2 nu r t 2 mu 1 6 exp 2 nu nu mu 1 6 exp 2 mu nu GG t T r 2 1 6 exp 2 mu nu 1 6 exp 2 mu nu mu 1 6 r Dr r T 2 1 weyl 1 6 r exp 2 mu 1 6 r exp 2 mu nu 1 6 0303 r 1 2 r exp 2 mu mu 1 6 exp 2 nu mu 1 6 exp 2 nu r t 62 mu 1 6 exp 2 nu nu mu 1 6 exp 2 mu nu tt t t r 2 1 6 exp 2 mu nu 1 6 exp 2 mu nir mu 1 6 r rr r r 2 1 weyl 1 6 r exp 2 mu 1 6 r exp 2 mh nu 1 1212 r 1 2 6 r exp 2 mu mu 1 6 exp 2 nu mu 1 6 exp 2 r t nu mu 1 6 exp 2 nu nu mu 1 6 exp 2 mu nu tt t t r 2 1 6 exp 2mu nu 1 6 exp 2 mu nu mu 1 6 rr r r 2 r 2 1 weyl 1 6 r exp 2 mu 1 6 r exp 2 mu nu 1 1313 r 1 2 6 r exp 2 mu mu 1 6 exp 2 nu mu 1 6 exp 2 r t nu mu 1 6 exp 2 nu nu mu 1 6 exp 2mu nu tt t t r 2 1 6 exp 2 mu nu 1 6 exp 2 mu nu mu 1 6 rr r r 63 gt r 2 1 gt weyl 1 3 r exp 2 mu 1 3 r exp 2 mu nu 1 3 2323 r 1 2 gt r exp 2 mu mu 1 3 exp 2 nu mu 1 3 exp 2 nu r t 2 gt mu 1 3 exp 2 nu nu mu 1 3 exp 2 mu nu tt t t r 2
4. der t V V der t R R riemann der t R R der tV V D F riemann der z V 2 riemann der t V V riemann V lt 1 14 k x X0 2 y YO 7 2 z ZO 2 riemann K lt LLC OR V 1 k der x V 2 Ader y V 2 D F der t R R rmargin 61 dont print messages agamma setq lower t 68 Notes This is the most general conformally flat nonstatic perfect fluid solution of Einstein s equations see Kramer et al Exact solutions of Einstein s field equations Cambridge Uni versity Press 1980 p 371 theorem 32 15 The matter density in it denoted 3C t here is a function of t only and is related to k F and Rby kc 1 F R The Einstein tensor is calculated here in addition to the other quantities and the messages about unsuccessful attempts at substitutions are suppressed Also printing the antisym metrized Ricci rotation coefficients agamma is suppressed In this example instead of substituting the explicit forms of the components of the metric tensor it is more reason able to feed the information about them into the formulae step by step Each substitution results in a partial simplification of the formulae The fourth substitution stems from the identity Va Vy Vi k V 1 The second substitution makes use of the marker m it results in replacing D D and D by the appropriate derivatives of F V V R R The first t
5. 2323 riemann calculated TIME 690 msec riemann completed TIME 740 msec 1 gt ricci 2 r exp 2 mu nu 00 r gt exp 2 mp nu mu t t t t gt nu exp 2 mu mu mu Tr E 1 gt ricci 2 r exp nu mu m 0 1 1 gt ricci 2r exp 2 mu mu 11 r gt exp 2 nu mu mu t t t t 60 2 exp 2 nu mu exp 2 nu mu t 2 exp 2 mu nu exp 2 mu r r t 2 exp 2 nu mu exp 2 nu mu t 2 exp 2 mu mu exp 2 mu r gt nu exp 2 mu nu mu YE r r 2 1 zd gt ricci r exp 2 mu r exp 2 mu n r exp 2 mu 22 r 2 gt mu r r 2 1 1 gt ricci r exp ie 2 mu r exp 2 mu n r exp 2 mu 33 r 2 gt mu r ricci calculated TIME 820 msec CURVATURE INVARIANT calculated TIME 850 msec 2 1 1 gt CURVATURE INVARIANT 2 r exp 2 mu 4r exp 2 m nu 4r r 2 gt exp 2 mu mu 2 exp 2 nu mu 2 exp 2 nu m r t tt 2 gt 2 exp 2 nu mu mu 2 exp 2 mu nu 2 exp 2 mu nu t t r 2 gt 2 exp 2 mu nw mu 2r rr r r 61 2 1 weyl 1 3 r exp 2 mu 1 3 r exp 2 m nu 1 0101 E 1 2 3 r exp 2 mu mu 1 3 exp 2 nu mu 1 3 exp 2 r t nu mu 1 3 exp 2 nu mu mu 1 3 exp 2 mu nu tt t t r 2 1 3 exp 2 mu nu 1 3 exp 2 mu nu mu 1 3 rr r r 2 r
6. If the printout is to be inserted in a text for publication then the user is advised to use the output for latex command described in Section 23 This untidiness of Ortocartan printouts is the price we had to pay for making the printing procedure more powerful Consider the following example these are copies of parts of the original input and output for the program calculate transferred here by a text editor with some irrelevant or empty lines removed calculate print example constants a b c d coordinates x functions f x operation deriv x a b der x f c d The output for this example will be print example I UNDERSTAND YOU REQUEST THE FOLLOWING EXPRESSION TO BE SIMPLIFIED gt deriv x a 50 THE RESULT IS b f d 1 c gt result a b c log a log b f 1 XX x I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 50 msec You have never seen a problem involving this kind of expression Well honestly neither have we But you can be safely assured that no matter how wildly complicated an ex pression is ortocartan and calculate will know how to handle and print it This generality made it difficult to instruct the program to end each line only at a or a in the base level but someday we may come back to the problem and solve it By the way try to print the same expression with Latex we dare say Ortocartan does it in a more reada
7. Note how the markers were used to simplify the substitutions the single eguation der t t M der M V represents the 3 equations d x dt 0V 0x for i 1 2 3 simultaneously The input data are setg lower nil squint a first integral of the Newtonian equations of motion constants m parameter t functions x t y t z t V x y z Q11 x y z Q12 x y z Q13 x y z Q22 x y z D23 x y z NER z L1 x y z L2 x y z L3 x y z E x y 2 J variables x y z integral Q11 der t x 2 2 Q12 der t x der t y 2 Q13 der t x der t z Q22 der t y 2 112 2 x Q23 der t y der t z Q33 der t z 2 L1 der t x L2 der t y L3 der t z E markers M substitutions maineq der t t M der M V m dont print maineq setq lower t and the results are a first integral of the Newtonian equations of motion gt integral E Q11 x 2 Q12 x y 2 Q13 x Zs Q22 t t t t t gt y 2 Q23 y Z 33 z Li x L2 y L3 z t t t t t t substitutions maineq gt THIS IS THE COEFFICIENT OF x t gt equation Q11 1 x 113 THIS IS THE equation 2 THIS IS THE equation 3 THIS IS THE equation 4 THIS IS THE equation 5 THIS IS THE equation 6 THIS IS THE equation T THIS IS THE COEFFICIENT OF x y t t 2342 ati x y 2 COEFFICIENT OF X Za t t 2013 Qit X Z
8. The single components specifically requested in this way are printed even if they are equal to zero Note that if the symmetries of some quantity like Rag are trivial in practical use then each set of indices referring to that quantity should be ordered in the proper way e g tensors ricci 1 0 will result in printing no components of Rag as the combination 1 0 of the indices of Rag is not considered at all by the program The correct request is tensors ricci 0 1 17 Expanding natural powers of sums For purposes of uniqueness natural powers of sums are automatically expanded with use of rules of the type a by a 2ab b This as well as automatic application of the rule of distributivity of multiplication is necessary to carry out all the algebraic simplifications that are possible Without this quite trivial flaws inescapably occur e g the program would not recognize that 1 2 1 22 2 0 However if the exponent is larger than 3 then the expansion is suspended and the expres sion is left in a non expanded form to avoid too large expressions which would probably 24 appear in such a case This might of course be inconvenient in some cases so the user may change this rule if necessary In order to do it one should insert into the data the item of the form expand powers from lt n1 gt to lt n2 gt where lt n1 gt and lt n2 gt are the minimal and the maximal value respec
9. ifold of constant positive curvature in fact a 3 sphere the connection coefficients used here as input data were previously calculated by another program of the Ortocartan set as the Christoffel symbols for the metric of a 3 sphere This example is so simple in order that the readers can look up the answer in textbooks and verify that it is correct The number of dimensions of the manifold n can be arbitrary and it is the first argument of the function curvature The connection coefficients are assumed symmetric and so there should be 3n n 1 of them The program checks whether the number of the connection coefficients actually given is equal to this The order in which the conection coefficients should be given is described in Appendix C 2 it is the obvious one The input data is here setq lower nil curvature 3 104 the curvature of christoffels of spherical space coordinates r th ph connection 0 0 0 sin r cos r 0 sin r cos r sin th 2 O cos r sin r 0 0 0 cos th sin th 00 cos r sin r 0 cos th sin th 0 setq lower t and the results are the curvature of christoffels of spherical space 0 gt connection cos r sin r 11 0 2 gt connection cos r sin r sin th 22 1 1 gt connection cos r sin r 01 1 gt connection cos th sin th 22 2 1 gt connection cos r sin r 02 2 1 gt connection cos th
10. landlagr eulagr and squint 40 Cor The program bllisevol s era xk fa Boa ee Elena Ier uud 40 22 The program curvature uoce pela Dre er AW ATH ee AL 43 C 3 The program landlagr a RE A ke Be RUE 44 C4 he program eulagr y aie pte E pte Ee a Sera ua 44 3 92 The program squitit sia pre we aaa Ye wot Rex AS 46 Versions of Ortocartan for different computers 49 E Sample prints 50 E 1 Example I The Robertson Walker Metric 51 E 2 Example II The most general spherically symmetric metric in the Schwarzschild coordinate system a A T An CECI we eS EOS 56 E 3 Example III The Stephani solution ya ui ea ea 22 Dr xc 68 E A Example IV The Nariai solution 4 4 2 RR 8 404 4 RAS T9 E 5 Example V The Laplace equation in the cylindrical coordinates 85 E 6 Example VI The spherically symmetric metric in the standard coordinates with the arguments of functions written out explicitly 87 E T Example VII Application of the program Ellisevol to check the Ellis evolu tion equations for the Lanczos metric a eee 91 E 8 Example VIII Application of the program curvature 104 E 9 Example IX Application of the program landlagr 2 106 E 10 Example X Application of the program eulagr 111 E 11 Example XI Application of the program squint ss 112 1 Application of the program The
11. marked by the message END OF WORK followed by the information RUN TIME lt n gt msec Note All the input data are processed by the procedure of algebraic simplification so that they acquire the standard form required by the program They are reprinted in the mathematical format after they are standardized Therefore they may differ from the original data by ordering the sums and products or by the replacement of cos x by 1 sin x and so on You should study this part of the printout carefully because it shows what the program is really processing 10 The frame for the data The first and the last item of the input data are always the same and for clarity are best placed in a separate line each The first item has the form 16 ortocartan L do not overlook the apostrophel The second item of the data is the title of the problem It is a completelv arbitrarv single S expression i e either a single atom or a single list The precise shape of the title has no meaning for the program it is simplv printed as the heading of the printout in order to mark it bv an easilv recognizable identifier chosen bv the user For example here are some possible titles CURVATURES SCHWARZSCHILD METRIC GENERAL SPHERICAL METRIC Note however that the title AXIAL METRIC would result in an error it consists of two S expressions In this case the program would understand that the atom AXIAL is the title while the atom METRIC is a p
12. or rmargin 132 Similarly you can adjust the left margin by inserting the item lmargin lt n 1 gt into the data This will have an effect only in those formulae that extend over more than one line the second and following lines will be printed starting at the n 1 st column The default value for lmargin is 8 The first line of each formula always begins at column 5 and this is not adjustable 34 22 Storing the results on a disk file and printing them In a normal run Ortocartan will show the results of the calculation only on the screen When the calculation is simple such as the Schwarzschild or the Robertson Walker metric the formulae will flash through the screen much faster than you can read them You can go back and forth along the results to see the details by pressing the buttons PgUp and PgDn on the keyboard However it is often convenient to print the results on paper In order to do so you have to click on the File in the upper left corner of the screen and then click the To file option in the dropdown menu Then you can choose the name and the path of the directory where the complete output from Lisp will be stored Later you can view it with any text editor edit and print it 23 Special forms of the output Ortocartan produces formulae that are easily readable However sometimes these formulae have to be used as input for another calculation or inserted in a text for publication Retyping
13. r C r psi 0 gt ematrix 1 0 0 gt ematrix Cr 1 l 1 2 gt ematrix psi 1 2 1 2 gt ematrix 1 2 exp 1 2 r psi 2 3 gt ematrix exp 1 2 r 3 ematrix completed TIME 210 msec 0 gt velocity 1 gt DETERMINANT EMATRIX 1 2 exp r DETERMINANT EMATRIX calculated TIME 270 msec 93 0 1 2 gt ie Cr psi 1 1 1 2 gt ie psi 1 2 1 2 gt ie 2 exp 1 2 r psi 2 3 gt ie exp 1 2 r 3 ie calculated TIME 450 msec 0 gt uvelo 1 gt lvelo 1 0 gt lvelo Cr 1 gt metric 1 00 gt metric Cr 0 1 2 2 gt metric psi cC r 11 94 1 1 4 exp r psi gt metric 22 gt metric exp r 33 metric calculated TIME 520 msec 00 2 2 sl gt invmetric 1 C r psi O1 1 gt invmetric Il Q H uo un H 11 1 psi gt invmetric 22 gt invmetric 4 exp r psi 33 gt invmetric exp r invmetric calculated TIME 560 msec gt agamma C exp 1 2 r 12 95 1 1 2 gt agamma 1 2 Lambda exp 1 2 r psi 1 2 C 12 1 2 gt exp 1 2 r psi 3 1 2 gt agamma 1 2 exp 1 2 r psi 23 agamma calculated TIME 750 msec agamma completed TIME 760 msec 0 gt gamma C exp 1 2 r 12 0 gt gamma C exp 1 2 r 21 1 gt gamma C exp 1 2 r 20 1 1 2 2 gt gam
14. r psi 22 gt projdd exp r 3 8 projdd calculated TIME 1110 msec 99 projdd completed TIME 1110 msec 0 gt projdu GT 1 1 gt projdu 1 1 2 gt projdu 1 2 3 gt projdu 1 3 projdu calculated TIME 1140 msec gt EXPANSION SCALAR 0 EXPANSION SCALAR calculated TIME 1150 msec sheardd calculated TIME 1160 msec SHEAR O lacce IS COVARIANTLY CONSTANT TIDAL MATRIX OF lacce completed TIME 1160 msec ALL THE ROTATION CONSTRAINTS ARE FULFILLED IDENTICALLY ROTATION CONSTRAINTS calculated TIME 1160 msec ALL THE SHEAR CONSTRAINTS ARE FULFILLED IDENTICALLY SHEAR CONSTRAINTS calculated 100 TIME 2500 msec ALL THE ROTATION EVOLUTION EQUATIONS ARE FULFILLED IDENTICALLY ROTATION EVOLUTION EQUATIONS calculated TIME 2800 msec 2 gt riemann C exp r 0101 2 gt riemann C exp r 0202 1 2 gt riemann C exp r psi 0212 1 2 gt riemann C exp r psi 0313 2 gt riemann Lambda 2 C exp r 1212 2 gt riemann Lambda C exp r 1313 2 gt riemann Lambda C exp r 2323 riemann calculated TIME 3050 msec riemann completed TIME 3090 msec 2 gt ricci 2 C exp r 00 101 2 gt ricci 2 Lambda 2 C exp r 11 2 gt ricci 2 Lambda 2 C exp r 22 2 gt ricci 2 Lambda 2 C exp r 33 ricci calculated TIME 312
15. tes contain sets of test examples for the corresponding programs The sequence of actions is now the following Copy all the lis files onto your disk Let the directory where the files are stored have the name akr ortocar this is my directory you can choose any name you like Call Lisp and then from within Lisp write rdf akr ortocar ortcsl lis rdf akr ortocar calcsl lis rdf akr ortocar elliscsl lis rdf akr ortocar ncurvcsl lis rdf akr ortocar lanlagcs lis rdf akr ortocar eulagcsl lis rdf akr ortocar squintcs lis Lisp will respond to each rdf by confirming that it has defined several functions whose names and meanings need not bother you Then write reclaim This command will cause that the Lisp system will clear the core of all useless data Next write preserve After this command Lisp will write the core image to the disk file and quit Next time when you call Lisp from the same image all the Ortocartan definitions will already be parts of the Lisp system At this point you do not know yet how to use the program Ortocartan This will be described in the sections that follow The thing to remember now is this Codemist Standard Lisp the same is true for Cambridge Lisp and for at least some other Lisp systems can either consider upper case letters identical to their lower case counterparts this is the default mode or can treat them as different symbol
16. 00 2 gt metric fi 11 2 gt metric f2 22 2 gt metric f3 33 metric calculated TIME 250 msec 00 gt invmetric 1 11 2 gt invmetric fi 22 2 gt invmetric f2 108 33 2 gt invmetric f3 invmetric calculated TIME 310 msec 1 1 gt agamma 1 2 f1 fi 0 1 t 2 1 gt agamma 1 2 f2 f2 02 t 3 1 gt agamma 1 2 f3 3 03 t agamma calculated TIME 520 msec agamma completed TIME 540 msec 0 1 gt gamma 1d Tl 11 t 0 gt gamma f2 22 2 2 t 0 el gt gamma f3 f3 3 3 t gamma calculated 109 TIME 590 msec gamma completed TIME 590 msec 0 gt christoffel fi fl 11 t 0 gt christoffel f2 f2 22 t 0 gt christoffel f3 f3 33 t 1 1 gt christoffel fi f1 Oi t 2 1 gt christoffel f2 f2 02 t 3 1 gt christoffel f3 f3 03 t CHRISTOFFEL SYMBOLS calculated TIME 710 msec CHRISTOFFEL SYMBOLS completed TIME 720 msec gt landlagr 2 fi f2 f3 2 f2 fl f3 2 f3 f1 t t t t 110 I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 870 msec This lagrangian can then be used as data for the program eulagr and the resulting Euler Lagrange eguations can be compared with the Einstein eguations derived for the same metric in the ordinary way E 10 Example X Application of the program eulagr The program will derive the Newtonia
17. 2 COEFFICIENT OF x y t t Q22 2 Q12 x y COEFFICIENT OF x y Z t t 2 Q23 2 Q013 2 Q12 x y Z 2 COEFFICIENT OF x Z t t Q33 2 013 x Z 3 COEFFICIENT OF y t Q22 y 2 COEFFICIENT OF y Zi t t 114 t equation 2 023 Q22 8 y THIS IS THE COEFFICIENT OF equation 033 2 023 9 y THIS IS THE COEFFICIENT OF eguation 33 10 z THIS IS THE COEFFICIENT OF equation L1 11 x THIS IS THE COEFFICIENT OF equation L2 L1 12 X y THIS IS THE COEFFICIENT OF equation L3 Ll 13 X z THIS IS THE COEFFICIENT OF equation I2 14 y VA VA y gt 115 THIS IS THE COEFFICIENT OF eguation L3 ub2 15 y Z THIS IS THE COEFFICIENT OF eguation L3 16 z THIS IS THE COEFFICIENT OF zd equation 2m Q11 V 17 THIS IS THE COEFFICIENT OF 1 equation 2m Q12 V 18 3 y THIS IS THE COEFFICIENT OF 1 eguation 2m Q13 V 19 X X X THESE ARE THE TERMS THAT ARE 2m y gt t 2m Z t 2m sf 1 1 012 V 022 V Q23 V y y y 2m 2m 2m 1 i 1 FREE OF THE DERIVATIVES 116 Q13 V Q23 V Q33 V Z Z Z E E E gt eguation m L1 V m L2 V m L3 V 20 x y z I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 820 msec Now we shall substitute the well known solution of these equations into the
18. 2 3 gt ematrix r sin theta 3 ematrix completed TIME 20 msec 2 gt DETERMINANT EMATRIX r exp nu mu sin theta DETERMINANT EMATRIX calculated TIME 40 msec 0 gt ie exp nu 0 1 gt ie exp mu 1 2 1 gt ie r 2 3 1 1 gt ie E sin theta 3 97 ie calculated TIME 200 msec 0 gt agamma 1 2 exp mu nu Oi T 1 gt agamma 1 2 exp nu mu 01 t 2 1 gt agamma 1 2 r exp mu 12 3 1 gt agamma 1 2 r exp mu 13 3 1 zu gt agamma 1 2 r cos theta sin theta 23 agamma calculated TIME 360 msec agamma completed TIME 360 msec 0 gt gamma exp mu nu 10 r 0 gt gamma exp nu mu 11 t 1 1 gt gamma r exp mu 22 58 gt gamma r exp mu 3 3 2 1 gt gamma r cos theta sin theta 3 3 gamma calculated TIME 380 msec gamma completed TIME 380 msec 2 gt riemann exp 2 nu mu exp 2 nu mu exp 2 nu 0101 t tt 2 gt nu mu exp 2 mi ni exp 2 mu mu exp 2 mu t t r rr gt nu mu r r 1 gt riemann r exp 2 mu nu 0202 r 1 gt riemann r exp nu mu m 0212 t 1 gt riemann r exp 2 mu nu 0303 r 1 gt riemann r exp nu mu mn 0313 t 59 gt riemann r exp 2mu mu 1212 r 1 gt riemann r exp 2 mu mu 1313 r 2 2 gt riemann r exp 2 mu r
19. 2a btp ZC ap t r 2 The missing simplifications are t r p 3 31 1 8 8 q t r cos phi p 4 1 3 1 1 1 log q 2 log ZC ap 1 3 ab hp ab Sp 4ap ZC 2b m 5 2 a i 6 After they are done the more neat result is obtained 3 1 1 4 dS dt 8abtp 8abp t r cos phi 4btp ZC 2 1 2 2btp ZC ap ht tSp 7 The net result of having the substitution 7 carried out may be effected by declaring the equation 1 in the symbols and then writing the equations 3 4 5 and 6 in turn in the substitutions Then however such a coupled series of substitutions would be performed a lot of times each time inside a large expression and this would make the calculation slow It is more economical to force the program to replace every occurrence of dS dt directly by 7 This can be done One should declare S as one of the functions and then write substitutions der t S deriv t lt the right hand side of 1 gt data substitutions der t S t 2 2 po 2 der t S q 1 t r cos phi 8 p 8 der t S log q 2 1og ZC a p 13 a h p 3 b ta S p b 4 a ZC p 2 m b der t S a 2 21 For brevity we omit the corresponding substitutions in dS d dS dr and dS dz The data substitutions specify the replacements to be made inside the substitu tions before the program starts to run They have the
20. 2xm r 1 2 CO CICR em 1 IO substitutions ematrix f ematrix g because after f and g are replaced by their values in the ematrix they will never appear later and the program need not look for any other opportunity to replace them This rather simple example was chosen just for its simplicity Ortocartan was success fully tested on several much more complicated metrics see Appendix E In general the user is advised to specify the places in which the substitutions should be carried out as precisely as possible It makes the calculation faster the program does not look for opportunities to make the substitutions in the formulae where these substitutions are not necessary Every unsuccessful attempt at performing a substitution is communicated to the user in the printout Suppose for instance that the user has written in the substitutions riemnn 0 101 0202 030 3 Cau sn CU r a Dez 2 but the variable x did not appear in 0202 at all Then the program will print the following message above Hos gt riemann DID NOT PROVIDE ANY OPPORTUNITY TO PERFORM 0202 gt THE FOLLOWING SUBSTITUTIONS 2 2 2 2 gt X R Y Z This means that in the next run of the program with the same data the address of this substitution should be riemann 0 1 0 1 0 3 0 3 This is not an error message and the program will continue to work These messages can be turned off altogether This happens when one inserts the atom messages into
21. A B would be understood as the sequence of 3 atoms the A the and the B However in other versions of Lisp A B would be understood as a single atom consisting of the 3 characters Therefore it is always safer to separate all signs of mathematical operations from their arguments by blanks like in all examples in this manual Division is denoted by the slash A B representing A B Ortocartan uses the same hierarchy of algebraic operations as is used in ordinary alge bra i e the algebraic operations are performed in the following sequence first exponen tiations then multiplications then divisions then additions with subtractions This order is changed by the use of parentheses just like in algebra Negative integers may be represented either as single atoms whose first character is the sign e g 2 or 10 or as a list of two atoms where the first atom is the sign and the second atom is the absolute value of the number e g 2 or 10 Both representations are correct For non numerical quantities however the second representation is the only correct one A means the negative of A while A depending on the Lisp system would be understood either as a sequence of two atoms or as a single atom which would be treated by the program Ortocartan as a single symbol of some quantity Non integer rational numbers are represented as two element lists the first element being the numerator and the second el
22. MODIFY WILL HAVE THE FOL LOWING FINAL FORM 4 Those substitutions which were modified are printed in their final working form One can avoid having the list of modified substitutions printed by inserting the atom modifications into the argument dont print see sec 20 20 Suppressing some parts of the printout Sometimes the user is not interested in having all the results of the routine run of the program In this case he or she may instruct the program not to print some intermediate results or to stop its work earlier than usually before calculating the Weyl tensor This results in saving paper as well as computer s time and core and conseguently user s time too The reguest to stop the work earlier is expressed by inserting into the data the item of the form stop after lt name gt where lt name gt is one of the names from the dictionary in section 8 only scalars and tetrad names will have the reguired result For example if the expression 33 stop after riemann is found in the data then the Ricci and Weyl tensors will not be calculated If the user is not interested in some intermediate results then the appropriate reguest iS dont print lt a series of names gt where lt a series of names gt consists of any number of names from the dictionary in section 8 again only scalars and tetrad names except einstein i e only the guantities that are calculated in the routine run will have the reguir
23. MU T R NU T R exp 2 MU IR 2 NU T R exp 2 MU T R MU T R 22 2 2 NU T RDD exp 2 NU T R MU T R 2 1 exp 2 NU T R MU T R exp 2 NU T 11 R MU T R NU T R 1 1 2 1 ricci R exp 2 MU T R R exp 2 MU T 22 90 sl gt R MU T R gt R exp 2 MU T R NU L 2 2 2 gt T R R 2 1 gt ricci R exp 2 MU T R R exp 2 MU T 33 1 gt R MU T R R exp 2 MU T R NU L 2 2 2 gt T R R ricci calculated TIME 689 msec I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 689 msec E 7 Example VII Application of the program Ellisevol to check the Ellis evolution equations for the Lanczos metric The Lanczos metric is ds dt Crdy do ean e dz where b C r A Ae7 References The original paper K Lanczos Zeitschrift f r Physik 21 73 1924 English translation Gen Rel Grav 29 363 1997 The input data is here 91 setg lower nil ellisevol LANCZOS METRIC coordinates t phi r z velocity 1 0 0 0 constants C Lambda symbols psi C 2 r Lambda Lambda exp r ematrix 1 C r O O 0 C 2 r Lambda Lambda exp r 12 0000 1 2 exp 1 2 r C 2 r Lambda Lambda exp r 1 2 0 000 exp 1 2 r substitutions C 2 r Lambda Lambda
24. computer at the N Copernicus Center and to buy a license for the Codemist Standard Lisp from Codemist Limited I A K am very grateful to Professor Trautman for including me in his research group A How to acquire Ortocartan In order to use Ortocartan one must first buy the Codemist Standard Lisp It can be bought from Professor John Fitch Director CODEMIST Limited Alta Horsecombe Vale Combe Down BATH Avon BA2 5QR England phone and fax 44 1225 837 430 email jpffitch maths bath ac uk Ortocartan is free of charge If you wish to have it just contact A Krasiriski I will send you a diskette How to install Ortocartan when Lisp is already working is described in sec 7 B How to use the program Calculate The definition of the function calculate is contained in the file calcsl lis in the distri bution diskette of Ortocartan see sec 7 In order to use the program you simply have to follow the instructions from sec 7 To make sure that the program is indeed at your disposal after starting Ortocartan write infocalculate and press RETURN The response should be reassuring With the exceptions of integration and factoring polynomials this program can carry out any kind of elementary algebraic operations but lacks the facility of writing programs in it without resorting to Lisp i e is not a programming language in itself The format for the user s data is nearly identical to the one described in
25. einstein 2CFD Gi 3 C 11 t 1 2 gt einstein 2CFD C 3C 22 t 1 2 gt einstein 2CFD C 300 3 3 t einstein calculated TIME 6061 msec weyl calculated TIME 6141 msec ALL COMPONENTS OF THE WEYL TENSOR ARE ZERO I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 6141 msec E 4 Example IV The Nariai solution ds P dt E r da dy dz where P a t cos log r L b t sin log r L L is a constant r 22 y 22 a t and b t are arbitrary functions of time Reference 75 A Krasifiski J Plebariski Rep Math Phys 17 217 1980 The input data is setg lower nil ortocartan Nariai solution coordinates t x y z constants L markers m ematrixP 0 0 O 0 L r 0 0 0 0 7 Y 0 0 0 0 L r symbols P a cos log r L b sin log r L substitutions agamma riemann der m r m r agamma riemann cos log r L P a b sin log r Da agamma riemann z 2 r 2 x 2 y 2 functions a t b t r x y z dont print messages rmargin 61 setq lower t Notes This is a solution of the Einstein s equations in empty space with the cosmological term note the form of the Ricci tensor in the output Note the use of markers which implies substitutions by pattern matching they are simpler than in the previous example so may be more readable It was not specified here
26. evolution equations of the kinematical tensors of fluid flow as defined by Ellis Since all these equations are conse quences of the Ricci identity Ua ay Uang Uu RF gro they will be fulfilled identically in many cases However they may fail to be identically fulfilled when we make assumptions about separate parts of the flow e g if we assume that the shear is zero As is well known such assumptions have consequences for the other characteristics of the flow and the Ellis equations will show what the consequences are Along the way the program calculates the expansion acceleration the shear tensor and scalar the rotation tensor and scalar and the electric and magnetic parts of the Weyl tensor with respect to the velocity vector In order to make sure that the program Ellisevol is indeed contained in the core image of Lisp that you use write infoellis and press RETURN Since the signature assumed in the calculation is the formulae may differ from those given in textbooks and so some of them are quoted below for reference The program is called by writing ellisevol G F R Ellis in General relativity and cosmology Proceedings of the International School of Physics Enrico Fermi Course 47 General Relativity and Cosmology Edited by R K Sachs Academic Press New York 1971 40 in the first line of the data file All the remaining parts of the data are the same as for the program Ortoc
27. gt You can prevent this expression from being printed it will likely be quite long by writing dont print maineq However it is not reasonable to allow the program to calculate d dt without taking into account the set of equations C 1 Hence you should insert the following item into the data 47 substitutions maineg der t t f1 maineq der t t f2 maineq der t t f3 the right hand side from C 1 gt the right hand side from C 1 gt the right hand side from C 1 gt In this way the program will eliminate second derivatives of the f thus making use of the set C 1 The maineq means that these three substitutions should be carried out within the d7 dt the main equation before the program goes on with the calculation see below Next the program finds and prints the coefficients of all the expressions df af dr I l dt dt dt y T and the terms that do not contain any derivatives of the f These coefficients are printed with the headings that have the following form gt THIS IS THE COEFFICIENT OF 2 f3 t t and the coefficients themselves are printed in the following form gt equation lt the appropriate expression gt 6 For a set of n equations to determine n functions there will be n 4 1 n 2 n 3 such coefficients altogether If you do not need some of them you can insert the following item into the data dont print equation lt n1 gt lt n2 gt lt n3 gt
28. gt weyl L y r Ss L 0202 2 2 gt weyl L yzr 0203 2 2 2 2 2 2 2 gt weyl L xr L yr 2 33 L 0303 2 2 2 2 2 2 2 gt weyl L xr L yr 2 33 L 1212 2 2 gt weyl L yzr 1213 2 2 gt weyl m h GT 1223 20 2 2 2 gt weyl L yr de GB L 1313 2 2 gt weyl L xyr 1323 2 2 2 2 gt weyl L xr 1 3 L 2323 weyl calculated TIME 3420 msec I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 3420 msec 84 E 5 Example V The Laplace equation in the cylindrical coordi nates The input data is here calculate Laplace eguation in cylindrical coordinates coordinates x y z functions F r phi z symbols re LLIS dig 2 E 2 phi arctan y x cosphi cos phi substitutions Qe coy edu emt i Roy Of x 2 cosphi 2 x r cosphi y r sin phi sin phi 3 sin phi 1 cosphi 2 cosphi cos phi operation deriv x x F deriv y y F deriv z z F Notes This example shows how to use the program calculate Since coordinate transfor mations as such are not available in Ortocartan the transformation to the cylindrical coordinates is achieved through a trick the unknown function F is defined to depend not directly on the cartesian coordinates x and y but on r and which are defined as explicit expressions in x and y In the result thus obtained z and y are replaced by the ap
29. has recognized the Greek letters and printed them in Latex s favourite way and what Latex will now do with them riccioo 2r exp 2u v r exp 2v ie exp 20 Hy exD 2 4p He exp 2j v exp 2p V rr EXP 24 Vyr Mar riccio 2r exp v LL i t ricci 2r exp 2u p r exp 2v iu exp 2v H exp 2v va pa exp 2 1 V r exp 2p Usrr exp 2u v r Mr riccigg r exp 2u r exp 2p v r Tr exp 2u pyr r ricci33 r exp 2u rl exp 2u v Lr exp 2n p Tr 18 19 21 22 The equation numbers above are continued from Section 1 because we have not readjusted the equation counter You can do it in your own Latex preamble 67 E 3 Example III The Stephani solution ds D dt R IVY dz dy dz where 1 V 1 KE 20 y go 2 zo D FV V R R Ro k o yo zo and F are arbitrary functions of t Reference H Stephani Commun Math Phys 4 137 1967 The input data is here setg lower nil ortocartan the Stephani solution coordinates t x y z functions C t k t R t XOCt YO t ZO t F t V bx ye DU om emtrixD 0 0 0 O R V 0000 R V o 0 0 0 R V tensors einstein markers m substitutions riemann O 1 0O f 0202 0303 der t D der t V der t D Vx D F der t R R agamma riemann der m D deriv m F
30. ie D 0 1 1 gt le R V 1 2 1 gt le R V 2 3 1 gt le R V 3 ie calculated TIME 760 msec agamma calculated TIME 1060 msec agamma completed TIME 1060 msec 71 Oops This is very untidy printing the t and x in the second derivative of V have been separated But you can easily learn to live with this and use the output for latex when you need a really neat printout 0 1 1 i gt gamma R D R V D V 11 t t 0 1 Hh lt l iL 1 gt gamma R FV D V V R FD V 2 0 t y t gt y 0 1 1 ced gt gamma R D R V D V 22 t t 0 Ex 1 ai E 1 gt gamma R FV D V V R FD V 30 t z t gt z 0 ed si s 4 gt gamma R D R V D V 33 t t 12 1 gt gamma oR 2 1 1 i gt gamma R 22 1 gt gamma oR 3 1 1 1 gt gamma R 3 3 2 gt gamma R 3 2 2 Fi gt gamma R 33 gamma calculated TIME 1160 msec gamma completed TIME 1170 msec gt riemann 01041 gt riemann 0202 73 gt riemann 0303 gt riemann 1212 gt riemann 1313 gt riemann 2323 riemann calculated TIME 5931 msec riemann completed TIME 6001 msec gt ricci 3CF 00 gt ricci CF 11 gt ricci CF 22 gt ricci CF 33 ricci calculated TIME 6021 msec CURVATURE INVARIANT TIME 6031 msec calculated 74 gt CURVATURE INVARIANT 6 C FD C 12C t 2 gt einstein 30C 00 1 2 gt
31. in which components of agamma and riemann the substitutions should be performed The command dont print messages suppresses the information about unsuccessful attempts at substitutions The output is Nariai solution symbols gt P a cos log L log r 4 b sin log L log 76 gt r substitutions agamma riemann 1 gt r mr m agamma riemann i gt cos log L log r a b sin log L log 1 gt r a P agamma riemann 2 2 2 2 gt Z x y r 0 gt ematrix P 0 1 1 gt ematrix Lr 1 2 1 gt ematrix Lr 2 3 f gt ematrix Lr 3 ematrix completed TT TIME 300 msec 3 3 gt DETERMINANT EMATRIX L r P DETERMINANT EMATRIX calculated TIME 340 msec P 0 1 L r 1 1 L r 2 1 L r 3 ie calculated TIME 490 msec gt agamma 1 2 L xr aP sin log L log r 1 2 L xr a b 1 2 L xr a b 2 1 P sin log L log r 78 agamma 1 2 L yr aP sin log L log 02 DIL yr b g a b 2 1 P sin log L log r 0 1 1 1 agamma 1 2 L zr aP sin log L log 03 r 1 2 L zr a b 1 2 L zr a b 2 1 P sain log L log r J 1 1 1 agamma 1 2 L yr 12 1 1 1 agamma 1 2 L zr 13 2 sd 1 agamma 1 2 L xr 12 2 1 1 agamma 1 2 L zr 23 3 1 i agamma 1 2 L xr 13 79 3 zd f gt agamma 1 2 L yr 23 agam
32. nil before the ortocartan line Without it all capitals will be mapped into lower case letters and printed as such Even worse things can occur if you do insert the setg lower nil command and then carelessly use capitals and lower case letters With this command inserted all function names should be written in lower case for example sin but not SIN the SIN would then be an unknown function for Ortocartan You can then use capitals only as names of the symbols introduced by yourself in the data for Ortocartan and in the title If the setg lower nil has been used then it is advisable to place the command setg lower t at the end of the data for Ortocartan i e behind the two right parentheses Then Lisp will revert to its default mode after it has finished the calculation otherwise it will continue to see capitals and lower case letters as different ACKNOWLEDGEMENTS This manual was rewritten into a computer file for its third edition thanks to the courtesy of Professor F Hehl We express our gratitude to him and to his collaborators at the University of Cologne who actually did the hard work of retyping Our thanks are due to Drs J Richer and A Norman who reworked Ortocartan into Cambridge Lisp and implemented it on an IBM computer This enabled one of the authors A K to rewrite Ortocartan into Slisp 360 now a defunct version The kind assistance of Professor J Fitch in this last task is hereb
33. of variables should be properly ordered for uniqueness The deriv functional expression will be ordered while der will be left intact literally as given by the user For instance if the user writes der X Y X F then such an expression will not be changed while if one writes deriv X Y X F then such an expression will be changed either to der X X Y F or to der Y X X F depending if X or Y has the higher priority 10 An indefinite integral should be represented as int lt the variable gt lt the integrand gt For example f Gdz should be represented as int x G and f ein 7d should be represented as int x exp sin x 2 The program is not able to process definite integrals If a definite integral must be introduced then it can be declared as a function of unspecified shape whose derivatives should be substituted by the required expressions supplied by the user on input how such substitutions may be requested see further Let us now give a few examples of more complicated expressions represented on input for the program Ortocartan 1 3 Lr r a r 2mr a should be represented as COE 3 PS e Uu ro Da DI PDS A E 2 1 1 2m r should be represented as 1 1 2 m r or as 1 1 2 m r oras 1 2 m r 1 a df dx f z V dz should be represented as a der x f int x x V The program prints the results in the mathematical format i e with superscripts su
34. same syntax as the substitutions themselves with the difference that here the addresses are the appropriate left hand sides of the equations from the substitutions and the replacements are performed in the cor responding right hand sides If no address is given then the appropriate data substitution is performed just everywhere in the substitutions 32 In the example given above the value of dS dt initially egual to the right hand side of 2 after being in turn transformed by 3 4 5 and 6 will finally change into the right hand side of 7 and then just this expression will be directly substituted for dS dt in the calculation proper The use of the data substitutions facility has the following result 1 The substitutions are standardized and printed as explained in section 19 just as if there were no data substitutions 2 The data substitutions are printed analogously immediately below with the heading THE SUBSTITUTIONS LISTED ABOVE WILL BE THEMSELVES TRANSFORMED BY THE FOLLOWING SUBSTITUTIONS Don t get confused if you had turned off printing the list of substitutions then they will not be listed above although the heading of data substitutions will say so Each eguation printed has its own sub heading which says either EVERYWHERE or gt IN THE VALUE OF THE EXPRESSION lt the appropriate left hand side from substitutions gt 3 Another heading is printed THE SUBSTITUTIONS YOU ASKED ME TO
35. sin th 12 CONNECTION COEFFICIENTS completed TIME 60 msec 0 2 gt ncurvature sin r 101 105 0 2 2 gt ncurvature sin r sin th 202 1 gt ncurvature 1 001 1 2 2 gt ncurvature sin r sin th 212 2 gt ncurvature 1 002 2 2 gt ncurvature sin r 112 ncurvature calculated TIME 190 msec I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 190 msec E 9 Example IX Application of the program landlagr The Landau Lifshitz lagrangian equals Y gR where g is the determinant of the metric tensor and R is the Ricci scalar with the derivatives of the Christoffel symbols dropped The example will be a diagonal Bianchi type I metric for which the reduced lagrangian is known to provide the correct Einstein equations The input data is landlagr lagrangian for a diagonal Bianchi I metric coordinates t x y z functions fl t f2 t f3 t ematrix 10000 f10000 f2 0 0 0 0 f3 106 rmargin 61 and the result is lagrangian for a diagonal bianchi i metric 0 gt ematrix 1 0 1 gt ematrix f1 1 2 gt ematrix f2 2 3 gt ematrix f3 3 ematrix completed TIME 20 msec gt DETERMINANT EMATRIX fi f2 f3 DETERMINANT EMATRIX calculated TIME 30 msec 0 gt ie 1 0 1 1 gt ie f1 1 2 1 gt ie f2 2 107 ie calculated TIME 220 msec gt metric 1
36. tensor Eag Capi du Ega under the name elweyl The shear evolution eguations Wr 2 haha 0 5 ha hg qs hala Way Gyd p 30008 1 z haola 0 t Esg 0 under the name shearevol The magnetic components of the Weyl tensor 1 v Hag 3V ge ayu C ayuu Hga under the name magweyl the 4 is the Levi Civita symbol The magnetic constraint equations 2U aWB v gho hg lwg og leia Hag under the name magcons where wg is the rotation vector field defined by a 1 aBy 2 9 Uu gts Each of the names of the quantities printed can be used as an address in the substitu tions and as an entry in the dont print or stop after just like in the main program Ortocartan For the print names that consist of two words like the raychaudhuri equa tion only the first word should be used as an address or as an entry in dont print and stop after 42 All the facilities described for Ortocartan exist also here including the tensors except that the Christoffel symbols and the metric are calculated in every run because they are needed at later stages of the calculation but their printing can be suppressed with dont print Please do not forget about the last line with the two right parentheses This applies to all the other programs of this appendix C 2 The program curvature This program calcul
37. the argument dont print see sec 20 and example IV in Appendix Sometimes the whole value of some guantity should be substituted by some other expression The user may in this case avoid the necessity to rewrite the whole expression by writing the atom actual as the left hand side of the appropriate eguation So for example the eguation 27 riemann 01 0 1 actual x 2 means from now on the tetrad component Roio of the Riemann tensor will be equal to x The previous value is forgotten The equations defining the substitutions are printed in the mathematical format above all the results after the symbols preceded by the heading substitutions and with the sub headings which either say everywhere or are the series of addresses as written by the user One can avoid having the list of substitutions printed sometimes it is long by inserting the atom substitutions into the argument dont print see sec 20 If some of the left hand sides is a sum or a product then the substitution required may be missed in some cases One of such cases is when the user wants the sum x y to be substituted by U and in the expression now processed there is the sum x 2y Then the program will not be able to recognize that x 2y should be replaced by y U The same concerns a product like ab the program will not recognize it as being a part of an expression like ab If this difficulty is likely to appear then it is advised t
38. with a number and does not contain any of the following special characters NAME PRINT NAME left parenthesis right parenthesis period comma blank dollar sign colon backslash X slash exponentiation sign 2 multiplication sign plus or minus or The user is warned that different keyboards are not necessarily compatible and various special signs on them often correspond to different symbols For safety the user is therefore advised to avoid special signs like or amp unless one is aware what one really does with them Any other characters available on the input equipment are permissible Each of the prohibited characters has a special meaning for the Lisp system and when used inside a string of characters would cause splitting the string into more atoms or other structures Examples of atoms The Codemist Standard Lisp has 72 characters per line on the screen Longer atoms may be tolerable within the program code but they would look untidy when showed on the screen A 12 DELTA X1 ATOM Examples of strings which are not atoms B A BC AT L 1 5 DOLLAR each of these contains forbidden characters 2DELTA this one starts with a number Floating point numbers which are atoms for the Lisp system are not used in precise calculations as their use introduces automatically decimal approximations of fractional numbers very much unwanted here Therefore f
39. 0 msec CURVATURE INVARIANT calculated TIME 3120 msec 2 gt CURVATURE INVARIANT 6 Lambda 4 C exp r 2 gt einstein 3 Lambda 4 C exp r 00 gt einstein Lambda 11 gt einstein Lambda 22 gt einstein Lambda 33 einstein calculated TIME 3140 msec gt RAYCHAUDHURI EQUATION O RAYCHAUDHURI EQUATION calculated 102 TIME 3150 msec 2 gt weyl 1 3 C exp r 0101 2 gt weyl 1 3 C exp r 0202 1 2 weyl C exp r psi 0212 2 gt weyl 2 3 C exp r 0303 1 2 gt weyl C exp r psi 0313 2 gt weyl 2 3 C exp r 1212 2 gt weyl 1 3 C exp r 1313 2 gt weyl 1 3 C exp r 2323 weyl calculated TIME 3250 msec 2 gt elweyl 1 3 C exp r psi 1 1 2 1 gt elweyl 1 12 C psi 22 103 2 gt elweyl 2 3 C 33 elweyl calculated TIME 3310 msec ALL THE SHEAR EVOLUTION EQUATIONS ARE FULFILLED IDENTICALLY SHEAR EVOLUTION EQUATIONS calculated TIME 3820 msec gt magweyl 1 2 C 23 magweyl calculated TIME 3890 msec ALL THE MAGNETIC CONSTRAINTS ARE FULFILLED IDENTICALLY magcons calculated TIME 5170 msec I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 5170 msec E 8 Example VIII Application of the program curvature In this example the program will calculate the curvature tensor for a 3 dimensional man
40. THE SYSTEM ORTOCARTAN USER S MANUAL Andrzej Krasinski N Copernicus Astronomical Center Polish Academy of Sciences Bartycka 18 00 716 Warszawa Poland email akr camk edu pl and Marek Perkowski Department of Electrical Engineering Portland State University P O Box 751 Portland Oregon 97 207 U S A email mperkows ee pdx edu FIFTH EDITION Warszawa April 2000 Contents 1 Application of the program 2 The algorithm modelled by the program 3 Atoms 4 Lists 5 The representation of mathematical formulae in Ortocartan 6 Typing the input 7 Starting Lisp and Ortocartan 8 The dictionary for communicating with the program 9 The printout 10 The frame for the data 11 Declaration of coordinates 12 Declaration of arbitrary functions 13 Declaration of the constants 14 Declaration of the tetrad 15 Symbols for sums and other expressions 16 Calculating coordinate components of various quantities 17 Expanding natural powers of sums 18 Substitutions 19 Substitutions in the data 20 Suppressing some parts of the printout 21 Formatting the output 22 Storing the results on a disk file and printing them 23 Special forms of the output 24 Errors 11 12 14 15 16 17 18 19 20 21 22 24 25 31 33 34 35 35 36 A How to acquire Ortocartan 38 How to use the program Calculate 38 A9 99 gt C The programs ellisevol curvature
41. TS metric tensor and its inverse g the arrav of coefficients of the tetrad of forms eta ematrix determinant of the matrix eta determinant the array of coefficients of the inverse tetrad e ie antisymmetrized Ricci rotation coefficients agamma full Ricci rotation coefficients gamma Christoffel symbols christoffel Riemann tensor riemann rie Ricci tensor ricci ric scalar curvature curvature Finstein tensor einstein ein Weyl tensor weyl C The program names are easy to remember because with two exceptions they are the same as the names that appear on output The exceptions are the determinant appears on output as determinant ematrix and the curvature appears on output as curvature invariant 14 9 The printout All the results of the program are printed in the mathematical many line format They are printed in the form of equations lt name of the quantity gt lt indices attached to the name sometimes none gt lt value of the appropriate component of the guantity gt The tetrad components of the Riemann Ricci Weyl and Einstein tensors are printed with all indices down The tensor components if requested by the user to be calculated are printed with the appropriate positions of the indices corresponding to the valence requested The tetrad and tensor components of the same quantity are printed under different names
42. an overtly absurd or unintelligible error message please check carefully the positions of all parentheses in your data There is one more possible error that is not signalized at all and may cause trouble Each item of the data described in one of the sections 11 to 21 may be used only once This is not really a limitation of the power of the program it is just a small piece of rigour forced upon the user as each item can hold many requests of its appropriate kind If any of the items is used twice or more times then only its last appearance has the expected results and the previous ones are simply ignored For example in the call ortocartan SPHERICAL METRIC IN STANDARD FORM functions MU T R NU T R coordinates T R THETA PHI ematrix exp NU 0 0 O O exp MU 0 O O 0 R 0 0 O O Cr sin theta tensors riemann ricci stop after ricci tensors riemann stop after weyl 23 only the components Rags of the Riemann tensor will be calculated and printed second appearance of tensors while the requests to calculate R 3 5 and Rag will be ignored Also the program will finish its work after calculating the Weyl tensor i e after completing the whole routine run while the request to stop after the Ricci tensor will be ignored We recall what has already been said before if you wish to use capitals and lower case letters as different symbols then you should insert the command 36 setg lower
43. ands for the whole old expression This is sometimes useful but the address es of this substitution must be specified or else all equations will be multiplied by the factor ab c The disadvantage of the substitutions by pattern matching is that they are slower than the literal substitutions and sometimes can be very much slower when the right hand sides are complicated functions of the markers Note that Ortocartan understands the functions quotient and remainder acting on the markers However it is the user s exclusive responsibility to make sure that the arguments of these functions will in each case turn out to be integer numbers Expanding integer powers of sums is a part of the substitutions For instance in order to expand A B one should write in the substitutions A B 4 expand Should just any power of A 4 B be expanded one writes A B M1 expand where M1 should be one of the markers The user writes the substitutions by pattern matching together with the literal ones in the argument substitutions without marking them in any way The presence of the argument markers is a sufficient warning for Ortocartan it will then be able to distinguish the two kinds of substitutions by itself 30 19 Substitutions in the data This facility was introduced long ago before easy to use text editors became commonly available and before the additional programs of the Appe
44. are all of second or first order To make sure that this program is in your core image write infosquint and press RETURN The program is called by writing as the first item of the data squint This is an abbreviation for square integral The second item of the data is the title The main part of the data are the following three items see example XI in Appendix E parameter an atomic name of the independent variable in the set of equations variables the list of names of the functions that should obey the set of equations integral the formula for the first integral to be tested by the program written in accordance with the rules of Section 5 The variables must be declared as functions in the functions list because the program allows them to depend also on other arguments in addition to the parameter specified in the data The integral may either be an explicit expression which is being tested by the program whether it is a first integral indeed or a polynomial of second or first degree in the first derivatives of the variables with unknown coefficients The unknown coefficients must then be declared as functions of the appropriate variables Example suppose you have a set of 3 second order ordinary differential equations to be fulfilled by the functions f t f t and f t of the form d f Ud am kad ia 5For an actual example of a simple application of this program
45. art of the relevant data see further sections Soon it would recognize that the atom METRIC has none of the required forms for the data and would print the message METRIC x x x IS AN ILLEGAL ARGUMENT FOR OUR SYSTEM SORRY CANT GO ON A convenient title is a reference to the paper from which the metric was taken The last item of the data is the set of two right parentheses They close the two parentheses from the first line All the other items of the data described in the sections that follow should be placed between the title and the last line with parentheses in an arbitrary order 11 Declaration of coordinates The user tells the program the names he has chosen for the coordinates by inserting into the data the item of the form coordinates atomic names for the coordinates The expected order is 29 z z x where 2 is the time coordinate Examples 17 coordinates X0 X1 X2 X3 coordinates T X Y Z coordinates t r theta phi coordinates T PH1 X1 X2 There are no default names for non declared coordinates so this piece of the data cannot be omitted If it is omitted then Ortocartan will print an error message and refuse to work 12 Declaration of arbitrary functions It has the following form functions name of function 1 list of arguments of function 1 gt name of function 2 gt list of arguments of function 2 gt For instance if the user wants to use th
46. artan described in the manual except that there is one additional item here velocity lt contravariant tetrad components of the velocity field u0 ul u2 u3 gt see Example VII in Appendix E This is the vector field for which all the kinematical guantities will be calculated This argument cannot be omitted the omission would result in an error communicated by the program Ellisevol Note It is implicitly assumed that this will be a velocity field of some continuous medium because this is the most freguent application of these formulae However the calculation makes sense for any timelike vector field whose length is normalized to 1 In particular this may the the unit vector collinear with a timelike Killing vector The second item of the data must be the title this rule applies to all the additional programs described in this appendix The guantities printed are the ematrix the velocity field tetrad components as given by the user the determinant of the ematrix the inverse matrix to the ematrix the contravariant coordinate components of the velocity field named uvelo the covariant coordinate components of the velocity field named lvelo the components of the metric tensor named metric the inverse metric named invmetric the agamma and the gamma and the Christoffel symbols named christoffel Then come The matrix of covariant derivatives of the covariant velocity fiel
47. as no special requests concerning the results All the pieces described further serve to adjust the results of the program to the momentary needs of the user 20 15 Symbols for sums and other expressions In by hand calculations it is sometimes convenient to develop a product involving a sum by applying the rule of distributivity of multiplication and sometimes it is not The choice reguires a small piece of intelligent thinking one must be conscious of the goal one wants to achieve But intelligence even in such small amounts is something as yet inaccessible to computers the program must follow an algorithm which specifies unigue decisions or unigue criteria of choice Conseguently in our program the intelligent pieces of work are left to the user himself The program applies the rule of distributivity always unless the user reguested this not to be done for a specific sum The format for such a reguest is described below If a sum which is present in the ematrix is not to be expanded when in a product then the user should introduce a separate symbol for the sum use the symbol in the ematrix and insert an additional item into the data of the form symbols lt the symbol gt lt the sum gt Actually one can declare in this way an arbitrary number of special symbols for sums symbols lt symbol 1 gt lt sum 1 gt lt symbol 2 gt lt sum 2 gt lt symbol n gt lt sum n gt Then in all the algebraic operations t
48. ates the curvature tensor from given connection coefficients in any number of dimensions see example VIII in Appendix E The connection coefficients are assumed symmetric i e torsion free but need not be metrical The program was written for one special application and hence the assumption of zero torsion it can be removed without any difficulty To make sure that this program is in your core image write infoncurva and press RETURN The first item of the data for this program is the line curvature n where lt n gt is the number of dimensions of the manifold on which the connection coefficients and the curvature are defined The next item should be the title of the problem just like in every other program of the Ortocartan collection The main part of the data is the specification of the connection coefficients that has the form connection list of components of the connection coefficients in the order D000 D001 D002 D003 DOOn DO11 D012 DOin Dn00 Dn01 Dnnn gt i e only the algebraically independent components of D with j k are given All the other parts of input like constants functions symbols substitutions etc can be used just like in Ortocartan if only they make sense here for example stop after or dont print would not work here because there is only one guantity that is being calculated The last item of the data are the two right parentheses 43 C 3 The prog
49. ble way Each base level line of the printout begins with the sign gt to facilitate reading those lines that were broken in unusual places For each example the input data was read from a disk file For each of the examples the input data as prepared by the user is shown separately The output is copied from disk files produced according to the instructions from sec 7 E 1 Example I The Robertson Walker Metric ds dt R dr 1 kr r d sin 9dp where R R t is an arbitrary function and k is an arbitrary constant Reference W Rindler Essential relativity Van Nostrand Reinhold Company New York Cincinnati Toronto London Melbourne 1969 p 234 The input data is here setq lower nil ortocartan Robertson Walker metric coordinates t r theta phi functions R t 51 constants k ematrix 10000 R 1 k amp amp r 7 2 42 0000 LY R 0000 r R sin theta setq lower t Notes The metric is conformally flat please verify that Ortocartan has recognized it Here we asked the Lisp system to treat upper and lower case letters as different symbols the first line of input The command in the last line of the input reverses the command from the first line The output is this some irrelevant or empty lines were deleted by the text editor The irrelevant lines are not parts of Ortocartan s output but are responses of the Lisp system that are not interesting for the use
50. bscripts and exponents all in their proper places Derivatives are printed according to the common convention in general relativity the name of the function is followed by the comma and by subscripts being the names of the variables of differentiation For example dF dx written on input as der x F will be printed as F while 04G 0z0y will be printed as Gay Unfortunately most Lisp systems do not support sophisticated text editing and cannot print more elaborate signs like the integral Therefore indefinite integrals are printed similarly as they stand in the input e g f Fdz will be printed as int x F See however section 23 in Ortocartan and in the associated programs one can write the output on a disk in the form of Latex code and then the printout may be passed through Latex with more satisfactory results in some cases 6 Typing the input The whole input described in the sections that will follow can be typed in directly from the keyboard However it will often contain misprints or will reguire adding new substitutions and resubmitting for calculation Therefore the user will usually want to prepare an input file and let Ortocartan read the data from there How to write the data is described in the next section You can save the data file using any text editor When you want Lisp to read the data from the file you have to write while working within Lisp rdf lt the name of the file gt 11 where lt the name of
51. d named vtida for tidal matrix of velocity The contravariant uacce and covariant lacce components of the acceleration field The rotation tensor wag named rotdd The mixed rotation tensor waf named rotdu The square of the rotation scalar The covariant projdd and mixed projdu components of the projection tensors Gap Halls and g u ug The expansion scalar The covariant sheardd and mixed sheardu components of the shear tensor and the square of the shear scalar The matrix of covariant derivatives of the covariant acceleration g called atida for the tidal matrix of the acceleration The rotation constraint equations Uo U oy Ug aWgy 0 square brackets on indices denote antisymmetrization round brackets on indices denote symmetrization The components of these equations are printed with the name rotcons The shear constraint equations A Dan nl 0 209 La Td 0 under the name shearcons The rotation evolution equations yp r yp r 2 ha hg wys a hg Haal 2050 p 30 a 0 under the name rotevol The tetrad components of the Riemann and Ricci tensors and the curvature scalar The Raychaudhuri eguation 1 2 a a a a B 0 30 Wa O Tag w Wag Rogw u U under the name raychaudhuri equation The tetrad components of the Weyl tensor The coordinate electric components of the Weyl
52. data and see what happens setq lower nil squint a first integral of the Newtonian equations of motion the final result constants m parameter t functions x t y t z t V x y z variables x y z integral 1 2 x m der tx 2 der t y 2 der tz 2 V markers M substitutions maineq der t t M der M V m dont print maineq setq lower t The result is a first integral of the Newtonian equations of motion the final result 2 2 2 gt integral V 1 2 m x 1 2 m y 1 2 m z t t t substitutions maineq 117 THE FIRST INTEGRAL IS ALREADY MAXIMALLY SIMPLIFIED AND IS EXPLICITLY CONSTANT gt maineg 0 I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 130 msec AA AE AA AE AAA KK K K K AA K K K K K K gt K gt K K K K AA K K K AAA AA AAA K 2K K K K K K K KK K You can produce many more examples by yourself if you use the sets of input data recorded in the tes files on the Ortocartan distribution diskette Do not rewrite them but use the editor to cut out single calls to Ortocartan or to the other functions Good luck and enjoy it THIS IS THE END OF THE MANUAL 118
53. dinate component R If an equation is not preceded by any address i e is immediately preceded by another equation or by the atom substitutions then the substitution it defines should be attempted just everywhere This is rarely reasonable and results in an unnecessary extension of the calculation time Note that those substitutions which should be carried out everywhere are attempted first before the addressed ones are considered Thus if some substitutions are addressed and some others are not the user does not have a full control of the order in which they will be executed The following example may show the usefulness of substitutions If the Einstein field equations are yet to be solved for a metric then the ematrix will contain some unknown functions After the Einstein tensor is calculated and the field equations are solved these functions become explicit expressions For instance let ds f r dt g r dr r dh sin hdp For this metric the data are coordinates t r h p functions f r g r ematrix f 1 2 0 0 0 o ea 0 0 r 0 0 0 0 rx sinh From the field equations G 0 it follows then that f 1 g 1 2m r where m is a constant In order to check that this is a solution it is enough to add just two more items to the data listed above 26 constants m substitutions f 8 1 2 m tr 1 2 1 7 01 E 439 2 Actually it would be more reasonable to write 1
54. e functions f x y z and g x y then the appro priate declaration is functions f x y z g x y If composite functions are to be used then for each function only those arguments must be given on which it depends directly The arguments which are themselves functions must be declared on their own For example if the user wants to use the functions g x y and f x y h t u where u is a function of z then the appropriate declaration is functions g x y f x yh h tu u z The order in which the declared functions appear is irrelevant from the point of view of syntax However the functions will be ordered by the program in sums and products according to the same order If the program Ortocartan is applied to some explicitly given tetrad which does not contain any arbitrary functions then this piece of data should be simply omitted Note The built in functions listed in section 5 must not be declared Of course their names should not be used for user defined arbitrary functions The reserved names not to be used as the names of functions include also the atoms nil times plus expt minus and In the formulae described in sections 14 and following only the names of the functions must be written their arguments do not have to be written out see examples I V in Appendix E However if the user wishes so the arguments can be written out Then 18 Ortocartan will automatically pick up this style and will write ou
55. ed TIME 980 msec 1 gt ricci 3R R 00 tt 2 2 2 gt ricci 2kR 2R R R T t 2 2 2 gt ricci 2kR 2R R R 22 t 2 2 2 gt ricci 2kR 2R R R 33 t ricci calculated TIME 1000 msec 59 CURVATURE INVARIANT calculated TIME 1000 msec gt CURVATURE INVARIANT 6kR 6R R 6R R weyl calculated TIME 1090 msec ALL COMPONENTS OF THE WEYL TENSOR ARE ZERO I REALLY LIKED THIS CAN I HAVE MORE PLEASE 717 END OF WORK RUN TIME 1090 msec E 2 Example II The most general spherically symmetric metric in the Schwarzschild coordinate system ds e dt e dr r dv sin vde where v v t r and u u t r are arbitrary functions Reference Most textbooks on general relativity e g J L Synge Relativity the general theory North Holland Publishing Company Amsterdam 1960 p 265 The input data is here ortocartan spherically symmetric standard coordinates t r theta phi functions nu t r mu t r ematrix exp nu 0 0 0 0 exp mu 0000 r0000 r sin theta rmargin 80 Notes nu stands for v mu stands for u rmargin 80 will suit the output to the pagewidth of this text Note the untidy linebreaks in the Riemann and Ricci tensors Some irrelevant and empty lines have been deleted again The output is 96 spherically symmetric standard 0 gt ematrix exp nu 0 1 gt ematrix exp mu 1 2 gt ematrix r
56. ed form like dF dx then the derivative should be written as der the variable the function In this example one should write der X F Multiple derivatives require only inserting the full series of variables between der and the name of the function e g F zey should be written as der X X Y F or der X Y X F or der Y X X F The program is not sensitive to such subtleties as commutativity of derivatives or even differentiability of the functions It just assumes that all the functions to be differentiated are differentiable and that in multiple derivatives all the differentiations commute Partial derivatives are denoted by the same symbol as total derivatives Sometimes it is hard to calculate by hand an explicit derivative of a large expression which must be inserted into the data In this case the task of calculating the derivative may be left to the program The format for writing such a derivative is the same as above only instead of the atom der one should write the atom deriv For instance suppose one is too lazy to calculate explicitly the expression d dx sin x ax sinf x ax log z sin x but such a derivative must be inserted into the data Then one should write deriv X sin X A 2 X sin X A 4 X log X sin X and the program will do the work Actually der may be always substituted by deriv and it is even advisable to do so in case of multiple partial derivatives In this case the series
57. ed result All the guantities that bear these names though calculated for the needs of further calculation will not appear in print For example if the reguest dont print ie determinant agamma gamma ricci is found in the data then none of the quantities e det e Ij I and Rj will be printed The series of names may also contain any of the atoms substitutions modi fications messages and timemessages Inserting them into the series will have re spectively the following results the list of substitutions will not be printed the list of substitutions modified by data substitutions will not be printed the messages about the unsuccessful attempts at substitutions will not be printed and the information about tim ings shown in sec 9 will not be printed The order of the atoms in the series is irrelevant 21 Formatting the output The linelength of the output on the screen is 72 characters However the user may wish to produce a narrower output e g in order to be printed and inserted in a typed manuscript or a wider output e g in order to use the paper more economically It is then possible to change the linelength of the output by inserting the following item into the data for Ortocartan rmargin lt n 1 gt where n is the required new linelength This will be useful mainly if the user wishes to print the results on wide paper with continuous feeding in this case the typical parameters are rmargin 120
58. elligible uniguely only when accompanied by some explanatory text which of course cannot be supplied to the computer in the form of English phrases Sums are represented exactly as in mathematics One can omit the sign before the first term of a sum or insert it both cases are legal One must only be careful to separate the and signs from neighbouring terms by blanks or parentheses so that they form separate atoms Also when a sum is too long to fit into one line of input and must be continued in the next line then the or sign may be placed either at the end of the preceding line or at the beginning of the following line but not in both these places simultaneously as this would cause an error the second sign would be understood as a symbol of some guantity Products are represented as in mathematics with the restriction that the multiplica tion sign must not be omitted and must be separated by blanks or parentheses from neighbouring factors Exponentiations are represented differently because writing them in coupled parallel lines would cause hard problems for the Lisp input procedure So exponentiation is written in one line format in the following form the base the exponent just like in FORTRAN For instance AP will be represented by A B In some versions of Lisp the symbol automatically splits any sequence of characters into separate atoms In those versions the symbol
59. ement the denominator of a fraction e g 1 2 will be represented as 1 2 Negative fractions may be also represented in two ways e g 2 3 may be written as 2 3 or as 2 3 Both representations are correct but the first one is recommended Functional expressions must be represented as lists the first element of the list being the name of the function and the following elements being consecutive arguments of that function Note each argument or the argument if it is single of a function must be a single S expression i e either a single atom or a single list For example the correct representation of sin xy is sin X Y while sin X Y would be understood as a functional expression in which the function sin is given three arguments X and Y Of course such a three argument sin would be processed by the program guite incorrectly The program knows and can process automatically the following functions exp log which stands of course for natural logarithms to base e cos sin tan ctan cosh sinh tanh ctanh arctan arcsin arsh the function inverse to sinh arch inverse to cosh and arth inverse to tanh All these functions are correctly differentiated and the most important simplifications like exp log X X log X Y log X log Y tan X ctan X 1 or sin X 2 cos X 2 1 are made on them If it is necessary to introduce a symbol for a derivative of some function of unspecifi
60. exp r psi dont print messages tensors einstein setq lower t Notes This is a stationary cylindrically symmetric solution of Einstein s equations with a rotating dust source and with a nonvanishing cosmological constant A The coordinates used in the metric shown above are comoving and the velocity vector field of the dust is one of the orthonormal tetrad vectors hence the tetrad components of velocity field are 100 0 Since this is a solution of Einstein s equations this vector field is uniquely determined by the metric and so as expected all the constraint and evolution equations will be identities However the acceleration 0 rotation expansion 0 and shear 0 are all calculated along with the electric and magnetic parts of the Weyl tensor This example is well suited to try out the output for latex option try it yourself The substitution in line 5 from the bottom was reguested to be done everywhere this is usually not a reasonable option because it makes the calculation slower It was done this way here in order to use it together with the dont print messages option that suppresses all the messages about unsuccessful attempts at substitutions The Einstein tensor calculated along the way makes it possible to easily see that it is a dust solution indeed LANCZOS METRIC symbols 2 gt psi Lambda Lambda exp r HC r substitutions everywhere 92 2 gt Lambda Lambda exp
61. gt 1 3 exp 2 mu nu 1 3 exp 2 mu m mu 1 3 r rr r r weyl calculated TIME 1430 msec I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 1430 msec This example provides a good opportunity to demonstrate the output in the Latex format Let us run the same example with the data modified as follows ortocartan spherically symmetric standard coordinates t r theta phi functions nu t r mu t r ematrix exp nu 0 0 0 O exp mu 0000 r0000 r sin theta output for latex dont print ematrix determinant ie agamma gamma riemann stop after ricci 64 The command rmargin is not reasonable here in fact with the output for latex command Ortocartan would ignore any rmargin command because the final layout will be made by Latex anyway In order to save space we have asked Ortocartan not to print anything before the Ricci tensor and to stop after calculating the Ricci tensor i e to give up on the scalar curvature and the Weyl tensor The output was produced in the Latex code that is first shown verbatim spherically symmetric standard ematrix completed TIME 240 msec DETERMINANT EMATRIX calculated TIME 270 msec ie calculated TIME 460 msec agamma calculated TIME 780 msec agamma completed TIME 780 msec gamma calculated TIME 830 msec gamma completed TIME 830 msec riemann calcula
62. hat instead of xz y U the substitution x U y be declared The same concerns products instead of a xb c one may write a c b or b c a The general principle is the left hand sides of the substitutions should be literally present in the expressions where they are to be substituted for Another example of a trouble if a sum is multiplied by a factor then the substitution for the non multiplied sum might be missed For example if X Y U then the program will not recognize that aX aY aU However the facility of substitutions is flexible enough to overcome such difficulties with a little help from the user The facility of substitutions may help to bypass some conventions of our system were they inconvenient in a particular case For example the program changes any even power of cos x to the appropriate power of 1 sin x just for uniqueness Were this inconvenient one may define an atomic symbol e g COSX to stand for cosa and then write the equation COSX cos x in the symbols and the equation cos x COSX in the substitutions Then COSX will appear instead of cos x in all the prints and will not be replaced by sin x See example V in Appendix E In addition to the literal substitutions described above one can also carry out another kind of substitutions in which the left hand side of an equation defines a pattern to be looked for and the right hand side says what to do with an expression t
63. hat matches the pattern For these substitutions one should insert an item of the form 28 markers lt a series of arbitrary atomic symbols gt into the data For example markers M1 M2 M3 M4 If such a command is found in the data then the program will understand that M1 M2 M3 and M4 will be used to stand for anything in the substitutions With the markers the user can define just a pattern to which an expression should conform in order to be replaced by some other expression For example if one writes in the substitutions sin x M1 1 CS x 2 x sin x 7 M1 2 then any power of sin z will be replaced by sin x th old power 2 1 78 7 where CS r is a user defined function note the danger if sin x happens to appear with an exponent that is negative or fractional or even is not a number at all then the substitution will be performed anyway If one would like such a substitution to be performed not only for sin z but for sin of any arbitrary argument then it is enough to use another of the markers instead of x e g sin M2 M1 1 CS M2 2 sin M2 Mi 2 One should not worry about using the same marker in different formulae each time with a different meaning the program assigns a meaning to each marker in each substitution anew and does not remember the meanings the markers could have had before Suppose the cartesian radius r 1 y 22 appears in a calculation One can t
64. he constants gt For example 19 constants M constants KAPPA RHO H The order of the constants has the same meaning as the order of the functions If no arbitrary constants appear in the metric then this piece of data should be omitted 14 Declaration of the tetrad Let us recall that the program Ortocartan can process only orthonormal tetrads in the signature The tetrad is declared in the form ematrix components of the matrix e in the order i 0 A 0 i 0 A 1 i 0 A 2 i 0 A 3 i 1 A O i 3 A 3 and in the notation described in section 5 gt For instance for the Schwarzschild metric ds 1 2GM c r dt 1 1 2GM c r dr r dd sin dy the declaration should look as follows ematrix 1 2 G M c 2 r 12 O O 0 Gard 2 FG M c 2 r 12 00 O 0 r 0 0 O O r sin theta Note that each component of e in the above list must be a single S expression i e must be embraced by parentheses if it is not an atom This piece of the data is the heart of the problem so needless to say its omission would push the program into a fatal error The omission of some relevant parentheses usually causes that the list becomes too long i e has more S expressions than the expected 17 This kind of error is directly communicated by the system Ortocartan The pieces of data described up to this place form a minimal set of data for Ortocartan This is all if the user h
65. he left hand sides of the eguations will be used while differentiation will operate on the right hand sides Example take the Nariai metric ds a t cos log r 1 b t sin log r 1 dt dr r d sin ddyp where is a constant and a and b are arbitrary functions of time Suppose you want the coefficient of dt not to be developed in products and you call it 4 Then the declaration should be symbols psi a cos log r 1 b sin log r 1 Of course this should be accompanied by 21 coordinates t r theta phi constants 1 functions a t b t ematrix psi 0 0 0 o a r 0 0 0 0 1 0 0 0 O 1 sin theta After such a call to Ortocartan psi will be used in all the algebraic operations but for Ow ot and Ov Or the program will automatically substitute respectively the quantities da t cos log r 1 0b 0t sin log r 1 and a r sinllog r l b r cosllog r l This example is shown in full in Appendix E The usefulness of symbols is most evident in the case of sums but the expressions represented by the symbols need not be sums The symbols declared on the left hand sides of the equations should not be declared separately as functions if they are this will have practically no result The equations listed in the symbols argument are printed in the mathematical format at the very beginning of the printout preceded by the heading symbols Note Before being prin
66. he program may also calculate the tetrad components of the Einstein tensor defined by Gy Ry 1 2 ng R 15 as well as the coordinate components of all the quantities including the metric tensor and Christoffel symbols With the exception of the Christoffel symbols for which the valence is fixed the indices of the tensor components may be in any desired positions e g for the Riemann tensor one may obtain Ragys and R g 5 and BP s and so on The tensor components are calculated as secondary objects by contractions of the sets of tetrad components of the appropriate guantity with the tetrad vectors given on input or with the inverse tetrad vectors found in the first step For example Fas eel gel yes Rosa 16 Wa ee gee s Rijk 17 Just how these and some other additional requests of the user may be communicated to the program will be explained further To use the program one does not need any special knowledge We only expect our users to be familiar with simple text editing and copying files All the remaining tiny amounts of information will be supplied by the present text In the first place the user must know two elementary notions of Lisp the atom and the list 3 Atoms The definition of an atom given below is with respect to the general Lisp definiton a restricted one so that it fits the needs of Ortocartan An atom is either an integer number or a continuous string of up to 72 characters which does not begin
67. he substitution may take guite a while It is more reasonable to direct those additional substitutions to the eguations This is done by writing equation lt n1 gt lt n2 gt lt n3 gt in front of the appropriate substitution where the numbers lt n1 gt lt n2 gt lt n3 gt each necessarily in parentheses refer to the eguation numbers Whatever devices of the main program Ortocartan make sense here can be used The last item of the data are the two right parentheses The integral need not necessarily be a polynomial of second degree in the f It can be a polynomial of first degree in f or it may be independent of the derivatives However the program squint will go wild and produce a nonsense result when you try an integral that is a polynomial in f of any degree higher than 2 or if the second derivatives of the f are not eliminated from dI dt D Versions of Ortocartan for different computers As mentioned in the introduction Ortocartan was originally written and implemented in the University of Texas Lisp 4 1 on a CDC Cyber 73 computer Those computers were scrapped in all sites where Ortocartan was used on them The file with the U T Lisp code of Ortocartan is still preserved and can be obtained from the author A K but since I do not have access to the U T Lisp myself this version will not be maintained further and has probably already become defunct which is a pity because the U T L
68. hen place r x y z in the functions and write substitutions der x r x r der y r y r der z r z r 3 With the markers the same effect can be achieved by writing just one equation substitutions der M1 r Mi r Let us stress that markers can symbolize not only atoms but any arbitrarily compli cated expressions For instance with 29 substitutions M1 1 2 Mi M1 120 also an expression like a b V will be replaced by a a DUS b3 a b M If M1 and M2 are markers then M1 M2 means any sum and M1 M2 means any product For instance the equation 2 log M1 M2 log M1 M2 2 means whenever a logarithm of any sum is multiplied by 2 sguare the sum and drop the coefficient 2 The markers can also be used to represent arbitrary parts of sums and products provided their meaning can be guessed uniquely For instance A B MI C M2 D means a sum in which anything stands between B and C and anything stands between C and D A B M1 means any sum starting with A B but A M1 M2 C is not unique if several terms stand between A and C then there will be no way to tell which of them should go into M1 and which into M2 Such a pattern will result in an error message The equation M lt M a b 7 2 c with M being a marker means multiply the equation whatever it was by ab c In this case M st
69. hird and fifth substitutions use the definition of D but apply it only in certain contexts For example in the first substitution V is replaced by V D F R R but only in those instances where V is multiplied by D Each consecutive substitution was guessed after inspecting the output obtained without it You are encouraged to repeat this procedure Run this example first without any substitutions then add the first one and see what has changed then add the second one and so on If you are clever with using your editor then you may use it to cut the data for this example out of the file containing this manual The output is here again with irrelevant lines deleted the Stephani solution substitutions riemann 0 1 0 1 0 2 0 2 0 3 0 3 1 1 gt V D R VR D F VDD t t t t t agamma riemann 1 1 gt D F V V R F R gt m t t m 69 riemann gt R RF D RV V t t riemann gt V k kV V V z X y riemann 1 1 1 gt V V R R F D riemann gt V 1 1 2 x k X0 1 2 y k Y0 gt 1 2 z k Z0 1 2 2 2 2 gt 4 k XO 1 4 k YO 1 4 k ZO 1 4 x k 2 2 gt 1 4 y k 1 4 z k riemann Il iw gt ematrix 1 1 gt ematrix R V 70 2 1 gt ematrix R V 3 1 Il ao lt gt ematrix 3 ematrix completed TIME 500 msec 3 3 gt DETERMINANT EMATRIX R V D DETERMINANT EMATRIX calculated TIME 520 msec 0 1 gt
70. is always annoying and carries the risk of typos This can be largely avoided thanks to the following two conveniences If the formulae produced by Ortocartan should be used as input for another run of Ortocartan then you should insert the following item into the data output for input Then the whole output will be written in the same notation as is used on input and every piece of it will be directly readable for Ortocartan as its data Usually you will want to use only parts of this output but then all you have to do is very simple editing cutting and pasting without any rewriting If the some of the formulae produced by Ortocartan should be used as a part of a text for publication then you should insert the following item into the data output for latex Then all the formulae will be written as expressions readable for the system Latex You will only have to add your favourite Latex preamble and run Latex on them they will then show up as a neat Latex output with all equations numbered Ortocartan will automatically replace the names of the Greek letters with their Latex codes e g if you use alpha as the name of the variable then it will be replaced by alpha in the Latex code and appear as a in the output from Latex The printing in Latex format is not 100 safe against Latex errors It may happen that the end of line in the output from Ortocartan will occur in a place that is not acceptable for Latex Then Latex wil
71. isp was a dialect of superb elegance The versions of Ortocartan written in an older implementation of Cambridge Lisp for IBM 360 370 computers and in the Slisp 360 version for Siemens 4004 computers have been defunct for some years already Information is missing on the Lisp 1108 version 49 for UNIVAC computers but it was not under our care anyway You may try to obtain information on it from Dr Gokturk Ucoluk Fizik Bolumu ODTU Ankara Turkey The Cambridge Lisp version for the Atari Mega ST computers still exists and works although the algebraic computing community seems to have taken divorce from these computers guite a while ago It will probably become defunct together with A K s Atari computer that may be among the last ones still working The previous edition of this manual describes how to use that version it can be obtained from A Krasinski The only version now under maintenance is the Codemist Standard Lisp version that will run wherever CSL can be used See Appendix A for more details E Sample prints In this appendix we present copies of original outputs from the computer The examples exhibit the various features of the program The printouts are broken into lines without obeying the standard rules of neat printing such as do not divide expressions of the form f x do not jump to the next line just before the right parenthesis or just after the left parenthesis do not separate the base from its exponent and so on
72. istance of your computer staff We assume that you have the Lisp already at your disposal and describe how to load the Ortocartan programs into it Since the system Ortocartan will be written into the computer s core on top of Lisp as a core image make sure first that you have a second copy of the pure Lisp core image to which you can revert if you wish to use Lisp without Ortocartan The diskette with the Ortocartan programs contains among other things the following files 1 Ortcsl lis containing the program for calculating the curvature tensors as described in Section 2 This is the main program and all the other programs make use of parts of this one The main program can be used alone the other programs must be loaded into core only together with the main one How to use the other programs is described in the Appendices B and C 2 Calcsl lis this is the algebraic abacus program 3 Elliscsl lis This is the program that calculates the Ellis evolution eguations 4 Neurvesl lis This is the program for calculating the curvature tensor for given connection coefficients in an arbitrary number of dimensions 5 Lanlages lis the program for calculating the lagrangian by the Landau Lifshitz method for a given metric 6 Eulagcsl lis the program for calculating the Euler Lagrange equations from a given lagrangian 12 T Squintcs lis the program for checking the first integrals quadratic in first derivatives The files
73. its inverse metric Christoffel symbols christoffel Riemann tensor riemann Ricci tensor ricci Weyl tensor weyl Einstein tensor einstein valence N required for quantity K gt is a list of the signs and the in the i th position of the list meaning that the i th index should be an upper one and the in the j th position meaning that the j th index should be a lower one For instance tensors riemann means that the user wants the components R g and R g 5 of the Riemann tensor to be calculated and printed If not all indices are on the same level i e are not all contravariant or not all covariant then the practical use of the symmetry properties may be difficult as lowering or raising an index with a nondiagonal metric often reguires much algebra In this case the symmetries which are not trivial i e do not amount just to eguality of some components or to the eguality of a component with the negative of some other component or to vanishing of a component are not taken into account and the program prints the dependent components too For instance when printing R g only the trivial symmetry Ra R 5 5 will be used while the symmetries corresponding to Ragy Rgays and Rassi Rage will be ignored i e 136 components if all nonzero will appear in print When printing R gy5 only R g 5 R gs will be used while Ragys Rysag and Rogys Rgays wi
74. l signal an error You may also prefer to have the Latex linebreaks in other places than Ortocartan will place them Such failures will have to be corrected by hand 35 24 Errors The program Ortocartan has some built in tests for correctness of the data If one of our tests finds an error then an error message is printed which should directly identify the kind of error that was made However only some of the possible errors can be identified uniquely For beginners in Lisp computing the most likely error is incorrect placement of paren theses If Ortocartan refuses to start working then almost surely some left parentheses in your data are unmatched If you have written the data directly from the keyboard you may try your luck by writing a few right parentheses and pressing RETURN again However if the parentheses were distributed incorrectly then a fatal error of one kind or another is sure to occur and the error message can be quite beside the point especially when the data are read from a disk There is no way for the Lisp system to recognize in correct distribution of parentheses It knows what you want it to do only by reading your data as lists and the parentheses divide the data into sublists Placing the parentheses incorrectly may turn some sublists into a mathematical nonsense but they will still be meaningful S expressions and Lisp will take them literally Hence whenever you get an obvious nonsense as a result or
75. ll be ignored i e 96 components if all nonzero will appear in print One can use the same name repeated a few times each time followed by a different valence e g tensors weyl weyl weyl but this is equivalent to the simpler expression where the name is used only once tensors weyl If a name of some quantity is not followed by a valence specification then it is understood that all the indices of the tensor should be down covariant There are two exceptions to this 23 1 The christoffel has an obvious valence which thus need not be specified 2 The einstein not followed by a valence specification is understood as a request to calculate the tetrad components of the Einstein tensor which are not calculated in the routine run of the program Each specification of valence or name of a guantity if no valence is present may be followed by a series of sets of indices each set defining a single component of the tensor Such a form will be understood as a reguest to calculate only the components thus defined For instance tensors riemann ricci 0 1 0 2 1 1 weyl 0 10 1 means that the user wants the program to print all the components R 5 5 of the Riemann tensor and all the components Rag of the Ricci tensor while of the components R g of the Ricci tensor and C545 of the Weyl tensor only RP R95 R and Coioi should be printed
76. loating point numbers are illegal in Orto cartan 4 Lists A list is a string of characters starting with the left parenthesis ending with the right parenthesis and containing atoms or other lists separated one from the other by single or multiple blanks or by single commas Many consecutive blanks have the same meaning as a single blank Two or more consecutive commas or commas separated only by blanks form in some Lisp systems an illegal character set and they result in an error Blanks are not necessary but allowed in front of and behind a parenthesis Examples of lists this is an empty list having no elements ATOM SOME MORE ATOMS GUITE A COMPLICATED LIST Examples of strings which are not lists SOMETHING IS MISSING one right parenthesis missing This is a very common type of error and one should be careful to avoid it It leads to unpredictable error messages that can be very misleading AGAIN SOMETHING MISSING one left parenthesis missing or one extra right parenthesis WHAT IS THIS this is a series of two lists Atoms and lists together are called symbolic expressions or S expressions 5 The representation of mathematical formulae in Or tocartan The notation reguired for Ortocartan is similar to the conventional mathematical notation The restrictions or changes result from the fact that the mathematical notation is in some points nonunigue and is int
77. ma Lambda exp 1 2 r psi C exp 1 2 r 21 1 2 gt psi 2 1 2 gt gamma exp 1 2 r psi 33 96 2 gamma calculated TIME 810 msec gamma completed TIME 810 msec 0 christoffel 02 0 christoffel 3 2 C r psi 1 1 christoffel 02 1 christoffel 1 2 1 2 C psi 2 christoffel 1 2 christoffel 1 r psi 2 1 2 C r psi 1 1 1 2 C 1 2 C Lambda r exp r psi 1 3 2 1 1 2 C r psi 1 1 2 C psi 1 2 1 1 2 Lambda exp r psi 1 2 C r psi 2 C exp r psi 2 2 Lambda psi 4C r exp r psi 2C 2 exp 97 2 1 gt christoffel 1 2 1 2 Lambda exp r psi 1 2 C 22 2 1 gt psi 2 gt christoffel 2 psi 33 3 gt christoffel 1 2 23 CHRISTOFFEL SYMBOLS calculated TIME 1000 msec CHRISTOFFEL SYMBOLS completed TIME 1000 msec gt vtida 1 2 C gt vtida 1 2 C TIDAL MATRIX OF lvelo completed TIME 1050 msec ACCELERATION Il o gt rotdd 1 2 C 1 2 98 rotdd calculated TIME 1060 msec rotdd completed TIME 1070 msec 2 gt rotdu 2 C exp r psi 1 0 2 1 gt rotdu 1 2 C r psi 2 1 zd gt rotdu 1 2 C psi 2 rotdu completed TIME 1100 msec 2 gt ROTATION SQUARED C exp r ROTATION SCALAR calculated TIME 1100 msec gt projdd psi 11 1 gt projdd 1 4 exp
78. ma calculated TIME 850 msec agamma completed TIME 860 msec 0 1 1 ml gt gamma L xr aP sin log L log r 10 1 ii Sl 1 d sel gt L xr a b L xr a b P sin C log gt L log r 0 zu 1 1 gt gamma L yr aP sin log L log r 20 si al Sl zu Ti 102 1 gt L yr cm b L yr a b P sin C log gt L log r 0 1 1 1 gt gamma L zr aP sin log L log r5 30 1 sd sei sl A2 Si gt L zr a b L zr a b P sin C log 80 gt L log r 1 1 1 gt gamma El cyr 21 1 1 1 gt gamma L xr 22 1 1 1 gt gamma iL ow 31 1 1 1 gt gamma L xr 33 2 1 1 gt gamma Sob zr 3 2 2 1 1 gt gamma L yr 3 3 gamma calculated TIME 970 msec gamma completed TIME 980 msec D 2 2 gt riemann L x r 01041 8l riemann riemann riemann riemann riemann riemann riemann riemann riemann riemann riemann 0 0 0 0 0 1 1 1 1 1 2 82 riemann calculated TIME 3260 msec riemann completed TIME 3300 msec 2 gt ricci L 00 2 gt ricci L 1 1 2 gt ricci L 22 2 gt ricci L 33 ricci calculated TIME 3330 msec CURVATURE INVARIANT calculated TIME 3330 msec 2 gt CURVATURE INVARIANT AL 2 2 2 32 gt weyl L xr 1 3 L 0101 2 2 gt weyl L xyr 0102 2 2 gt weyl X Z r 0103 83 lt 2 2 2 2
79. n eguations of motion for a point particle of mass m in the cartesian coordinates x y z from the lagrangian 1 L sme 9 2 V e y 2 where V is a potential and z t y t z t are the equations of a trajectory of the particle The input data are setq lower nil eulagr The lagrangian for the Newtonian eguations of motion in 3 dimensions constants m parameter t functions x t y t z t V x y 2 variables x y z lagrangian 1 2 m der t x 2 der t y 2 der t z 2 y setq lower t and the results are The lagrangian for the Newtonian equations of motion in 3 dimensions 2 2 2 gt lagrangian V 1 2 m x 1 2 m y 1 2 m z t t t 111 THIS IS THE VARIATIONAL DERIVATIVE BY x gt eulagr m x V 0 tt x THIS IS THE VARIATIONAL DERIVATIVE BY y gt eulagr my V 1 tt y THIS IS THE VARIATIONAL DERIVATIVE BY z gt eulagr m z V 2 tt z I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 100 msec E 11 Example XI Application of the program squint In order to make the result easy to verify we shall use the program squint to find a first integral of the eguations found in the previous example We shall at first pretend that we do not know what the integral should be and will assume that it is a general polynomial of second degree in the first derivatives by t of the functions x t y t and z t
80. ndices B and C were written It is unlikely that any user would find this useful today However the facility still exists so its description is retained As mentioned before the data written by the user are before applying processed by the procedure of algebraic simplification so that the user may write them in a form that is more compact than the final one e g one may leave the task of carrying out a multiplication by a sum or calculating a derivative to the program itself For this reason however the final form of the data may appear inconvenient e g clumsy or too extended In such a case the result may be smoothed out by the user either by rewriting the input data or by applying the substitutions Consider however the following example taken from a paper by J Kowalczynski Let SE Dm 2blog ZC ap blog q a p 4bZC p 1 3 hp 1 where a 1 or a 1 b c h and m are constants t 6 r and z are coordinates and ZC z e p PyP q m i t reos The svmbol S appears in the metric and so is differentiated during the calculation How ever the automaticallv calculated derivatives of S are rather messv for example 2 2 2 1 2 2 2 2 1 2 dS dt abtp t r log q 2abtp t r log ZC ap 6 1 8 2 2 1 2 8abtp q t r cos phi tr 7 1 9 2 2 2 1 2 8abp q t r cos phi 2amtp t r 3 2 2 1 2 2 2 1 2 8btp ZC t r 2 3 htp t r 2 a il 1 2 2 1 2
81. o then the arguments of functional expressions can be written out explicitly Most of the output is suppressed The output is SPHERICAL WITH ARGUMENTS 0 gt ematrix exp NU T R 87 1 gt ematrix exp MU T R 2 gt ematrix Il DI 3 gt ematrix R sin THETA ematrix completed TIME 50 msec 2 gt DETERMINANT EMATRIX R exp MU T R NU T R sin gt THETA DETERMINANT EMATRIX calculated TIME 90 msec ie calculated TIME 180 msec agamma calculated TIME 290 msec agamma completed TIME 300 msec gt gamma exp MU T R NU T R 10 2 gt gamma exp NU T R MU T R 11 1 88 1 1 gamma R exp MU T R 22 1 1 gt gamma R exp MU T R 3 3 2 1 1 gt gamma R cos THETA sin THETA 3 3 gamma calculated TIME 340 msec gamma completed TIME 340 msec riemann calculated TIME 530 msec riemann completed TIME 560 msec 1 gt ricci 2R exp 2 MU T R NU T R exp 00 7 2 gt 2 MU T R NU T R exp 2 MU T R 2 gt NU T R exp 2 MU lt TD MU T R 25 2 2 NU T R exp 2 NU T R MU T R 2 1 89 exp 2 NU T R MU T R exp 2 NU T 11 R MU T R NU T R 1 1 1 ricci 2R exp MU T R NU T R MU T R 0 1 1 1 ricci 2R exp 2 MU T R MU T R exp 11 2 2 2
82. or turns out to be zero then again it makes no sense to go on calculating the other guantities which are all zero so the work is stopped after printing the appropriate message If the Ricci tensor turns out to be zero then the Weyl tensor will be just egual to the Riemann tensor so also in this case the work is stopped after printing the message about the Ricci tensor being zero The message END OF WORK marks the proper end of the printout unless an error occured during the calculation see below The printout are the results of the calculation interspersed with the time messages of the form ematrix completed TIME lt ni gt msec DETERMINANT EMATRIX calculated TIME n2 msec ie calculated TIME lt n3 gt msec agamma calculated TIME lt n4 gt msec gamma completed TIME lt n5 gt msec and so on where lt n1 gt to lt n5 gt are times in milliseconds elapsed from the start of the work on the current problem until completing the appropriate step of the calculation A quantity is calculated when all its independent components were found and it is completed when also the dependent components were found with use of the symmetry properties The time messages are not always useful and they can be suppressed see sec 20 If an error occurred during the calculation then the printout is just suddenly termi nated and followed by an error message If no error occurred then the end of the printout is
83. program Ortocartan is devised for automatic calculation of the curvature tensors and some related guantities in general relativity from a given orthonormal tetrad representa tion of the metric tensor It was originally written in the University of Texas Lisp 4 1 programming language and implemented on a CDC Cyber 73 computer That version and a few later ones went out of operation together with the computers on which they were implemented As for today the only existing versions are 1 In Cambridge Lisp on the Atari Mega STE computers and 2 In Codemist Standard Lisp for the Windows 98 and Linux operating systems The latter is now the main version and this description will deal only with this one An older edition of this manual is available for the Cambridge Lisp version Additional programs based on Ortocartan that perform other kinds of calculation are described in the Appendices B and C Please send all correspondence concerning the program to A Krasinski 2 The algorithm modelled by the program The input data for the program is the tetrad of differential forms e E e dr 1 i 0 1 2 3 a 0 1 2 3 summation over all the values of a repeated index is implied The forms e represent the metric tensor according to the formula gagdz dx nijte 2 where 7 is assumed to be the matrix Tij 3 GO GO GO ked o oo eS O oOo i e the tetrad e is orthonormal The program calculates the determinant of the matrix e
84. propriate functions of r and Note how the program was prevented from replacing cos by 1 sin at a too early stage of the calculation this last trick is described in sec 18 The output is laplace equation in cylindrical coordinates symbols 2 2 1 2 gt r x y 85 1 gt phi arctan x y gt cosphi cos phi substitutions everywhere 2 2 2 gt x T everywhere everywhere gt X r cosphi everywhere gt y r sin phi everywhere 3 2 gt sin phi cosphi sin phi sin phi everywhere gt cosphi cos phi I UNDERSTAND YOU REQUEST THE FOLLOWING EXPRESSION TO BE SIMPLIFIED gt deriv x x f deriv y y f deriv z z f THE RESULT IS 86 i gt result r f tr f t f T 1 phi phi T Z Z rr I REALLY LIKED THIS CAN I HAVE MORE PLEASE END OF WORK RUN TIME 550 msec E 6 Example VI The spherically symmetric metric in the stan dard coordinates with the arguments of functions written out explicitly The input data here is setg lower nil ortocartan SPHERICAL WITH ARGUMENTS coordinates T R THETA PHI functions MU T R NU T R ematrix exp NUT R 0 O O O exp WU T R O 0 0 OR 0 0 O O R sin THETA dont print ie agamma riemann stop after ricci rmargin 61 setq lower t Notes This example is in fact a duplicate copy of the example II it is only meant to show that if the user wishes s
85. r Robertson Walker metric 0 gt ematrix 1 0 1 2 1 2 gt ematrix R 1 kr 1 2 gt ematrix T H 2 3 gt ematrix r R sin theta 3 ematrix completed TIME 190 msec 2 3 2 1 2 gt DETERMINANT EMATRIX r R 1 kr sin theta DETERMINANT EMATRIX calculated TIME 260 msec 52 gt ie 1 0 1 1 2 1 2 gt des R 1 kr 1 2 1 1 gt ie r R 2 3 1 1 1 gt ie r JR sin theta 3 ie calculated TIME 400 msec 1 1 gt agamma 1 2 R R 01 t 2 1 gt agamma 1 2 R R 02 t 2 1 1 2 1 2 gt agamma 1 2 r R 1 kr 12 3 1 agamma 1 2 R R 03 t 3 ci 2 172 gt agamma 1 2 r REC n 13 3 at i 1 gt agamma 1 27 R cos theta sin theta 23 agamma calculated 53 TIME 590 msec agamma completed TIME 590 msec 0 sl gt gamma R R 11 t 0 si gt gamma R R 2 2 t 0 1 gt gamma R R 33 1 1 2 1 2 gt gamma r R kr 2 2 1 epe 2 1 2 gt gamma r R kr 33 2 nis AL sl gt gamma r R cos theta sin theta 33 gamma calculated TIME 680 msec gamma completed TIME 690 msec gt riemann R R 0101 tt 54 gt riemann R R 0202 tt 1 gt riemann R R 0303 tot 2 2 2 gt riemann kR R R 1212 t 2 2 2 gt riemann gt kR R R 1313 t 2 2 2 gt riemann kR R R 2323 t riemann calculated TIME 930 msec riemann complet
86. ram landlagr This program calculates the reduced lagrangian for the Einstein eguations by the Landau Lifshitz method This is the Hilbert lagrangian with the divergences containing second derivatives of the metric already removed In short this noncovariant lagrangian is obtained by deleting from the scalar curvature those terms in which the Christoffel symbols are differentiated and taking the remaining part with the reverse sign To make sure that this program is in your core image write infolandlagr and press RETURN The program is called by writing at the beginning of the list of data landlagr The remaining data are exactly like for ortocartan The main part of the data is the ematrix and the quantities printed are the ematrix and its inverse the metric and its inverse the agamma the gamma the Crhistoffel symbols and the lagrangian You can direct substitutions to it with the address la grangian See example IX in Appendix E The last item of the data are the two right parentheses Users are warned that deriving the Einstein equations from a variational principle with assumptions limiting the generality of the metric is tricky and requires detailed knowledge about the problem It may happen that the Euler Lagrange equations will have nothing to do with the Einstein equations this is known for example to occur for certain Bianchi type models Therefore the user must make sure befo
87. re using the program landlagr that the situation is appropriate for using the lagrangian methods C 4 The program eulagr This program calculates the Euler Lagrange equations starting from a lagrangian specified by the user It is assumed that the resulting E L equations will be ordinary differential equations i e that there is only one independent variable in the lagrangian action integral To make sure that this program is in your core image write infoeulagr and press RETURN The program is called by writing as the first item of the data L D Landau and E M Lifshitz Teoria polya 6th Russian edition Izdatelstvo Nauka Moskva 1973 sec 93 See M A H MacCallum in General relativity An Einstein centenary survey Edited by S W Hawking and W Israel Cambridge University Press 1979 p 552 44 eulagr The second item of the data is the title The main part of the data are the following three items see example X in Appendix E parameter lt an atomic name of the independent variable gt variables lt the list of names of the lagrangian variables an arbitrary number of them gt lagrangian lt the formula for the lagrangian written in accordance with the rules of Section 5 gt The variables must be declared as functions in the functions list because the program allows them to depend also on other arguments in addition to the parameter specified in the data For example yo
88. s If you wish the upper and lower case letters to be considered different then write setq lower nil From now on however be careful to write all commands in lower case letters or else Lisp will not recognize them In particular you should write rdf lt filename gt if you write RDF then Lisp will protest that the function RDF is undefined When you wish to give up this additional convenience write setq lower t 13 When you wish to finish the session with Lisp click on the cross in the upper right corner of the screen Should you wish to preserve the Lisp output on a disk click the File in the upper left corner of the screen and then click To file in the dropdown menu Then you will be guided by the icons so that you can choose the directory where the output should be written When you wish to stop writing the output to the file click File again and then click Terminate log Note if you choose to write anything to the same file again after you have terminated the old contents of the file will be overwritten with the new data 8 The dictionary for communicating with the pro gram For the purposes of communicating the user s requests to the program all of the quantities that the program can calculate were assigned their unique names The dictionary of those names is given in the table below STANDARD NAME OF THE QUANTITY PROGRAM NAMES TETRAD TENSOR COMPONENTS COMPONEN
89. see Example XI in Appendix E U C1 46 the coefficients W V and U are functions but their exact forms are irrelevant here Suppose you expect the following expression to be a first integral of this set PP x ini ag where Q L and E are functions as yet unknown of the variables f Then the integral in the data for squint should be integral Q11 der t fi 2 2 Q12 x der t fi der t f2 2 Q13 der t fi der t 3 Q22 der t 2 2 2 Q23 der t 2 der t 3 Q33 x der t 3 2 Li der t fi L2 der t f2 L3 der t f3 E In this case the other parts of the main data should be parameter t variables f1 f2 f3 functions fi f2 t f3 t D11 f1 f2 f3 Q12 1 f2 f3 QIS 1 f2 f3 Q22 fl f2 f3 Q23 1 f2 f3 Q33 f1 f2 f3 L1 f1 f2 f3 L2 f1 f2 f3 L3 f1 f2 f3 E f1 f2 f3 you may allow all the functions to depend on more variables if you wish for example when you suspect that the coefficients of the first integral will explicitly depend also on t The number of the variables and consequently of the equations in the set is arbitrary The program will then print the resulting partial differential equations to be fulfilled by Qij L and E but it is up to you then to solve them First the program calculates and prints the total derivative d dt in the form of the equation maineq lt the calculated derivative dI dt
90. see the table in section 8 so that one cannot get confused about distin guishing them However the tetrad e and the inverse tetrad e have each one tetrad and one tensor index To denote which is which a dot is placed above or below the ten sorial index in print For instance the component e of the matrix e will be printed as 0 EMATRIX appropriate expression 0 while the component e of the inverse matrix e will be printed as 0 IE appropriate expression 0 The program follows a general convention used in relativity theory for all the quantities only their independent nonzero components are printed For instance advantage is taken of the symmetry of the Ricci tensor and of the components Ri only those with j will appear if nonzero in print The same is true for the Riemann tensor of the components Rijki only those will be printed for which simultaneously i lt j i k ll and j lt 1 while k and whose value is nonzero If for some quantity all the components happen to be zero then in order to avoid the user s confusion the message is printed ALL COMPONENTS OF THE name of the quantity ARE ZERO Now if all the Ricci rotation coefficients turn out to be zero then everyone can guess that the Riemann tensor and all its concommitants will also be zero In that case the program stops after printing the message ALL COMPONENTS OF THE AGAMMA ARE ZERO 15 If the Riemann tens
91. t the arguments of the functions throughout the whole calculation see Example VI in Appendix E The derivatives of functional expressions are then printed differently on the assumption that the arguments are not necessarily atoms but can be functional expressions themselves For example let f be a function of x y 2 and z One can then write in the input data PRIDE 1 2 z and the derivatives of F will be printed as 2 2 1 2 2 2 1 2 f 1 2 x x y f Qa y z X 1 2 2 1 2 2 2 1 2 f 1 2 Y LR y f x ty z y 1 2 2 1 2 i Sify Mx y y 2 z 2 the subscripts on the right hand sides referring to the consecutive arguments of f Such an output is less easily readable however and we do not recommend this style The function f from the last example should be declared as being dependent on x y and z Note the correct format for writing the functional expression in the input data it must be a list of the form lt function name gt lt argument 1 gt lt argument 2 gt lt argument n gt The whole functional expression must be enclosed in parentheses the series of arguments must not be enclosed in an extra pair of parentheses and the arguments must be separated from each other only by spaces do not use any commas 13 Declaration of the constants This piece of the data tells the program which symbols should produce zero when differ entiated The declaration has the form constants lt names of t
92. ted TIME 1070 msec riemann completed TIME 1110 msec begin equation ricci_ 00 2r 1 exp 2 mu nu _ r Vexp 2f mud a 6H 2 exp E 2 nu mu _ t t Nexp E 2 nu nu _ t mu _ t exp 2 mu 65 nu _ r r exp 2 mu nu _ r mu _ r end equation beginfequation ricci 01H 2r 1 exp au ma mu _ t end equation begin equation ricer 11 2r 1 exp 2 mu mu _ r exp 2 nu mu 16 42 exp 2 nu mu _ t t exp 2 nu nu _ t mu _ t exp E 2 mu nu _ r r exp 2 mu nu _ r mu _ r end equation beginfequation ricci_ 22 r 2 exp 20 mui 271 17 exp 2 mu nu x3 r 1 exp E 2 mu mu _ r r 2 end equation begin equation ricci_ 33 r 2 exp 2 mu r 1 exp 2 mu nu tr r 1 exp E 2 mu mu _ r r 2 end equation ricci calculated TIME 1250 msec I REALLY LIKED THIS CAN I HAVE MORE PLEASE 66 END OF WORK RUN TIME 1250 msec Yes some lines are too long to fit into this page This output was not meant to be shown to humans but to be read by Latex But verbatim is verbatim we do not cheat Now the same output will be inserted into this text as part of the Latex code with the time messages deleted Note how Ortocartan
93. ted and applied they are standardized by the simplifying procedure See therefore if their standardized forms are convenient for you If not make the appropriate adjustments in your data 16 Calculating coordinate components of various quan tities It is sometimes useful to calculate also or just the tensor components of some quantities e g the metric tensor to check the correctness of the tetrad components or the Christoffel symbols to solve the equations of a geodesic The program Ortocartan can do that it can calculate the tensorial components of the Riemann Ricci Einstein and Weyl tensors with any required positions of their indices the metric tensor the inverse metric and the Christoffel symbols In order to have some tensorial components calculated and printed one should insert into the data the item of the form tensors name of quantity 1 valence I required for quantity 1 gt valence II required for quantity 1 gt valence n required for quantity 1 gt name of quantity 2 valence I required for quantity 2 where the name of the quantity should be the identifier of the quantity to be calcu lated and printed in its tensorial form Essentially with two exceptions it is the name of 22 the tetrad guantity which is the source to calculate the relevant tensor The dictionary of these names is THE TENSOR WANTED NAME TO BE INSERTED IN tensors metric tensor and
94. the file gt should include the whole path of access unless the data file is in the same directory as the Lisp core image Remember to write the apostrophe it means that Lisp should not look for a value of lt the name of the file gt but just take the name literally The guotation marks are necessary in order that Lisp tolerates untypical characters like the backslashes or slashes or dots Without the guotation marks each such untypical sign would have to be preceded by the exclamation mark For example this is how I myself ask Lisp to read data for the Robertson Walker metric to be included in this manual further on from a disk file rdf akr ortocar robwal dat After reading such a command Lisp will read the file akr ortocar robwal dat and carry out whatever task it was given in those data Then it will automatically go back to the keyboard for more input When typing the input directly from the keyboard be careful not to continue any atom across the right margin the Lisp system might split one of the next atoms into two If an atom seems too long to fit into the current line then simply press Return or Enter The program will not start running until you close all parentheses and press Return after the last one 7 Starting Lisp and Ortocartan Codemist Standard Lisp must be bought from its owners the Codemist Limited see Appendix A Installing it on your computer will most probably require the ass
95. the inverse matrix e the antisymmetrized parts of the Ricci rotation coefficients ua defined by de T gyje A ef 4 and the full Ricci rotation coefficients T jk defined by Pije Lia jp Ux 5 where Lia Nis T GEE 6 and MU nT sik 7 The matrix 77 is the inverse matrix to 7 numerically identical to nij Further the program calculates the tetrad components of the Riemann tensor Rijki defined by dT T ATS 1 2 R jue A e 8 where pope 9 Rijki nis PU gu 10 the tetrad components of the Ricci tensor Rij defined by Ry Hu 11 the scalar curvature R defined by R n Ri 12 and finally the tetrad components of the Weyl tensor C j defined by C9 RY a 1 2 68 R 1 3 62 R 13 where Ciki Heise us 14 and 577 and 07 are multiple Kronecker deltas defined by the following properties 1 is equal to 1 when none of the values of the upper indices is repeated while the set of lower indices is an even permutation of the upper ones 2 is equal to 1 when none of the upper indices is repeated while the set of lower indices is an odd permutation of the upper ones 2 0 0 in all other cases i e if either any of the values of upper or lower indices is repeated or if the lower indices are not just a permutation of the upper ones The quantities defined above are calculated in every run of the program However on special request of the user t
96. the manual see example V in Appendix E The following differences with respect to the description in the manual must be observed here Section 1 The program can simplify symbolic expressions of in principle arbitrary degree of complexity In fact it gives the user direct access to the operations which are carried out as intermediate tasks in the program Ortocartan Section 2 38 Does not apply here at all Section 7 Whenever you start Ortocartan the function calculate is ready for your use too Section 8 Does not apply here at all Section 9 Several expressions can be simplified in one run of the program First the heading is printed I UNDERSTAND YOU REOUEST THE FOLLOWING EXPRESSION TO BE SIM PLIFIED Then the program prints the expression as written by the user on input translated from the input format of sec 5 to the normal mathematical format without yet simplifying it Next the program prints THE RESULT IS and the simplified expression follows The simplified expression is printed as the eguation result lt the expression gt lt i gt where lt i gt is a subscript running consecutively from 1 for the first expression Such labeling of the printed results makes it possible to direct the substitutions to different results No time messages are printed for single operations only the total run time for the call to calculate will be displayed Section 10 The first line here should be
97. tion to be done the left hand side is the old expression which should be replaced by the right hand side The addresses specify in which formulae the substitution that follows should be carried out The form of the addresses can be best explained on examples If the address is an atomic name of a quantity e g gamma riemann or rie then the substitution that follows should be attempted in each component of the quantity If the name is followed by sets of indices then the substitution should be attempted only in the components defined by these sets of indices For example if the addresses are gamma 0 O 2 00 3 riemann 030 3 rie L R 25 then R should be substituted for L in the components I o and Io3 of the gamma in the tetrad component A393 of the Riemann tensor and in all the coordinate components of the Riemann tensor The atomic names can be also followed by symbols defining the valence the ones described in sec 16 For some guantities e g for all the tetrad components or for the Christoffel symbols the valence is fixed forever and so need not be specified For some others it is relevant For instance rie 0101 030 3 ricci ric 01 L R means that R should be substituted for L in all the coordinate components R9 of the Riemann tensor in the coordinate components Ro and R So3 and in all the coordinate components Rasys then in all the tetrad components of the Ricci tensor and in the coor
98. tively of the exponents with which the expansion should be automatically done everywhere If n1 n2 then only the single exponent n nl n2 will be processed in this way Any value nl O is equivalent to nl O all powers up to n2 th will be expanded If n2 O or n2 nl then no powers will be expanded If one wants some definite expressions which appear in some definite places to be ex panded even if their exponents exceed the upper limit whether changed before by expand powers or not then one should use the facility of substitutions described in the next section 18 Substitutions As already mentioned before the program cannot be as intelligent as a human being Consequently a human being can use much of his her ingenuity to simplify the process of calculation or the results while a program will always follow rigid rules and it will miss some simplifications if they are not of an algebraic nature or if they involve a calculation with rational functions For instance Ortocartan cannot recognize that sin z cos x tan x what is sometimes necessary Therefore the user may introduce some simplifications by his her own The request to make such user defined substitutions is expressed by inserting into the data the item of the form substitutions lt a series of addresses equation 1 gt lt a series of addresses equation 2 lt a series of addresses equation n gt Each equation defines a substitu
99. u may wish to differentiate some of the E L equations with respect to another variable The consecutive Euler Lagrange equations are printed with the headings THIS IS THE VARIATIONAL DERIVATIVE BY lt the name of the lagrangian variable gt as the equations eulagr lt the appropriate equation gt lt index gt The names and indices are needed to address the substitutions Whatever devices of the main program Ortocartan make sense here can be used and all of Ortocartan s conventions apply If you wish to stop the calculation at an earlier stage you can write stop after lagrangian this would make sense if you want to just check whether the lagrangian you wrote in the data has no errors in it or stop after eulagr lt n gt where lt n gt is the index of the last Euler Lagrange equation that you want to have Then the whole expression eulagr lt n gt has to be in parentheses The last item of the data are the two right parentheses 45 C 5 The program squint This program verifies whether a given expression is a first integral of a given set of ordinary differential equations The program was written for a specific application therefore it is rather limited in its abilities It is assumed that the hypothetical or actual first integral is a polynomial of first or second degree in the first derivatives of the functions that should obey the set of equations It is also assumed that the equations in the set
100. y gratefully acknowledged The program Calculate see Appendix B and two other now defunct programs were written and implemented as a part of a project supported by the A von Humboldt Foundation at the Max Planck Institute in Garching Germany The Slisp 360 version of Ortocartan was implemented as a part of a project supported by the Deutsche Forschungsgemeinschaft at the Konstanz University Germany The pattern matching substitutions were added together with several other improvements as a part of a project supported by the Deutsche Forschungsgemeinschaft at the University of Cologne A K is grateful to Professors J Ehlers H Dehnen and F Hehl who were the respective hosts of those projects for their kind hospitality The Atari computer on which a newer Cambridge Lisp version of Ortocartan was implemented was bought for funds kindly provided by Professor M Demianski from his grant The implementation of Ortocartan on it was facilitated by the assistance of Professor M A H MacCallum The programs described in Appendix C were first written and implemented with that version The most recent implementation in the Codemist Standard Lisp to be run under the Windows 98 and Linux operating systems was made possible thanks to the grant no 2 PO3B 060 17 awarded by the Polish Research Committee to a research group at the Institute of 37 Theoretical Physics led by Professor Andrzej Trautman The funds from the grant were used to upgrade A K s

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