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ABSTRACT

U>1b rep<a*t !•*• \ui®r'» nanuad tor ALBERT, a package of foW Caiteeh oaiputet prograoe i#hlch calculate analytle expreasicma for

gonaral raiattâUo equations of aotloa.tor. for to. oov,rl.ot «>a

and flw The aotpvt.

The Input for eontraearlant

the olxed co^poiHMttii atreas*«aar(Dr tMsor, •variable irt the discratloa of tha user, Includes nalytle ajqprca^sioRs for ê equations of aotlon, Qtfletetffsl

c

*i

lyabola^ âbc ^ Rieaann tensor,tensor, ^ ^ Eêln tensor, G*^,* and for the Med scalar,H. An ojtlOnil l^lafld for cxMifdiog the anawors In a powwr series

In s^ ptBrfMrlMaPr *, aj|»eara in the liqsut tensors; and foriMortag only teraa of saro, flivt, and second order. ALBERT 1« srtttsf

^ in the IBM language fOOUC, which Is quits slailar to FORfRAS;which, uollks FORTRAH, provides fOr algebraic naaipulatlon of ajnAdip' and tor aaal^^lc dlffereatlatlon of fonetlons. ALBERT should be

UMl^, dtiWFit ■odlfiesUcn| on caoputer which has a FOHHAC

;|êciiliar. A co^lete ^ vrotsnm In ALBERT Is inclMdsdthis rôrt."

m

PBttpswrted In part by the Istlonal Sdenoa FCubtetlOft------«w - -

Office of Maval Raeeareh [lonr-220(tt)J.

ad P. Sloan Foundation Raeeareh Fellow.

^ Df'crrpl!?^rr’n'^

^ THIS TECHNICAl REPORT IS 0M£ OF THE GRAPE AIP SER ‘ SCIENTIFIC MEMORANOA, TlOttflCH REPGfTS < HlECElt

iri

J;WI TINGS.l9Ptm-*v

Blank Page

TABLE OF COWTENTS

Page

I. Introduction and Motivation 1

*II. Brief Description of ALBERT 2

III. How to Use the Prograas In ALBERT •*

A. Brief Description of the FORMAC Language ^ * B. EINSTEIN, the Program for Calculating Rlcd and

Einstein Tensors 9

1. Input for EINSTEIN 9

2. Coiplete Listing of EINSTEIN 13

3. Output for EDiSTEIN 21

k. Sample Problems for EINSTEIN 25

C. RUMAHN, the Program for Calculating Rienann Tensors ... 2h

1. Input for RIEMANN 34

2. Conplete Listing of RUMANN 3E

3. Output for RBMANN 40

k. Saaple Problems for RIEMANN 41

D. MOTION, the Program for Calculating the livergence of the Stress-Energy Tensor 42

1. Input for MOTION 42

2. Conplete Listing of MOTION 43

3. Output far MOTION ^

k. Sample Problems for MOTION 51

E. SUBSTITUTE, the Program for Making Changes of Variables . . at

1. Input for SUBSTITUTE 56

# The reader interested In setting up, as rapidly as possible, a working program for calculating Einstein and Rlcd tensors need read only these sections.

I ■'!-

Complete listing of SUBSTITUTE 59

3. Output of SUBSTnwrE 62

4. Sanple Problems for SUBSTITUTE 6k

IV. A Brief Guide to FORMAC Prograsndng 69

A. The FORMAC System 69

B. FORMAC Executable Statements 70

C. Ways to Avoid Exhausting the Computer's Memory 75

V. References 76

11

5i

i. nrraoiwcTioN AND MOTIVAIION

The initial step In nany general-relativity resenrch projects Is to

select a coordinate system and corresponding analytic forms for the metric

tensor, g^, and the stress-energy tensor, W ; and then to conpute the equa-

tions of motion, W ^ b " 0, and the Einstein tensor, Oa , for use in the field

equations 0 ^ » 8« H 1). At later stages, one also has occasional need for the

Riemaan and Ricci tensors, Rabca and R*^, and for the Rlcol scalar, R.

So long as one deals with sinple systems vhlch possess a high degree of

symsetry, the conputation hy hand of W^.^, 0^, Rabcd, Rab, and R is not too

tedious. However, in recent years, the systems under study have become more

and more conplexj and the task of computing the necessary equations of motion

and curvature tensors has beconta more and more formidable. Recent computations

of curvature tensors have sometimes required many weeks of tedious labor, and

the possibilities of mistakes of minus signs and factors 2 in the final results

have been very disturbing.

In response to this situation, we have developed the ccoputer programs

described here for calculating curvature tensors and equations of motion.

These programs have been used extensively over the last seven months in rela-

tivistic astrophysics research at Caltech. (See, e.g., the analysis of the

effects of rotation on the structure of general relativistic stellar models by

Hartle (1967); also, the analysis of nonradlal pulsations of relativistic

stellar model« by Thome and Canqpolattaro (1967).)

After these programs were ccopleted and in use at Caltech, we learned that

Fletcher (1965, 1966) and Clemens and Matzner (1967) have developed independently

cocputer programs for doing the same Job. The three different systems are

described -riefly in Fletcher, Clemens, Matzner, Thome, and Ziameraan (1967).

jn

ZI. BRIEF DESCRIPTION OF ALBERT

AIBERT la a package consisting of four iodapendent eooputer prograw —

EIBSTEIN, RIEMANN, MOTION, and SUBSTITUTE — vhlch perform four Independent

tasks.

EINBTBXN Is a program vhlch takes a metric tensor, g ., and caioulates

for It the Chrlstoffel symbolB, IV^ and rabc, the nixed conponents of the

Rlccl tensor, Ra , the Rlcd scalar, R, and the mixed coqionents of the

Einstein tencor, 0ab - Ra

b - i R t*. The Input to EISSIEIN are functional

forms fcr g^ and g ; and the output are printed expressions for all the

above quantities.

RBMAHN Is a program for calculating the corariaat conponents, R ^ ... aocd'

of the Rlenun curvature tensor for any given metric tensor, g . . The Input

to RIEMANN are functional fonas for g^ and gtb; and the output are printed

expression, for rabc, 1^°, and R^.

MOTIO» Is a program for calculating the divergence, W b b, of a

stress-energy tensor, W^. (We denote the stress-energy tensor hy W b rather

than Ta because the symbol T Is reserved for the tin» coordinate.) The Input

to MOTION are functional forms for gab, gab, and Wab; and the output are

printed expressions for T . , r . CJ and W b . aoc ab ' ■ »o

aiBSTlTOTB Is a program for making changes of variables in long, ca^H-

cated expressions. The Input to SUBSTITUTE are the changes of variables

desired, e.g.

X - - In (l-Sm/x)}

and the expressions in vhlch those changes are to be »de, e.g.

V ■ •■'(■^H •

«II

The output are printed expressions for the desired quantities In terns of the

new variables, e.g.

%k m 2^x

The prograaB EIN8TEIH, RIEMANN and MOTION ore severely limited hy core

storage problems: They are Incapable of calculating curvature tensors and

equations of motion for metrics and stress-energy tensors which are too

coopllcated. (Exaqples of input which will and will not coqpute are given In

ii IH.B.It, Hl.C.lt, HI.D.U, and III.E.4.) However, complicated problems can

be solved by putting Into EINSTEIN, RIBtAHN, and MOTION slnpllfled metric forms,

e.g.

g^ - A(X,Y,T)

rather than

g,^ - ev(x) (1. + A^X,!) P^cos Y) + AgCX.T) PgCcos Y)}}

and by then using SUBSTITUTE to make a change of variables In the resultant

curvature tensors and equations of motion, e.g.

A(X,Y,T) - eVM {!. + A^X.T) P^COS Y) + A2(X,T) P2(oon Y».

An option is Included In EINSTEIN, RIIMANN, MOTKW, and SUBSTITUTE for

expanding the output in powers of a small parameter and retaining only terms

of zero, first, and second order.

EINSTEIN, RIEMANN, MOTION, and SUBSTITUTE are all written in the IEH

language FORMAC. POBMAC is an extension of FORTRAN IV, but unlike FORTRAN IV,

FORMAC can currently be used only on ISM 7090 and 7094 canputers. Hopefully

an Improved version of POBMAC will be available for the IBM 360 system some-

time In 1966.

5 in.A

lU. HOW TO USB THE PKWRAMB H ALBBHT

DeBcrlptlon of the FORMAC language A. Brief

In this section we present only enough Information on lOBNAC to enable

the reader to use the prograns In ALBERT. A more detailed description of

FORMAC, intended to help the reader vfao wishes to «rite f^gjlg IQRMAC prograw

for himself or to modify ALBERT to suit his own individual needs, will he

found in 5 IV of this report. For a cooplete description of the FOHKAC system

and for guidance in writing coqpllcated FORMAC prograav, the reader is re-

ferred to the standard IBM FORMAC manual (UM 1964).

The FORMAC system is capable of performing formal algebraic aanipulations.

and differentiations of algebraic expressions. It can also evaluate numerically

the algebraic expressions which it creates, but no such numerical evaluations

are used in the programs described in this report,

FORMAC uses notation vhich is almost identical to that of FORTRAN IV:

1. All algebraic quantities -- constants, paramsters, spaeetime coordi-

nates, functions, algebraic expressions — are represented by "words"

vhich are constructed from 6 or fewer Latin letters and digits. For

example. In ALBERT we use X,Y,Z, and T to represent the k spacetlms

coordinates; and we use EP to represent a small parameter in texos

of which power series expansions are made. As further examples, in

the saqple problem given in i UI.B.U, ANU represents the function

v(X) and HI represents the function KAx,T). AS a final exaqple,

with the above definitions of X, Y, Z, T, SP, ANU, and HI, the state-

ment

m PAM3 - T * ANü + m/y,

tells the coaputer to use the word PAM3 to represent the algebraic

i III.A

expression T v(x) + llx(X,T)/l. (When defining an algebraic expres-

sion, one Bust always use the word UJT In front of the definition.

Othervlse, the coopvrter vtll atteapt to evaluate the expression

numerlcally. In the usual FORTRAN nanner.)

2. Algebraic quantities can be subscripted In the same nanner as Is done

In FORTRAN« For exaiqile, we denote the covarlant coavonents of the

metric tensor, gj,, by the words Ol(l,j); Ol(l,3) Is g13, 01(2,2)

18 g22» etc•

3. The cotquter distinguishes constants and parameters from functions

and from algebraic expressions by means of "ATfiMIC" and "DEPEND"

statements, which appear near the beginning of the program. (These

statements are similar to the EXIEFNAL statement of FORTRAN IV.)

a. The ATONIC statement identifies all algebraic quantities ("atomic

variables") which do not represent internally constructed alge-

braic expressions. All nonnumerical constants, parameters,

spacetime coordinates, and functions must appear in the ATOMIC

statement. For example, for a program involving only the quanti-

ties described in L above, all quantities except PAM3 appear in

the ATOMIC statement, and the atomic statement takes the form;

ATOMIC X, Y, Z, T, EP, ANU, HI.

b. The DEPEND statement identifies all functions which are not

algebraic expressions — i.e., which are atomic —; and it states

the functional dependences of the functions. For example, one

identifies v(X) and H1(X,T) as atomic functions by the above

AT0MIC statement plus the DEPEND statement

DEPEND (AMJ/X), (Hl/X,T).

Note that PAM3 as defined above is not an atonic function; it

i in.A

is an algebraio expression Bade up of etonde fuactlons and other

atonic variables, and it therefore does not appear in either an

AtyJKIC Btatenant or a VBPESD statenent.

4. In constructing algebraic expressions one can perform the elaoentary

operatlins of addition, subtraction, multiplication, division, and

exponentiation (l«e., raising to a power). The notation for these

operations is identical to that of FORTRAN IV:

+ denotes addition

denotes subtraction

* denotes aultipllcation

/ denotes division

** denotes exponentiation.

As in FORTRAN IV, the order in vhlch these operations are performed

is **, followed by * and /, followed by + and -. For exaople, con-

sider the following FORMAC statements and their translation to

ordinary algebraic language:

FORMAC LANQUAQE CQNVEKFIOHAL NOTATION

MT PAMS - T»ANlWaA this sets PAMS equal to

TvHj^

Y

lET SS - PAlO«W2.+3.*(X-y)*«T this sets SS equal to

(7)2**-)1

In constructing algebraic expressions one can use numbers which are

floating point (i.e. have decimals like 2. and 3. above) or which

are integers (e.g. 2 and 3). When doing floating point nanipulatlons

i ni.A

the computer generally outputs exponential notation — e.g. "2.5E-1"

(a.SxlO-1) Instead of l/k. Vhen doing Integer algebraic manipula-

tions, PORMAC uses 2**(-Z) Instead of 2.5E-1. When Integer (i.e.

rational) expressions are mixed together vith floating point expres-

sions hy the user, the computer converts them all to floating point

before manipulating.

5. The conputer recognizes and knows how to differentiate the following

elementary functions:

PORMAC LANGUAGE CONVENTIONAL NOTATION

FMCEXP(ANU) exponential: e

FMCSIN(2.«T) sine: sln(2T)

IMCC(te(Hl**2) cosine: cosC^ )

FMCI^O(Y) natural logarithm: log(Y)

FMCATN(raCEXP(AlIU)) arctangent: tan" (ev)

niCHrN(3*Z+7) hyperbolic tangent: tanh(3Z+7)

These functlona can be used freely in algebraic expressions.

6. FORMAC is capable of differentiating any algebraic expression con-

structed according to the above rules. If A is a function of X, Y,

T, but not of Z

DEPEND (A/X,Y,T),

then various derivatives of A are denoted as follows

FORMAC LAMOUAOE CONVENTIONAL NOTATION

FMCiaF(A,{X,l)) dA/äX

I>ICDIF(A,(X,2)) d2A/äX2

IMCDIF(A,{X,l),(y,l)) a2A/öxaY

FMCDIF(A,{X,2),(T,1)) öVÄ

i IZI.A

In constructing algebrmlc expressions, one can use FORMAC differentia-

tion freely, and the requested differentiation will alvays be performed

to conpletlon. For exanple, consider the following sequence of

ooonandst

FORMAC COMMNS RESULT OF COMMAND

M&dC X, Y, Z, T, EP, ANU, HI Defines the atonic variables for the program.

DBPEMD (A1IU/X),(HVX,T) Defines the atomic functions and their dependences.

LET PAM3 - T*Airo«Hl/Y Sets PAM3 equal to TvH^Y.

LET Ql - FMCDIF(PAM3,T,1) Performs the required differentiation on EAM3, yielding

01 - vH^Y + (Tv/YjÖHj/dT.

UT CC - PMCDIF(FMC8IH(Hl)-tT»X, (T,1),(X,1))

Performs the required differentiation yielding

cc - (cos HjKa^/aräx)

- (sin ^Kayamayax) +1

I£T CD - rMCDIF(AlW,Y) Performs the required differentiation yielding

CD - 0.

There are coimands In the FORMAC language which remove parentheses from

algebraic expressions, which factor terms out thereby inserting parentheses,

which truncate power series, and which perform other useful aatnipulatlons.

However, none of these caanmnds are necessary to the use of the prograum in

ALBERT, so we shall delay describing them until § IV.B.

§ III.B.l

B. J3J1STEIN, the Program for Calculating Ricol and Einstein TensorB

1. Input for EINSTEIN

EINSIEIN Is the program \rtiich calculates Ricci and Einstein tensors.

In order to use EINSTEIN, one must supply the following!

1. An ATftalC statement identifying all atomic variables for the

problem. The AxfalC statement takes the form

ATOMIC X, Y, Z, T, EP, . . . .

The user Inserts after EP all functions and parameters which

appear in his metric tensor.

2. A DEPEND statement identifying all atomic functions which appear

in the metric tensor and specifying their functional dependences.

For the form of the DEPQJD statement, see page 5.

3. A statement declaring the value of the integer EPTERM. EPTERM is

used to tell the coaputer whether or not to expand the curvature

tensors to second order in the parameter EP and truncate thereafter

(i.e., keep only terms in EP»«o, EP»*1, and EP»«2).

If EPTERM - 0 terms in all powers of EP will be retained.

If EPTERM ji 0 only terms of order 0,1, 2 In EP will be retained.

k, "lEi" statements defining all nonzero covarlant conponents of the

metric tensor, gj,, in terms of atomic variables, g., is denoted

in ALBERT by <;i(l,j). The metric components 1, 2, 3, k refer to the

spacetime coordinates X, Y, Z, T respectively.

5. "I£r" statements defining all nonzero contravariant components of

the metric tensor, g ^ in terms of atomic variables, g1"* is

denoted In ALBERT by G2(l,j). The matrix G2(l,J) must be, of course,

the algebraic Inverse of Gl(l,j) - at least up to second order in

the expansion parameter EP.

i m.B.i

These 5 statemente are Inserted toy the user directly Into the body of the

program EDBTEIH at the points dlllneated toy eonuent cards. (See listing In

the next section).

For example (another example Is given In i Ul.B.k), to calculate Wed

and Einstein tensors for the standard, static, spherically symnetrlc aetrlc

ds2 - -e^dr2 - r2(dö2 + sin2« d,2) + ev(r)dt2 (i)

one must:

1. Mentally replace r toy X, 9 toy Y, «p toy Z, t by T, v toy AMI, and X toy

ALAM;

2. Supply the following cards at the required points In KDBTEIH

AltfMIC X, Y, Z, T, EP, ABU, ALAM

DEFEND (AIV,ALAM/X)

EPTERM - 0

LET 01(1,1) m -IMCEXP(ALAM)

IfiT 01(2,2) » -X**2

MT 01(3,3) - -(X*n4CSIN(Y))«*2

1EI 01(4» - niCEXP(AMü)

LET 02(1,1) - -IMCEXF(-ALAM)

ISI 02(2,2) . -iyx**2

LET 02(3,3) - -l/(X«niCSIN(Y))**2

LET 02(4,4) - IMCEXP(-AHU)

As In FOBTHAN all cards supplied toy the user must have their first

entry In or after column 7 and their last entry in or toefore oolumi

72. Long statements can toe continued over as may as nineteen

contlnuatloa cards. Any card with a nonzero entry In colun 6 <«

considered a continuation card.

10

5 III.B.l

3. Run the program on an IBM-7090 or 7094 computer using a IB8YS-F0RMAC

operational tape (obtainable from IBM), which contains the PORMAC

coqpller.

The user Is free to give the functions and variables entering into his metric

tensor any names constructed from 6 or fewer characters, except the following:

NAMES WHICH MUST NOT BE USED AS

VARIABLE NAMES IN ALBERT

1. X, Y, Z, T except for use as spacetime coordinates.

2* ^ exce|rt for use as a constant (not ftmctlonal) expansion parameter; and EPTERM, except for use to control expansions in EP.

3. 01 or 02 except for use as metric tensors.

h. W except for use as stress-energy tensor.

5. Any name beginning with a digit (0, ..., 9); I.e., all names must begin

with Latin letters.

6. Any name beginning with the following (these are forbidden in all

F0RMAC programs):

AIOCtfN D(ÄJBLE I^GICA

ATOMIC END j^CU

AOTSIM ERASE ^Ej,

BCDC0N EVAL PARAM

CENSUS EXPAND PART

C^W« FIND REAL

C#»l IMC SÜBST

CtfEEF lUHCTI HJHEtfü

DEraUD DC SYMARG

DIMENS INTBOE

LET

11

9 UI.B.l

7. The following naam, which «re used for various other purposes by ALBERT

ASS, AA, AAA HBUBB, HEX?

BBtHH, ELAMK PP, Q, R

CARD, CCS RICCI, RTE», BUT

Di, ua, HEFF s, so, si, ...., ss Jti, HP SCAL

I, J, K, L, M, H SUM

II, JJ, JJJ, U. Tl, T2, T3, IT

UHB, LBL ZO, ZI, Z2

MCS, «J, MU

FOHMAC requires a huge aoount of core storsge for Its operations. Con-

sequently, If one Inputs metric tensors vhlch are too conpllcated, be will

exhaust the aemory capacity of the computer before finishing the conputatloo.

When this happens, the only way of using the ALBERT programs as they stand Is

to simplify the form of the metric tensor. The slu^llfled Rlccl and Einstein

tensors vhlch result therefrom can be reexpressed In terms of the more coa^ll-

eated metric by means of the program SUB9TIIUTE. For examples of metrics which

did and did not compute without simplification, and of the simplifications

used for certain conpUcated metrics, see {} III.B.U, Ul.C.k, JH.V.k, and

III.E.1».

One of the most effective ways to avoid exhausting the computer's memory

Is to avoid using complicated denominators In the expressions for the metric

tensor. FORMAC does not handle denominators efficiently. For exaaqple

01(1,1) - 1./(1.-8»B«/X),

where m depends on X, Is much more likely to produce trouble than

01(1,1) - fMCEXP(ALAM),

where ALAM depends on X.

12

S III.B.2

2. Complete Listing of EIMSTEIN

In the listing of Einstein which follows, and in all other program

listings in this report, all control cards are omitted. (Control cards

are cards which precede and follow each program and which tell the computer

such things as who the user of the program Is and what language the program

is written in.) Because control cards vary in format fron one computer

Installation to another, the user should consult his friendly computer

representative for special Instructions on their use. The user should

warn his computer representative that he is using PORMAC, which requires

somewhat different control cards than other languages.

13

8 m.B.2

»IBFMC ONE NODECK SYMARC

C

C EINSTEIN, A PRIIGRAM TO CALCULATE THE RICCI AND EINSTEIN TENSORS

c 0EPfcND IANU'ALAM/X) J with your oim Input.

C C ATOMIC AND DEPtNO STATEMENTS MUST BE SUPPLIED BY USER C IMMEDIATELY PR6CEOING THIS COMMENT. C THE ATOMIC STATEMENT HUST LIST ALL FUNCTIONS ANÜ INDEPENDENT C VARIABLES USED IN A PARTICULAR PROBLrM. C THE DEPEND STATEMENT DEFINES ALL THE FUNCTIONAL DEPENOPNCIES C FOR THE PROBLEM, I.E. WHICH FUNCTIONS DEPEND ON WHICH VARIABLES.

REAL MCS(4,4,4» LOGICAL Q

DIMENSION CARDn2),LlNen2»,CCS(A,4,4l,Cim,4),G2(4,4),Iim DIMENSION RICCKAI

C INTEGER EPTERM

C

C IF EPTERM = 0 TERMS CONTAINING ALL POWERS OF EXPANSION C PARAMETER WILL BE RETAINED. C IF EPTERM .NE. 0 TERMS CONTAINING THE EXPANSION PARAMETER r . r.*r rn> T0 * P3HER GREATER THAN 2 WILL BE DISCARDED. C A CARD CONTAINING THE INTEGER VALUE OF EPTERM MUST FOLLOW c THIS COMMENT.

EPTERM = o 1 IL8!" Repl-<* thiB card c i vlth yoirr own Input.

DO I I - 1,* DO 1 J = 1,4 LET G1(I,JI = 0. LET G2(I,JJ = 0.

1 CONTINUE C

C ALL ELEMENTS OF THE COVARIANT METRIC TENSOR, C1(I,J), AND THE C CUNTRAVARIANT METRIC TENSOR, G2n.J», HAVE BEEN SET EUUAL

C USER MUST SUPPLY "LET" STATEMENTS IJEFININü ALL NON ^FRO C ELEMENTS OF THtSE TENSORS IMMEDIATELY FOLLOWING THIS COMMENT.

C LET Gl(l,l) = -FMCEXP(ALAM) LET Gl(2,2) = -X**2 LET 01(3,3) » -tX*FMCSIN(Y))**2 LET 01(4,A) = FMCEXP(ANU) I User: Replace these eards LET 02(1,1) = -FMCEXPI-ALAM) '--"•- *—.-"'* CmrQ* LET 02(2,2) = -1/X«*2 LET 02(3,3) » -l/(X*FMCSIN(Y))«i*2 LET G2(4,4) = FMCEXP(-ANU)

with your own Input.

11+

i ni.B.2

c c C THE COVARIANT METRIC TENSOR — OUTPUT C

WRITE(6,S03) 503 FORMAT« 51HIC0VAR[ANT METRIC TENSOR HOU ZERO ELEMENTS (INPUT) I

DO 2 I = lf^ DO 2 J^lt* LET Q * MATCH ID, G1«I,J),0. IFIOI GO TO 2 HRITE(6,800II,J

800 FORMATdH I5,1HL I5,1HL ) BEGIN = 0.

Kb LET BEGIN » BCDCUN OK I • J ) , L INE, 12 MRITE(6t200MLINE(LI,L*2,12l IF(BEGIN.NE.O.)GO TU 46

2 CONTINUE C C THE CONTRAVAR1ANT METRIC TENSOR — OUTPUT C

WR(TE(6,700I 700 FORMAT«S'iHOCONTRAVARIA^T METRIC TENSOR NON ZERO ELEMtNTS « INPUT)»

DO 47 I = 1,4 UO 47 J = l,<t LET Q « MATCH 10, G2«I,J),C. IF(Q)GO TO 47

WRTTE«6,iOO) I,J 500 F0RMAT«1H I5,IHU I5,1HU )

BEGIN * 0. 48 LET BEGIN = BCDCON G2«I,J),LINE,12

WRITE«6,200)«LINE(L),L-2,12) IF(BEGIN.NE.O.)00 TO 48

47 CONTINUE C C THE COVARIANT CHRISTflFFEL SYMBOLS C

WRITE(6,201) 201 FORMAT«34H1THE COVARIANT CHRISTOFFEL SYMBOLS )

00 3 I = 1,4 DO 3 J « 1,4 00 3 K = 1,4 LET Tl = 0. GO TO «31,32,70,71),!

31 LET Tl = FMCDIF«G1«J,K),X,1) GO TO 33

32 LET Tl = FMC0IF«G1(J,K),Y,1) GO TO 33

70 LET Tl = FMCOIF«Gl(J,K),Z,l) GO TO 33

15

{in.B.2

LET Tl = FMCDIF(GHJ,K>,T,1) CONTINUE LET T2 = 0. GO Tn(35,36,72,n)tJ

LET T2 - FMCDIF(Gl{I,KI,X,l) GO TO 3*

LET T2 = FMCDIF(Gl(I,K),y,l) GO TO 34

LET T2 « FMCDIF(Gin,K»,Z,l) GO TO 34

LET T2 = FMCDIF(Gl(ItK),T,l> CONTINUE LET T3 = 0, GO TO (38,3t>,74,75),K

LET T3 = FMCDIF(Gl(I,J)fX,l) GO TO 37

LET T3 = FMCDIF(Gl(I,J)tVtU GO TO 37

LET T3 « FMCOIF(Gl(I,J),Z,l) GO TO 37

LET T3 = FMCOIF<GHI,J),T,l» LET CCSd.J.K) - 0.5*(Tl*T2-r3» LET CCSU.I.K) « CCS(I,J,K) LET Q « MATCH ID. CCS(I.J.K),0. IF«Q)GO TO 3 WR1TEI6.203)1.J.K

203 FORMATdH I5.2HLS I3,2HLS IA.1HL» LET ANS = EXPAND CCSd.J.K) LET ANS * ORDER ANS.INC.FUL BEGIN » 0.

5 LET BEGIN ■ BCDCtIN ANS,LINE.12 MRITk(6.200>(LINK(L).L>2.12)

200 FORMAT!1H 5X 12A6> IFtBEGIN.NE.O.IGO TO 5 ERASE ANS

3 CONTINUE ERASE Tl,T2,n

C C THE MIXED CHRISTOFFEL SYMBOLS C

WRITE(6.205) 205 FORMAT! 30H1THE MIXED CHRISTOFFEL SYMBOLS )

00 6 I = 1,4 00 6 J s 1,4 DO 6 K > 1,4 LET MCS(I.J.K) » 0. DO 7 L = 1,4 LET TT = G2(KtL»*CCS(I,J,L) LET TT = EXPAND TT IF(BPTERH)506t507.506

507 LET SUM = TT

16

i m.B.s

ERASE TT GO TO 514

506 CONTINUE LET SI « COEFF TT,EP««0,R LET S2 * COEFF TT,EP«»liR LET S3 ' COEFF TTfEP**2tR ERASE TT LET SUM = SI ♦ S2*EP ♦ S3*EP*EP ERASE Sl,S2iS3

51* CONTINUE LET MCS(I,J,K> = MCSIIfJiK) ♦ SUM ERASE SUM

7 CONTINUE LET MCSdfJfKI = EXPAND MCS(IiJtK) LET MCS(J,I,K) = MCS(IfJiK) LET Q = MATCH If), MCS( I . JiK) ,0. IFIOIGO TO 6

WRITE(6,769) I.J.K 769 rORMATUH I!),2HLS H,2HLS U.IHU )

LET ANS * ORDER MCSU , J ,KI, INC.FUL BEGIN * 0.

8 LET BEGIN « BCDCflN ANS,LINEil2 WRITt(6,?00)(LINE(L),L=2,12) IF(BEGIN.NE.O.ICO TO B ERASE ANS

6 CONTINUE C C

UO 350 1=1,*> DO 350 J*l,4 DO 350 K=l,* bRASE CCS(I,J,K)

350 CONTINUE C THE RICCI TENSOR C

WRITE(6,2I7) 217 F0RMAT(36HlTHt RICCI TENSOR, MIXED COMPONLNTS I

DO 50 N = 1,4 DO 50 K * 1,4 LET SI = 0. DO 51 I = i,U DO 51 M = 1,4 LET Tl = 0. GO TO (21,22,23,24),I

21 LET Tl = FMCDIF(MCS(N,M, I),X,U GO TO 19

22 LET Tl » FMCDIF(MCS(N,M,n,Y,ll GO TO It

23 LET Tl = FMCDIF(MCStN,M, I),Z,l) GO TO 19

2* LET Tl - FMCDJFIMCSIN.M, I >,T,1I

17

( nz.B.2

19 CONTINUE LET Tl " EXPAND Tl LtT T2 = 0. GO TO (52,53,54,551,N

= -FMCOIF|MCS(ItM,I),X,l) 56 » -FMCOIF(MCS( I,MtI),Y,lI 56 « -FMCUIF(MCS(I,M,n,Z,n 56 = -FMCDIF(MCSJI,M,I),T,II JE

LET T2 « EXPAND T2 LET TT = G21K,MI*»T1*T2» ERASE Tl,T2

LET TT = EXPAND Tl IF(EPTeRM)5(H,5l3,50<,

513 LET SI = SI ♦ TT ERASE TT

504

52 LET T2 GO TO

53 LET T2 GO TO

54 LET T2 GO TO

55 LtT T2 56 CONTIN

51

511

510

LET SI » EXPAND SI GO TO 51 CONTINUE LET SO = CDEFF TT,EP**0,R LET S3 = CDEFF TT,EP**1,R LET S4 * COEFF TT,EP*«2,K ERASE TT LET SI = SI ♦ (SO ♦ S3*eP ♦ S4*EP*EPI ERASE SO ,S3,S4 LET SI = EXPAND SI

CONTINUE LET S2 = 0. DO 84 I = 1,4 DO 84 J = 1,4 DO 84 M » 1,4

LET TT G2(KfM)*(-MCS(J ,H,I)*MCS MCS(1,J,JJJ LET TT = EXPAND Tl IF(ePTeRM)510,51l,t)lü

LET S2 = 52 + TT ERASE TT LET S2 = EXPAND S2 GO TO 84

CONTINUE LET SO « COEFF TT,EP**0,R LET S3 = COEFF TT,EP«*1,R LET S4 » COEFF TT,EP**2,R ERASE TT LET S2 = S2 ♦ (SO * S3*EP ♦ S4*EP*fcP) ERASE S0,S3,S4 LET S2 = EXPAND S2

84 CONTINUE

16

{ III.B.2

LET RTEN «= SI ♦ S2 ERASE SliS2 LET RTEN= EXPAND RTEN IF (N-K) 983,98^,983

98* LET RICCKN) » RTEN 983 CONTINUE

LET 0 = MATCH ID,RTEN,0. IF(0)GO TO 50 LET ANS = ORDER RTEN,INC,FUL WRITE(6,216IK,N

216 FORMATdH I5,IHU I5,1HL I 501 rüRMAT(2I5)

BEGIN = 0. 58 LET BEGIN = BCDCON ANS,LINt,12

WRITE«6,200)(LINE(«L),IL « 2,12) 502 F0RHAT(12A6)

IF(BEGIN.NE.O.)CO TO 58 ERASE ANS

50 CONTINUE

c c

92 CONTINUE

THE RICCI SCALAR c

LET SCAL = Riccim ♦ Riccn2i LET SCAL = EXPAND SCAL LET SCAL « SCAL ♦ aiccim LET SCAL « EXPAND SCAL LET SCAL = SCAL + RICCI I*) LET SCAL * EXPAND SCAL LET ANS = = ORDER SCAL,INC,FUL WRITE (6v233»

233 FORMAT (ITHlTHt RICCI SCALAR) BEGIN = n.

234 LrT BEGIN = BCDCÜN ANS,LINE,12 WRITE (6,200) (LINE (ID, I L = 2,12) IF (BEGIN.NE.O.) GH TO 234 ERASE ANS

C C THE EINSTFIN TENSOR, DIAGONAL COMPONENTS C

221 FORMAT! SfiHITHE HINSTEIH TENSOR. MIXED COMPONFNTS, DIAGONAL ELEMKN ITS )

DO 516 L=l,4 LET ANS = RICCI(L) - 0.5*SCAL LET ANS = EXPAND ANS

LET ANS = ORDER ANS, INC, FUL WRITE(6,219)L,L

219 FORMATdH I5,1HU I5,1HL) «EG IN = 0.

342 LET BEGIN = BCDCON ANS, LINE, 12

19

ft nx.B.2

WRITE(6.200I (LINE(IL),IL»2,12) IFCBEGIN.NE.O.I GO TO 342 ERASE ANS

516 CONTINUE 93 CONTINUE

STOP END

20

I III.B.3

3. Output for EJUBTEIH

The printed output for EXNEKEEIN includes: the metric coopooentB, g^.

and g1'', which wore input hy the user; the Christoffel symbols, which are

calculated from the fanulas

rijk " * (8lk,J + gJlc,l " hi,**' (2)

m „mit _ (%)

(note that the Christoffel symbols are symaetrlc In the first two indices);

tht «nixed components of the Ricci curvature tensor, calculated from

^n ■ ^{Cl " Cn + C ri/ - ^m1 ^ } ' w

the ftLccl scalar,

R - ^ J (5)

and the diagonal mixed cooponents of the Einstein tensor,

O11 =Rk .iRBk . (6) n n * n

(Recall that the off-diagonal continents of the Einstein tensor are Identical

to the off-diagonal cooponents of the Ricol tensor.) Conponente which are

zero are not printed out; and when two cooponents are equal by symnetry con-

siderations (e.g., T^j - r213), only one of them is printed.

The format of the printed output is self-explanatory except for the

following; Before each component of a quantity the computer prints the

numerical values of the Indices, plus syitibola to identify whether the indices

are contravarlant (U for upper) or covariant (L for lower). For exaaple, in

21

5 ZII.B.3

the Rlod tenaoor

UJ 3L «-» R^ .

Nareorer, in the Cbrlrtoffel Bytfboli an 8 Is used to Identify the indices

«hieb are sywatric:

1I£ SU3 kh *-* !%_., synaartrlc under interchange of 1 and 3.

For metrics «faich are too conplloated, the ccs^uter nill exhaust its

meaoory capacity hefore ccspleting the cooputatioDS. When this occurs, the

coq^uter prints out "ERROR TRACE. CAUfi IN REVERSE ORDER.", followed by a

list of routines being used at that ■cment, and then followed by "FOBMAC

ERROR NUMBER '7537' HAS OCCURRED. EXECUTION TERMINATED." The eaqnxter then

texninates the ccsputation. Other error messages may occur as a result of

faulty input. In order to interpret these error messages the reader should

refer to pp. 213-222 of the FORMAC nanual (DM 1961*).

As an exaaple of successful output fi-om a coqputation by EINUTEIH, we

give below the results for the static, sphericaUy-sygnstric mstrio of equa-

tion (l). The input which genera'Jd these results vas given in { m.B.l

and also in the listing of EINSTEIN in { ni.B.2.

22

COVAKIANT METRIC THNSDR NON ZtRO ELEMENTS (INPUT) 5 I1I.B.3 1L 1L -FHCEXPIALAMI«

2L 2L -X**2.0S

3L 3L -X**2.0*FMCSIN(Y»**2.0t

4L <*l FMCEXP(ANU)»

CONTRAVAKIANT METRIC TFNSOk NON ZERO ELEMENTS (HPUT> 1U 1U -FMCEXP(-ALAM)t

2U 2U -X**{-2,0)%

3U 3U -X**(-2.Ü»*FMCSIN(Y»**(-2.0)$

4U 4U FMCEXP(-ANU»t

THE COVARIANT CHRISTOFFEL SYMBOLS 1LS 1LS 1L -5.0E-l*rMCEXPlALAM)*FMCDlF(ALAM,(X,n)»

1LS 2LS 2L -X$

US 3LS 3L -X*FMCSlN(Y)**2.0t

1LS 4LS 4L 5.0H-1*FMCEXP(ANU>*FMC0IF(ANU,(X,1)l$

2LS 2LS 1L X*

2LS 3LS 3L -X**2.0*FMCSIN(Y)*FMCCOSm$

3LS 3LS IL X«FMCSlN(Y)**2.0t

3LS 3LS 2L X**2.0*FMCSIN(Y)*rMCCOS(Y)t

41S 4LS IL -5.ÜE-mMChXPUNU)*FMCDIF(ANlMX,l))t

THE MIXED CHRISTCIFFEL SYMRÜLS 1LS US IU b.OC-l*FMCOIFIALAM,(X,ll)t

1LS 2LS 2U X^*(-l,0>t

US 3LS 3U X**(-l.Ü)t

1LS 4LS 4U 5.0t-l*eMCniF(ANU,(X,llli

2LS 2LS Ul -X*FMCEXPI-ALAM)$

2LS 3LS 3U FMCSIN(YI»*«-l.r »♦FMCCOS(Y)t

3LS 3LS IU -X*FMCSIN(YI**2.0*FMCEXP(-ALAM)t

3LS 3LS ?U -FMCSIN(YI*FMCC()S(Y>i

4LS 4LS IU 5.Üb-l*FMCEXP«-ALAM*4NUI«FMCDIF<ANUfIX.l))«

23

S ZZX.B.S

THE RICCI TENSOR, MIXED COMPONENTS 1U 1L U U -X**(-l.0)*FMCEXP(-ALAM)«FMCDlF(ALAM,(X,n»-2.5E-l*FMCEXP«-ALAMI« FMCDIF<ALAM,IX,1II»fMCDIF(ANU. IX, IM♦2.5E-1»FMCEXP(-ALAM»♦FMCDIFI ANU,IX,l)l**2.0+5.OL-l*FMCEXP|-ALAM»*FMCDIF(ANU,(X,2lli

2U 2L X**|-2.0)*FMCEXPI-ALAM)-X**(-2.O»-5.OE-l*X»*l-l.0l*FMCEXP«-ALAM»* FMCDIF(ALAM,(Xtll)*5.0E-l*X**(-1.0>*FMCtXPI-ALAM)«FMCDIFIANU,IX,l) )$

3U 3L X**l-2.0)*FMCEXP(-ALAM)-X**(-2.0l-5.0E-l*X*«l-l.OI*FMCEXPI-ALAMJ« FMCUIF(ALAM,(X,n)*5.0E-l*X**«-l.0)*FMCEXPI-ALAMI*FMC0IF1 ANU,I X,H >»

4U ^L X**(-l.OI*FMCEXP|-ALAMI*FMCDIF(ANU,IXfl»»-2.5e-l*FMCEXP(-ALAMI» FMCDIF(ALAM,|Xfn»«FMCD|F(ANU,(X,lll*2.5E-l«FMCEXP(-ALAM)*fHCÜIFI ANU,(X.D)♦♦2.0+5.OE-l*FMCEXP(-ALAM)*FMCDIF(ANU,(X,2>IS

THE RICCI SCALAR

2.O^X^«(-2.0l*FMCEXP(-ALAM)-2.O^X«««-2.OI-2.O^X^^(-l.O)^FMCEXP(- ALAM>^FMCDIFIALAM,<X,UI+2.0^X*«(-l.O)*FMCEXP|-ALAM»^FMCDlr(ANU,lX ,n)-5.0E-l^FMCEXP(-ALAM)^FMCDlFIALAM,(X,l)|*FMCOIF(ANU,IX,l)) + b.0t-l^FMCEXP(-ALAMI^FMCDIF(ANU,IX,ni« + 2.0*FMCEXP(-ALAH)*FHC0IF( ANU,IX,2))t

THE EINSTEIN TENSOR, MIXED COMPONENTS, DIAGONAL ELEMENTS 1U U -X«^(-2.0)*FMCEXPI-ALAM)+X+^l-2.OI-X^«(-l.0)^FMCEXP(-ALAH)^FMCDIFI ANU.Ixailt

2U 2L 5.0E-l^X^^(-l.0»*FMCEXP|-ALAM)^FMCDIF(ALAM,|X,lM-5.0E-l»X*^l-l.C) ♦ FMCEXPI-ALAMMFMCOIHANU, 1X,1M*2.5E-1^FMCEXP|-ALAM)«FMCDIF(ALAM, (X.K)^FMCOIFIANU,(X,l))-2.bE-I^FMCEXP(-ALAM)♦FMCDIF ( ANU, (X,1»)*♦ 2.0-5.0t-l*FMC£XP(-ALAM)*FMCOIF(ANU,IX,2)lt

3U 3L 5.0£-l»X^«(-l.0)»FMCFXP(-ALAMI^FMCDIFIALAM, I X, 11 l-5.0E-14>X»*|<-UO) ♦FMCEXP(-ALAM)^FMCDIFIANU,(X,in*2.5E-l«FMCEXP(-ALAMI*FMCDIFIALAM, (Xtl)l*FHCOIFIANU,(X,l) )-2.!iE-l*FMCEXP|-ALAM ) ♦FMCOIF i ANU, ( X, 1 !)«• 2.0-b.0E-l*FMCEXP(-ALAM)«FMCDIF«ANlJ,IX,2Mt

4U hi -X*^l-2.0»*FMCEXP(-ALAM)*X^^I-2.O)*X + «(-l.0»*FMCEXP(-ALAMMFMCDIF( ALAM,(X,l>)t

24

} ni.B.u

1». San>le Problene for HHBTEIH

Input and output for one sanple problem, the static spherically Byimetrlc

nstrle of equation (l), were given in the Xaat two aeetiona. As a second and

auch more complicated exjunple, consider the metric for a star in even-parity,

nonradial pulsation, as discussed by Thome and Caapolattaro (1967):

ds' ev (1 + HQ P^dt2 + 2 ^ Pi dtdr - eA (1 - Hg P|) dr^

rZ (1 -K?t) (Ar + sin20d(p2).

(7)

Here v and X are functions of r; H-, H^^, Hg and K are functions of r and tj

and F.(COBO) is the Legendre polynonial of order I.

To prepare input for computing the Ricci and Einstein tensors of the

metric (2), we make the following changes to FORMAC notation;

t ♦ T

(these are the standard coordinates as accepted by ALBERT).

r ♦ X

9 ♦ Y

<P ♦ Z

A ♦ ALAM

v ♦ AMU

«0* HO

Hl* HI

H2* H2

K * AK

P/* AL

(these names were chosen arbitrarily, subject only to the rules outlined in § III.B.l.)

y

25

i JJJ.B.k

Also, we make the declsloo to let the ccnputer treat P.(cos9) ■ AL(Y)

as an unapedfled function. Finally, because we want the Rlccl and Einstein

tensors only to first order (linear) In the perturbation fron equlllbrlun,

we attach the expansion parameter EP»*2 to the metric perturbations, AX, HO,

HI, and H2, wherever they appear. The resultant input and output are as

follows:

INPUT:

ATOMIC X.Y.?tT.PP,ALtANU.ALAM.AK»H0»Hl.H2 rFPFND (ANU.ALAM/X). (AL/Y) OFPFND (H0.HliH2.AK/X,T)

FPTERM = 3

LET Gl(l.l) = -(-AL*EP»*2«H2+1.)*FMCEXP(ALAM) LET Gin.<t) = AL»EP»*2*H1 LFT Gl(4.1 ) = Gl(1.4> LFT Gl(?,2) = -n.-AK»AL*EP»*2)»X«*2 LFT Gl(3»3) = -n.-AK*AL»EP*»2)»X»«2»FMCSIN(Y»»»2 LFTGKA.A) = (AL»CP**2»H0 + 1. )»FMCEXP(ANU) LFT G2(l.l) = -(AL*EP*»2*H2+1.)»FMCEXP(-ALAM) LFT G2(1.A) = AL»FP»*2»H1«FMCEXP(-ALAM-ANU1 LET G2(4,l ) = G2(1.4) LFT G2(?.2) = -(AK»AL»EP*»2+l.)»X*»(-2) LET G2(3.3) = -(AK»AL*EP»»2 + 1.)*(X»FMCSIN(Yd*•(-2 I LET G2(4.A) = (-AL»EP»»2*H0+1.)»FMCEXP(-ANU)

OOTCTT;

See following pages

26

§ zn.B.it

COVARIANT METRIC TENSOR NUN ZERO ELEMENTS «INPUT) 1L 1L -(-AL*EP**2.0*H2+1.0)*FMCEXP(ALAM)t

1L 4L AL«EP**2.0*Hlt

2L 2L -(-AK*AL*EP**2.O*l.O)*X**2.0t

3L 3L -(-AK*AL*EP*«2.6+1.0)*X**2.0*FMCSIN<YI*»2.0*

4L 1L AL*EP**2.0«Hi»

AL AL (AL*EP**i:.0*H(m.O)*FMCEXP(ANim

CONTRAVAKIANT METRIC TENSOR NON ZERO ELEMENTS (INPUT) 1U 1U -(AL*EP**2.0*H2*1.0)*FMCEXP«-ALAM)$

1U AU AL*EP**2.0*Hl*FHf.tXP(-ALAM-ANU)$

2U 2U -(AK»AL*fcP**2.O+l.O)*X**(-2.0lt

3U 3U -<AK*AL*EP**2.0*1.0)*X**(-2.0)*FMCSIN(Y)**(-2.0)»

AU iU AL*hP**2.U*Hl*FKCtXPt-ALAM-ANU)$

AU AU (-AL*EP**2.O*H0+l.O)«FMCEXP(-ANU)$

THE COVARIANT CHRISTOFFEL SYMBOLS 1LS US 1L 5.0E-l*AL*EP**2.0*H2*FMCEXP(ALAW)«FMCDIF(ALAM,(X,l))*-5.0E-l*AL*EP ♦*2.0*FMC£XP(ALAM)*FMCOIF(H2,U,l))-5.0E-l*FMCEXP(ALAM)*FMCDIF( ALAH,(X.l ) )S 1LS ILS 2L -5.0E-l*EP**2.0*H2*FMCEXP(ALAM)*FMCDIF(ALi<Y,l))$

ILS ILS AL -!>.ÜE-l*AL*EP**2.0*FMCEXP<ALAM)*FMCDIF(H2,(T,l))♦AL*EP**2.O^FMCDIF «HI,(X,l))«

ILS 2LS 1L 5.0t-l*EP**2.0*H2*FMCEXP(ALAM)*FMCDIF(AL,IY, l))t

ILS 2LS 2L AK*AL*EP*«2.0*X+'3.0E-l*AL*EP*«2.O*X«*2.O*FMC0IF(AK,(X,l))-Xt

ILS 2LS AL 5.0E-l*EP**2.0*Hl*FMCDIKALi (Y,l))» US 3LS 3L AK*AL*EP**2.0*X«FMCSTN(Y)«*2.0+5.0E-1*AL*EP**2.0*X**2.0*FMCSIN(Y)

♦•2.0*FMC01F(AK,(X,1))-X*FMCSIN(Y)**2.0» ILS ALS IL 5.0E-l*AL*EP**2.0*FMCEXP«ALAM)*FMCOIF(H2,(T, l))t

ILS ALS 2L -5.0E-l*EP**2.0*fU*FMCDlF(AL,IY,l))»

27

i III.B.U

1LS <rLS 4L 5.0E-1*AL*EP«*2.Ü»H0»FMCEXPI ANU»*FMCI)IFIANU, (X.l) )*5. Ob-i*/VL*EP** 2.0*FMCEXP(ANlJl*f MCDlF(H0.(X,l))+5.lJC-l»FMCtXP(ANlJI»rMCDIF(ANU,(X,

1)1» 2LS 2LS 1L -AK*AL*tP**2.0*X-4.0F-l*AL*EP**?.Ü*X«*2.0*FMCOIF(AK,I X, I)(»Xt

2LS 2LS 2L 5.0b-l*AK*EP**2.0*X**2.0*FMCr)IF(AL,(Vil))t

2LS 2LS ^L -5.0fc-l*AL*EP**2.a*X**2.ü*FMCnlF<AK,l1,1)(t

2LS 3LS 3L AK»AL*EP**2.0*X»«2.0*FMCSIN1 V)*hMCC(lSlY)+t).0F-l«'K*EP**2.0*X**2.0* FMCSIN(Y)«*2.0*FMC0IF<AL,(Y,l))-X«*2 .ü*fMCSINlY)»FMCCOS(Y) I

2LS ^LS 1L 5.0L-l*EP**2.0*Hl*FMr,DIF(AL, (Y,l ) ) t

2LS 4LS 2L 5.0E-l*AL*EP**2.r.*X**i.C*FMCDIF( AK, (),!))»

2LS 4LS tl 5.0t-inP*«2.0*HC*FMCEXP( ANLIEF MCD IF (AL, (Y,l ) )$

3LS 3LS 1L -AK*AL*EP«*2.0*X*FMCSINIY)♦♦2.0-6.OE-l*AL*EP**2.0*X*»2.0*FMCSlN(Y) ♦♦2.Ü^FMCl)rFlAK,(X,l))+X^FMCSINlY)^+2.0t

3LS 3LS 2L -AK*ALÊP^*2.O^X*^2.Ü^FMCSIN(Y)^FMCCOS(Y)-5.0E-l«AK^FP^^2.^X^^2.O ♦FMCSIN(Y)^^2.O*FMCDIF(AL,(Y,l))+X*»2.0*FMCSrN(Y)^FMCCllS(Y)»

3LS 3LS 4L -5.C/E-lÂL^FP*^2.0+X^^2.0^FMCSIN(V)^»?.J*FMCniF(AK,(T,l))t

3LS 4LS 3L 5.0b-l*ALÊP^2.0 + X^2.O^FMCSINm*^2.0^FMCDIF(AK,(T,l) )»

4LS 4LS 1L -5.0E-l*ALÊP^2.0^HO^FMCEXPUNUI^FMCOIF(ANU,(X,l))-i.üE-lÂL^tP^ 2.0^FHCEXP(ANlJ)+FMCDIF(HO,tX,l))*AL^£P^^2.0^FMCüIF(Hl,(T,l))- 5.0E-1^FMCEXP(ANU)*FMCDIF(ANU,(X,1) )t US 4LS 2L -5.üE-l*fcP^^2.O^H0^FMCEXP(ANU)^FMCOIF(AL,(Y,l ))t

4LS ALS 4L 5.0E-l*ALÊP^^2.0^FMCEXP(ANU)^FMCOIF{HO,(T,l ))1

THE MIXED CHRISTOFFEL SYMBOLS 1LS 1LS 1U -5.0E-14'AL*EP**2.0^FMCDIF(H2,IX«1))♦b.OF-l*FMCDI F ( ALAM,(X,l))»

1LS 1LS 2U 5.0fc-l*fcP*^2.O«H2^X^^(-2.0)«FMCFXP(ALAM)^FMCOIF(AL,(Y,l))»

1LS 1LS 4U -5.0E-l*AL«EP**2.0^Hl*FMCEXP<-ANU)*FMCDIF(ALAM,(X,l)l-5.0fc-l«ALÊP ••2.0^FMCEXP(ALAM-ANU»*FMCDlF«H2,(T,n)*ALÊP*»2.0*FMCEXP(-ANU)« FMCOIFCHl,IX.lll» ILS 2LS 1U -5.ÜE-l«EP*«2.0«H2^FMCOlFIAL,(Yfl))» ILS 2LS 2U -5.0E-1*ALÊP**2.0«FMCOIF1AK,«X11))*X^<-1.0)»

28

S IIZ.B.It

1LS 2Lb 4U !).0C-l*tI'»«2.0*Hl*FMCEXP(-ANU)*FMC0IF(AL,(V,l»)i

1LS 3LS 3U -5.0b-l*AL*bP**i.0*FMCDIF<AK,(X,n)*X**(-1.0»i

US <»LS IIJ 5.0l:-l*AL*EP«*2.ü*Hl*rMCfcXP(-&LAf«(*FMCDIF(ANU,«X,l)»-5.0b-.*AL*tP ♦ •2.0*FMCOIHH2tjr,ll)$

US MLS ZU 5.0L-l«£P**2.0*hl*X*«(-2.0)*FMCDrF(AL,(V,l))i

US 4LS AU 5.0E-l*AL*FP**2.0*FMr.OIF(HOi(Xf l))t5.0E-l*FHCDIF(ANU,(X,in»

2LS 2LS 1U AK*AL*EP**2.0*X*FMCK<P{-ALAM»-AL»tP*»2.O«H2»X«FMCeXP«-ALAM)* 5.0E-J*AL«LP**2.0*X**2.0*FMCEXP(-ALAM|*hMCt1IF(AK,(X,l»)-X*rKCEXP(- ALAK)S

2LS 2LS 2U -5.üE-l*AK*CP**J.0*FMCDIF(AL,(Y,l))t

2LS 2LS *U AL*LP**2.0*Hl*X*rKCEXP(-ALAM-ANU|-5.ÜF-l*AL*EP**2.ü*X**2.O*FMCEXP( -ANU»*FMCD!F1AK,<T,1 )lt

2LS US 3U -5.Ot-l*AK*CP*»2.ö*FMCOIF(ALi(Y,II)»FMCSIN(Y)*♦(-!,0)*FMCCOS(Ylt

2LS '♦LS 1U -5.üC-l«EP*^2.0*lll«FMCEXP(-ALAM)*FMCOIF<AL,(Vtll)t

2LS UlS 2U -5.Ut-l*AL*£P«*2.0*F«1CDlF( AK,(T,1I It

2LS <.LS 4U 5.0L-l*£P»*2.0*H0*FMCDlF(AL,(Y,int

3LS 3LS 1U AK*AL*EP**2.0*X*FMCSIN(Y)**2.0*FMCEXP(-ALAMI-AL*EP**2.0*H2*X* FMCSIN(YI**2.0*FMCEXP(-ALAM)*5.0E-l»AL*EP»*2.0*X«*2.0*FMCSIN<Y>** 2.0*FMCtXP(-ALAMI*FMCDIF(AK,(Xfl)l-X*FHCSIN(YI«*2.0*FMCEXP(-ALAN)t

3LS 3LS 2U 5.0d-l*AK»EP**2.0*FMCSIN(Y)**2.0*FMCDIF<ALt(Y,in-FMCSIN(YI*FMCCOS (Ylt

3LS 3LS 4U AL*tP*«2.ü*Hl*X*rMCSIN(Y)**2.0*FMCEXP(-ALAM-ANU)-5.0E-l*AL*EP**2.0 •X**2.0*FMCSIN(Y)**2.0«FMCEXP(-ANUI*FMCDIF(AKf«T,l))t

3LS 4LS 3U -5.CE-l*AL*tP**i.O*FMCOIF(AK,(T,llIt

4LS «fLS 1U 5.0t-l«AL*EP*»2.0*HO*FMCEXP(-ALAM+ANUI*FHCDIFUNU,«X,ll>t 5.0fc-l*AL*EP*«2.0*H2*FMCEXP(-»LAM*ANUI«FMCDIF(ANU,«X,lll-AL*tP*» 2.0*FMCtXP(-ALAM)*FMCOIF(Hl,(T,in+5.0E-l*AL»EP**2.0*FMCtXPJ-ALAM* ANU)«FMC01F(HO,(X,l) I♦5.0E-l*FMCtXPl-ALAM*ANU»*FMCDIF(ANUt(X, IM $

4LS 4LS 2U 5.0t-l*EP*«2.&*HO»X**(-2.OI*FMCEXIMANUI*FMCniF«ALtlYfI)I»

4LS 4LS 4U -5.uE-l*AL*EP**2.0»Hl*FMCEXP(-ALAMI*FMCI)IF(ANUt«X,ll)*5.0t-l*AL*EP **2.0*FMCOIF|HO,IT>llit

29

THE RICCI TENSORt MIXED COMPONENTS lU IL-AL*EP**2.0*H2*X** C-I»0)♦FMCEXPI-ALAM)*FMCOI FIALAMtlX»l)l-2.5E-l*AL*EP**2.0*H2*FMCEXP(-ALAM)*FMCDIF(ALAM,IX.l)»*FMC0IF<ANU,(X,in*2.5E-l*AL*EP**2.0*H2*FMCEXP(-ALAM»*FMCOIFCANU,IX,in**2.O*5.0E-1*AL*EP**2.0*H2*FMCEXP(-ALAM|*FMC0IF(ANU,IX,2H-2.0*AL*EP**2.0*X**(-l.O»*FMCEXP(-ALAM»*FMCOIFIAK,(X,in^AL*EP**2.0*X**I-l.0)*FMCEXPI-ALAMI*FMCDIF(H2,IX,1)I^5.0E-1*AL*EP*»2.0*FMCEXPI-ALAMI*FMCOIFIAK,(X,l))*FMCDIFIALAM,(X,nl-AL*EP**2.0*FMCEXPl-ALAMI♦ FMCDIF(AK,(Xt2 I)-2.5E-l*AL*EP**2.0*FMCEXPI-ALAM|*FMC0IFIALAM#(X# 1 I )*FMCDIF(HOt(XtlII*5.0E-l*AL*EP**2.0*FMCEXP(-ALAM)*FMCOIF(ANU#IXi11) *FMCDIF(HO»IXtlII»2.5E-l*AL*EP**2.0*FMCEXP(-ALAM)*FMC0IF(ANU#CXf I I)*FMCOIFIH2 »IXf111♦5.0E-l*AL*EP**2.0*FMCEXPI-ALAM)*FMCDIF(HOf(Xf2) )♦5.0E-l♦AL♦EP♦♦2.0*FMCEXPI-ALAM-ANU)*FMCDIF(ALAM,(XtIM♦FMCOIFIHI, (T,in-AL*£P**2.0*FMCEXPI-ALAM-ANUl*FMC0IFCHlt (T,l If (X,in* 5.0E-l*AL*EP**2.0*FMCEXPC-ANUI*FMCDIF(H2,(T,2II-5.0E-l*EP**2.0*H2* X**(-2.0I*FMCSINlYl**I-l.0I*FMCC0SCYI*FMCDIF(AL,IVtlII-5.0E-IPEP** 2.0*H2*X**(-2.0I*FMC0IFCAL,(Y,2II-X**(-l.OI*FMCEXPI-ALAMI*FMCOIFI ALAM,(X,III-2.5E-1*FMCEXP(-ALAMI*FMC0IFIALAM.IX,1I1*FMC0IFIANU,IX, III*2.5E-1*FMCEXPI-ALAMI*FMC0IF(ANU,IX,III**2.0»5.0E-1*FHCEXPI-ALAMI*FMCOIF(ANU,IX,2IIS

2U IL-5.0E-l*EP**2.0*H0*X**I-3.0I*FMC0IFUL, lY.lI I*2.5E-l*EP**2.0*H0*X **(-2.0I*FMC0IF(AL,IY,lI I*FMCOIF(ANU,(X,lIK5.OE-l*EP**2.0*H2*X**l -3.0I*FMCDIFCAL.IY,lII*2.5E-l*EP**2.0*H2*X**(-2.0I*FMCOIFCAL,IY.ll I*FMCDIF(ANU,IX,lII-5.0E-l*EP**2.0*X**I-2.01♦FMCEXPI-ANUI♦FMCDIFf AL,lYfllI^FMCDIFIHl,CT,lII-5.0E-l*EP**2.0*X**(-2.01•FMCDIFIAK,(XtI II*FMCOIF(AL,IY,lI I♦5.0E-l*EP**2.0*X**I-2.01•FMCOIFIAL,lY,111* FMCOlFtHO,(X,llIS

AU ILAL*EP**2.0*HI*X**I-1.0I*FMCEXPC-ALAM-ANUI*FMC0IF(ALAM,(X,III*AL*EP **2.0*Hl*X**(-l.0I*FMCEXPI-ALAM-ANUI*FMC0IFIANU,(X,1Il♦AL♦EP*♦2.0* X**C-l.OI*FMCEXPI-ANUI*FMCOIFIAK,IT,llI-AL*EP**2.0*X**I-I.0l* FMCEXPI-ANUI*FMC0IFIH2tlT,lII-5.0E-l*AL*EP**2.0*FMCEXPI-ANUl* FMCDIFIAK,IT,III*FMC0IF(ANU,IX,1II*AL*EP**2.0*FMCEXPC-ANUI*FMC0IFI AK,(T,lI,IX,lII^5.0E-l*EP**2.0*Hl*X**C-2.0I*FMCSIN(YI**f-1.0I* FMCCOSIYI*FMCEXPC-ANUI*FMCOIFIAL,(V,111 + 5.0E-l*EP**2.0*Hl*X**C-2.0 I*FMCEXPI-ANUI*FMCDIFCAL,(Y,2IIS

lU 2L-5.0E-l*EP**2.0*H0*X**l-l.0I*FMCEXPI-ALAMI*FMCDIFIAL,(YfII!♦ 2.5E-IPEP**2.0*H0*FMCEXP(-ALAMI*FMCDIFIAL,(Y,III*FMCOIFIANU#IX,llI ♦5.0E-l*EP**2.0*H2*X**l-1.0I*FMCEXPI-ALAMI*FMC0IFCAL,IY.lll* 2.5E-l*EP**2.0*H2*FMCEXPI-ALAMI*FMCD|F(AL,IY,lI I♦FMCDIFIANUtIX,l11 -5.0E-l*EP**2.0*FMCEXPI-ALAMI*FMCOIFIAK,(X,lII*FMCOIFIALtIY,III* 5.0E-l*EP**2.0*FMCEXPI-ALAMI*FMC0IFIALtlY,llI*FMCDIFIH0tlX#lll- 5.0E-l*EP**2.0*FMCEXP(-ALAM-ANUI*FMCDIFIALfIY,lII*FNCD1F(Hl»(T1111

5 III.B.U

THE RICCI TENSOR (continued)

2U 2L -AK*AL*EP**2.0*X*«(-2.0)-5.0E-l*AK*EP**2.C»X**(-2.0l*FMCSlN<YJ**« -1.0l*FMCCOS(Y)*FHCDIF(AL,(Y,l))-5.0E-i»AK*rP**2.0*X»*(-2.0l* FMCDIFJALl(Yf2)|tAL*EP**2.0*H2*X*»|-2.0»*FMCEXP«-ALAM)-5.0E-l*AL^ EP**2.0*H2*X**(-l.O»*FMCEXP(-ALAM)*FHC01FULAM,(X,l|)*5.0E-l*AL*EP »♦2.0*H2*X««(-1.0»*FMCEXP(-ALAM)*FMCDIF(ANU. (Xf1))-2.0»AL*EP«*2.0* X«*(-l.O)*FMCEXP(-ALAM»*FMCDIF«AK,(Xtint5.0E-l*AL*EP**2.O*X**( -1.0)*FMCEXP«-ALAMI*FMCDIF(HOiUtin*5,OE-l*AL*EP**2.0*X**(-1.0)* FHCEXP<-ALAM)*FHCDIF(H2,(X,n»-AL*EP**2.0*X*«(-l.OI*FMCEXP(-ALAM- ANU)*FMCDlF(Hl,(Tfl))+2.5E-l*AL*eP**2.0*FMCEXP(-ALAMt«FMC0IF(AK,(X ,l))*FMCDIF(ALAMf(X,l))-2.5E-l*AL*EP»*2.0*FMCEXP(-ALAMI*FMCDIF(AK, (X,1)I*FMCDIF(ANU,(X,1)I-5.0E-1*AL*EP**2.0*FMCEXP(-ALAM)*FMCDIF(AK ,(X,2»)+5.OE-l*AL*EP«»2.O«FMCEXP«-ANU»*FMCOIF(AK,(T,2»l*5.0E-l*EP **2.0»HO*X«^l-2.0)*FMCDIF(ALf(Y,2))-5.0E-l»EP**2.0*H2*X**(-2.0»* FMCDIF«AL,(Y,2)»*X**(-2.O)*FMCEXP(-ALAM)-X*«(-2.0l-5.OE-l*X**l-l.O )*FMCEXP(-ALAM)*FHCDlF(ALAMt(X,ni»5.0E-l*X«*(-l.OI*FMCEXP(-ALAMI* FHCDIF(ANU.<Xtll)$

30a

5 m.B.if

2.5r-l*fcP**2.0*Hl*FMCEXP(-ALAM-ANI »♦FMCI)IF(4L,(y,l))*FMCDir(ALAM, ( X,l))-2.5E-l*fiP**2.0*Hl*FMCEXP(-ALAM-ANU)*FMCDIF(AL.(Y,ll)*FMCDIF( ANU,(X,n|-5.0E-l*EP**2.0*FMCEXP(-ALAM-ANU»*FMCDIF(AL,(Vf!»)♦ FMCüIF(Hl,(X,l))*5.0E-l»EP**2.0*FMCEXP(-ANU)*FMCDIF(AK,<T,l)»« FMC0IF(AL,(Y,l))t5.0E-l*EP**2.0*FMCEXP(-ANUI*FMC0IF(AL,(y,l)>» FMCDIF(HZ,(Ttl))$

3U 3L -AK«AL«EP**2.C*X*«(-^.0»-5.0E-l*AK*EP^»2.0*X*»(-2.0»*FMCSINU)**< -1.ÜI*FMCC0S(Y)*FMCD1F(AL,(Y,l»I-5.0E-1*AK*EP**2.0*X**(-2.0)♦ FMCDIFULI (Yi2) I ♦ AL*EP**2.0*H2*X** (-2 .0) ♦FMCEXPi-ALAM )-5. 0E-1*AL« EP**2.0*H2«X**(-i.0)*FMCKXP(-ALAM)*FMCDIF(ALAM,«Xlll)*5.UE-l*AL*£P ♦♦2.0*H2*X**{-l.OI*FMCEXP(-ALAM(«FMCOIF(ANU,(X,lI)-2.0*AL*EP**2.0* X**(-1.ÜI*FMCEXP(-ALAM)*FMCDIFJAK,(X,U)+5.0E-1*AL*EP**2.0*X**( -l.ü)*FMCEXP(-ALAM)*FHCDIF(HO,(X,ll )+5.0E-l*AL«EP**2.0*X**(-1.0)» FMCEXP(-ALAM)*FMCUIF(H2t(X,l)»-AL«EP**2.0*X**(-l.O»*FMCEXP(-ALAM- ANUI*FMCDIF<H1,(T,1))+2.5E-1*AL*EP**2.0*FHCEXP|-ALAM)»FMC01F(AK,(X ,ll)*FMCDIF(ALAK,(X,l))-2.5t-l*AL*EP**2.0*FMCEXP(-ALAM)«FMCOIF(AK, (X,l>>*FMCOIF(ANLJ,(X,l()-5.0E-l*AL*eP*^2.0*FHCEXP(-ALAM»*FMCDIF<AK f (X,2) ) + 5.Ob-l*AL*EP**2.0*FMCFXP(-ANli)*rMCDtF{AKf(T,2)»+5.0E-1*EP ♦♦2.0*H0*X*«(-2.0)*FMCSi.(Y)*«(-1.0)*FMCC0S(YI*FMCDIF(AL,(Y.l»)- 5.0E-l»EP**2.0»H2*X**(-2.0)*FMCSIN(Y)**(-l.O)*FMCCOS(V)*FHCOIF(AL, (Y,l)(♦X**(-2.O)*FMCtXP(-ALAMI-X**(-2.O)-5.0E-l*X**(-l.O)*FMCEXP(- ALAM)*FMCDIF(ALAMf<X,in*5.0E-l*X*«t-1.0)*FHCEXP(-ALAM)*FMCDIF(ANU ,<X,1I )%

HI ^L -AL»EP«*2.0*X**(-1.0)*FMCEXP(-ALAM)*FMCüIF{AK,(T,ll)*AL*EP**2.0*X ♦♦I-1.0)»FMCEXPI-ALAM) ♦FMCOIF«H2,(T,1)1*5.0E-1*AL*EP**2.0*FMCEXP(- ALAM)*FMCDIF(AK,(T,1 ) )♦FMCOIF(ANU,(X,1))-AL*EP**2.0*FMCEXP(-ALAM)♦ FMCDlF(AKr<T,l).(X,l))-5.0E-l*EP«*2.0*Hl*X**(-2.0)*FMCSISlY)*»< -l.J)*FMCCOS(Y)«FMCEXP(-ALAM)*FMCOIF(ALi«Yfl))-5. 0E-1*EP**2.0*H1*X ♦♦(-2.ü)*FMCEXP(-ALAM)*FMCOIF(AL,<Y,2))»

2U «L -2.5E-l*EP**2.0»Hl*X**(-2.0)*FMCEXP(-ALAM)«FMC01F(ALi(Y,l))*FMCOIF IALAM, (X,l))+2.5E-l*EP»*2.0*Hl*X**(-2.0)*FMCEXP(-ALAM)*FMCDIF(AL,( Y,l))*FMCDIF(ANU,(Xll))+5.0E-l*EP**2.0«X**(-2.0)«FMCEXP(-ALAM)« FMCüIF(AL,IY.1))*MCD1F(H1,«X,1))-5.0E-l*EP»*2.0*X*«(-2.0)*FMCDIF( AK,(T,1))»FHCDIF(AL,(Y,l))-5.0E-l*EP**2.0*X*«(-2.0)*FMCOIF<ALt(Y.l ) )*FMCDIF(K2,(T,lt)t

4U 'tL AL*EP**2.Ü*H2*X**(-1 .0)*FMCEXP(-ALAM)*FMCDIFtANU,(Xtl))-2.5E-l*AL* EP**2.0*H2*FMCEXP(-ALAM)♦FMCOIFi ALAM, (X,l))*FMCOIF(ANU,U,i) »♦• Z.5E-1*AL*EP*«'2.0*H2*FMCEXP(-ALAM)«FMCDIF(AMU,(X,1))**2.0* 5.UE-1*AL*EP**2.0*H2*FMCEXP(-ALAM)*FMC0IF(ANU,(X,2))♦AL*EP**2.0*X •♦(-l.O)*FMCEXP(-ALAM)*FMCDIF(HO,(X,l))-2.0»AL*EP**2.O*X«*{-1.0)• FMCEXP(-ALAM-ANU)*FMCOIF(Hlf(T,l))-5.0E-l*AL*EP**2.0*FMCEXP(-ALAM) ♦FMCDIF(AK,(Xfl))«FMC0IF(ANU,(X,l))-2.5E-l*AL*EP*«2.0*FMCtXPi-ALAM )*FMCniF(ALAMf(X,l))♦FHCOIF(HO,<X,1))♦5.0E-l♦AL*EP♦♦2.0*FMCEXP(- AL4rt) ♦FMCDIF UNU , ( X, 1) ) *FMCDIF ( HO, ( X, I ) ) +2.5E-l*AL*£P*»2. 0*FMCEXP( -ALAM)*FMCDIF(ANU,(X,1))«FMC0IF(H2,(X,1))+&.0E-l*AL*EP**2. ()*FMCEXP (-ALAM)*FMCOlFIHO,(X,2))+5.0E-l*AL*tP*«2.0*FMCEXP«-ALAM-ANll)* FMCÜIF(ALAM,(X,1))*FMC01F1H1,(T,1))-AL*EP«*2.0*FMCEXP(-ALAM-ANU)* FMC0lF(Hl,(T,l),(X,l))+AL«EP**2.0*FMCEXP(-ANU)*FMCDIFIAK,(r,2))* 5.0E-l*AL*EP**2.0*FMCEXP(-ANU)*FMCDlF(H2,(T,2))*5.0E-l»EP**2.0«H0* X**(-2.Ü)*F»1CSIN(Y)«*(-1.0)«FMCCOS<Y)*FMCDIF(AL,(Y,l))*5.0E-l*EP** 2.0»H3*X**(-2.0)*FMCI)1F(AL,(Y,2))*X*««-1.0)*FMCEXP(-ALAM)♦FHCOIF« ANU,«X,l))-2.5E-l*FMC£XP(-ALAM)*FMC0IF(ALAH,IX,l))*FMCDIF<ANU,(X,l ))+2:.5E-l*FMCEXPl-ALAM)*FMCDIF(ANU,«X,l))**2.0+5.OE-l*FMCEXPl-ALAM )*FMCDIF(ANU,«X,2))t

51

i m.B.u

THE RICCI SCALAR -2.0» AK*AL*EP**2.0*X**>(-2. 0)-AK*hP**2.0*X**(-i.U)*FMCMN(Y 1**1-1.;.. )*FMCCQStY)*(;MCI)IF(AL,(Y,l) )-AK*tP**i .0*X**1-2.0 )*FMCDIF «I, ( Y,2 ) ) ♦ 2.C*AL*fcP**2.0*H?*X**(-2.0)*FMCEXP(-ALAM|-2.0*AL*EP**2.ÄH2*X**( -l.0)*FMCEXP{-ALAM)*FMCOIF(ALAM,(X,l))+2.0*4L*EP**2.0*H2*X**(-l.Ül ♦ FMCeXP(-ALAM|*FMCDIFlANlJ,(X,m-5.0t-l*AL*[ P**2.0*H2*FMCEXPt-ALAM )*F^CDIF(ALAM,(X,1))♦FMCDIF(ANU,(X,1I)♦••5.0E-1*AL*EP♦»2.0*H<*FMCEXP (-ALAM)*FHCDlF(ANU,(X,n)**2.0*AL*EP**2.0*H2*FMCEXP(-ALAM):FMCDIF( ANU,(X,2»)-6.C*AL*EP**2.3*X**(-l.0)*FMCfcXP(-ALAMI*FMr,DIF(AK, (X.ll ) +2.0*AL*CP*»2.0*X**(-1.0)*FMCFXP(-ALAM)*FMCDIF(HO,(X,1))t2.;j*AL*tP **2.0*X**(-l.0)*FMCEXP(-ALAM)*FfCDIF(H2f(X,l> >-4.0*AL♦EP**^.0*X**( -1.0)*FMCfcXP(-ALAM-ANU)*FMCOlF(Hl,(T,ni*AL*EP**2.0*F^CtXP(-ALAM)* FMCÜIF(AK,(X,1))*FMCDIF(ALAM,(X,1I)-AL*LP**2.0*FMCEXP(-ALAM)* FMCniF(AK,(X,l))*FMCOIF(ANU,(X,n)-2.C*AL*EP**2.Ü*FMCEXP(-->LAM)* FMCOIF(AK,(X,2)l-5.0.-l*AL*EP**2.U*FMCFXP(-ALAMI*FMCDIF(AL.\M,(X1l) )*FMCDIF(H0,(X,1)(♦AL*EP**2.0*FKCFXP(-ALAM)*FMCOIF(ANU,(X,!))* FMCUIF(HÜ,(X,l))+5.0r-l*AL*EP**2.0*FMCEXP(-ALAM|*FMCDIFtANiif(X.ll) ♦ FMCDIF(H2,(X,n)+Al-*EP**2.0*FMCbXP<-ALAM)*F«COIF(HO,{X,2) UAL'EP ♦ *2.Ü*FMCFXP(-ALA>'-ANIJ)*FMCÜIF(ALAM,(X,1))*FHCDIF(H1,(T,1) I-2.0* AL *EP**2.0*FMC£XP(-ALAh-ANU)*FKCDIF(Hl,(T,l),(X,l)l+2.Ü*AL*E"**2.0* FMCCXP(-ANU)*FMCDIF(AK,(T,2)UAL*tP**2.0*FM:fcXP(-ANU)*F»'CDIF(H2,(T ,2))+£P**2.0*HO*X**(-2.0l*FMCSIN(Y)**(-1.0)*FMCCOS(Y)*FMCUlF(AL,(Y ,in+cp«*2.o*HO*X**I-2.Ü)*FMCDIF(AL,(Y,2l)-EP**2.0*H2*X**(-2.0)* FMCSIN(Y)**(-l.ül*FMCCnS(Y)*FMCDIF(ALfIYtl)l-EP**2.0*H2*X*-(-2.ü)* FMCt)IF(AL,(Y,2))*2.C*X**(-2.0>*FMCEXP(-aLAM)-2.Ü»X**(-2.U)-2.0*X** (-l.O)*FMCEXn(-ALAM)*FMCDIF(ALfi.,1',(X,n )+2 . 0*X** (-1.0 I *FMCt *P ( - AL AM )*FMCÜIF(ANU,(X,i))-5.0e-l*FMCEXP(-ALAM)*FMCDIFlALAM,(X,n(*FMCDIF (ANU,(X,l)) + l5.01.--l*FMCEXP(-ALAM)*FMCDIF(AMU,(X,ll)**2.0*FMf.:-XP«-

ALAMI*FMCDIF(ANU,(X,?l )t

32

i III.B.U

THF EINSTEIN TENSOR, MIXED COMPONENTS, DIAGONAL ELEMENTS IU U

AK*AL»EP»*2.O*X**(-2.O)»5.0E-1*AK*EP<*2.0*X«-*(-2.O)*FMCSIN(YI**( -1.0)*FMCCOSlY)*FMr,OIF(AL,(Y,] ) 1 ♦5.0E - I ♦ AK »E P**2. ü*X** (-2 .0 I ♦ FMCDIF(AL,(Y,2)»-AL«EP**2.0»H2*X**(-2.0I*FMCEXP|-ALAH)-AL*EP»«2.0* ^♦X**(-1.0)*FMCEXPl-ALAMI*rMCI)IFIANU,IX,l)|fAL*EP**2.0*X**(-|.0)* FMCEXP(-ALAM)»FMCDIF(AK,(X,llI -AL♦EP**2.Ü*X*«(-i.0)»FMCEXP(-ALAMI * FMCDIF(H0,(X,l))f2.C*AL*HP**7.',*X*«(-1.0)*FMC£XPt-ALAM-ANU)*FMCOIF (M1,(T,I))*5.0E-1»AL ■'EP«'*?.0<'FMCEXP(-ALAM)*FMCOlF(AK,(X,m*FMCDIF (ANU, (X.l) )-AL*EP**2.0*FMCEXPl-ANU)*FMCDIF(AK,(T,2ll-5.0£-l*EP** 2.ü<H0*X**(-2.0(*FMCSlNlY)**(-1.0)*FMrX0S(YI*FMC0IF(AL,(Y,ll)- 5.OE-l»EP**2.O«H0*X*«(-2.O)*FHCn!F(AL,(Y,2))-X«*(-2.O)»FNCEXP(- ALAM)+X*«(-2.0I-X»«(-1.0)*FMCEXP(-ALAM)*FMCDIF(ANU,1X,1)I$

2U 2L

5.0£-l*AL*EP**2.0*H2*X*«(-I.0)«FMCEXP(-ALAN|*FMCDIF(ALAM,<X,in- 5.0E-l*AL»EP**Z.O*H2*X**(-1.0)*FMCEXP(-ALAM)*FMCDIF(ANU,(X,m + 2.5E-l*AL*FP**2.0*H2*FMCEXP(-ALAMI*FMCDIF(ALAM,(X,l)l«FMC0IF(ANU,( X,l))-2.5E-l*ALtEP**2.0*H2*FMCEXP(-ALAMI*FMCDIF(ANU,(X,n»*«2.0- &.ÜE-1*AL*CP**^.0*H2*FMCEXP1-ALAM)«FMCD1F(ANU,(X,2))*AL*EP**2.0*X ♦♦(-1.0I*FMCEXP(-ALAM)*FMCDIF(AK,(X,1I)-5.0E-1*AL*EP**2.0*X*«|-1.0 )*FMCEXP(-ALAM)«.FMCDIF(H0,(X,l)l-5.0E-l*AL*EP**2.O»X»*(-l.O)* HMCtXP(-ALAM|*FMCDIF(H2,(X,1)I♦AL»EP**2.0*X**(-1.0|«FMCEXP<-ALAM- ANUI«FMCDIF(Hl,(T,in-2.5E-l*AL*EP**2.0*FMCEXPI-ALAMI«FMCDIF(AK,(X ,l))*FMCf)IF(ALAM,IX,n) + 2.5E-I*AL*EP**2.0*FMCEXP(-ALAM)*FMC0IF(AK, (X,l) l*FMr.DIFtANU,{X,l))+5.0E-l*AL*EP«*2.0*FMCEXP(-ALAMI»FMCl)IF«AK ,(X,2I)*2.5E-l*AL*£P*»2.0*FMCEXP(-ALAM)«FMC0lFIALAMf(x,n)*FMCDIF( HO,(X,l)|-5.0E-i*AL*EP*«2.0*FMCEXP(-ALAM)*FMCDIF(ANU,(X,l)l«FMCOIF (H0,(Xfl)l-2.5E-l*AL*EP**2.0*FMCEXP(-ALAM)*FMCDIF(ANU,(X,ll)* FMCDIFlH2,U,in-5.0E-l»AL*EP«-*2.0*FMCEXP(-ALAM)*FMCDIF(H0,(X,2))- 5.0E-l*AL*EP**2.0*FMCEXP(-ALAM-ANU)*FMCDIF(ALAM,(Xll)|*FMCDIF(Hl,I T,ll)»AL*EP**2.0*FMCEXP(-ALAM-ANU)*FHCDIF(Hl,(T,l),(X,l))- 5.0E-l*AL*EP**2.0»FMCEXP(-ANU)«FMCDlF(AK,(T,2))-5.bE-l*AL»EP**2.0* FMCEXP(-ANU)♦FMCDIF(H2,(T,2))-5.0E-l*EP«*2.0*HO*X»*(-2.0|*FMCSIN(V • ♦♦(-1.0)«'FMCCf)S(YI»FMCDIF(AL,(Y,n)+5.0E-l«EP**2.0»H2*X*»(-2.0)* FMCbIN(Y)**(-1.0)*FMCCO5(V)*FMCDIFIAL,(Y,U)+5.0E-l*X**(-l.OI* FMCEXP(-ALAM)«FMCDIF(ALAM,1X,1))-5.0E-1*X**(-1.0)*FMCEXP(-ALAM)* FMCDIF(ANU,(X,l)U2.5E-l*FMCEXP(-ALAMl*KMC0IFIALAMt(X,n)*FMCDlF« ANU,(X,l)l-2.5E-l*FM(:EXP(-ALAM)*FMC0IKANU, (X,II»♦♦2.0-5.OE-1* FMCEXP(-ALAM|<'FMCDIF(ANU,(X,2I)$

33

5 III.B.U

THE EINSTEIN TENSOR (continued)

3U 3L 5.0E~MAL*EP**2.0*H2tX**l-l.[J)*FMr,EXP(-AL4HI*FMCDIH(ALAM, ( X,i) )- 5.0E-l*AL*EP**2.0*H2*X**(-1.0)*l:M(,eXP(-ALAM| ♦FMCDI F ( ANIJ, ( X , 1 I ) ♦ i.l3E-l*AL*EP**2.ü*H2*FMCEXP(-ALAM)*FMCDIF(ALAM,(X,l) )*FMCDIKANU, ( X,il)-2.5E-l*AL*EP**2.0*H2*FMCEXP(-ALAM)*FMCOIFlANU,IXt 1 ))**2.0- rJ .OE-l ♦AL*EP**2. 0*H2 *FMCEXP(-ALAM) »FMCOIFI ANU, tX,2l)*-AL*EP«;*2.0*X **(-1.0)*FMCEXP(-ALAM)#FMCOIF(AK,(Xf1) I-5.OE- 1 *AL*EP**2.?♦X«*(-1.0 l*FMCEXP(-ALAM)*FMCDtF{HO, (X,lI)- 5.OE"1*AL*EP*«2.0*X** ( - I.0)* FMCEXP1-ALAM)*FMC0IF(H2, (X,ll I ♦AL*f-P*»2 .0»X** (-i . 0 ) *FMC EX P (-AL AM- ftNU)*FMCDIF(Hl,(r,l) l-2.5£-l«AL*tP*<!2.0«FMCEXP(-ÄLAMI*f MCOIF(AK,(X ,l>)*FMCOIF(ALAMf(X,l))+2.5E-l*AL*EP*«2.0*FMCEXP(-ALAM)tFMCDIF(AK, (X,l) )*FMCOIF(ANU, tX,l))t!>.0E-l*AL«EP**2.O*FMCEXP(-ALAMI*FMCOIF(AK f (X,2) ) + 2.5E-l*AL<=EP**2.0*FM:EXP(-ALAM)-FMCDIF(ALAM, (X, I) XFMCDIFl HO,(X,l)l-5.0E-l*AL*EP**2.0*FMCEXP(-ALAM|*FMC0IFtANU, ( X , 1 1 I*FMCDIf (HO, (X, 1 ) )-2.5E-l*AL*EP**2.0*FMCF.XP(-AlAM)*FMCDlF(ANU,( X, 1 I I*

(■MC0IF(H2f(X,l))-5.0E-l*AL»EP**2.0*FMCEXP(-ALAM)*FMCDIF(H0,(X,2))- 5.0E-1*AI.*EP**2.0*FMCEXP(-ALAM-ANU)«FMCDIF(ALAM, ( X, 1) I »FMCDI F ( HI , ( T,l))tAL*EP**2.0*FMCEXP(-ALAM-ANU»«FMCÜIF(Hl,(T,1),(X,1))- 5.0E-1*AL*EP**2.0«FMCEXP(-ANU)»FMCDIF(AK,(T,2II-5.0E-1 * AL♦EP**2. 0* FMC£XP(-ANU)*FMCDir-(H2,(T,2))-5.CE-l<-EP**2.0*H0*X**(-2.0l*FrCDtF( AL,(Y,2Il + 5.0E-l*EP»*2.0*H2*X**(-2.0)*FMCDIF(AL,(Y, 2))f5. OE-l*X**( -1.0)*FMCEXP(-ALAM)*FMCOIF(ALAM,(X.l))-5.OE-1»X*»(-1.üI♦FMCEXI» (- ALAM)*FMCDIF|ANU, (X,l»)♦2.5£-l*FMCEXP<-ALAM)*FMCnIF IALAM,U,!»)* FMCOlFlANU.IXi1)l-2.5E-l*FMCEXP(-ALAM)*FMCOIF(ANU,(X,ll )*»2.0- 5.0E-l*FMr.EXP<-ALAM)*rMCOrF(ÄNU,(X,2) )t

4U 4L AK*AL*EP**2.0*X<-*(-2.0)+5.0E-l*AK*EP**2.ü*X**(-2.0)*FMCSIN(VI**( -1.0)*FMCCnSm<FMC0IFUL,(Y,l))+5.OE-l*AK*EP**2.0*X*«(-2.0l* FMC0IF(AL,(Y,2I)-AL*EP**2 .0*H2*X**(-2.0(*FMCEXP(-ALAM)♦AL«EP**?.0* H2*X**(-1.0)*FMCEXP(-ALAM)*FHCDIF(ALAM,(X,11 I♦3.0*AL*EP**2.0*X«*( -1.0)*FMCEXP(-ALAM)*FMCCIF(AK,(X,1))-AL*EP**2.0*X**(-1.0)*FMCEXP(- ALAM)*FMCDIF(H2,(X,1))-5.OE-I*AL*EP**2.0*FMCEXP(-ALAM»*FMCDIF(AK,( X.ll)*FMCDIF(ALAM,(K,1)I♦AL*EP**2.0*FMCEXP(-ALAM)*FMCDI FIAK,(X,2)) ♦ 5.0E-1«£P**2.0»H2*X»*(-2.0»*FMCSIN(Y)*<(-I.0 )*FMCCQS(Y)♦FMCOIF{AL ,(Y,ll)+5.0E-l»eP»*2.0*H2*X**(-2.0)*FMCDIF(AL,IY,2l)-X**(-2.0»* FMCEXP(-ALAM|tX««(-2.0l+X**(-1.0)*FMCEXP(-ALAMI*FHC0IFIALAM,(X,l))

33a

§ III.C.I

C. RUMANH. the Program for Calculating Rlemann Tensors

1. Input for RnMANH

RiaiANN is the program which calculates the covarlant conponents,

Rabcd' ot the Rien"um tensor. The Input for MEMAMH Is Identical to that

for EIMSTEIM! The user must supply ATOMIC and DEPEND statements, an EPTERM

declaration, and LET definitions of the metric conponents Ol(l,J) and G2(I,J),

(See i III.B.l.) A listing of RIQUNN showing where to Insert the required

cards follows.

34

i m.c.s

2, Conpiete Ustlng of gjgWjM

SIBFHC TWO NOOECK SYMARG

C C R1EMANN, A PROGRAM TO CALCULATE THE R1EMANN TENSCR

ATOMIC XiY,ZtTt£PtANUtALAM \ User: Repiace these cards DEPEND (ANUtALAM/X) J with your own Input.

C C ATOMIC AND DEPEND STATEMENTS «UST BE SUPPLIED BY USER C IMMEDIATELY PRECEDING THIS COMMbNT. C THE ATOMIC STATEMENT MUST LIST ALL FUNCTIONS AND INDEPENDENT C VARIAÖLES USED IN A PARTICUl*R PROBLEM. C THE DEPEND STATEMENT DEFINES ALL THE FUNCIICNAL CEPENDENCIES C FOR THE PROBLEM, l.fc. WHICH FUNCTIONS DEPEND ON WHICH VARIABLES. C

REAL MCS(4,4,4) LOGICAL Ü DIMENSION CARD(12),L|NE(l2)fCCS(4,4,4l,GlU,4)iG2(4f4l

C INTEGER EPTERM

C C IF EPTERM = 0 TERMS CONTAINING ALL POWERS OF EXPANSION C PARAMETER WILL BE RETAINED. C IF EPTERM .NE. 0 TERMS CONTAINING THE EXPANSICN PARAMETER C TO A POWER GREATER THAN 2 WILL BE DISCARDED. C A CARD CONTAINING THE INTEGER VALUE OF EP1ERM MUST FOLLOW THIS C COMMENT. C ^ User: Replace this card

EPTERM »0 J vith your own Input. C

00 1 I = li« DO I J = li4 LET GlllfJ) = 0. LET G2«I.J) =« 0.

1 CONTINUE

C ALL ELEMENTS OF THE COVARIANT METRIC TENSOR, GKI.J), AND THE C CONTRAVARIANT METRIC TENSOR, G2(I,JI, HAVc BEEN SET EQUAL TO C ZERO. , .„„ C USER MUST SUPPLY ••LET" STATEMENTS DEFINING ALL NON ZtRO C ELEMENTS OF THESE TENSORS IMMEDIATELY FOLLCWING THIS COMMENT. C C

LET GIU,1) = -FMCEXPIALAM» ^ LET G1I2.2) = -X**2 LET 01(3,31 = -(X*FMCSIN(Y)l**2 LET G1U,4) = FMCEXP(ANU) LET G2I1,11 = -FMCEXPI-ALAM) LET G2(2,2> = -1/X*»2 LET G2(3,3) « -l/(X*FMCSIMY I )**2 LET G2(*,AI = FMCEXP(-ANU) 7

35

User: Replace these cards with your own Input.

l m.c.2

c c C THE COVARIANT METRIC TENSOR — OUTPUT C

501 FORHAT( 51HIC0VAR1ANT METRIC TENSOR NON ZERO ELEMENTS (INPUT» ) DO 2 I = It* 00 2 J = IfA LET U = MATCH ID» GKIIJIIO. IF(g) GO TO 2 WRITE(b.aOOM*J

800 FORMATUH I5,IHL I5,IHL ) BEGIN = 0.

46 LET BEGIN « BCOCON Gl(I»J)iLINE,12 WRI TE (6,200 MLINE(L)tL=2» 12) IF(BEGIN.NE.O.)GO TÜ 46

2 CONTINUE C C THE CONTRAVARIANT METRIC TENSOR — OUTPUT C

700 F0RMAT?54H0C0NTRAVARIANT METRIC TENSOR NON ZERO ELEMENTS (INPUT))

DO 47 1=1,4 00 47 J = I,4 LET Q = MATCH ID, G2(I,J)»0. IH(Q)GO TO 47

MRITE(6,500) I,J 500 FORMATUH I5,1HU I5,1HU )

BEGIN = 0. 48 LET BEGIN « 8CDC0N G2(I,J)«LINE, 12

WRITE(6,200)(LINE(L),L=2,12) IFIBEGIN.NE.OGO TO 48

47 CONTINUE C C THE COVARIANT CHRISTOFFEL SYMBCLS C

WRITE(6,201) „, r , 201 F0RMAT(34H1THE COVARIANT CHRISTOFFEL SYMBOLS )

00 3 I = It* DO 3 J =« I»A DO 3 K = 1,4 LET Tl = 0. GO TO (31,32,70,71),I

31 LET Tl = FMCOIF(Gl(J,K),Xfl) GO TO 33

32 LET Tl = FMCDIF(G1(J,K),Y,1) GO TO 33

70 LET Tl » FMCOIF(Gl(J,K),Z»l) GO TO 33

71 LET Tl = FMCDIF(G1(J,K),T,1) 33 CONTINUE

36

5 III.C.2

LET T2 = 0. GO TO(35f36,72t73).J

35 LET T2 * FMCD1F(G1(11K ) ,X,I) GO TO 34

36 LET T2 « FHCOIF(Gl(ItK),V,I) GO TO 3A

72 LET T2 = FMC0IF(G1(I,K»,Z,1)

GO TO 34 73 LET T2 » FMCOIFlGl(11K) ,T,1) 34 CONTINUE

LET T3 » 0. GO TO <38f39,74t75)tK

38 LET T3 = FMC0IF1GH 11 J ) tX, U GO TO 37

39 LET T3 * FMC01F<Gl(I.J)»Y,1)

GO TO 37 74 LET T3 = FMCDIF(G1(11J ) tZ11)

GO TO 37 75 LET T3 = FMC0IF(Gl(ItJ)fTt1) 37 LET CCSd.J.K» = 0. 5*( T1 + T2-T3)

LET CCS(Jtl.K) = CCS(IiJ.K) LET Q = MATCH IDt CCS1I?J.K),0.

IFCü)GO TO 3 HRlTE(6,203)IiJ.K

203 FORMATUH 15, 2HLS I3.2HLS I4,1HL) LET ANS = EXPAND CCS(I,J,KI LET ANS = ORDER ANS,INC,FUL BEGIN = 0.

5 LET BEGIN = BCDCON ANS,LINE,12 WRITE(6,200»(LINE(L),L=2,12)

200 FORMATUH 5X 12A6) IFIBEGIN.NE.O.IGO TO 5

ERASE ANS 3 CONTINUE

ERASE T1,T2,T3

C THE MIXED CHRISTOFFEL SYMBOLS

C

205 FORMATI 30HITHE MIXED CHRISTOFFEL SYMBOLS >

Ü0 6 I = 1,4 DO 6 J = 1,4 DU 6 K = 1,4 LET MCSd.J.K) = 0. DO 7 L = 1,4 LET TT = G2(K,L)*CCS(I,J,L)

LET TT = EXPAND TT IF(EPTERM)506,507,506

507 LET SUM * TT ERASE TT GO TO 514

37

COEFF TT,EP**0,R COEFF TT,EP**1,R COEFF TT,EP*»2,R

SI ♦ S2*EP ♦ S3*EP*EP

i III.C.2

506 CONTINUE LET SI = LET S2 = LET S3 » LET SUM ERASE TT

ERASE SltSZiSB 51A CONTINUE

LET MCS(I,J,K) = MCSII.J.K) ♦ SUM ERASE SUM

7 CONTINUE LET MCSdtJiKI « EXPAND MCSHtJ.K) LET MCSIJ.I.K) » MCS(ItJfK) LET 0 - MATCH IOt HCS(I.JtK)«0. IFtQIGO TO 6

WRITE(6,769) I,J,K 769 FORMATdH I5.2HLS I3.2HLS K.IHU )

LET ANS * ORDER MCS(I,J,K) , INC.FUL BEGIN - 0.

8 LET BEGIN * BCOCON ANS.LINE,12 WRITE(6t200)(LINE(L).L>2«12) IF(BEGIN.NE.O.)GO TO 8 ERASE ANS

6 CONTINUE C C C THE RIEMANN TENSOR C C

MRITE(6,215) 215 F0RMATK1H1THE RIEMANN TENSOR« COVARIANT COMPONENTS )

DO 15 I « 1,4 LL = 1*1 iF(LL.GT.4)LL=4 DO 15 J =LL,4 DO 17 K = 1,4

DO 17 L = J,<» LET SUM = 0. DO 18 MU = 1,4 LET TT = (KCS(J,K,MUI*CCS(I,L,MU)-MCS(J,L,MU>*CCS(I,K,MU)

1 ) LET TT = EXPAND TT IF(EPTERM)520,521,S20

521 LET SUM =■ SUM + TT GO TO 18

520 CONTINUE LET SI » COEFF TT,EP»*0,R LET S2 ■ COEFF TT,£?♦♦!,R LET S3 = COEFF TT,EP<"*2,R ERASE TT LET SUM = SUM ♦ SI ♦ S2*EP ♦ S3*EP*EP

38

i in.c.2

ERASE Sl,S2tS3 18 CONTINUE

LET ül = 0. GO TO (41,42,80.81),K

41 LET Dl = FMCOIF(CCS(J,L,n,X,l) GO TO 40

42 LET 01 » FMCOIF(CCSCJ,L,U,Y,l) GO TO 40

80 LET 01 = FMC0IF(CCS(J,L,I),Z,1) GO TO 40

81 LET 01 = FMCOIFJCCS( J,L,n,T,l) 40 LET 02 * 0.

GO TO (44,45,82,83),L 44 LET 02 * FMC0IF(CCS(J,K,I),X,1)

GO TO 43 45 LET 02 » FMCDIF(CCS{J,K,l),Yll)

GO TO 43 82 LET 02 = FMCDIF(CCS(J,K,I),Z,1I

GO TO 43 83 LET 02 - FHCDIF(CCS(J,K,n,T,l( 43 CONTINUE

LET OIFF » 01-02 ERASE 01,02 LET RMT = OIFF ♦ SUM ERASE SUM,OIFF LET ANS * EXPAND RMT ERASE RMT LET U > MATCH 10, ANS,0. IF(Q)GO TO 17 WRITE(6,206)1,J,K,L

206 FORMATdH 5X I5,1HL I5,1HL 15, 1HL IS,1HL ) LET ANS = ORDER ANS,INC,FUL 6EGIN » 0.

25 LET BEGIN = BCDCON ANS,LINE,12 MRITE(ü,200)(LINE(IL),IL*2,12) IF(BEGIN.NE.0.)GO TU 25 ERASE ANS

17 CONTINUE 15 CONTINUE

STOP END

39

{ III.C.3

3. Output for MBKAMN

The printed output for RZBiANN Includes: the metric ccaq^onentb, g.. and

g '') the Chrlstoffel symbols (eqs. (2) and (3))) and the covarlant cooponents

of the Rleoum curvature tensor

Babcd ' rbc radl " rbd raol + rbda,c " rbca,d * (8)

The format Is similar to that for the program EINSTEIN, and error messages are

Identical to those for EINSTEIN. The computer prints only the nonzero conpo-

nents of the Rlenann tensor; and of those components which can be obtained from

each other by synnetry operations, It prints only one. For example, R^gsb lB

printed If It Is nonzero; but R^^» ^s» R21fc5' R3lH2' RU312' R3l»21' "^

\,22 are nevep Prt11**^ because

R123l» " " R213l* " " R12lf3 ' + R21^3 " + R3lH2 " " ^312 " " R3U21 " + ^312*

As an example of output from a computation by RIEHANN, we give below the

results for the static, spherlcaUy-syoaetrlc metric of equation (l). The

Input which generated these results was given In 5 III.B.l and also In the

listing of RID4ANN In $ 111.(3.2.

METRIC TENSOR C0MPOHEMT8 AND CHRISTOFFEL SINBOIfi

- output Identical to that trcn EINSTEIN as given on page 23.

mE RIEHANN TENSORf CCVAPIANT COPCNEMTS U 2L U 2L

-S.0E-1*X*FNCCIF(ALAHI(X,1))S 1L 3L U 3L

-5.0E-l*X«Ff<CSIN(Y)«*2.C*FPCC(F(ALAH,IX,ll)t 1L 4L U 4L

2.5E-l*FHCEXP(ANt»*FHCCIF(ALAH.lXtnt*FMC0IF(ANUtlX.ni-2.5E-l* FMCEXP(«NL)«FHCCIFMNtt(X.l))**2.O-5.0F-l*FPCEXP(«Nll*Fh'CCIFIANLf ( X>2)IS

2L 3L 2L 3L X*e2.C»FHCSIN(V)*«2.0«FHCEXP(-ALAM)-X*»2.0«FfCSIN(V)«»2.CJ

2L 4L 2L 4L -S.OE-l^X^FPCEXPC-ALAH + AND^FHCCIFIANU.Uil»)»

3L 4L 3L 4L -5.0E-l«X*FPCSIN(Y)4*2.C«FPCEXP(-ALAM«ANU)*FPCCIF(«NLi(X,int

1*0

{ zu.c.it

k. gHflfl P^*>1— fear RnMAHH

Input «ad output far one unple problra, the atetle apberloaUy Bynnetrlc

Mtrlo of evintlon (l), vere given In the lut two seotlcna. Aa a aecood

axa^le oonalder the metric (7) for a atar In eren-parlty, nonradlal pulaatlon.

The Input to RXENAHR for tbla metric la Identical to the Input to EZNSTEIN

(gee page 25 ). The output from RIBdAMH la partlaHjr Identical to that ircm

gUMTRm (pagea 26-88 are Identical); hut Inatead of outputtlng Rlecl and

Einateln tenaora aft-ir the Chrlatoffel syatoola, RIBMHM outpwta HleBann tenaor

ccnponenta, of which we present only five:

THE R1SMANN TENSOR, COVARIANT COMPONENTS

5#0E-l*AK*AU*EP»*2.0*X*FMC0IFULAM,(XflH-AL«EP**2.0*X*FMC0IFJAK,< X,in*5.0E-l*AL*fcP»*2.O*X*FMCDIFIH2,(X,U>*2.5E-l*AL*EPM2.0»X«* 2.0*FMCüIF«AK,(X,l))»FMCDIF(ALAM,«X,H»-5.0E-l*AL*EP*«2.0*X«*2.0« FMCDIFJAK,«X,2)»-5.0E-l*EP**2.0eH2*FMCEXPtALAM)*Ft«C0IF(AL,<Y,2J)-

5.0E-l»X*FMCDIF(ALAM,(X,l))» IL 2L IL AL

-5.0E-l*EP**2.0«'Hl*X**(-1.0)»FMCOIFIAL,<Ytin-2.5E-l«EP*»2.0«Hl* FMC01F(AL,«Y,in*FMCOIF<ALAM,tX,lH*2.5E-l*EP**2.0«Hl*FMCDIF(AL, Y fl))«FMCDIF{ANU,«X,lH-5.0E-l«EP«*2.0*FMCEXP«ALAM»*FMCDIF«AL,(Y,l )*FMCüIF(H2,(T,l)l*5.0E-l*EP**2.0*FHCDIF«AL,(Y,l))*FMC0IF«Hl,(X,l)

)i IL 2L 2L 4L

5.0E-l«AL*EP**2.0*Hl*X«FMCEXP<-ALAM)*FMCDIF(ANU,(Xfin+5.0E-l*AL* EP**2.0*X*FMCDIF(AK,(T,1)1-5.0E-l*AL*EP«e2.0«X*FMCDIF(H2,( T,U)- 2.5E-l*AL»EP»»2.0*X**2.0«FMCDIF(AK,(Tfin*FHC01F(ANU,(X,l)»* 5.0E-l*Al.*EP»*2.0*X**2,0*FMCDIF(AK,(T,l»,<X,l»»+5.0E-l*EP**2.0*Hl*

FHCUIFIAL,IY,2ll»

-5,0E-l*AL*EP**2.0*Hl*X*FMCEXP(-ALAM>eFMCOIF(ANU,(X,U)-5.0E-l*AL* EP**2.0*X*FMCDIF(AK,(T,in»5.0E-l*AL*EP**2.0*X«FMCDIF(H2,CT,ni* 2.5E-l*AL»EP**2.0*X*«2.0*FMCDIF(AK,«T,l)l*M'C0IF(ANU,IX,in- 5.0E-l«AL»EP**2.0*X*«2.0*FMCDIF(AK,«T,ll,IX,l»)-5.0E-l*£P**2.0*Hl«

FMCDIF(AL,lYf2)»$

5.0E-l*AX*AL*EP**2.0*X*FMCSIN(Y)**2.0»FHC0IF(ALAM,(X,in-AL*EP** 2.0*X»FMCSIN(YI**2.0*FMCDIFlAK,(X,l)»*5.0E-l*AL*EP**2.0«*FMCSINtY )*«2.0*FMCDIF(H2,«X,lM*2.5E-l*AL*EP*»2.OeX«»2.0*FMCSIN«YI*»2.O* FMCDIFIAK,(xa)H'FMCDIF(ALAM,(X,l)»-5.0E-l*AL*EP"2.0*X**2.0« FMCSIN(Yt**2.0»FMC0IF(AK,<X,2>)-5.0E-l*EP**2.0«H2*FMCSlNlY»«FMCC0S «Y)*FMCEXP(ALAM)*FMCDIF(AL,IYtlll-5.r(E-l*XeFNCSINIYI«2.0*FMCDIF(

ALAM,(X.1))$

hi

j in.D.i

D. MOTION, the Program for Calculating the Divergence

-Energy Tensor of the Stress-

1. Input for MOTION

MOTION is the program vfalch calculates the divergence of the stress-energy

tensor, Wa b. The Input for MOTION Is Identical to that for EINSTEIN

(page 9) except for this; One must also supply

6. "LET" statements defining all nonzero mixed components of the

stress-energy tensor, W.% In terms of atomic variables. W."'

Is denoted in MOTION hy W(l,j) - i.e., the first index is co-

variant; the second is contravariant.

A listing of MOTION showing where to Insert the required cards follows:

k2

i m.D.2

2. Conplete Ll»tlrw of MOTIOW

SIBFMC C C C C C

C C C C C C C c c

EOM NOÜECK

MOTION, A PROGRAM TO CALCULATE STRESS-ENERGY TENSOR

THE DIVERGENCE OF THE

SVMARG

ATOMIC DEPEND

X,Y,Z,T,eP,ANU,ALAH,PfRHO (ALAM.ANUfP,RMO/XI } User: Replace theae oerde

with your own Input.

ATOMIC AND DEPEND STATEMENTS MUST BE SUPPLIED BY USER IMMEDIATELY PRECEDING THIS COMMENT. THE ATOMIC STATEMENT MUST LIST ALL FUNCTIONS AND INDEPENDENT VARIABLES USED IN A PARTICULAR PROBLEM. THE DEPEND STATEMENT DEFINES ALL THE FUNCTIONAL DEPENDENCIES FOR THE PROBLEM. I.E. WHICH FUNCTIONS DEPEND ON WHICH VARIABLES.

REAL MCSU,4I4I LOGICAL 0 DIMENSION CAR0<12l,LINE(12),CCSU,<>t4)fGlUt4>,G2U.4l DIMENSION WU,4)

INTEGER EPTERM

IF EPTERM = 0 TERMS CONTAINING ALL POWERS OF EXPANSION PARAMETER WILL BE RETAINED.

IF EPTERM .NE. U TERMS CONTAINING THE EXPANSION PARAMETER TO A POWER GREATER THAN 2 WILL BE DISCARDED.

A CARD CONTAINING THE INTEGER VALUE OF EPTERM MUST FOLLOW THIS COMMENT.

}UBer: Replace tbla card vlth your own input. EPTERH = 0

DO I I = 1,4 DÜ 1 J = I,It LET GKI.JI = 0.

LET G2(1,J) = 0. CONTINUE

ALL ELEMENTS OF THE COVARIAMT METRIC TENSOR, GKI.J», AND THE CCNTRAVARIANT METRIC TENSOR, G2(I,J), HAVE BEEN SET EQUAL TO ZERO. USER MUST SUPPLY "LET" STATEMENTS DEFINING ALL NON ZERO tLEMENTS OF THESE TENSORS IMMEDIATELY FOLLOWING THIS COMMENT.

LEI LET LET LET LET LET LET LEI

Gl(l,l) G1I2,2) Gl<3,3» G1U.AI G2I1,1I G2I2,2) G2I3,3) G2U,<>)

-FMCEXPIALAM) -X**2 -(X*FMCSIN(YII«»2

FMCEXP(ANU) -FMCEXP(-ALAM) -1/X**2 -1/(X*FMCSIN(Y))**2

FMCEXPI-ANU)

User: Replace theae cards vlth your own Input.

Il3

i III.D.2

WftlTE«6,501l t>01 Fn«MATI 51HlC0VARlANr METRIC TENSOR NCN ZERU ELEMENTS (1NPUTI I

00 2 I 3 1,4 DO ? J = 1,4 LET Q = MATCH 10, GUI,J),0. IFIQI GO TO 2 WRITE(6,eOO)I»J

Ö00 FUKMATUH I5,1HL I5,IHL ) BEGIN = Ü.

40 LET BEGIN = 3CDCUN Gl(ItJ)iL INE,12 WRITE(6,200((LINE(L),L=2,12» IF(BEGIN.NE.0.)GO TU 40

2 CONTINUE WK1TE(6»700»

700 FORMATI54H0C0NTHAVARI ANT METRIC TENSUR NON ZERO ELEMENTS (INPUT) I 00 42 I = 1,4 00 42 J = 1,4 LET Ü = MATCH ID, G2(I.JI.O. IFIÜIGÜ TO 42

MRITE(6,500) I.J 500 FORMATUH I5,1HU I5,1HU I

BEGIN = 0. 41 LET BEGIN = BCOCON G2(I,J),LINE,12

UKITE(6,20ai(LINEIL),L=2,12) IKBEGIN.Nb.O.tGO TO 41

42 CONTINUE

20 C C ALL ELEMENTS OF THE STRESS-ENERGY TENSUR, MIXED CQMPCNENTS C WU.J), ARE SET EQUAL TO ZERO. NOTE THE FIRST INDEX OF W C IS DOWN (CUVARIANT) AND THE SECOND INDEX IS UP (CONTRAVARIANT). C USER MUST SUPPLY '«LET" STATEMENTS DEFINING ALL NON ZERO C ELEMENTS IMMEDIATELY FOLLOWING THIS COMMENT C

LET Wd.ll = -P -j LET w(2,2) » -P I user: Replace these cards LtT W(3,3I = -P f with your «m input. LET W(4f4) = RHO J

C WRITE(6.208)

208 F0RMAT155H1STRESS-ENERGY TENSOR NON ZERO ELEMENTS, MIXED COMPONENT IS (INPUT) )

DO 20 1 = 1,4 DO 20 J = 1,4 LET M(I, J) = 0.

CONTINUE

ALL ELEMENTS OF WU.J), ARE SET

kh

5 III.0.2

COVARIANT CHRISTOFFEL SYMBOLS >

ÜO 16 I*liA DO 16 J * li<» LET Q « MATCH 10,Wl1,J),0. IFIQIGO TO 16

WRIIE{6,505) I,J 505 FURMATdH 15, 1HL U, IHU )

BEGIN s 0. LET ANS>W(I.JI

96 LET BEGIN = BCDCON ANS ,LINE,12 WRITEI6,200)(LINE(L),L*2,12) IFIQEGIN.NE.O.I GO TO 96

ERASE ANS 16 CONTINUE

HRITE(6,878) 878 FORMATI 35H1THE

DO 3 I = 1,4 DO 3 J = I ,A 00 3 K = 1 ,4 LET Tl = 0. GO TO (31,32,70,71),!

31 LET Tl « FMCÜ!FJGl(J,KI,X,ll GO TO 33

32 LET Tl x FMC0IF«G1(J,KI,Y,1I GO TO 33

70 LET Tl = FMCOIF(Gl(J,K),Z,ll GO TO 33

71 LET Tl = FMCDIF(G1(J,K1,T,11 33 CONTINUE

LET T2 = 0. GO T0(35,36,72,73I,J

35 LET T2 » FMCDIFIGl( I ,K),X,11 GO TO 34

36 LET T2 = FMCDIFIGl( I ,KI,V,11 GO TO 34

72 LET T2 = FMCDIF(Gl( I ,K),Z,11 GO TO 34

73 LET T2 = FMCDIF!Gl(I,KI,T,11 34 CONTINUE

LET T3 = 0. GO TO (38,39,74,75),K

38 LET T3 « FMCDIF!Gl ( I,J).X,11 GO TO 37

39 LET T3 >= FMCDIF ( Gl! I, J) ,V, 11 GO TO 37

74 LET T3 = FMCDIF!Gl! I,J),Z,11 GO TO 37

75 LET T3 = FMCDIF!G1!I,J),T,11 37 LET CCSII,J,K) = 0.5*!T1*T2-T3)

LET CCS!J,I,K) = CCS(I,J,K1 LET 0 = IF(U)GO

MATCH TO 3

10, CCS(!,J,K),0.

45

ft IU.D.2

WRlTE{6t2ü3)I>JtK 203 FQRMATdH I5,2HLS I3f2HLS U.IHL»

LET ANS = EXPAND CCS(I,J,K) LET ANS = ORDER ANS.INCiFUL BEGIN = 0.

5 LET BEGIN * BCOCON ANS,LINEtl2 WRITEIb.200)(LINE(LI. 1 = 2,12 I

200 FORHATdH 5X 12A6) IFIBEGIN.NE.OGO TO S ERASE ANS

3 CONTINUE ERASE T1,T2,T3 MRITE(ö,209)

209 FORMAT(1HÜ) WRITE(6,21ü)

210 F0RMAT(28H0ALL OTHER COMPONENTS VANISH I WRITE(6,205)

205 FüRMAT( 30HITHE MIXED CHRISTOFFEL SYMBOLS DO 6 I - 1,4 DO 6 J = 1,4 00 6 K « 1,4 LET MCS(I,J,KI < 0. 00 7 L = 1,4 LET TT = G2(K,L)«CCS(I,J,L) LET TT = EXPAND TT LET SUM = TT IFIEPTERM.EQ.OIGO TO 502 LET SI = COEFF TT,EP«*0,R LET S2 « COEFF TT,EP**1,R LET S3 » COEFF TT,EP«*2,R ERASE TT LET SUM » SI ♦ S2*EP ♦ S3*EP*EP ERASE S1,S2,S3

502 LET MCS(I,J,K) - MCS(I,J,K) ♦ SUM ERASE SUM

7 CONTINUE LET MCS(I,J,K) « EXPAND MCS(I,J,KI LET MCS(J,I,K) * MCS(I,J,K) LET 0 • MATCH ID, MCS(I,J,K),0. IF(ü)GO TO 6 WRITE(6,204)1,J,K

204 FORMATI1H I5,2HLS I3,2HLS I4,1HU ) LET ANS * ORDER MCS ( I, J,K ), INCFUL BEGIN - 0.

8 LET BEGIN • BCDCON ANS,LINE,12 MRITE(6,200)(LINE(L),L<2,12) IF(BEGIN.NE.O.)GO TO 8 ERASE ANS

6 CONTINUE DO 350 I » 1,4 DO 350 J - 1.4

46

i m.D.a

00 350 K = !,<► ERASE CCS( I,J,KI

350 CONTINUE MRITE(6,209) HRITE(6,210)

C C THE EQUATIONS OF MOTION EHIMOI C

HRITE(6t207) 207 FÜRMAT(47HITHE EQUATIONS OF MOTIONf COVARIANT CCMPCNENTS )

00 21 MU = li<t LET SI » 0. 00 22 NU * lf4 LET Tl » 0. GO TO (23.24,25.261.NU

23 LET Tl « FMCDIF(W(MU.NU).X.l) GO TO 27

24 LET Tl - FMC0IF(W(MU.NU),Y,11 GO TO 27

25 LET Tl * FMCDIF(HIHO.NUl.Z.l) CO TO 27

26 LET Tl = FMCOIFlW(MU,NU).T.ll 27 CONTINUt

LET SI « SI ♦ Tl LET SI » EXPAND SI

22 CONTINUE LET S2 « 0. LET S3 « 0. DO 28 I « 1,4 00 2 6 NU - 1.4 LET Tl - MCS(NU.I.NUI*h(MU,n LET Tl ■ EXPAND Tl IF(EPTERMJ508,509,508

509 LET S2 - S2 ♦ Tl ERASE Tl

GO TO 512 508 CONTINUE

LET SO « COEFF Tl,fcP**0,R LET S4 « COEFF T1,EP»«1,R LET S5 * COEFF Tl.EP**2,R ERASE Tl LET S2 « S2 ♦ (SO ♦ S4«EP ♦ S5*EP*EP) ERASE S0,S4,S5

512 CONTINUE LET S2 - EXPAND S2 LET T2 - HCS(NU,MU.n*M(I,NU) LET T2 » EXPAND T2 IF(EPTERHI511.510.511

510 LET S3 « S3 ♦ T2 ERASE T2 LET S3 ■ EXPAND S3 GO TO 28

511 CONTINUE

hi

i III.D.2

LET SO = COEFF T2,tP**0,R LET S4 = COEFF TZ.EP^liR LET S5 = CÜEFF T2,LP**2tR ERASE T2 LET S3 = S3 ♦ (SO ♦ SA*EP ♦ S5«EP*tP) ERASE S0,S4,S5 LET S3 = EXPAND S3

28 CUNTINUc LET EM = SI+S2-S3 ERASE Sl,S2iS3 HRITE(6,206»MU

206 FÜRMATdHO 15, 1HL I 563 FURMAT(15I

LET ANS = EXPAND EM ERASE EM LET ANS = ORDER ANS.INC.FUL

BEGIN = 0. 29 LET BEGIN = BCDCON ANSiLlNE,12

WRITE16,200)(LINE«IDfIL"2f12» 562 FORMAT«12A6I

IF(BEGIN.NE.0.)GÜ TO 29 ERASE ANS

21 CONTINUE 250Ü CONTINUE

STOP END

he

§ III.D.5

3. Output for MOTION

The printed output for MOTION Includes: the metric components, gj, and

g , and the stress-energy tensor, Wj - all of vhich are Input by the user -;

the Chrlstoffel symbols (eqs. (2) and (3)); and the divergence of the

stress-energy tensor ("equations of motion")

^a " ^l = «a',! + V V " ralJ V ' (9)

The format Is similar to that of EINSTEIN; and error messages are Identical

to those for EINSTEIN.

As an exan?>le of output from a confutation by MOTION, we give below the

results for the Interior of a static, relatlvlstlc stellar model. The metric

Is the static, spherically symmetric metric of equation (l), and the stress-

energy tensor Is

T11 = T2

2 = T33 = - p, T^ = p .

Here p (pressure) and p (density of mass-energy) are functions of r ■ X

only. This metric and stress-energy tensor are precisely the ones Included

in the listing of MOTION in the last section. The output follows:

1*9

§ UI.D.3

METRIC TENSOR COMPONEMTS (INPWT)

- Identical to thoae given for this metric on page 23.

STRHSS-ENEKGY TENSOR NON ZERO ELEMENTS, MIXED COMPONENTS (INPUT»

-P» 2L 2U -P$

3L iU -P$

RHC$

CHRISTOFPEL SWlBOIß (OUTPUT)

- Identical to thOEe given for this metric on page 22.

THE tUUATIONS ÜF MOTION, COVARIANT COMPONENTS

1L

-5.0E-l*P*FMCDIH4NL,IX,n)-5.0E-I»RHn*FMCO.F(ANO.(X,n|.(:MCDIF(P,

iL 0.0»

3L 0.0*

-.L O.Oi

50

{ III.DA

I». Sample Problem for MOTION

Input and output for one sample problem, the Interior of a static relatlv-

Istlc stellar model, vere given In the last two sections. As a second example,

consider a star in even-parity, nonradial pulsation. The metric for such a

star was given in equation (7), and the stress-energy tensor is as follows

(cf. Thorne and Caapolattaro 1967):

"l1 " W22 - W3

3 - - p + r-2 e'^2 W P^äp/dr) - rp(Wn),

WUU = p - r-2 e-V2 W P/öp/ör) + (p + p)(Wn)

W^ - (p + p) e-v P/ [^ - r-2 eV2 (dw/ät)l ,

w/- (p+p)r-2e-V2P/(aw/dt),

w^ - (p+p) e-v (öp//ae)(äv/öt),

(10) Wu2 - - (p + p) r"2 (ÖPf/äö)(öv/dt).

Here p (density), p (pressure), and 7 (adlabatic index) are functions of r

only; dn/n is defined by

(Wn) • [- r"2 e"^2(äw/dr) - /(/+l) r"2 V + K+H^fil P^j (U)

/ la the integer order of P (cosö); W and V are functions of r and t associated

with the displacement of the fluid from equilibrium; and the remaining quanti-

ties vere discussed on pages 25 and 26.

51

i III.D.U

To prepare Input for cooputlng the divergence of the stresa-energy

tensor, we nüie the following changes to FjJRMAC notation in addition to

those discussed on page 25 :

i ♦ AJ, P ♦ P

V ♦ V, p ♦ BH^

W ♦ WF, 7 ♦ OAM

Also, because ve need ansvers only to first order in the perturbation from

equilibrium, ve attach the expansion parameter KP**2 to the perturbation

quantities AK, HO, Hl, H2, V, and WF, wherever they appear. The resultant

input and output are as follows:

52

ft m.D.u

ZHFUTt

ATOMIC XtYfZ,T,EP.AL.AJ,ANU.ALAM,AK,H0iHl.H2fViWF,GAMtP.RHO DEPEND (ANU,ALAM,GAM,P,RHO/XI,UL/VI DEPEND IH0fHl.H2tVtWF.AK/X.TI

EPTERM

LET GUI.11 LET GUI.*) LET GlUtll LET GU2.2) LET GU3.3I LEI GU<i,*) LET G2(l,l) LET G2(1,AI LET G2(4I1) LET G2(2,2I LET G2I3,3) LET 02(4,4)

-«-AL*EP»*2*H2*1.)*FMCEXP«ALAM) AL«EP**2»H1 GUI,4»

-«l.-AK«AL*EP**2)*X«*2 -(l.-AK*AL*EP**2)»X*»2*FMCSIN(YI**2 (AL*EP**2«H0*l.)^FMCEXP(ANU)

-(AL*EP*«2*H2*l.)«FMCEXP(-ALAM) AL»EP**2*Hl*FMCEXP«-ALAH-ANU) G2(l,4)

-(AK*AL«EP**2+l.)*X»«(-2) -(AK*AL*EP*»2*l.)*(X»FMCSIN(V))**(-2) (-AL*EP««2*H0*l.»»FMCEXPl-ANU)

LET W(l,l) = -»-AJ*UJ-fl.)*V*X«M-2)*AK*H2/2.-X**«-2l* 1 FMCEXP(-ALAH/2.)*FMCDIF(WF,(X,l)))*AL*EP«*2*GAM« 2 P*AL*EP»*2 ♦MF*X««(-2)«FMCEXP«-ALAM/2. )♦ 3 FMCDIF(P,(X,l))-P

LET M(l,4) « AL*EP*«2*Hl«(P*RHO)«FHCEXP(-ANU)-AL*EP«»2* 1 (P*RH0)*X««(-2)*FMCEXP(ALAH/2.-ANU)* 2 FMC0IF(WF,(T,1I)

LET W(2,2) = W(l,l) LET W(2,4) = EP**2*(P*RH0)«FHCEXP(-ANU)«FMCDIF«AL,( V, 1))♦

I FHC0IF(V,(T,1)) LET H(3,3) = M(l,l) LET W(4,n « AL*EP**2*(P+RH0)*X«*<-2)*FMCEXPI-ALAM/2.)♦

1 FMCDIF(HF,(T,1)) LET W(4,2) * -EP**2*(P*RH0l»X*««-2)*FMC0IF(AL,(Y,l))*

1 FMCDIF(V,(T,U) LET W(4,4) = (-AJ»(AJ*l.)*V*X**(-2)+AK*H2/2.-X**l-2)*

1 FMCEXP(-ALAM/2.)*FMCDIF(WF,(X,1)))*AL*EP«*2* 2 »P+RHO)-AL*EP**2*«F«X**(-2)*FMCEXP(-ALAM/2. )♦ 3 FMCOlFIRHOt(X,1))+RH0

OUTPUT: } ni.D.lf

COVARIANT METKIC TENSOR NON ZERO ELEMENTS (INPUT) 1L U -(-AL*EP**2.0*H2*l.CI*FMCEXP(ALAM)»

1L AL AL*EP**2.0«Hl*

2L 2L -(-AK*AL*EP**2.0+l.Ct*X»«2.0»

3L 3L -(-AK«AL*EP**2.0+l..C»#X»*2.0»FMCSIN(YI**2.0i

AL AL ( AL»£P**2.0*H0n.0»*FHCEXPlANU»$

CDNTRAVARIANT METRIC TENSOR NON ZERO ELEMENTS (INPUT) 1U It -(AL*EP*»2.0*H2*1.0)*EMCEXP(-ALAM|$

1U AU AL*EP**2.0*Hl«FMCEXP(-ALAM-ANU)t

2U 2U -(AK*AL*EP**2.0+l.0)*X**(-2.0)»

3U 3U -(AK*AL*EP**2.0+1.0)*X*»(-2.0)*FMCSIN(Y)**(-2.0)»

AU AU (-AL*EP**2.O*H0+l.O)*FMCEXP(-ANU)t

STRESS-ENERGY TENSOR NCN ZERO ELEMENTS, MIXED COMPONENTS (INPUT) 1L 1U -(-AJ*(AJ+1.0)*V*X**(-2.O)+AK+H2*5.0E-l-X'**(-2.0)*FMCEXP(ALAM*( -5.0E-1))«FMCOIF(HF,(X,1)))*AL*EP**2.O*GAM*P+AL*EP»*2.0«HF«X«*( -2.0)«FMCEXP(ALAM*(-5.0E-l))*FMCCIF(P, !X,l))-P$

1L AU AL*EP»*2.Ü*H1*(P+RHC)*FMCEXH(-ANU)-AL*EP**2.0*(P*RHO)«X«*(-2.0)* FMCEXP(ALAM*5.0E-l-ANU)*FMCDIF(WF,(T,l))t

2L 2U -(-AJ*(AJ«-l.0)*V*X**I-2.O)+AK*H2*5.0E-l-X«*(-2.O)*FMCEXP(ALAMM -5.0E-l))*FMCDIF(WF,(X,l)))*AL*EP**2.0*GAM»r*AL*EP**2.0*WF»X*»( -2.0)*FMCEXP(ALAM*(-5.0E-l))*FMCDIF(Pf(X,l))-PI

2L AU EP»*2.0*(P*RH0)*FMCEXP(-ANU)*FMCDIF(AL,(Y,1))*FMCCIF(V,(T,I))»

3L 3U -(-AJ*(AJ*1.0)»V*X**(-2.O)+AK+H2*5.0E-l-X**(-2.0)*FMCEXP(ALAM*( -5.OE-l))*FMC0IF(WF,(X,l) ) )*AL*EP**2.O*GAM«P*AL«EP**2.0»WF*X»*( -2.0)*FHCEXP(ALAM*(-5.0E-l)»*FMCDIF(P,(X,U)-P$

AL 1U AL«EP**2.0*(P+RHO)*X*»(-2.O)*FMCEXP(ALAM*(-5.OE-l))*FMCClF(WF,(TfI

) )S AL 2U -EP**2.0*(P*RHC)»X**(-2.0I*FMCD1F(AL,(Y,1))*FMCDIF(V,(T,1))»

AL AU (-AJ^(AJ+l.U)*V*X^»(-2.0)*AKtH2*5.0E-l-X**(-2.0)«FMCEXP(ALAM*( -5.0E-l))*FMCDIF(WF,(X,l)))*AL*EP**2.0*(P*RHO)-AL*EP*«2.0*WF»X**« -2.0)*FMCEXP(ALAM*(-5.0e-l))*FMC01F(RH0,(X,l))*RHC$

CHRISTOFFEL SYMBOLS

— Identical to those on pages 27-29.

51*

5 in.DA

THE EQUATIONS OF MOTION, COVARIANT COMPONENTS

a -2.0*AJ*AL*EP**2.0«GAM*P*V^X«*l-3.0l+5.0E-l*AJ*AL*EP**2.0*GAM*P*V* XMI^.OCtFMCOIHANClX.l) ) ♦AJ*AL*EP**2.0*GAH«P*X*»(-2.0 •♦FMCDIFIV i(X,l)»♦AJ«AL*EP**2.0*GAM*V*X**(-2.0)*FHCOIF(P,(X,llI♦5.0E-l*AJ*AL ♦EP**2.0*P«V»X«*(-2.0l«FMCDIF(ANU,(X,l))*AJ*AL»EP«*2.0»P*V*X**l -2.0)^FMCDlFlGAM,(X,in+5.0t-l*AJ*AL*EP*«2.0*RHO*V*X«*(-2.0)* FMCDIFIANU,lX,l))-2.0*AJ**2.0*AL*EP**2.0*GAH«P*V*X**(-3.0)* 5.0E-l*AJ««2.O«AL*EP*«2.0*GAM*P*V*X«*(-2.0)«FMC0IFUNUt(X,l)J+AJ** 2.0«AL*£P*»2.O*GAM*P*X»*(-2.OI*FMCDIF(V,(X,11)*AJ««2.O*AL*EP«*2.0* GAH«V«X*«(-2.0)*FfCCIFIP,IX,l))*-5.0E-l*AJ«»2.0»AL«EP**2.0*P«V*X**( -2.0)*FMCDIF(ANUr(X,l))+AJ**2.0*AL*EP**2.O*P*V*X**(-2.0l*FMCDIF( GAM,(X,1)»+5.0E-1^AJ**2.0*AL*EP**2.0*RHO»V*X**(-2.0»*FMCCIF(ANU,(X ,n)-5.0E-l*AK*AL*EP**2.0*GAM*P*FMCOIF(ANU.(X,l)l-AK*AL»EP»*2.0* GAM*FMCOIF(P,IX,n)-5.0E-l*AK*AL«fcP**2.O*P*FHCDIF(ANUf IX,m-AK*AL • EP**2.0*P*FMCniF(GAM,(X,n l-5.OE-l*AK«AL*EP**2.0*RHO*FMC0 IF ( ANU, ( X,n)-2.5E-1*AL*EP«*2.0*GAM*H2*P*FMCDIF{ANU,(X,1>1-5.OE-1*AL*EP*« 2.0*GAM*H2*FMC0IF(P,IX,1I)-2.0»AL*£P**2.0»GAM*P*X**I-3.0)*FMCEXP(- 5.0E-l*ALAM)*FMC0IF(WF,(X,l))-5.OE-l*AL*EP**2.O*GAM*P*X**(-2.0)* FMCEXP(-5.0E-l«ALAM)*FMC0IF(ALAM,IX.l»I♦FMCDIFIWF,(X,11 )♦ 5.0E-l*AL*EP**2.0*GAM*P»X«*(-2.O)*FMCEXP(-5.0E-l*ALAM)*FMC0IF(ANU, (X,ll)«FMCDIFIWF,(X,l))+AL*EP**2.0*GAM«P*X**(-2.0)*FMCEXP(- 5.0E-l*ALAM)*FMCDIF(WF,(X,2))-AL*EP**2.0*GAM*P*FMCCIFIAK,(X,l))- 5.0E-1*AL*EP**2.0*GAM*P»FMCD1F(H2,(X,11)♦AL*EP^*2.0*GAM*X**1-2.0)♦ FMCEXPI-5.0E-l*ALAHJ*FMCDIF(P,(X,l) l*FMCOIF(HF , ( X, 1) J-2.5£-MAL«EP ♦♦2.0*H2*P*FMCDIF(ANU,(X,l))-5.OE-l«AL«EP**2.0*H2*P*FMCCIF(GAM,(X, l))-2.5E-l*AL*EP«*2.0*H2*RH0*FMCDIF(ANU,(X,l))*5.0E-l*AL*EP**2.0*P *X*«{-2.O)*FMCEXP(-5.0E-l*ALAM)*FMC0IF(ANU,(X,ll)*FMCDIF(k(F,(Xllll ♦AL*EP*«2.ü*P»X**l-2.0l*FMCEXP(-b.0E-l*ALAH»*FMCCIF(GAM,(X,l))* FMCDIFIWF,(X,1)I-AL*EP**2.0*P*X*«<-2.0)*FMCEXP(5.0E-1*ALAM-ANUI« FMCDIF(WF,(T,2)) + AL*EP**2.0*P*FMCEXP(-ANUI«FMCDIF I hi,(T,l)I- 5.0E-l*AL*EP**2.O*P*FMC0IF(h0,(X,l))+5.OE-l«AL*EP«*2.O*RHO«X«*l -2.0)*FMCEXP(-5.0t-l*ALAM|*FMCDIF(ANU,(X,in*FMC0IF(MF,{X,l))-AL« EP**2.0*RhQ*X**(-2.0)*FMCEXP(5.0E-l*ALAM-ANU)*FMC0IF«WF,(T,2»)*AL* EP«*2.0«RH0*FMC£XP(-ANU»*FMC0IF(Hl,(T,I))-5.0E-l«AL«EP»«2.0*RH0* FMCDIF(HO,(X,l)l-2.0«AL*EP«*2.0*WF*X**(-3.0»*FMCEXP(-5.0E-l*ALAM)* FMCüIF(P,(X,l))-5.0c-l«AL*EP*«2.0*WF*X**(-2.0)*FMCEXP(-5.0E-l*ALAM )*FMCOIFlALAM,(X,n)«FMC0IF(P,(X,l))+5.0E-l*AL*EP**2.O*WF*X««l-2.0 )*FMCfcXPl-5.0E-l*ALAM)*FMCDIF(ANU,(X,ll)*FMC0IF(P,(X,l))* 5.0E-1«AL*EP*«2.0«WF*X**(-2.0)*FMCEXP(-5.0E-1*ALAM)«FMCCIFIANU,(X, l))*FMCDlF(RHU,(X,n)+AL«EP*»2.0*WF*X**(-2.0)«FMCEXP(-5.0E-l«ALAMJ ♦FMCDIF(P,IX,2M*AL*EP»*2.0*X**(-2.0)«FMCEXP(-5.ÜE-l»ALAM)*FMCDIF{ P,(X,l))*FMCDIF(WF,(X,l)»-5.0E-l*P*FMCDIF(ANU,IX,l))-5.0E-l*RH0» FMCD1F(ANU,(X,U)-FKCÜIF(P,(X,1) )t

55

§ III.D.J»

THE EQUATIONS OF MOTION (continued)

2L AJ*£P»*2.0*GAM*P*v*X**(-2.O)*FMCUIHAL,(Y,l))*AJ**2.0*EP**2.O»GAM* p«V*X**(-2.0l*FMCUIf;lAL,(Y,n)-AK*EP**2.0«GAM»P»FMCDrFlALf (y,l))- 5.0E-l♦tP♦♦2.O*GAM♦H2♦P♦FMCDIF(AL,(Ytl) (♦eP**2.0*GAM*P«X**J-2.0»* FMCtXP(-5.0E-l*ALAM(*FMCDIF(ALf(Y,l))*FMCDIF(WF,(X,ll)-S.OE-l#EP** 2.0*HO*P*FMCüIF(AL,(Y,1)I-5.0E-l*tP**2.0*H0*RHO*FKC0lF(AL,«Y,1))♦ EP**2.0*P*FHCEXP(-ANU)*FMCCIF(AL,(Y,llI«FMCDIF(V,(T,2)»♦EP«*2.0* RHO*FMCEXP(-ANtil*FHCDIF(AL, (Y, 1) ) «FMCDIF I V, (T , 2 ) ) ♦EP*«2.0*WF«X**( -2.0l*FMCEXP(-5.Ub-l»ALAM)*FMC0IF(AL,<Y,in*FMCDIF{P,(X,n)$

U 0.0»

<.L -AJ*AL*£P«*2.0*P*X*»(-2.0)»FMCDIF(V,(T,1))-AJ*AL*EP**2.0*RHQ^X**| -2.0l*FMCDIF(V,(r,l))-AJ*«2.O*AL»EP**2.O*P*X**(-2.0)*FCCClF(V,(T,l ))-AJ**2.0*AL*, P«*2.i)*RHO*X»*(-2.0)*FMCÜlFlV,(T,1)) + 5.0E-1*AL*EP«* 2.3»P*X*«t-2.ü)*F^CfiXPl-5.0b-l*ALAM)*FMCDIF(ANUi(X,1))♦FMCOIF(WF,( T,lI)+5.0£-l*AL*EP**2.0*RHU*X**(-2.0)*FMCEXP(-5.0E-l*ALAI')*FHCüIFl ANU, (X,l( (♦Ff'CDIFIk.F, ( r,l) ) ♦AL*EP**2. 0«X*» (-2. 0 ) ♦FI'CEXP (- 5.0E-l*ALAMI*FKCDIFiP,(X,l) ) *(-MCüI F I WF , ( T , I I |-EP**2.0*P*X«* (-2,0) ♦ FMCSIN{YI»*l-l.o)*FI'CCOS)(Y)*FMCüIFIAL,(Y,U l*FMCOIF(V, « T, 1) )-EP** 2.0*P*X**l-2.0)*Ft'CtIF(AL. ( Y, 2 ) l*FMCU IF I V, ( I, U I-EP**2. ü*RHa*X**( -2.0)*FMCSIN(YI**(-1.0)*FMCCOS(Y)»FMCDIF(AL,(Yt1)I*FMCDIF(V,(T,l)) -EP*»2.0*RHÜ*X**(-2.0)*FMCDIF(AL,(Y,2)1*FHCCIF(V,(T,I))*

55a

""ff"9!

BLANK PAGE

■• ■■^m sw* ■ 'WX^r&HvimfimmmmmmmimmiF- !wqm:iiwm

$ III.B.l

1. Input for SUBSTITOTE

SUBSTIIOTE is the program which takes long, complicated expressions and

makes changes of variables In them. The Input for SÜBSTITOTE Is handled In

a slightly different manner than that for EINSTEIN, RIB1A1IN, and MOTION:

Four statements are supplied by the user In the body of the program, and the

remaining input information is supplied on "data cards" at the end of the

program.

The statements supplied in the body of the program are these:

1. ATfftac and DEPEND statements. In these statements must be listed

all atomic variables for the original, unsubstltuted expressions

and for the substitutione which are to be made. ?or example, if

X(X) in the expression

G4 B e (-?-^)'? Is to be replaced by

X = - In (1-2VX), (13)

then \ X, m must all be declared in the ATOMIC statement; and

A and m must both be declared In DEPEND.

2. EPTERM statement. EPTERM mist be defined, as in the program

EINSTEIN, to tell the computer whether or not to truncate the

answers at second order in EP. (of. page 9.)

3. A "LBL PARAM" statement. This statement contains a list of the

names of all atomic variables for which substitutions are to be

made. For example, in the situation described above, X appears

56

5 III.E.l

In the LBL PARAM etatement. The entries In the LBL PARAM list must

appear In precisely the same order as Is used for the Input data

(cf. 4b below). The form of the LBL PARAM statement Is as follows:

col.3 col.7

LBL PARAM (NAME1, AA(l)), (NAMES, AA(2)),

(NAMEN, AA(N)).

Here, NAME1, —, NAMEN are the names of the atomic variables for

which substitutions are being made. No more than 50 changes of

variables can be made unless the user redlmenslons AA.

k. The data supplied by the user at the end of SUBSTITUTE Is punched

on cards with the first entry of each card In column 1, and the last

entry before or In column 72. The data which must be supplied are

as follows:

a) Two Integers, on a single card, giving the number of

substitutions, "NSUB", to be made, and the number of

complicated expressions, "NEXP", to be expanded with

the substitutions. The first of these Integers, "NSUB",

must end In column 5 and the second, "NEXP", must end In

column 10. In this card, and this card only, the data

need not start In column 1.

b) Next come the NSUB substitutions - I.e., expressions for

the old atomic variables In terms of the new ones. These

are read In one at a time, each preceded by a card listing

a label or title of the quantity being read In, (these

labels can be 72 characters In length), and each followed by a

blank card. The substitution expressions must be read in

57

5 III.E.l

with the sane order as Is used to list them In the

LBL PARAM statement (see above). Each substitution

expression can be many cards long -with each card's

entries beginning in column 1 and ending before or in

column 72. The conputer identifies the end of a sub-

stitution expression by a dollar sign, $f with which

the user must terminate each substitution,

c) The last quantities to be read in are the NEXP con^Ucated

expressions which are to be expanded in terms of the new

atomic vurlables. These are read in one at a time, each

preceded by a label card, each terminating in a dollar

sign, and each followed by a blank card.

A listing of SUBSTITUTE, including sasple data which illustrate the

above instructions, follows.

56

i 1II.E.2

2. A Conplate IlBtlng of SUiJB'iTXVrE

SIBFMC ONE NODECK 5YMARG

C C PLACE ATOMIC AND DEPEND STATEMENTS FOLLOWING THIS COMMENT C

ATOMIC X«Y.ZtT.ALAM.ANU»FM»PHI "l User: Replace these cards DEPEND (ANU.ALAM»PHI .EM/X) J vlth your own Input.

C C

DIMENSION CARD{6nn)fAA(5n) DATA BLANK /6H /

C INTEGER EPTERM

C C IF EPTFRM = 0 TERMS CONTAINING ALL POWERS OF EXPANSION C PARAMETER WILL BE RETAINED. C IF EPTFRM .MF. 0 TERMS CONTAINING THE EXPANSION PARAMETER C TO A POWER GREATER THAN 2 WILL HE DISCARDED, C A CARD CONTAINING THF INTEGER VALUE OF EPTFRM MUST FOLLOW THIS C COMMENT. C m-rrm. * i User: Replace this card c FPTERM = o } ^^ your own lnpirt#

WRITE(6.?n2) 202 FORMAT(57H1SUBST!TIJTIONS TO BE MADE IN CURVATURE TENSOR EXPRESSIO

INS ) WRITE(6.2nO)

200 FORMAT (IHfl) READ!5,101INSURS.NEXP

101 FORMAT(2I5) DO 10 II = l.NSUPS READ(5,]02)(CARD(J),J=1.1?)

102 FORMÄT(12A6) WRITE(6»105)(CARD(J)»J=l ,12)

105 FORMATdH 5X i2A6) I = 0

1 CONTINUE JJ = 1+1 JJJ = 1+12

READ(5,102 MCARD( J) ,J = JJ.JJJ) IFtCARDUJl.EO.RLANOGO TO ?

WRITF(6iin6)(CAPD(J),J=JJ,JJJ) 106 F0RVAT(10X,12A6)

1=1+1? GO TO 1

3 K = n LET AA(II) = ALGCON CARD,K

10 CONTINUE C C PLACE THE LBL PARAM STATEMENT FOLLOWING THIS COMMENT C

^ , User: Replace this card LBL PARAM (ALAM,AA(in, (ANU,AA(2)) ) „m yav am ilVXt.

59

I ni.E.2 DO ?0 II = l.NEXP WRITF(6.104)

10A FORMAT(lHl) WRITF(6,?00)

RFAD(5,??2)(CARO(J)fj=lfl?) 722 FORyAT(12A6)

WRITF(6i223)(CARD (J)fJ=1.12) 223 FORMAT(5X.12A6)

I = 0 1? CONTINUF

JJ = I + 1 JJJ =1+12 RFAD(5.]02)(CARD!J)tJ=JJ.JJJ) lF(CARD(JJI.eQ.BLANK)GO TO 2 WRITE(6.105)(CARD! J)»J = JJ»JJJ) 1=1+1? GO TO 12

2 K = f) LET EXP = ALGCON CARD.K LET ANS = SUBST EXPtLBL IF(EPTFRM.FQ.0)GO TO 32 LET ANS = EXPAND ANS ERASE FXP LFT ZO = COFFF ANS,FP**niiPP LFT 21 = COFFF ANS,FP»»1.PP LFT Z2 = COEFF ANS,EP**2.PP ERASF AMS

LFT FXP = FP**0»ZO + EP*»1«Z1 + FP»*2*72 GO TO 33

32 CONTINUE LET EXP = ANS

33 CONTINUE LET EXP = EXPAND FXP

WRITE(6t]07) 107 FORMAT(3AHnTHF FXPRFSSION AFTER SUBSTITUTION )

WRITE(6.2nO) S = 0.

11 LFT S = RfDCON EXP ♦ CARD ♦ 12 WR1TE(6.105)(CARD!J)»J=2.12) IF(S.NF.O.)GO TO 11

20 CONTINUE STOP END

60

S ni.E.s

Bf far SUBSTTIVIE

(Usert Replace these data cards ty your own data cards.)

7 4 ALAM - FMCLOG(l-2»EM/X) $

ANU 2»PHI $

1U 1L EINSTfIN TENSOR COMPONENT -X»»(-2)»FMCEXP(-ALAM) + X»»(-2) - X»»(-1)»FMCEXPI-ALAM»»FMCDIF(ANU.Xt1) 5

2IJ 2L EINSTEIN TENSOR COMPONFNT l./(X»?.)«FMCFXP(-ALAM) • FMCD I F ( ALAM,X.1) - l./(X»2.) *FMCEXP(-ALAM)*FMCDIF(ANU.Xtl) + FMCFXP(-ALAM)»FMCDIF(ALAM,X. 1)/4. «FMCDIFIANU.X.D - FMCEXP(-AL AM)»FMCDIF(ANU.X♦1)»*2/4, - FMCEXP (-ALAMi •FMCDIF(ANU»X,2)/2. $

31J 3L EINSTEIN TENSOR COMPONENT l./(X»?.)»FMCEXP(-ALAM)»FMCDIF(ALAM,X.l) - l./(X#2.)»FMCEXP(-ALAM) »FMCDIF(ANUiX.l) + FMCEXPt-ALAM)«FMCDIF(ALAM,X,1)*FMCDIF(ANÜ.X.1)/4. -FMCEXP(-ALAM)»FMCDIF(ANU.X,1)»»2M. - FMCEXP(-ALAM)»FMCDIF(ANU,X.2) /?. $

*U 4L EINSTEIN TENSOR COMPONENT -l./(X»X)»FMCEXP(-ALAM) + l./(X»X) + 1,/X»FMCEXP(-ALAM)»FMCDIF( ALAMtX,1 ) $

61

i III.B.3

3. Outpirt of SUDBTriVrE

The output for SUBSTITOTK begins vltta a reproduction of all the sutoatltu-

tlone (changes of variable) to be made. Then follow, In the order In which

they vere input, the various ccag)llcated expressions to be expanded. These

are printed, first in their original form, and Imedlately thereafter expanded

In terms of the new atomic variables.

As an example, here is the output generated by the data which were given

in the last section (NSUB - 2, HEXP - k)t

SUBSTITUTIONS TO DE MADE IN CURVATURE TENSOR EXPRESSIONS

ALAM -FMCL0&(1-2*EM/X» t

A NU 2»PHI i

IU 1L bINSTtIN TENSOR COMPONENT _X**(_2)«FMCEXPI-ALAHI ♦ X**(-?) - X**(-1)*FMCEXP«-ALAN»*FMCDIF(ANUtX.lI t

THE EXPRtSSION AFTER SUBSTITUTION

EM«X**(-3.0)*2.ü+eM«X**(-2.0)«FMCDlF(PHI,(X,l))*4.0»X«*I-1.0)* FMC0IF(PHl,(X,in*(-2.0)t

62

9 III.E.3

2U ^L EtNSTF.IN TENSOR COMPONENr l./(X*2.)*FMCEXP(-ALAM) ♦ FMCOIF(ALAM.X,1» - l./(X*2.) »FMCEXP(-ALAMI«HMCDIF(ANU,X,l) ♦ FMCEXP (-AL AM) ♦FMCDIF ( AL AM,X, I »M. ♦ FMCDIF(ANU,X,1) - FMCEXPt-ALAM)*FMCOIF ( AMUt X , I )*«2/'». - FMCEXP«-ALAM) ♦ FMCDIFUNUtX,2)/2. t

THE EXPRfcSSION AFTER SUBSTITUTION

-EM«X»*(-3.0l*EM*X«*(-2.0l*FMCOIF(PHl,(X,l))ttM*X«*(-1.0)*rMCDIF( PHI,(Xll))**2.0*2.0*tM*X**(-1.0)*FMCDIF(PHI,(X,2n*2.0*X**(-2.0»* FMCDIF(tM,{X,l))*X*»(-l.O)*FMCDIF(EM,tX,l))<'FMCDIF(PHI,(X,l)l-X**( -l.O)*FMCDIF(PHI,(X,I)l-FMCOIF(PHI,(X,ll)**2.0-FMCDIFIPHI,(X,2l)t

3U 3L EINSTEIN TENSOR COMPONENT l./(X*2.)*FMCEXP(-ALAM)*FMCOIF(ALAM,X,l) - 1./(X*2.»»FMCEXP(-ALAM ) *FMCDIFlANU,Xtll ♦ FMCEXP(-ALAM»OFMCDIF(AL AM,X,1)*FMCOIF(AMU,X,lI/4. -FMCEXP(-ALAHI*FMCOlF(ANU,X,l)**2/4. - FMC6XP(-ALAM)»FMCOIF(ANU,X,2 I

/2. «

THE EXPRESSION AFTER SUBSTITUTION

_EM«X**{-3.0)+EM*X**(-2.0)*FMCOIF(PHI,(X,U)+EM*X**t-1.0)*FMCDIF( PHI,(X,l)l**2.O»2.O+bM*X**(-1.0)*FMCniFtPHl,(X,2H*2.OtX**(-?.0)* FMCOIFIEM,(X,l))*X**(-l.O)*FMCDIKEM,(X,l))*FMCDIF(PHI,(X,il)-X*«( -l.u)*FMCDIF(PHI,(X,ll)-FMCDIF(PHI,(X,l))*»2.0-FMCDIF(PMI,(X,2nt

4U <»L FINSTEIN TENSOR COMPONENT -l./(X*X)*FMCEXP(-ALAMI + l./(X*XI + 1./X^F^CEXPI-ALAM»»FMCUIFIALAM,X,1» t

THE EXPRrSSIDN AFTER SUBSTITUTION

X*«(-2.())*FMCDIF(rM, IX,1> )*2.0»

63

} III.E.U

k. Sanple Problen» for SUBgrnWTE

Input and output for one sample problem were given In the last two sec-

tions. As a second exanqple we give Input and output for a problem In which

United core storage actually forced us to use SUBSTITUTE:

Chltre, Hartle, and Thome (1967) have been studying recently the funda-

mental modes of pulsation for slowly rotating, rfllatlvlstlc stellar models.

To second order (e ) In the rotation and first order {(,) in the pulsation,

the metric for such a star can be put Into the form

gii ° eX I1 + 2 €2 f'1 eX + Vs) + 2 S'"1 *\ - «^(«o + Vs^

gl4 " «Ul • - ^ V2

g^ = r2 {1 + 2 £2 (T2 - h2)P2 - e2t K£z]

833 " r2 Bln20 {1 + 2 e2 (v2 - h2)P2 - 625 K2P2)

&3k ' gU3 ' r2 sin2e W* " 0) " «Cli)

«1^ " - «V [1 + 2 e2 (hQ + h2P2) - e2 e"vr2 sln28 (S - fl)2

+ €2t (K0 + N^)) (1U)

\*ere x - r, x2 » 9, x3 ■ q), x - t, and

P2 - i (3 cos2« - 1). (15)

In this metric A, v, i3, hp, hg, n^, Bg, and Tg are functions of r only; p.,

Jl' Ho* H2» N0' N2' aad S "^ functlon8 of r and *» »n«1 e, S, fl are constanta.

This metric is so coaplicated that EDGTEIN was unable to calculate the

61»

i m.EA

correspoodlng alxed Qurlstoffel syAols or Rlcci tensor or Einstein tensor.

An error mntoer 7537 (ca^uter's memory exhausted) was obtained when the

eaqpatatloa yma attempted.

To get around this difficulty, a slupllfled metric, equivalent to

expression (Xk) hat containing fever terne, vas put Into EINSTEIN:

«11 " ^ U + e2 fi] ,

g^ - r2 {I + e2 C} ,

gjj - r2 sln20 {1 + e2 C} ,

gj^ = - ev (1 + e2 0] + c2 (r2 Bln2e)1/2 ;

811 » «» (1 " ^ A) ,

14 41 2 -v _ _ g . g - - € e «^ »q,

g22 - r-2tl.620).

(16a)

(16b) g33 - (r sine)-2 {1 - c2 C} - e2 e'V2 ,

34 43 -v .u/- g =g - - e e >ir,

g44 • - e-v {1 - €2<7} ,

Here the new variables 9^, 9e, and 7/ are functions of r and t; while a, B,

C and 9 are functions of r, 6, and t. EINSTEIN was able to compute the

Bled and Einstein tensors for this sl^llfled metric to order e2 without

65

5 in.E.4 I i

(17)

exhausting the computer's nenory. The resultant Riccl and Einstein tensors

were then put through SUBSTITOTE, which made the following changes of variable

- accurate to first order in ( - to return them to the original notation:

^.S [1 + 2 Cr"1 eA u0] ,

*e - e-X {1 - 2 tr"1 eA M0} ,

-^ - o - 3 + 5J1 ,

a '2 Ze^i^a - Zftr2 ßln2ö + 2 h0 + WQ

+ (2 l^ + 5N2) i (3 co828 - 1) ,

fl - 2 eX (ny/r + (m^/r) J (3 cos2« - l)} {1 - 2 letyr]

- 5 (HQ + H2 i (3 cos2© - 1)) ,

& . 12 (v2 - hg) - tK2) i (3 cos2Ö - 1) ,

In making these changes of variable by means of SUBSTITUTE, we exhausted the

computer's memory on a few of the coqronents of the Einstein tensor. When

that happened, we broke the difficult components into pieces and performed

the substitutions piece by piece.

As a guide to the user of SUBSTITOTE who has a long problem to do, we

reproduce here the input to and the output from SUBSTITOTE for part of the

above problem. Note that SUBSTITOTE does not truncate in powers of (

(denoted EZ below), so that truncation must be done by hand.

66

fi in.E.if

ATOMIC XtY.Z.T.EPtEZ ATOMIC ALAM.ANU.PIK.PIE.A.B.CtFO.WR ATOMIC AMU0.AJltH0.H2fCH0fCH?,FNn.FN2.EWn.EM2iAK2.V2.O2.OM,OMB DEPEND (ALAM»ANU,OMn.HO»H2.FMO»EM2«V2/X) DEPEND (PlK.PIE»WRtAMUO.AJl.CHn.CH2.EN0,EN2.AIC2.O2/X.T) DEPEND (A.B.C»FO/X,Y.T)

({ m.x.i.i)

EPTERM (8 IIX.E.1.2)

LBL PARAM (PIKtAA(1) ). (PIE iAA(2 ) 11 (WRtAAl 1 (BiAA(5l)t (CtAA(6))t (FO.AA(7l)

3) ) . ! A.AAC») ) »

(ft m.E.i.s)

PIK I 2 (5 in.K.iA (.)) FMCEXP(ALAM) » <1. + 2. • EZ «FMCEXPCALAM) » AMUO/X)$

PIE - FMCEXP(-ALAM) • Il.-2.«EZ»FMCEXP(ALAM)»AMUO/X) $

WR = OM - 0*8 + EZ » AJ1 $ (ft in.x.iA (b))

?. • EZ • FMCFXP(-ANU) • AJ1 » lOV - OMR) » X » X » FMCSIN(Y) *» 2 + 2. • HO ♦ EZ ♦ ENO + (2. • H2 + FZ * EN2) ♦ 0.5 » (3. • FMCCOS(Y) »• 2 - 1.1 $

R = 2. • F»«CEXP(ALAMJ • « EMO/X + EM2/X • 0.5 * (3. » FMCCOS(Y) »«2 -1.1) • II. - 2. » EZ • FMCEXPJALAM) • AMUO/X) - E2#(CHO+CH2» 0.5» n.«FMCC0S(Y)»«7 - ].)) I

C = (7. • (V? - H2) - FZ • AK7) • 0.5 • (3. »FMCCO?(Y) ** 2 1.) S

FO = EZ • 02 • 0. 5 » 13. • FMCCOS(Y) *• ? 1.) S

1 3 RICCI TENSOR COMPONFNT 5.0E-l»EP»PIE»X»»2.0»FMCSIN(Y)»«2.0»FMCFXP(-ANU)*FMCÜIKtwR,(T.l)•( X.1J »-2.5E-l«EP»PIE«»2.0»X»*2.0»FMC.SIN(Y)»*2.0*FMCEXP(-ANU)»FfiCDIF IPIK«ITtl))»FMC0IF«WR»(X.1))S (ft III B 1 U (c))

k 5 RICCI TFNSOR COVPONFMT -2.0»FP»PIE*X»FMC.C:NIY)»»2.0»FI«,CFXD(-ANIJ)»FMCDIF(,A'R,(X.1 ) ) + 7.5F-1»EP«PIK»X»«2.0»FVCSIN(Y)«»?.0«FMCFXP(-ANI))»F,,C1)IF( /'MJ,(X»1 ) ) • F^OIFfVRflX,! ) )-5,OF-1»FP*PIF«X*»2.n»FMCSIN( Y)*»?.n«F''CFXP(-AMU) •F^OIFIWRtlX,?) )-2.?>F-l»FP*PIF»»2.n»X»*2.0»F"C5iM(V)**?.f)*"MCLXP( -A»£IJI»FMCDIF(PI'' tlX,] ) )»FN<riIF(WR,(X,l I )-5 . ClE-l »EP» X**2 . 0-F./CS I N ( Y )»«?.f)»Fvcrxp(-ANin«FMCiJir(PiF.<x.i ))«FWCDIF(V,R. (X.D II

67

i ZII.E.4 OOTm

4 3 RICCi TENSOK CÜMPÜNENT -2.0*EP*PIE*X*FMCSIN(y)»*2.0*FMCEXP(-ANU)*Fh»CCIF(KR,(X,l))♦ 2.5E-l*cP*PIE*X**2.0*FMCSIN(Y)*'*2.0*FMCfcXP(-ANU)*FMCOIF(ANU,(X,n» ♦FMCDIFCWR,(X,l()-5.0E-l*EP*PIE*X**2.0*FMCSIN(Y)«*2.0*FMCEXP(-ANUI ♦ FMCDIFIHR, (X,2))-2.5E-l*tP*PIE**2.O*X*«2.0*FMCSIN(YI♦♦2.0»F>'CEXP« -ANUI*FMCDIF(PIKf(X,ll)*FMCL>IF(WR,(X,1 I )-b.OE-l*£P*X**2.0*FMCSIN(Y »♦*2.0*FMCEXP(-ANU)*FHCOIF(PIEfCX,ll)*FMCOlFCWR,(X,in»


AMU0*EP*>EZ*X*FMCSINm**2.O»FMCEXP(-ANUI*FMCDIFUNU,(X,l))*FMC0m OMB.lX.l))*5.0E-l-AMUO*£P*£Z*X*FMCSIN(Y)**2.0*FMCEXP(-ANU»*FMCOIF( OMBi1X,2))+AHU0*EP*Ei»FMCSIN(Y)**2.O*FMCEXP(-ANU)*FMCDIF<OMB,(X,H )*{-3.4S99S998)*AMU0*EP*EZ**2.0*X*FMCSlN(YI**2.0*FMCEXP(-ANUJ» FMCDIF(AJl,(X,l()*FMCOIF(ANU,(X.l))*(-5.0£-l)+AMUO*EP*EZ**2.0*X» FMCSIN(Y)*«2.0*FMCEXP(-ANU)*FMCD(F(AJi,(X,2I1♦AMU0«EP*EZ*«2.0* FMCSIN(Y)**2.0*FMCEXP(ALAM-ANU)*FHCDIF(AMU0f {X,m*FMCDIFIOMö,U,l ))*(-2.01*AMLO*bP*EZ**2.0*FMCSIN(Y)*«2.U*FMCEXP(-ANU)*FMCDlF«AJl,« X,l)»♦3.49999S9b*AMU'J*tP*EZ**3.O*FMCSIN(Y)**2.0*FCCEXP(ALAH-ANU»* FMCDIFIAJl,(X,l))*FMCDIF(AMUO, I X,1)1«2.O + AMU0**2.0*tP*EZ**2.O»X**J -l,0l*FMCSIN(Y)**2.0*FHCEXP{ALA*-ANU)*HMrDIF(0MB,(X,l»)*2.0*AMUÜ*« 2.0*EP*cZ**2.0*Ff«CSIN(Y)**2.0*FMC£XP(ALAM-ANUI*FMCDIF(ALAM,(X,l|»« FHCÜIFIOHÖ,(X,l)l*{-3.U)+AMU0**2.O*EP*EZ**3.O«X**(-l.Ü)*FMCSIN(Y) ♦ ♦2.0«FMC£XPlALAM-ANU)*FMCDIF(AJlf(X,l) I *(-2.0)♦AMUO**2.0*EP«EZ«» 3.O»X*«(-l.OI*FMCSIN(Y)**2.O*FMCEXP(ALAM*2.0-ANUI*FMCüIF(AMU0f(Xtl »»♦FMC0IF1QMD,(X, I)) ♦2.0*AMUO**2.0*EP*tZ**J.O*FMCSIN(Y)**2.0» FMCEXP(ALAM-ANU)*FMCOIF(AJl,(X,l))*FMCDIF(ALAM,(X,l))*3.0*AHU0** 2.0»EP*£Z**4.0*X«*(-1.0)*FMCSIN(Y)**2.0*FMCEXP(ALAI»*2.0-ANU»* FMCOIF(AJl,(X,l))*FMC0IF(AMüO,(X,l))*(-2.0)*AMU0**3.n«EP*E/>*3.ü»X ♦♦(-2.0)*FMCSIN(Y)«*2.0*FMCEXP(ALAM*2.O-ANUI*FMC0IF(uMB,(X,ll)»I -2.0)*AMUü**3.0»EP*EZ«*3.0*X**(-l.OI*FMCSIN(y)**2.0*FMCEXP(ALAM* 2.0-ANU)*FMCDlF(ALAM,(Xtl))♦FMCOIF{QMh,(X,1))»«.O*AMU0**3.0«EP*tZ »♦4.0*X**(-2.0»*FMCSIN(YI**2.Ü*FMC£XP(ALAM*2.0-ANL)*FMC01FIAJI,(X, 1))*2.0<-AMUO**3.0*EP*EZ**4.Ü*X**«-l.O)*FMCSIN(Y)**2.0*FMCEXP(ALAH» 2.0-ANUI*FMCDIF( AJl,{X,n)*t-MCUIFIALAM, 1 X, 1 I ) ♦(-^.0 ) »EP^EZ^X» FMCSIN(YI*«2.0*FKCEXP(-ALAM-ANU)*FMCDIF(AJl,(X,l))*l-2.0l*EP«EZ*X* FMCSIN(YI**2.0*FMCEXP(-ANU)*FMC0IF(AMU0,(X,1))*FWCC1F(0MB,(X,ll>«I -5.0E-l)tEP*£Z*X**2.0*FMCSIN(Y)**2.0*FMCEXP(-ALAM-ANU»*FHC0IF(AJlt

(X,l))*FMCDIFIALAM,(X,U(*2.5t-l«-EP*tZ*X**2.0«FMCSIN(Y)*«2.0« FMC£XP(-ALAM-ANUI*FMCOIF(AJl,(X,in*FMCCIF(ANU,IX,l) I »2. 5E-UEP*EZ *X**2.0*FMCSIN(Y)**2.0*FMC£XP(-ALAM-ANU)*FMCDIF(AJl,(Xf2n*( -5.0E-ll+£P*EZ*^2.0*X*FMCSIN(Y)**2.0*FMCEXP(-ANU)*FMCDIFUJlf (X,l) )*FMCDlF(AMU0,(X,U)*5.0E-ltEP*X*FMCSIN(Yl**2.0*F(»CEXP(-ALAM-ANU»* FMC0IF(OMB,(X,ln*2.0+£P*X**2.O*FHCSIN(Y)*^2.O*FHC£XP<-ALAH-ANU)* FMCDIFIALAM,(X,1))«FMCDIF(OMB,(X,I))♦(-2.5E-1)♦£P*X*«2.0«FMCSINIY) ♦♦2.0*FMCEXP(-ALAM-ANU)*FMCDIF(ANU,(X,1))*FMCDIF(0MB,(X,1)!♦( -2.5E-l)*tP*X**2.0*FMCSIN(Y)**2.0*FMCEXP(-ALAM-ANU)*FMCCIF(OMDf(X, 2)1*5.0E-1$

G8

(m.E.v

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69

§ IV .A

IV. A BRIEF GUIDE TO FORMAC PROGRAMMIMG*

A. The FORHAC System

In writing a program using FORMAC it is helpful to consider the FORMAC

compiler as an extension of the facilities of FORTRAN IV which allows for the

aRnlpulatlon of algebraic symbols. The FORTRAN IV features, such as integer

and floating point arithmatlc, type statements, data statements, and logical

operations, all exist and can be used in FORMAC programs.

When a FORMAC program is submitted for compilation the machine Imnediately

produces a purely FORTRAN program, and the FORMAC conuands are rendered as

cannent statements.

For further detail« see the PORMAC mnual (IB4 1962).

70

5 rv.B

B. FORMAC Executable Statements

Because algebraic quantities rather than numbers are usually being

manipulated by a PORMAC program. Input data are considered as alphanumeric

Information and are read Into the machine by the FORTRAN IV "A" format. A

thlrty-slx bit computer word In core storage can contain 6 alphanumeric

characters. Thus 72 columns on em IEM data card represent 12 computer vordc.

The program SUBSTITOTE Illustrates how an algebraic expression up to 50 cards

In length can be read Into memory.

Once an expression, say ALGEXP, has been entered. It must be converted

to a form that can be manipulated conveniently by the compiler. This Is done

by the PORMAC executable statement ALGC0N

K = 0

LET A = ALGC0N ALGEXP, K

It must be emphasized that all variables which appear In expressions that are

read In as data must be declared In the ATOMIC and DEPEND statements.

When a quantity, such as A above, has been put Into proper form by means

of an ALGC0N statement, It can then be operated on by other FORMAC executable

statements. The following executable statements are available In FORMAC.

(Note that some of these statements have been described In earlier sections.)

AUTSIM Used to evaluate elementary functions that have

constants or FOBTHAN-type expressions as arguments.

BCDCjiN To be described below.

CENSUS Used to count the number of computer words, terms,

or factors in a FORMAC expression.

71

J IV.B

C(5EFF

ERASE

EVAL

EXPAND

FIND

LET

MATCH

PART

SUBST

Used to obtain the coefficient. In an algebraic

expression, of an atomic variable or an atomic variable

raised to a specified power.

Used to erase a FORMAC quantity in core storage.

Used to evaluate a FORMAC expression using supplied

values of the variables. This results in a numeric

quantity.

Used to simplify, collect terms, and remove parentheses

in algebraic expressions.

Used to establish a function relation between a FORMAC

expression and prescribed atomic variables.

Used to define algebraic expressions.

See below.

See FORMAC manual.

See below.

Used to specify the sequence in which factors in products,

and terms in suns are to appear in expressions.

These comnani . are described in more detail in the FORMAC manual, and

most of them were used in the ALBERT programs.

It is useful to note that most operations performed on numeric quantities

have their FORMAC analogs. For example, logical operations can be perfomsd

on FORMAC variables Just as they can be on FORTRAN variables: One algebraic

expression can be caqpared to another by the FORMAC function MATCH. The

sequence

ATOMIC X, Y

LOGICAL Q

72

5 IV.B

LET C = X*»2 + Y«*2

LET Q = MATCH ID, B, C

IF(Q) GO TO 2

Q6NTIHüE

vould compare B to Cj and if they vere eqiilvalent, It vould set the value of

Q to the TORTBAN logical TRUE and then transfer to statement number 2 for its

next instruction.

Once a desired algebraic quantity has been constructed, it can be evalu-

ated numerically, or substitutions can be made that vill convert it to yet

another symbolic quantity. The POFMAC operation

LET Z = SUBST B, LIST

vill result in a new algebraic quantity, Z. LIST is a listing of the syinbols

In B and the names of new expressions that are to take their places. The

actual fonmtlon of UST is done by the PORMAC function, PARAM. For example,

consider the following:

ATOMIC X, Y ALPHA, BETA

IBT C = X*«2 + Y«*2

UX A * AIiPHA«*2

LET B = BETA**2

LIST PARAM (X,A), (Y,B)

LET Z » SUBST C, LIST

Z now has the value ALPHA*«* + BETA*«*. Notice that the statements on cards

begin in colvann 7 but the word, LIST, is placed in columns 1 * k.

Assuming a computation has been coiqpleted, the information, other than

numeric, must be put into a form suitable for printing. This is accomplished

by the PORMAC conmrnd BCDQÄN.

75

§ IV.B

niMENSIjJN CARD (12)

ANS > 0

LET ANS ° BCOC^N Z, CARD,12

will cause alphanumeric Information to be etored In the 12 conputer uorts of

the array CARD. This can In turn be vrltten out In FOHTRAll'a "A" formt.

74 (

i IV.C

C. Ways to Avoid Exhausting the Computer's Memory

What has been attempted with the ALBERT prograns is to make them as

simple and easy to use as possible. As mentioned earlier, FORMAC on the

191 709k suffers from a severe lack of core storage. The program EINSTEIN

plus system overhead took in the neighborhood of 5/7 of the 32K memory

capacity of the machine. Reading the Input metrics in on data cards would

conserve seme core space; but for clarity we chose to define the metric by

means of LET statements in the body of the program. We have circumvented

the core problem to some extent by simplification of the metrics themselves,

by punching the output from EINSTEIN on cards, and by expanding that output

in terms of new variables using the SUBSTITUTION program (cf. } III.E.l*).

FORMAC provides for the use of an overlay feature, which might be

another alternative for solving core problems. Yet another, is to purge

from the system portions of the user's program after they have been executed

(see Clemens and Matzner 1967). However, the only truly satisfactory solu-

tion to the core problem will be to go to a computer with e larger memory —

e.g. the ia( 360 series.

75

V. RKFERENCES

Chitre, A. M., Hartle, J. B., and Thome, K. S. 1967, "The Pundaaental

Mode of Pulsation of Slowly Rotating, Relatlvietlc Stellar Model«,"

In preparation.

Clemens, R. and Matzner, R. 1967, "A System for Symbolic Computation of

the Rlemann Tensor," Technical Report Bo. 635, University of Maryland,

Department of Physics and Astronony, College Park, Maryland.

Fletcher, J. 0. 1965, "Grad Assistant - A Program for Symbolic Algebraic

Manipulation and Differentiation," UCRL Report 1^62U-T.

196G, "Grad Assistant - A Program for Symbolic Algebraic

Manipulation and Differentiation," Comnunicatlons of Assoc. for Compnt.^

Machinery. £, 552.

Fletcher, J. G., Clemens, R., Matzner, R., Thorne, K. S., and ZiMaerman, B. A.

1967, "Computer Programs for Calculating General Relativlstic Curvature

Tensors," Ap. J. Letters. IJjS, L91.

Hartle, J. B. 1967, "Slowly Rotating Relativlstic Stars, I. Equations of

Structure," Ap. J.. to be published December 1967.

IEM 196k, User's Preliminary Reference Manual for the Experimental FORmul/»

MAnlpulation Compiler (FOBMAC). (international Business Machines

Corporation, Boston Advanced Prograiming, 545 Technology Square (3rd

floor), Cambridge, Massachusetts 02139).

Thorne, K. S. and Campolattaro, A. 1967, "Nonradial Pulsation of General-

Relatlvistic Stellar Models: I - Analytic Analysis for / ;. 2,"

AP» J«. to be published September 1967.

76

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