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General Relativistic Calculations inMathematica

By George E. Hrabovsky

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What I Will Cover0000 What is xAct?0000 What Do We Do Next?0000 Contraction0000 Covariant Derivatives0000 The Second Bianchi Identity0000 Perturbative GR

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What is xAct?

This is a very complete system for doing tensor analysis in the Mathematicasystem.

All you need do is open the system:

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What Do We Do Next?

Now that we have started things up, there are some things we need to do.

1. We need to define the underlying manifold we are working in. We do this with the

DefManifold command: DefManifold[M, dim, {a, b, c,...}] defines M to be an ndimensional

differentiable manifold with dimension dim (a positive integer or a constant symbol) and

tensor abstract indices a, b, c, ... . DefManifold[M, {M1, ..., Mm}, {a, b, c,...}] defines M to be the

product manifold of previously defined manifolds M1 ... Mm. For backward compatibility dim

can be a list of positive integers, whose length is interpreted as the dimension of the defined

manifold. Here we define a fourdimensional manifold having the indices

a,b,c,d,e, f,g,h,i, j,k,l:

DefManifold[M, 4, {a, b, c, d, e, f, g, h, i, j, k, l }]

**DefManifold: Defining manifold M.

**DefVBundle: Defining vbundle TangentM.

In defining the manifold, we have also defined thevector bundlecalled TangentM, : TMM. This is where our tensors will be.

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2. We need to define the metric using the DefMetric command: DefMetric[signdet, metric[a,b],

covd, covdsymbol] defines metric[a, b] with signdet 1 or 1 and associates the covariant

derivative covd[a] to it. Note that we have the convention [a] has a as the subscript, and [a]

has it as the superscript.

DefMetric[- 1, metric[- i, - j], cd, {";", ""}, PrintAs "g"]**DefTensor: Defining symmetric metric tensor metric[-i, -j].

**DefTensor: Defining antisymmetric tensor epsilonmetric[-a, -b, -c, -d].

**DefTensor: Defining tetrametric Tetrametric[-a, -b, -c, -d].

**DefTensor: Defining tetrametric Tetrametric[-a, -b, -c, -d].

**DefCovD: Defining covariant derivative cd[-i].

**DefTensor: Defining vanishing torsion tensor Torsioncd[a, -b, -c].

**DefTensor: Defining symmetric Christoffel tensor Christoffelcd[a, -b, -c].

**DefTensor: Defining Riemann tensor Riemanncd[-a, -b, -c, -d].

**DefTensor: Defining symmetric Ricci tensor Riccicd[-a, -b].

**DefCovD: Contractions of Riemann automatically replaced by Ricci.

**DefTensor: Defining Ricci scalar RicciScalarcd[].

**DefCovD: Contractions of Ricci automatically replaced by RicciScalar.

**DefTensor: Defining symmetric Einstein tensor Einsteincd[-a, -b].

**DefTensor: Defining Weyl tensor Weylcd[-a, -b, -c, -d].

**DefTensor: Defining symmetric TFRicci tensor TFRiccicd[-a, -b].

**DefTensor: Defining Kretschmann scalar Kretschmanncd[].

**DefCovD: Computing RiemannToWeylRules for dim 4

**DefCovD: Computing RicciToTFRicci for dim 4

**DefCovD: Computing RicciToEinsteinRules for dim 4

**DefTensor: Defining weight +2 density Detmetric[]. Determinant.

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3. We can define tensors using the DefTensor command: DefTensor[T[a, b, c, ...], {M1, ...}]

defines T to be a tensor field on manifolds and parameters M1,... and the base manifolds

associated to the vector bundles of its indices. DefTensor[T[a, b, c, ...], {M1, ...}, symmetry]

defines a tensor with symmetry given by a generating set or strong generating set of the

associated permutation group. In fact we can define any order of tensor just by specifying theindices. A scalar has no indices so we can define the scalar

DefTensor[s[], M]

**DefTensor: Defining tensor s[].

s

s

a tangent vectorDefTensor[tv[i], M]

**DefTensor: Defining tensor tv[i].

tv[i]

tvi

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a covectorDefTensor[cv[- i], M]

**DefTensor: Defining tensor cv[-i].

cv[- i]

cvi

and so on.

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We can define tensors by their symmetries or antisymmetries, too. Here we define the tensor

Tijthat is antisymmetric:

DefTensor[T[-i, - j], M, Antisymmetric[{- i, - j}]];

**DefTensor: Defining tensor T[-i, -j].

T[- i, - j]

Tij

T[- j, - i]

Tji

To make this work, sinceMathematicadoes not enforce our symmetry rules, we need to make

it do so by using the ToCanonical command. Lets try it againT[- j, - i] // ToCanonical

-Tij

We can add two such tensors in a way that gives us a 0 result,T[- i, - j] + T[- j, - i]

Tij +Tji

T[- i, - j] + T[- j, - i] // ToCanonical

0

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Contraction

We can do some operations. Lets say we have the Riemann tensor Rijkland we want to

contract it using the metric gik. we use the ContractMetric command,metric[i, k]Riemanncd[- i, - j, - k, - l] // ContractMetric

R[]jlMathematicaalso understands that this is the Ricci tensor, Rjl,

metric[i, k]Riemanncd[- i, - j, - k, - l] // ContractMetric // InputForm

Riccicd[-j, -l]

Contracting it again gives us the Ricci scalar,metric[j, l]Riccicd[- j, - l] // ContractMetric // InputForm

RicciScalarcd[]

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Covariant Derivatives

The covariant derivative of our tensor, iTjl, is input

cd[- i][T[- j, - l]]

iTjlIf we have multiple covariant derivatives, we would enter them as follows, where @ is the

Map command:cd[- a] @ cd[-b] @ cd[- c] @ T[- d, - e] // ToCanonical

abcTdeTo evaluate this we use the command SortCovDs

cd[- a] @ cd[-b] @ cd[- c] @ T[- d, - e] // ToCanonical // SortCovDs

-R[] l$1032cbe aTdl$1032 -R[] l$1032cbd aTl$1032e -Tl$1030e

bR[

]

l$1030cad

-Tdl$1030

bR[

]

l$1030cae -R[

]

l$1030cae

bTdl$1030 -

R[] l$1030cad bTl$1030e -R[] l$1028bae cTdl$1028 -R[] l$1028bad cTl$1028e +cbaTde -R[] l$1028bac l$1028Tde -R[] l$1032cba l$1032Tde

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This is correct, but too difficult to interpret, we add the command ScreenDollarIndices to

put it into a lnaguage we can read instead of just the computer reading itcd[- a] @ cd[-b] @ cd[- c] @ T[- d, - e] // ToCanonical // SortCovDs // ScreenDollarIndices

-R[] fcbe aTdf -R[] fcbd aTfe -Tfe bR[] fcad -Tdf bR[] fcae -R[] fcae bTdf -R[] fcad bTfe -R[] fbae cTdf -R[] fbad cTfe + cbaTde -R[] fbac fTde -R[] fcba fTde

Instead of having to write all of this all of the time we make a sessionwide command$PrePrint= ScreenDollarIndices;

cd[- a] @ cd[-b] @ cd[- c] @ T[- d, - e] // ToCanonical // SortCovDs

-R[] fcbe aTdf -R[] fcbd aTfe -Tfe bR[] fcad -Tdf bR[] fcae -R[] fcae bTdf -R[] fcad bTfe -R[] fbae cTdf -R[] fbad cTfe + cbaTde -R[] fbac fTde -R[] fcba fTde

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The Second Bianchi Identity

Here we seek to reproduce the Second Bianchi identity. We redefine our covariant

derivative

DefCovD[CD[- a], {";", ""}]**DefCovD: Defining covariant derivative CD[-a].

**DefTensor: Defining vanishing torsion tensor TorsionCD[a, -b, -c].

**DefTensor: Defining symmetric Christoffel tensor ChristoffelCD[a, -b, -c].

**DefTensor: Defining Riemann tensor

RiemannCD[-a, -b, -c, d]. Antisymmetric only in the first pair.

**DefTensor: Defining non-symmetric Ricci tensor RicciCD[-a, -b].

**DefCovD: Contractions of Riemann automatically replaced by Ricci.

We begin by establishing a termterm1 = Antisymmetrize[CD[- e][RiemannCD[- c, - d, -b, a] ], {- c, - d, - e} ]

1

6cR[] adeb - cR[] aedb - dR[] aceb + dR[] aecb + eR[] acdb - eR[] adcb

We sum these,bianchi2= 3 term1 // ToCanonical

cR[] adeb - dR[] aceb + eR[] acdb

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We expand the covariant derivatives in Christoffel symbols:exp1 = bianchi2 // CovDToChristoffel

[]aefR[] fcdb -[]febR[] acdf -[]adfR[] fceb +

[

]fdbR[

]

acef +

[

]fdeR[

]

acfb -

[

]fedR[

]

acfb +

[]acfR[] fdeb -[]fcbR[] adef -[]fceR[] adfb -[]fecR[] afdb -[]fcdR[] afeb +[]fdcR[] afeb + cR[] adeb - dR[] aceb + eR[] acdb

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We expand the Riemann tensors in Christoffel symbolsexp2 = exp1 // RiemannToChristoffel

-[]febc[]adf +[]fdbc[]aef +[]aefc[]fdb -

[

]adf

c

[

]feb -

c

d

[

]aeb +

c

e

[

]adb +

[

]feb

d

[

]acf -

[]feb []adg []gcf -[]acg []gdf - c[]adf + d[]acf -[]fcbd[]aef -[]aefd[]fcb +[]aef []fdg []gcb -[]fcg []gdb - c[]fdb + d[]fcb +[]acfd[]feb + dc[]aeb - de[]acb -[]fdb e[]acf +[]fdb []aeg []gcf -[]acg []gef - c[]aef + e[]acf +[]fcbe[]adf -[]fcb []aeg []gdf -[]adg []gef - d[]aef + e[]adf +[]adf e[]fcb -

[]adf []feg []gcb -[]fcg []geb - c[]feb + e[]fcb -

[

]a

cfe[

]f

db +

[]acf []feg []gdb -[]fdg []geb - d[]feb + e[]fdb - ec[]adb +ed[]acb +[]fde []afg []gcb -[]acg []gfb - c[]afb + f[]acb -[]f

ed []afg []gcb -[]acg []gfb - c[]afb + f[]acb -

[]fec-[]afg []gdb +[]adg []gfb + d[]afb - f[]adb -[]fce []afg []gdb -[]adg []gfb - d[]afb + f[]adb -[]f

cd-[]afg []geb +[]aeg []gfb + e[]afb - f[]aeb +

[]fdc

-[]afg []geb +[]aeg []gfb + e[]afb - f[]aeb

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We canonicalize this to enforce symmetries and antisymmetries,exp3 = exp2 // ToCanonical

0

Which is what we expected.

cR[]a

bde +dR[]a

be c+eR[]a

bc d = 0

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Perturbative GR

We begin this by opening the perturbative package:

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We must first establish our metric perturbation, this is because the metric is a vacuum

metric; thus all small perturbations are gravitational waves. We define the metric perturba

tion using the DefMetricPerturbation[the existing metric, name of the perturbation, pertur

bation parameter]DefMetricPerturbation[metric, pert,]**DefParameter: Defining parameter .**DefTensor: Defining tensor pert[LI[order], -a, -b].

We can tellMathematicato write the perturbation using the traditional notation h,PrintAs[pert] ^= "h";

A secondorder perturbation in the metric is then writtenpert[LI[2], - a, -b]

h2ab

So the firstorder perturbation of the metricgabisPerturbation[metric[- a, -b], 1]

h1ab

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for gabwe have,Perturbation[metric[a, b], 1]

gab What? What is ? It turns out that Mathematicadoes not know how to perform a perturba

tion on the inverse metric. We need to use the ExpandPerturbation command,Perturbation[metric[a, b], 1] // ExpandPerturbation

-h1ab

This is not too impressive, lets try a fourthorder perturbationPerturbation[metric[a, b], 4] // ExpandPerturbation

24h1ac

h1 dc h

1 ed h

1 be - 12h

1 dc h

1 bd h

2ac + 6h2ac

h2 bc -

12h1ac

h1 bd h

2 dc - 12h

1ach1 dc h

2 bd

+ 4h1 bc h

3ac + 4h1ac

h3 bc -h

4ab

Lets examine the thirdorder perturbed metric

Perturbed[metric[- a, -b], 3]

gab + h1ab +

1

22 h2ab +

1

63 h3ab

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and the inverse metricPerturbed[metric[a, b], 3] // ExpandPerturbation

gab - h1ab + 12

2 2h1ac h1 bc -h2ab +

1

63 -6h1ac h1 dc h1 bd + 3h1 bf h2af + 3h1ae h2 be -h3ab

Jolyon Bloomfield [5] produced a method of extracting only the firstorder terms of the

expansion, since is assumed to be small, where we can write the metric as the sum of a

linearized metric and a perturbed metric, gab= g0

ab+ habwe adapt it for our usefirstorderonly= pert[LI[n_], __] 0 /; n > 1;

Perturbed[metric[- a, -b], 3] /. firstorderonly

gab + h1ab

Given an action we can derive an equation of motion. First we need to establish our scalar

field,DefTensor[sf[], M, PrintAs ""]**DefTensor: Defining tensor sf[].

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We now define the perturbations of the scalar field,DefTensorPerturbation[pertsf[LI[order]], sf[], M, PrintAs ""]**DefTensor: Defining tensor pertsf[LI[order]].

We now define the potential and the Planck mass,DefScalarFunction[V]

DefConstantSymbolmassP, PrintAs " mp"

**DefScalarFunction: Defining scalar function V.

**DefConstantSymbol: Defining constant symbol massP.

Now we need to construct the LagrangianL = Sqrt[- Detmetric[]]massP22 RicciScalarcd[] - 1 / 2 cd[-b][sf[]]cd[b][sf[]] -V[sf[]]

-g mp2 R[]

2- V[] - 1

2b b

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We vary the fieldsvarL = L // Perturbation

- [g] mp2 R[]2

- V[] - 12

b b 2 -g +

-g 12mp

2 [R[]] + 12

-b b -1;b b -1 V[]

We expand thisvarL = L // Perturbation // ExpandPerturbation

- gh1 aa

mp2 R[]2

- V[] - 12

b b 2 -g + -g 12mp

2 -h1ac R[]ac +

gac 1

2-h1dd;c;a -h1dc;d;a +h1 ;dc d ;a +

1

2h1dc;a;d +h1da;c;d -h1 ;dc a ;d +

1

2-1;bb - b gbe 1;e -h1be e -1 V[]

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We perform the metric contractionsvarL = L // Perturbation // ExpandPerturbation // ContractMetric

-1

2mp

2 -g

h1ab R[]ab +

1

4mp

2 -g

h1 aa R[] -

1

2-g

h1 aa V[] -

1

4mp

2 -g h1b ;ab ;a - 14mp

2 -g h1ba;b;a + 14mp

2 -g h1a ;bb ;a +1

2-g

h1ba a b + 1

2mp

2 -g

h1ba

;a;b -1

4mp

2 -g

h1a ;b

a ;b -

1

2-gb 1;b - 1

2-g 1;b b -

1

4-g

h1 aa b b - -g 1 V[]

? RicciTo*

xAct`xTensor`

RicciToEinstein RicciToTFRicci

RicciToEinstein[expr, covd]expands expr expressingall Ricci tensors of covd in terms of the Einstein and RicciScalar tensors

of covd. If the second argument is a list of covariant derivatives the command is applied sequentially on expr. RicciToEinstein [expr]expands all Ricci tensors.

Then we enforce the symmetriesvarL = L // Perturbation // ExpandPerturbation // ContractMetric // ToCanonical

-1

2mp

2 -g

h1ba R[]ba +

1

4mp

2 -g

h1b

bR[] -

1

2-g

h1b

bV[] - 1

2mp

2 -g

h1b ;a

b ;a +

1

2mp

2 -g

h1ba

;b;a -

-gb 1;b + 1

2-g

h1baa b -

1

4-g

h1a

ab b - -g 1 V[]

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The scalar equation of motion is then found by taking the variational derivatives of the

scalar field and set them equal to 00 == VarD[pertsf[LI[1]], cd][varL] / Sqrt[- Detmetric[]]

0

1

-g 1

1

-g

a

a

-

1

1

-gV[

]

we enforce the Kronecker delta0 == VarD[pertsf[LI[1]], cd][varL] / Sqrt[- Detmetric[]] /. delta[- LI[1], LI[1]] 1

0 -gaa - -gV[]

-g

We canonicalize it0 == VarD[pertsf[LI[1]], cd][varL] / Sqrt[- Detmetric[]] /.

delta[- LI[1], LI[1]] 1 // ToCanonical0

aa

- V[

]

There we have it.

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We can get the tensor equation of motion,vartf = 2 (-VarD[pert[LI[1], a, b], cd][varL] / Sqrt[- Detmetric[]])

1

-g2

1

2mp

2 11 -gR[]ab -1

4mp

2 11 -ggabR[] +

1

211 -ggabV[] +

1

411 -ggab c c -

1

211 -ggacgbd c d

This is a mess, so we begin enforcing the Kronecker deltasvartf =

2(-VarD[pert[LI[1], a, b], cd][varL] / Sqrt[- Detmetric[]] /. delta[- LI[1], LI[1]] 1)1

-g2

1

2mp

2 -gR[]ab -

1

4mp

2 -gg

abR[] +

1

2-gg

abV[] +1

4-gg

ab c c -1

2-gg

acg

bd c d

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We now expand by expressing all Ricci tensors of covariant derivatives in terms of Einstein

tensors and Ricci scalarsvartf =

2(-VarD[pert[LI[1], a, b], cd][varL] / Sqrt[- Detmetric[]] /. delta[- LI[1], LI[1]] 1 //

SeparateMetric[metric] // RicciToEinstein)1

-g2 -

1

4mp

2 -gg

abR[] +1

2mp

2 -g G[]ab +

1

2gbaR[] +

1

2-gg

abV[] -1

2-ga b + 1

4-gg

abgcd c d

We expand thisvartf =

2(-VarD[pert[LI[1], a, b], cd][varL] / Sqrt[- Detmetric[]] /. delta[- LI[1], LI[1]] 1 //SeparateMetric[metric] // RicciToEinstein) // Expand

mp2 G[

]ab -

1

2

mp2 g

abR[

] +

1

2

mp2 g

baR[

] +gabV[

] -

a

b

+1

2

gabgcd

c

d

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Then we enforce the symmetries,vartf =

2(-VarD[pert[LI[1], a, b], cd][varL] / Sqrt[- Detmetric[]] /. delta[- LI[1], LI[1]] 1 //SeparateMetric[metric] // RicciToEinstein) //

Expand// ContractMetric // ToCanonical

mp2 G[]ab +gabV[] - a b +

1

2gab c c

We set this equal to 0,0 vartf

0mp2 G[]ab +gabV[] - a b +1

2gab c c

Here we get the conservation of energyeterm = IndexCollect[cd[a] @ vartf // ToCanonical, CD[-b][sf[]]] // Simplify

b(- aa + V[])

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then0 eterm

0 b(- aa + V[])this is degenerate with the scalar equation of motion

0 aa - V[]

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Bibliography

[1] M. Nakahara, (2003), Geometry, Topology, and Physics, 2nd Edition, Taylor and Francis

Group.

[2] Theodore Frankel, (2012), The Geometry of Physics, 3rd Edition, Cambridge UniversityPress

[3] Jose M. MartinGarcia, (2009), xAct Intro

[4] B. F. Schutz, (1984), The Use of Perturbation and Approximation Methods in General

Relativity. In: Relativistic Astrophysics and Cosmology: Proceedings of the GIFT Interna

tional Seminar on Theoretical Physics, (Eds.) Fustero, Xavier; Verdaguer, Enric; Singapore:

World Scientific 3598 (1984)

[5] Jolyon Bloomfield, (2013), xAct Tutorial

xAct can be found at the website: http://www.xact.es/index.html

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Thank You for Attending!

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