DifferentialGeometry/Tensor/EnergyMomentumTensor - Maple Help

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Tensor[EnergyMomentumTensor] - find the energy-momentum tensor for various matter fields

Tensor[MatterFieldEquations] - find the field equations for various matter fields

Tensor[DivergenceIdentities] - check the divergence identities for the energy-momentum tensor field for various matter fields

Calling Sequences

     EnergyMomentumTensor(FieldType, g, F1, F2, ...)

     MatterFieldEquations(FieldType, g, F1, F2, ...)

     DivergenceIdentities(FieldType, g, F1, F2, ... , T, E1, E2,...)

Parameters

   FieldType  - a string, one of "DiracWeyl", "Dust", "Electromagnetic", "PerfectFluid", "Scalar", "NMCScalar"

   g          - a metric tensor

   F1, F2,..  - scalars, tensors or spinors, defining the fields needed for the field theory designated by FieldType

   T          - a rank 2 tensor (the energy-momentum tensor)

   E1, E2,..  - scalars, tensors or spinors, defining the field equations for the field theory designated by FieldType

 

Description

Examples

Description

• 

The energy momentum tensor is a symmetric, rank-2 contravariant tensor T which determines the right-hand side of the Einstein field equations.

• 

If FieldType = "DiracWeyl", then the additional arguments for EnergyMomentumTensor are: a solder form (compatible with the metric g), a rank 1 covariant spinor ψ, and the complex conjugate ψ.

• 

If FieldType = "Dust", then the additional arguments for EnergyMomentumTensor are: a vector field U, a scalar μ (energy density).

• 

If FieldType = "Electromagnetic", then the additional arguments are either: a 1-form A (the electromagnetic 4-potential), or a skew-symmetric rank 2 tensor F (the field strength tensor).

• 

If FieldType = "PerfectFluid", then the additional arguments for EnergyMomentumTensor are: a vector field U, and scalars μ (energy density) and p (pressure).

• 

If FieldType = "Scalar", then the additional argument for EnergyMomentumTensor is a scalar φ.

• 

If FieldType = "NMCScalar", then the additional argument for EnergyMomentumTensor is a non-minimally coupled scalar φ.

• 

See the Details help page for the explicit formulas used to calculate the various energy-momentum tensors, the matter field equations and the divergence identities.

• 

These commands are part of the DifferentialGeometry:-Tensor: package, and so can be used in the form EnergyMomentumTensor(...), MatterFieldEquations(...), DivergenceIdentities(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. They can always be used in the long form DifferentialGeometry:-Tensor:-EnergyMomentumTensor, DifferentialGeometry:-Tensor-MatterFieldEquations, DifferentialGeometry:-Tensor:-DivergenceIdentities.

Examples

withDifferentialGeometry:withTools:withTensor:

 

Example 1. "DiracWeyl"

First create a vector bundle N with base coordinates t,x,y,z and fiber coordinates z1,z2,w1,w2.

DGsetupt,x,y,z,z1,z2,w1,w2,N

frame name: N

(2.1)

 

Define a metric of signature 1,1,1,1 and an orthonormal tetrad.

N > 

g1evalDGx4dt&tdtdx&tdxdy&tdydz&tdz

g1x4dtdtdxdxdydydzdz

(2.2)
N > 

OTetradevalDG1x2D_t,D_x,D_y,D_z

OTetrad1x2D_t,D_x,D_y,D_z

(2.3)

 

Calculate the solder form.

N > 

σ1SolderFormOTetrad

σ1x222dtD_z1D_w1+x222dtD_z2D_w2+22dxD_z1D_w2+22dxD_z2D_w1I22dyD_z1D_w2+I22dyD_z2D_w1+22dzD_z1D_w122dzD_z2D_w2

(2.4)

 

Define a rank 1-spinor field ψ1 and its complex conjugate.

N > 

ψ1evalDGhxdz1fxdz2

ψ1hxdz1fxdz2

(2.5)
N > 

barpsi1evalDGhxdw1fxdw2

barpsi1hxdw1fxdw2

(2.6)

 

Calculate the Dirac-Weyl energy momentum tensor T.

N > 

T1EnergyMomentumTensorDiracWeyl,g1,σ1,ψ1,barpsi1

T12fx2hx2x3D_tD_y+2hxfx+fxhxD_xD_y2fx2hx2x3D_yD_t+2hxfx+fxhxD_yD_x

(2.7)

 

Evaluate the Dirac-Weyl field equations E1 for the given spinor field ψ.

N > 

E1MatterFieldEquationsDiracWeyl,g1,σ1,ψ1,barpsi1

E1I22fxx+fxxD_w1I22hxx+hxxD_w2,I22fxx+fxxD_z1+I22hxx+hxxD_z2

(2.8)

 

Check the divergence identity for the dust energy momentum tensor T. The LHS is the covariant divergence of the energy momentum tensor and the RHS is a combination of the field equations.

N > 

Div1,RHS1DivergenceIdentitiesDiracWeyl,g1,σ1,ψ1,barpsi1,T1,E1

Div1,RHS12xhxfx+xfxhx+2fxhx2hxfxxD_y,2xhxfx+xfxhx+2fxhx2hxfxxD_y

(2.9)
N > 

Div1RHS1

0

(2.10)

 

We note that fx=hx=1x is a solution of the Dirac-Weyl field equations:

N > 

mapDGsimplify,evalE1,fx=1x,hx=1x

0D_z1,0D_z1

(2.11)

 

 The covariant divergence of the energy momentum tensor vanishes on this solution:

N > 

DGsimplifyevalDiv1,fx=1x,hx=1x

0D_t

(2.12)

 

Example 2. "Dust"

First create a manifold M with base coordinates t,x,y,z:

N > 

DGsetupt,x,y,z,M

frame name: M

(2.13)

 

Define a metric.

M > 

g2evalDGdt&tdtt2dx&tdxdy&tdydz&tdz

g2dtdtt2dxdxdydydzdz

(2.14)

 

Define the normalized 4-vector representing the 4-velocity of the dust.

M > 

u2evalDGcoshftD_tsinhfttD_x

u2coshftD_tsinhfttD_x

(2.15)
M > 

TensorInnerProductg2,u2,u2

1

(2.16)

 

Define the energy density.

M > 

μ2ht

μ2ht

(2.17)

 

Calculate the dust energy- momentum tensor T2.

M > 

T2EnergyMomentumTensorDust,g2,u2,μ2

T2htcoshft2D_tD_thtcoshftsinhfttD_tD_xhtcoshftsinhfttD_xD_t+htcoshft21t2D_xD_x

(2.18)

 

Evaluate the dust field equations E2 for the given u2 and μ2.

M > 

E2MatterFieldEquationsDust,g2,u2,μ2

E2htcoshft+h.tcoshftt+htsinhftf.ttt,coshft21+coshftsinhftf.tttD_tcoshftsinhft+coshftf.ttt2D_x

(2.19)

 

Check that the following values for ft and ht solve the dust field equations.

M > 

Solnht=_C21+t2_C1212,ft=arcsinh1t_C1

Solnht=_C21+t2_C12,ft=arcsinh1t_C1

(2.20)
M > 

simplifyevalE2,Soln,symbolic

0,0D_t+0D_x

(2.21)

 

Check the divergence identity for the dust energy-momentum tensor T2. The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the field equations.

M > 

Div2,RHS2DivergenceIdentitiesDust,g2,u2,μ2,T2,E2

Div2,RHS22htcoshft2ht+h.ttcoshft2+2htcoshftsinhftf.tttD_t2htcoshftsinhft+h.tcoshftsinhftt+2htcoshft2f.tthtf.ttt2D_x,2htcoshft2ht+h.ttcoshft2+2htcoshftsinhftf.tttD_t2htcoshftsinhft+h.tcoshftsinhftt+2htcoshft2f.tthtf.ttt2D_x

(2.22)
M > 

Div2&minusRHS2

0D_t

(2.23)

 

Example 3. "Electromagnetic"

First create a manifold M with base coordinates t,x,y,z.

M > 

DGsetupt,x,y,z,M

frame name: M

(2.24)

 

Define a metric.

M > 

g3evalDGx2dt&tdtdx&tdxdy&tdydz&tdz

g3x2dtdtdxdxdydydzdz

(2.25)

 

Define an electromagnetic 4-potential A3.

M > 

A3evalDGf1xdt+f2xdy

A3f1xdt+f2xdy

(2.26)

 

Calculate the electromagnetic energy-momentum tensor T3.

M > 

T3EnergyMomentumTensorElectromagnetic,g3,A3

T3f2x2x2+f1x22x4D_tD_t+f1xf2xx2D_tD_y+f2x2x2+f1x22x2D_xD_x+f1x