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  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 ), a rank 1 covariant spinor , and the complex conjugate .

• 

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

• 

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

• 

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

• 

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

• 

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

• 

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

 

Example 1. "DiracWeyl"

First create a vector bundle  with base coordinates  and fiber coordinates .

(2.1)

 

Define a metric of signature  and an orthonormal tetrad.

N > 

(2.2)
N > 

(2.3)

 

Calculate the solder form.

N > 

(2.4)

 

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

N > 

(2.5)
N > 

(2.6)

 

Calculate the Dirac-Weyl energy momentum tensor .

N > 

(2.7)

 

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

N > 

(2.8)

 

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

N > 

(2.9)
N > 

(2.10)

 

We note that  is a solution of the Dirac-Weyl field equations:

N > 

(2.11)

 

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

N > 

(2.12)

 

Example 2. "Dust"

First create a manifold  with base coordinates :

N > 

(2.13)

 

Define a metric.

M > 

(2.14)

 

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

M > 

(2.15)
M > 

(2.16)

 

Define the energy density.

M > 

(2.17)

 

Calculate the dust energy- momentum tensor .

M > 

(2.18)

 

Evaluate the dust field equations  for the given  and .

M > 

(2.19)

 

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

M > 

(2.20)
M > 

(2.21)

 

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

M > 

(2.22)
M > 

(2.23)

 

Example 3. "Electromagnetic"

First create a manifold  with base coordinates .

M > 

(2.24)

 

Define a metric.

M > 

(2.25)

 

Define an electromagnetic 4-potential .

M > 

(2.26)

 

Calculate the electromagnetic energy-momentum tensor .

M > 

(2.27)

 

Note that the energy-momentum tensor can also be computed from the field strength tensor .

M > 

(2.28)
M > 

(2.29)

 

Evaluate the electromagnetic field equations  for the given 4-potential .

M > 

(2.30)

 

Note that the electromagnetic field equations  can also be computed from the field strength tensor .

M > 

(2.31)

 

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

M > 

(2.32)
M > 

(2.33)

We note that is a solution of the electromagnetic field equations:

M > 

(2.34)

 

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

M > 

(2.35)

 

Example 4. "PerfectFluid"

First create a manifold  with base coordinates :

M > 

(2.36)

 

Define a metric.

M > 

(2.37)

 

Define the normalized 4-velocity.

M > 

(2.38)
M > 

(2.39)

 

Define the energy density.

M > 

(2.40)

 

Define the pressure.

M > 

(2.41)

 

Calculate the perfect fluid energy-momentum tensor .

M > 

(2.42)

 

Evaluate the fluid field equations  for the given fluid.

M > 

(2.43)

 

We can use the dsolve command to find the energy density  and the pressure  which satisfy the field equations.

M > 

(2.44)
M > 

(2.45)

 

Example 5. "Scalar"

First create a manifold  with base coordinates .

M > 

(2.46)

 

Define a metric.

M > 

(2.47)

 

Define a scalar field.

M > 

(2.48)

 

Calculate the energy- momentum tensor  for the scalar field .

M > 

(2.49)

 

Evaluate the matter field equations  for the given scalar field .

M > 

(2.50)

 

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

M > 

(2.51)
M > 

(2.52)

 

Example 6.  "NMCScalar"

First create a manifold  with base coordinates .

M > 

(2.53)

 

Define a metric.

M > 

(2.54)

 

Define a scalar field

M > 

(2.55)

 

Calculate the energy-momentum tensor  for the non-minimally coupled scalar field .

M > 

(2.56)

 

Evaluate the matter field equations  for the given scalar field .

M > 

(2.57)

 

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

M > 

(2.58)

 

M > 

(2.59)

See Also

DifferentialGeometry

Tensor

 


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