The EnergyMomentum tensor - Maple Programming Help

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Physics[EnergyMomentum] - The EnergyMomentum tensor

Calling Sequence

EnergyMomentum[μ,ν]

EnergyMomentum[keyword]

Parameters

mu, nu

-

the indices, as names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves

keyword

-

optional, it can be definition, array or matrix, nonzero

Description

• 

The EnergyMomentum[mu, nu], displayed as T__μ,ν, is a computational representation for the EnergyMomentum tensor. The components of this tensor are initially set equal to zero. To define these components in any particular way use the Define command passing to it an equation with EnergyMomentum[mu, nu] on the left-hand side and the desired definition, as a symmetric tensorial expression or a symmetric matrix on the right-hand side.

• 

When the metric is set to represent any curved spacetime, the EnergyMomentum tensor is the source of the gravitational field, entering Einstein's equations

Gμ,ν=8πΤμ,ν

  

where G__μ,ν is the Einstein tensor, expressed in terms of the Ricci tensor R__μ,ν as

Gμ,ν=Rμ,ν12gμ,νRαα

• 

The EnergyMomentum tensor also satisfies 𝒟μΤμ,ν=0, where 𝒟μ is the covariant derivative operator D_, and this 4D divergence is entered as D_[mu](EnergyMomentum[mu,nu]).

• 

When the indices of EnergyMomentum assume integer values they are expected to be between 0 and the spacetime dimension, prefixed by ~ when they are contravariant, and the corresponding value of EnergyMomentum is returned. The values 0 and 4, or for the case any dimension set for the spacetime, represent the same object. When the indices have symbolic values EnergyMomentum returns unevaluated after normalizing its indices taking into account their symmetry property.

• 

Computations performed with the Physics package commands take into account EnergyMomentum's sum rule for repeated indices - see `.` and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.

• 

Besides being indexed with two indices, EnergyMomentum accepts three keywords:

– 

definition: returns the definition of the EnergyMomentum tensor in terms of the Ricci tensor.

– 

matrix: (synonyms: Matrix, array, Array, or no indices whatsoever, as in EnergyMomentum[]) returns a Matrix that when indexed with numerical values from 1 to the dimension of spacetime returns the value of each of the components of EnergyMomentum. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting matrix takes into account the character of the indices.

– 

nonzero: returns a set of equations, with the left-hand-side as a sequence of two positive numbers identifying the element of G__μ,ν and the corresponding value on the right-hand-side. Note that this set is actually the output of the ArrayElems command when passing to it the Array obtained with the keyword array.

• 

Some automatic checking and normalization are carried out each time you enter EnergyMomentum[...]. The checking is concerned with possible syntax errors. The automatic normalization takes into account the symmetry of EnergyMomentum[mu,nu] with respect to interchanging the positions of the indices mu and nu.

• 

The %EnergyMomentum command is the inert form of EnergyMomentum, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

EnergyMomentumμ,ν

Τμ,ν

(2)

When Physics is loaded, all the components of Τμ,ν are set equal to 0

EnergyMomentumdefinition

EnergyMomentumμ,ν=0000000000000000

(3)

In order to set these components in any particular way use the Define command passing to it an equation with EnergyMomentum[mu, nu] on the left-hand side and the desired definition, as a symmetric tensorial expression or a symmetric matrix, on the right-hand side. For example, the general form of Τμ,ν in terms of the energy density W, the flux density Sμ and the stress tensor σi,j of a system can be entered as

DefineSμ,σμ,ν,symmetric

Defined objects with tensor properties

γμ,σμ,Sμ,μ,gμ,ν,σμ,ν,εα,β,μ,ν

(4)

EnergyMomentumμ,ν=Matrix4,μ,ν→ifμ=4thenifν=4thenWelseSνend ifelifν=4thenSμelseσμ,νend if

EnergyMomentumμ,ν=σ1,1σ1,2σ1,3S1σ2,1σ2,2σ2,3S2σ3,1σ3,2σ3,3S3S1S2S3W

(5)

You can now set these to be the components of Τμ,ν

Define

Defined objects with tensor properties

γμ,σμ,Sμ,μ,gμ,ν,σμ,ν,Τμ,ν,εα,β,μ,ν

(6)

After this definition, you can query about the definition, the nonzero components, any particular covariant or contravariant component via

EnergyMomentumdefinition

EnergyMomentumμ,ν=σ1,1σ1,2σ1,3S1σ2,1σ2,2σ2,3S2σ3,1σ3,2σ3,3S3S1S2S3W

(7)

EnergyMomentum1,2

σ1,2

(8)

In the definition above, all the components of Τμ,ν are constant. To set part or all of them as depending on the coordinates, for instance in a generic coordinate system and using spherical coordinates,

Setupcoordinates=spherical

Default differentiation variables for d_, D_ and dAlembertian are:X=r,θ,φ,t

Systems of spacetime coordinates are:X=r,θ,φ,t

_______________________________________________________

coordinatesystems=X

(9)

you can indicate the functionality in the definition. For example, set W to be constant (i.e. no functionality) but the flux density Sμ and stress sigmaμ,ν tensors depending on X

EnergyMomentumμ,ν=Matrix4,μ,ν→ifμ=4thenifν=4thenWelseSνXend ifelifν=4thenSμXelseσμ,νXend if

EnergyMomentumμ,ν=σ1,1r,θ,φ,tσ1,2r,θ,φ,tσ1,3r,θ,φ,tS1r,θ,φ,tσ2,1r,θ,φ,tσ2,2r,θ,φ,tσ2,3r,θ,φ,tS2r,θ,φ,tσ3,1r,θ,φ,tσ3,2r,θ,φ,tσ3,3r,θ,φ,tS3r,θ,φ,tS1r,θ,φ,tS2r,θ,φ,tS3r,θ,φ,tW

(10)

CompactDisplay

SXwill now be displayed asS

sigmaXwill now be displayed asσ

(11)

Define now Τμ,ν with these components

Define

Defined objects with tensor properties

γμ,σμ,Sμ,Xμ,μ,gμ,ν,σμ,ν,Τμ,ν,εα,β,μ,ν

(12)

To see the continuity equations for the components of Τμ,ν, use for instance the inert version of the covariant derivative operator D_ and the TensorArray command

%D_μ=D_μEnergyMomentumμ,ν

%D_μEnergyMomentum~mu,ν=0

(13)

VectorcolumnTensorArray

%d_1σ1,1r,θ,φ,t,r,θ,φ,t+%d_2σ1,2r,θ,φ,t,r,θ,φ,t+%d_3σ1,3r,θ,φ,t,r,θ,φ,t+%d_4S1r,θ,φ,t,r,θ,φ,t=0%d_1σ1,2r,θ,φ,t,r,θ,φ,t+%d_2σ2,2r,θ,φ,t,r,θ,φ,t+%d_3σ2,3r,θ,φ,t,r,θ,φ,t+%d_4S2r,θ,φ,t,r,θ,φ,t=0%d_1σ1,3r,θ,φ,t,r,θ,φ,t+%d_2σ2,3r,θ,φ,t,r,θ,φ,t+%d_3σ3,3r,θ,φ,t,r,θ,φ,t+%d_4S3r,θ,φ,t,r,θ,φ,t=0%d_1S1r,θ,φ,t,r,θ,φ,t+%d_2S2r,θ,φ,t,r,θ,φ,t+%d_3S3r,θ,φ,t,r,θ,φ,t+%d_4W,r,θ,φ,t=0

(14)

value

rσ1,1r,θ,φ,tθσ1,2r,θ,φ,tφσ1,3r,θ,φ,t+tS1r,θ,φ,t=0rσ1,2r,θ,φ,tθσ2,2r,θ,φ,tφσ2,3r,θ,φ,t+tS2r,θ,φ,t=0rσ1,3r,θ,φ,tθσ2,3r,θ,φ,tφσ3,3r,θ,φ,t+tS3r,θ,φ,t=0rS1r,θ,φ,tθS2r,θ,φ,tφS3r,θ,φ,t=0

(15)

In curved spaces, EnergyMomentum enters Einstein's equations as the source of the gravitational field

g_sc

_______________________________________________________

The Schwarzschild metric in coordinates r,θ,φ,t

Parameters: m

_______________________________________________________

g_μ,ν=rr+2m0000r20000r2sinθ20000r2mr

(16)

EnergyMomentumdefinition

Τμ,ν=Gμ,ν8π,Τμ,ν=σ1,1Xσ1,2Xσ1,3XS1Xσ2,1Xσ2,2Xσ2,3XS2Xσ3,1Xσ3,2Xσ3,3XS3XS1XS2XS3XW

(17)

Take the first of these two equations and compute a tensor array for it

1

Τμ,ν=Gμ,ν8π

(18)

TensorArray1

σ1,1r,θ,φ,t=0σ1,2r,θ,φ,t=0σ1,3r,θ,φ,t=0S1r,θ,φ,t=0σ1,2r,θ,φ,t=0σ2,2r,θ,φ,t=0σ2,3r,θ,φ,t=0S2r,θ,φ,t=0σ1,3r,θ,φ,t=0σ2,3r,θ,φ,t=0σ3,3r,θ,φ,t=0S3r,θ,φ,t=0S1r,θ,φ,t=0S2r,θ,φ,t=0S3r,θ,φ,t=0W=0

(19)

This is the expected result for the Schwarzschild metric: in vacuum, the components of the Einstein tensor, and with them those of Τμ,ν are all equal to zero.

See Also

ADMEquations, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, value

References

  

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

Compatibility

• 

The Physics[EnergyMomentum] command was introduced in Maple 2017.

• 

For more information on Maple 2017 changes, see Updates in Maple 2017.