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tensor

  

displayGR

  

display the nonzero components of a specific general relativity tensor

  

display_allGR

  

display the nonzero components of all of the general relativity tensors and quantities calculated by tensor[tensorsGR]

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

displayGR(GR_name, object)

display_allGR(coord, cov_metric, con_metric, det_met, C1, C2, Rm, Rc, R, G, C)

Parameters

GR_name

-

name of a specific GR-related object where the name is one of coordinates, cov_metric, contra_metric, detmetric, Christoffel1, Christoffel2, Riemann, Ricci, Ricciscalar, Einstein, or Weyl

object

-

object to display

coord

-

list of coordinate variable names (for example, [t, x, y, z])

cov_metric

-

covariant metric tensor (index_char=[-1,-1])

con_metric

-

contravariant metric tensor (index_char=[1,1])

det_met

-

determinant of metric tensor components

C1, C2

-

Christoffel symbols of first and second kind

Rm, Rc, R

-

Riemann tensor, Ricci tensor, Ricci scalar

G, C

-

Einstein and Weyl tensors

Description

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

• 

The function displayGR(Einstein, G) displays the nonzero components and the index character of the tensor_type G using the symmetries of the Einstein tensor to reduce the number of components shown. G is expected to be a tensor of rank 2 with the proper symmetries. It can be calculated using tensor[Einstein].

• 

The function display_allGR(coord, cov_met, cont_met, det_met, C1, C2, Rm, Rc, R, G, C) displays all of the GR-related objects passed in the parameter list. They must be passed in the following order: the coordinates list, the covariant metric tensor ([-1,-1]), the contravariant metric tensor ([1,1]), the determinant of the metric tensor components (algebraic type), the Christoffel symbols of the 1st kind ([-1,-1,-1]), the Christoffel symbols of the 2nd kind ([1,-1,-1]), the Riemann tensor, the Ricci tensor, the Ricciscalar ([]), the Einstein tensor, the Weyl tensor.  Each of the quantities is displayed by making the appropriate call to tensor[displayGR]. See tensor[tensorsGR] for the calculation of these quantities.

• 

For displaying general tensor_type objects not listed above, use the 'display' option of tensor[act].

• 

These functions are part of the tensor package, and so can be used in the form displayGR(..) / display_allGR(..) only after performing the command with(tensor), or with(tensor, displayGR) / with(tensor, display_allGR).

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

Define the coordinates and covariant metric for the Schwarzschild metric:

withtensor:

coordst,r,θ,φ:

garraysymmetric,sparse,1..4,1..4:

g1,112mr:g2,21g1,1:g3,3r2:g4,4r2sinθ2:

metriccreate1,1,evalg

metric:=tablecompts=12mr0000112mr0000r20000r2sinθ2,index_char=1,1

(1)

Compute the curvature (without the print option)

tensorsGRcoords,metric,contra_metric,det_met,C1,C2,Rm,Rc,R,G,C:

Use displayGR to show that it is a vacuum solution of the Einstein field equations:

displayGREinstein,G

The Einstein Tensor

non-zero components :

None

character : [-1, -1]

(2)

Use displayGR to show the nonzero Christoffel symbols of the first kind:

displayGRChristoffel1,C1

The Christoffel Symbols of the First Kind

non-zero components :

[11,2]=mr2

[12,1]=mr2

[22,2]=mr+2m2

[23,3]=r

[24,4]=rsinθ2

[33,2]=r

[34,4]=r2sinθcosθ

[44,2]=rsinθ2

[44,3]=r2sinθcosθ

(3)

Display everything using display_allGR:

display_allGRcoords,metric,contra_metric,det_met,C1,C2,Rm,Rc,R,G,C

_________

The coordinates variables are :

x1=t

x2=r

x3=θ

x4=φ

_________

The Covariant Metric

non-zero components :

cov_g11=12mr

cov_g22=112mr

cov_g33=r2

cov_g44=r2sinθ2

_________

Determinant of the covariant metric tensor :

detg=r4sinθ2

_________

The Contravariant Metric

non-zero components :

contra_g11=rr+2m

contra_g22=r+2mr

contra_g33=1r2

contra_g44=1r2sinθ2

_________

The Christoffel Symbols of the First Kind

non-zero components :

[11,2]=mr2

[12,1]=mr2

[22,2]=mr+2m2

[23,3]=r

[24,4]=rsinθ2

[33,2]=r

[34,4]=r2sinθcosθ

[44,2]=rsinθ2

[44,3]=r2sinθcosθ

_________

The Christoffel Symbols of the Second Kind

non-zero components :

{1,12}=mrr+2m

{2,11}=r+2mmr3

{2,22}=mrr+2m

{2,33}=r+2m

{2,44}=r+2msinθ2

{3,23}=1r

{3,44}=sinθcosθ

{4,24}=1r

{4,34}=cosθsinθ

_________

The Riemann Tensor

non-zero components :

R1212=2mr3

R1313=r+2mmr2

R1414=r+2mmsinθ2r2

R2323=mr+2m

R2424=msinθ2r+2m

R3434=2rmsinθ2

character : [-1, -1, -1, -1]

_________

The Ricci tensor

non-zero components :

None

character : [-1, -1]

_________

The Ricci Scalar

R=0

_________

The Einstein Tensor

non-zero components :

None

character : [-1, -1]

_________

The Weyl Tensor

non-zero components :

C1212=2mr3

C1313=r+2mmr2

C1414=r+2mmsinθ2r2

C2323=mr+2m

C2424=msinθ2r+2m

C3434=2rmsinθ2

character : [-1, -1, -1, -1]

_________

(4)

See Also

Physics

tensor(deprecated)

tensor(deprecated)[act]

tensor(deprecated)[tensorsGR]

 


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