Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
ComputationWithMetric
2
S
Some computation examples are done based on manifold.
2
S
Geodesic Equations
Geodesic Equations
Set metric and general vector tensor:
In[4]:=
g=
"g",{θ,ϕ},DiagonalMatrix[{R^2,R^2Sin[θ]^2}],
True;T=
["T",{1,0},{θ,ϕ},Array[[θ,ϕ]&,{2}],g]
 | CreateTensor |
 | IsMetric |
 | CreateTensor |
"T"
##
Out[5]=
STensor
 |  |
|
Compute geodesic expression:
In[6]:=
geo=T["a"]Grad[T["b"],"a"]
Out[6]=
STensor
[b,]
| |
|
Give geodesic equations:
In[7]:=
Column[Thread[Equal[geo["Components"],0]]]
Out[7]=
T 2 T 2 (0,1) T 1 T 1 (1,0) T 1 |
T 2 (0,1) T 2 T 1 T 2 (1,0) T 2 |
Compute Killing Vector Condition
Compute Killing Vector Condition
Set metric and a general vector field:
In[8]:=
g=
"g",{θ,ϕ},DiagonalMatrix[{R^2,R^2Sin[θ]^2}],
True;T=
["T",{1,0},{θ,ϕ},Array[[θ,ϕ]&,{2}],g]
| CreateTensor |
| IsMetric |
| CreateTensor |
"T"
##
Out[9]=
STensor
| |
|
Compute Killing vector field equation:
In[10]:=
ξ=
[Grad[g["bc"]T["c"],"a"],Symmetric[{"a","b"}]]
| Symmetrize |
Out[10]=
STensor
[,ba]
|  |
|
Make every component equal to zero and simplify with coordinate conditions:
In[11]:=
Simplify[And@@Thread[Flatten[ξ["Components"]]0],R>0&&0≤θ<Pi&&0≤ϕ<2Pi]
Out[11]=
Sin[θ]Cos[θ][θ,ϕ]+Sin[θ][θ,ϕ]0&&[θ,ϕ]0&&[θ,ϕ]+[θ,ϕ]0
T
1
(0,1)
T
2
(1,0)
T
1
(0,1)
T
1
2
Sin[θ]
(1,0)
T
2
Schwarzschild Metric
Create Schwarzschild metric:
In[12]:=
g=
"g",{t,r,θ,ϕ},DiagonalMatrix-1-,,,,
True
| CreateTensor |
2M
r
-1
1-
2M
r
2
r
2
r
2
Sin[θ]
 | IsMetric |
Out[12]=
STensor
| |
|
Tensors for General Relativity
Tensors for General Relativity
Compute Riemann tensor:
In[13]:=
Rie=
[g]
| RiemannTensor |
Out[13]=
STensor
|  |
|
In[14]:=
Rie["Components"]//MatrixForm
Out[14]//MatrixForm=
|
|
|
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|
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Compute Ricci tensor, which ends up to be zero tensor:
In[15]:=
Ric=
[g]
| RicciTensor |
Out[15]=
STensor
| |
|
In[16]:=
Ric["Components"]
Out[16]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
So does the Einstein tensor:
In[17]:=
| EinsteinTensor |
Out[17]=
{{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
Friedmann–Lemaître–Robertson–Walker metric
Set FLRW metric:
In[40]:=
FLRW=
"g",{t,r,θ,φ},DiagonalMatrix-1,,a[t]^2*r^2,a[t]^2*r^2*Sin[θ]^2,
True
| CreateTensor |
a[t]^2
1-kr^2
| IsMetric |
Out[40]=
STensor
| |
|
Conformally Flat
Conformally Flat
Compute Weyl tensor, which ends up to be zero tensor:
In[59]:=
c=
[FLRW]
| WeylTensor |
Out[59]=
In[60]:=
Simplify[c["Components"]]//MatrixForm
Volume of the Universe
Volume of the Universe
When k = 1, the r is defined as Sin(ψ) where 0 ≤ ψ ≤ π. Transform coordinate system first:
Then, taking k as 1, the universe is close and computable in terms of its volume. We need to extract induced metric for space part, and simplify with coordinate conditions:
Integrate for the whole coordinate system:
Vaidya Metric
From given Schwarzschild metric:
Do transformation to tortoise coordinate system:
Make constant M into a function of coordinate u, then we obtain the Vaidya Metric:
Compute its Einstein tensor and momentum-energy metric: