In[]:=
ResourceFunction["MetricTensor"]["Kerr"]
In[]:=
MetricTensor
Type: Covariant
Symbol:
g.
μ.ν.
Dimensions: 4
Signature: Indeterminate
["MatrixRepresentation"]
Out[]=
-1+
2r.M.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
,0,0,-
2r.J.
2
Sin[θ.]
2
r.
+
2
J.
2
Cos[θ.]
2
M.
,0,
2
r.
+
2
J.
2
Cos[θ.]
2
M.
2
r.
+
2
J.
2
M.
-2r.M.
,0,0,0,0,
2
r.
+
2
J.
2
Cos[θ.]
2
M.
,0,-
2r.J.
2
Sin[θ.]
2
r.
+
2
J.
2
Cos[θ.]
2
M.
,0,0,
2
Sin[θ.]
2
r.
+
2
J.
2
M.
+
2r.
2
J.
2
Sin[θ.]
M.
2
r.
+
2
J.
2
Cos[θ.]
2
M.

In[]:=
MetricTensor
Type: Covariant
Symbol:
g.
μ.ν.
Dimensions: 4
Signature: Indeterminate
["Properties"]
Out[]=
{MatrixRepresentation,ReducedMatrixRepresentation,Coordinates,CoordinateOneForms,Indices,CovariantQ,ContravariantQ,MixedQ,Symbol,Dimensions,SymmetricQ,DiagonalQ,Signature,RiemannianQ,PseudoRiemannianQ,LorentzianQ,RiemannianConditions,PseudoRiemannianConditions,LorentzianConditions,MetricSingularities,Determinant,ReducedDeterminant,Trace,ReducedTrace,Eigenvalues,ReducedEigenvalues,Eigenvectors,ReducedEigenvectors,MetricTensor,InverseMetricTensor,LineElement,ReducedLineElement,VolumeForm,ReducedVolumeForm,TimelikeQ,LightlikeQ,SpacelikeQ,LengthPureFunction,AnglePureFunction,Properties}
In[]:=
MetricTensor
Type: Covariant
Symbol:
g.
μ.ν.
Dimensions: 4
Signature: Indeterminate
["LineElement"]
Out[]=
2
d.s.

2
d.r.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
2
r.
+
2
J.
2
M.
-2r.M.
+
2
d.θ.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
+
2
d.t.
-1+
2r.M.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
-
4r.J.d.t.d.ϕ.
2
Sin[θ.]
2
r.
+
2
J.
2
Cos[θ.]
2
M.
+
2
d.ϕ.
2
Sin[θ.]
2
r.
+
2
J.
2
M.
+
2r.
2
J.
2
Sin[θ.]
M.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
In[]:=
MetricTensor
Type: Covariant
Symbol:
g.
μ.ν.
Dimensions: 4
Signature: Indeterminate
["LengthPureFunction"][{1,2,3,4}]
Out[]=
-1+
2
r.
+
9
2
J.
2
Cos[θ.]
2
M.
+
2r.M.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
+
4
2
r.
+
2
J.
2
Cos[θ.]
2
M.
2
r.
+
2
J.
2
M.
-2r.M.
-
16r.J.
2
Sin[θ.]
2
r.
+
2
J.
2
Cos[θ.]
2
M.
+16
2
Sin[θ.]
2
r.
+
2
J.
2
M.
+
2r.
2
J.
2
Sin[θ.]
M.
2
r.
+
2
J.
2
Cos[θ.]
2
M.
In[]:=
CoordinateChartData["ParabolicCylindrical","Metric"]
Out[]=
{{
2
#1〚1〛
+
2
#1〚2〛
,0,0},{0,
2
#1〚1〛
+
2
#1〚2〛
,0},{0,0,1}}&
In[]:=
CoordinateChartData["ParabolicCylindrical","Metric",{a,b,c}]
Out[]=
{{
2
a
+
2
b
,0,0},{0,
2
a
+
2
b
,0},{0,0,1}}
In[]:=
CoordinateChartData[{"Toroidal",r},"Metric",{a,b,c}]
CoordinateChartData
:r is expected to be a string, a pair of the form {_String, _}​, or the symbol All.
Out[]=
CoordinateChartData[{Toroidal,r},Metric,{a,b,c}]
In[]:=
CoordinateChartData[{"Toroidal",1},"Metric",{a,b,c}]
CoordinateChartData
:Evaluation point {a,b,c} has dimension 3​, which does not match dimension specification 1.
Out[]=
CoordinateChartData[{Toroidal,1},Metric,{a,b,c}]
In[]:=
CoordinateChartData[{{"Toroidal",r},"Euclidean",3},"Metric",{a,b,d}]
Out[]=

2
r
2
(Cos[b]-Cosh[a])
,0,0,0,
2
r
2
(Cos[b]-Cosh[a])
,0,0,0,
2
r
2
Sinh[a]
2
(Cos[b]-Cosh[a])

In[]:=
CoordinateChartData[{"Stereographic","Sphere",3},"Metric",{a,b,d}]
Out[]=

4
4
R.
2

2
R.
+
2
a
+
2
b
+
2
d

,0,0,0,
4
4
R.
2

2
R.
+
2
a
+
2
b
+
2
d

,0,0,0,
4
4
R.
2

2
R.
+
2
a
+
2
b
+
2
d


In[]:=
ResourceFunction["MetricTensor"]["EddingtonFinkelstein"]["MatrixRepresentation"]
Out[]=
-1+
2M.
r.
,±1,0,0,{±1,0,0,0},0,0,
2
r.
,0,0,0,0,
2
r.
2
Sin[θ.]

In[]:=
ResourceFunction["MetricTensor"]["OutgoingEddingtonFinkelstein"]["MatrixRepresentation"]
Out[]=
-1+
2M.
r.
,-1,0,0,{-1,0,0,0},0,0,
2
r.
,0,0,0,0,
2
r.
2
Sin[θ.]

In[]:=
ResourceFunction["MetricTensor"]["OutgoingEddingtonFinkelstein"]["LorentzianConditions"]
Out[]=
Indeterminate
In[]:=
ResourceFunction["MetricTensor"]["Schwarzschild"]["LorentzianConditions"]
Out[]=

2
r.
>2r.M.,2r.M.<
2
r.
,
2
Sin[θ.]
>0
In[]:=
InputForm[%25]
Out[]//InputForm=
{r.^2 > 2*r.*M., 2*r.*M. < r.^2, Sin[θ.]^2 > 0}
In[]:=
ResourceFunction["MetricTensor"][{"Schwarzschild",q},{a,b,c,d}]["LorentzianConditions"]
Out[]=
{
2
b
>2bq,
2
b
>0,2bq<
2
b
,
2
b
2
Sin[c]
>0}
In[]:=
Reduce[%]
Out[]=
(q≤0&&((b<2q&&(Sin[c]<0||Sin[c]>0))||(b>0&&(Sin[c]<0||Sin[c]>0))))||(q>0&&((b<0&&(Sin[c]<0||Sin[c]>0))||(b>2q&&(Sin[c]<0||Sin[c]>0))))
In[]:=
ImageSynthesize["three black holes gravitationally bound"]
Out[]=