In[]:=
ResourceFunction["MetricTensor"]["Kerr"]
In[]:=
MetricTensor["MatrixRepresentation"]
Out[]=
-1++,0,0,-+,0,++-2r.M.,0,0,0,0,+,0,-+,0,0,++
2r.M.
2
r.
2
J.
2
Cos[θ.]
2
M.
2r.J.
2
Sin[θ.]
2
r.
2
J.
2
Cos[θ.]
2
M.
2
r.
2
J.
2
Cos[θ.]
2
M.
2
r.
2
J.
2
M.
2
r.
2
J.
2
Cos[θ.]
2
M.
2r.J.
2
Sin[θ.]
2
r.
2
J.
2
Cos[θ.]
2
M.
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["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["LineElement"]
Out[]=
2
d.s.
2
d.r.
2
r.
2
J.
2
Cos[θ.]
2
M.
2
r.
2
J.
2
M.
2
d.θ.
2
r.
2
J.
2
Cos[θ.]
2
M.
2
d.t.
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["LengthPureFunction"][{1,2,3,4}]
Out[]=
-1++++++-2r.M.-++16++
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.
16r.J.
2
Sin[θ.]
2
r.
2
J.
2
Cos[θ.]
2
M.
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[]=
{{+,0,0},{0,+,0},{0,0,1}}&
2
#1〚1〛
2
#1〚2〛
2
#1〚1〛
2
#1〚2〛
In[]:=
CoordinateChartData["ParabolicCylindrical","Metric",{a,b,c}]
Out[]=
{{+,0,0},{0,+,0},{0,0,1}}
2
a
2
b
2
a
2
b
In[]:=
CoordinateChartData[{"Toroidal",r},"Metric",{a,b,c}]
Out[]=
CoordinateChartData[{Toroidal,r},Metric,{a,b,c}]
In[]:=
CoordinateChartData[{"Toroidal",1},"Metric",{a,b,c}]
Out[]=
CoordinateChartData[{Toroidal,1},Metric,{a,b,c}]
In[]:=
CoordinateChartData[{{"Toroidal",r},"Euclidean",3},"Metric",{a,b,d}]
Out[]=
,0,0,0,,0,0,0,
2
r
2
(Cos[b]-Cosh[a])
2
r
2
(Cos[b]-Cosh[a])
2
r
2
Sinh[a]
2
(Cos[b]-Cosh[a])
In[]:=
CoordinateChartData[{"Stereographic","Sphere",3},"Metric",{a,b,d}]
Out[]=
,0,0,0,,0,0,0,
4
4
R.
2
+++
2
R.
2
a
2
b
2
d
4
4
R.
2
+++
2
R.
2
a
2
b
2
d
4
4
R.
2
+++
2
R.
2
a
2
b
2
d
In[]:=
ResourceFunction["MetricTensor"]["EddingtonFinkelstein"]["MatrixRepresentation"]
Out[]=
-1+,±1,0,0,{±1,0,0,0},0,0,,0,0,0,0,
2M.
r.
2
r.
2
r.
2
Sin[θ.]
In[]:=
ResourceFunction["MetricTensor"]["OutgoingEddingtonFinkelstein"]["MatrixRepresentation"]
Out[]=
-1+,-1,0,0,{-1,0,0,0},0,0,,0,0,0,0,
2M.
r.
2
r.
2
r.
2
Sin[θ.]
In[]:=
ResourceFunction["MetricTensor"]["OutgoingEddingtonFinkelstein"]["LorentzianConditions"]
Out[]=
Indeterminate
In[]:=
ResourceFunction["MetricTensor"]["Schwarzschild"]["LorentzianConditions"]
Out[]=
>2r.M.,2r.M.<,>0
2
r.
2
r.
2
Sin[θ.]
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[]=
{>2bq,>0,2bq<,>0}
2
b
2
b
2
b
2
b
2
Sin[c]
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[]=