What is nr for the Schwarzchild metric?
What is for the Schwarzchild metric?
n
r
Ricci scalar term vanishes, but the next-order correction is there.
Kretschmann scalar
What isn’t volume of a ball ∫ up to specified distance (which depends on the line element) integrated over solid angle?
g
grr[r_]:=(1-((sRadius*r^2)/(gRadius^3)))^(-1)
In[]:=
grr[r_]:=(1-((a*r^2)))^(-1)
In[]:=
Because it is a spherically symmetric metric, the surface area is still the same....
Integrate[4Pir^2Sqrt[grr[r]],{r,0,1}]
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Out[]=
Assuming[{a>0},%]
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Out[]=
FullSimplify[%]
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Out[]=
Plotππ+
1
2
a
-
1
a
3/2
-
a
a
π+21-a
-
-a
-1+a
-(-1+a)a
+ArcSin[a
],{a,0,2}In[]:=
Out[]=
As a function of radius away from the origin, what is the volume of a ball?
n
r
4
r
The lack of an r^2 term is a consequence of only having tidal deformation
depending on R vs.
R
S
48/
2
G
2
m
4
c
Integrate[1/Sqrt[1-2m/r],r]
In[]:=
1-
r+2mArcTanh2m
r
1-
2m
r
Out[]=
Limit[%158,r0]
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$Aborted
Out[]=
Limit
1-
r+2mArcTanh2m
r
1-
,r02m
r
In[]:=
$Aborted
Out[]=
$Version
In[]:=
12.1.0 for Mac OS X x86 (64-bit) (December 24, 2019)
Out[]=
On a sphere, in any number of dimensions, the formula is
On a sphere, in any number of dimensions, the formula is
n
r
4
r
https://www.wolframscience.com/nks/notes-9-15--sphere-volumes/
https://www.wolframscience.com/nks/notes-9-15--sphere-volumes/
s[d]=(d/2)!
d/2
π
s[d](1-RicciScalar/(6(d+2))+…)
d
r
2
r
For a sphere:
RicciScalar=d(d-1)/
2
a
Exact result for d-dimensional sphere surface
d(d/2)!Integrate[,{θ,0,r/a}]
d/2
π
d
a
d-1
Sin[θ]
In[]:=
$Aborted
Out[]=
-Cos[x]Hypergeometric2F1[12,(2-d)2,32,]/.xr/a
2
Cos[x]
In[]:=
-CosHypergeometric2F1,,,
r
a
1
2
2-d
2
3
2
2
Cos
r
a
Out[]=
-Cos[x]Hypergeometric2F1[12,(2-d)2,32,]/.x0
2
Cos[x]
In[]:=
-
π
Gammad
2
2Gamma+
1
2
d
2
Out[]=
%165-%
In[]:=
π
Gammad
2
2Gamma+
1
2
d
2
r
a
1
2
2-d
2
3
2
2
Cos
r
a
Out[]=
d(d/2)!%
d/2
π
d
a
In[]:=
ballonsphere=!d-CosHypergeometric2F1,,,
1
d
2
d
a
d/2
π
π
Gammad
2
2Gamma+
1
2
d
2
r
a
1
2
2-d
2
3
2
2
Cos
r
a
In[]:=
1
d
2
d/2
π
d
a
π
Γd
2
2Γ+
d
2
1
2
r
a
2
F
1
1
2
2-d
2
3
2
2
cos
r
a
Out[]=
Volume of a circle on a 3D sphere
ballonsphere/.d2
In[]:=
2π1-Cos
2
a
r
a
Out[]=
Series[%,{r,0,8}]
In[]:=
π-+-+
2
r
π
4
r
12
2
a
π
6
r
360
4
a
π
8
r
20160
6
a
9
O[r]
Out[]=
ballonsphere/.d3
In[]:=
4π-Cos+
3
a
π
4
r
a
3/2
1-
2
Cos
r
a
1
21-
2
Cos
r
a
ArcSinCosSec
r
a
r
a
2
3/2
1-
2
Cos
r
a
Out[]=
FullSimplify[%]
In[]:=
2πArcCosCos-Cos
3
a
r
a
r
a
2
Sin
r
a
Out[]=
FullSimplify[%175,r/a>0]
In[]:=
2πArcCosCos-AbsSinCos
3
a
r
a
r
a
r
a
Out[]=
FullSimplify[%177,0<r/a<Pi/2]
In[]:=
2
a
2r
a
Out[]=
Series[%,{r,0,8}]
In[]:=
4π
3
r
3
4π
5
r
15
2
a
8π
7
r
315
4
a
9
O[r]
Out[]=
Series[ballonsphere,{r,0,8}]
In[]:=
Out[]=
FullSimplify[%,0<r/a<Pi/2]
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Out[]=
Scharzchild
Scharzchild
http://pi.math.cornell.edu/files/Research/SeniorTheses/rudeliusThesis.pdf
https://projecteuclid.org/download/pdf_1/euclid.mmj/1029001150 [ Alfred Gray ]
https://projecteuclid.org/download/pdf_1/euclid.acta/1485890018
Given nr what can we deduce about a space?
Given what can we deduce about a space?
n
r
See paper by Gray + ....
Flamm paraboloid
Flamm paraboloid
http://inspirehep.net/record/1707969/files/1812.03259.pdf
Reconstructing from random points
Graphics3D[Point[RandomVariate[MultinormalDistribution[IdentityMatrix[3]],1000]]]
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LightConeQ[{t0_,x0__},{t1_,x1__}]:=EuclideanDistance[{x0},{x1}]<=(t1-t0)
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pts=RandomVariate[MultinormalDistribution[IdentityMatrix[3]],50];
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CausalGraphClosure[pts_]:=Graph[If[LightConeQ[#[[1]],#[[2]]],#[[1]]#[[2]],Nothing]&/@Tuples[pts,2]]
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CausalGraphClosure[pts]
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Out[]=
CausalGraphClosure[RandomReal[10,{20,2}]]
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Out[]=
LayeredGraphPlot[%]
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Out[]=
Goal : reduce
TransitiveReductionGraph
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Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[RandomReal[10,{50,2}]]]]
In[]:=
Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[RandomReal[10,{50,2}]]]]
In[]:=
Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[RandomReal[10,{50,3}]]],AspectRatio1/2]
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Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[RandomVariate[MultinormalDistribution[IdentityMatrix[2]],50]]],AspectRatio1/2]
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Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[RandomVariate[MultinormalDistribution[IdentityMatrix[3]],50]]],AspectRatio1/2]
In[]:=
Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[Flatten[Array[List,{10,10}],1]]],AspectRatio1/2]
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Out[]=
LayeredGraphPlot[TransitiveReductionGraph[CausalGraphClosure[Flatten[Array[List,{7,7}],1]]],AspectRatio1/2]
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Out[]=
GraphPlot[TransitiveReductionGraph[CausalGraphClosure[Flatten[Array[List,{7,7}],1]]],AspectRatio1/2]
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Out[]=
GraphPlot3D[TransitiveReductionGraph[CausalGraphClosure[Flatten[Array[List,{5,5,5}],2]]]]
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Position[CellularAutomaton[90,{{1},0},20],1]
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GraphPlot[TransitiveReductionGraph[CausalGraphClosure[Position[CellularAutomaton[90,{{1},0},40],1]]],AspectRatio1/2]
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Out[]=
With[{w=TransitiveReductionGraph[CausalGraphClosure[Position[CellularAutomaton[90,{{1},0},40],1]]]},GraphPlot[w,AspectRatio1/2,VertexCoordinatesVertexList[w]]]
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With[{w=TransitiveReductionGraph[CausalGraphClosure[{1,-1}#&/@Reverse/@Position[CellularAutomaton[90,{{1},0},40],1]]]},GraphPlot[w,AspectRatio1/2,VertexCoordinatesVertexList[w]]]
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With[{w=TransitiveReductionGraph[CausalGraphClosure[Position[CellularAutomaton[90,{{1},0},20],1]]]},GraphPlot[w,AspectRatio1/2,VertexCoordinates({1,-1}#&/@Reverse/@VertexList[w])]]
In[]:=
Out[]=
GraphPlot3D[TransitiveReductionGraph[CausalGraphClosure[Position[CellularAutomaton[90,{{1},0},40],1]]],AspectRatio1/2]
In[]:=
Out[]=