What is
n
r
for the Schwarzchild metric?

Ricci scalar term vanishes, but the next-order correction is there.
Kretschmann scalar
What isn’t volume of a ball ∫
g
up to specified distance (which depends on the line element) integrated over solid angle?
In[]:=
grr[r_]:=(1-((sRadius*r^2)/(gRadius^3)))^(-1)
In[]:=
grr[r_]:=(1-((a*r^2)))^(-1)
Because it is a spherically symmetric metric, the surface area is still the same....
In[]:=
Integrate[4Pir^2Sqrt[grr[r]],{r,0,1}]
Out[]=
In[]:=
Assuming[{a>0},%]
Out[]=
In[]:=
FullSimplify[%]
Out[]=
In[]:=
Plot
1
2
a
π
-
1
a
3/2
-
a

π+
a
π+2
1-a
-
a
-1+a
-
-(-1+a)a
+ArcSin[
a
],{a,0,2}
Out[]=
As a function of radius away from the origin, what is the volume of a ball?
n
r
=4π/3r^3(1-
4
r
XXXX)
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
In[]:=
Integrate[1/Sqrt[1-2m/r],r]
Out[]=
1-
2m
r
r+2mArcTanh
1-
2m
r

In[]:=
Limit[%158,r0]
Out[]=
$Aborted
In[]:=
Limit
1-
2m
r
r+2mArcTanh
1-
2m
r
,r0
Out[]=
$Aborted
In[]:=
$Version
Out[]=
12.1.0 for Mac OS X x86 (64-bit) (December 24, 2019)

On a sphere, in any number of dimensions, the formula is

n
r
=4π/3r^3(1-
4
r
XXXX)

https://www.wolframscience.com/nks/notes-9-15--sphere-volumes/

s[d]=
d/2
π
(d/2)!
s[d]
d
r
(1-RicciScalar
2
r
/(6(d+2))+…)
For a sphere:
RicciScalar=d(d-1)/
2
a
Exact result for d-dimensional sphere surface
In[]:=
d
d/2
π
(d/2)!
d
a
Integrate[
d-1
Sin[θ]
,{θ,0,r/a}]
Out[]=
$Aborted
In[]:=
-Cos[x]Hypergeometric2F1[12,(2-d)2,32,
2
Cos[x]
]/.xr/a
Out[]=
-Cos
r
a
Hypergeometric2F1
1
2
,
2-d
2
,
3
2
,
2
Cos
r
a


In[]:=
-Cos[x]Hypergeometric2F1[12,(2-d)2,32,
2
Cos[x]
]/.x0
Out[]=
-
π
Gamma
d
2

2Gamma
1
2
+
d
2

In[]:=
%165-%
Out[]=
π
Gamma
d
2

2Gamma
1
2
+
d
2

-Cos
r
a
Hypergeometric2F1
1
2
,
2-d
2
,
3
2
,
2
Cos
r
a


In[]:=
d
d/2
π
(d/2)!
d
a
%
In[]:=
ballonsphere=
1
d
2
!
d
a
d
d/2
π
π
Gamma
d
2

2Gamma
1
2
+
d
2

-Cos
r
a
Hypergeometric2F1
1
2
,
2-d
2
,
3
2
,
2
Cos
r
a


Out[]=
1
d
2
!
d
d/2
π
d
a
π
Γ
d
2

2Γ
d
2
+
1
2

-cos
r
a
2
F
1
1
2
,
2-d
2
;
3
2
;
2
cos
r
a
Volume of a circle on a 3D sphere
In[]:=
ballonsphere/.d2
Out[]=
2
2
a
π1-Cos
r
a

In[]:=
Series[%,{r,0,8}]
Out[]=
π
2
r
-
π
4
r
12
2
a
+
π
6
r
360
4
a
-
π
8
r
20160
6
a
+
9
O[r]
In[]:=
ballonsphere/.d3
Out[]=
4
3
a
π
π
4
-Cos
r
a

3/2
1-
2
Cos
r
a

1
21-
2
Cos
r
a


+
ArcSinCos
r
a
Sec
r
a

2
3/2
1-
2
Cos
r
a


In[]:=
FullSimplify[%]
Out[]=
2
3
a
πArcCosCos
r
a
-Cos
r
a

2
Sin
r
a

In[]:=
FullSimplify[%175,r/a>0]
Out[]=
2
3
a
πArcCosCos
r
a
-AbsSin
r
a
Cos
r
a

In[]:=
FullSimplify[%177,0<r/a<Pi/2]
Out[]=
2
a
π2r-aSin
2r
a

In[]:=
Series[%,{r,0,8}]
Out[]=
4π
3
r
3
-
4π
5
r
15
2
a
+
8π
7
r
315
4
a
+
9
O[r]

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
See paper by Gray + ....

Flamm paraboloid

http://inspirehep.net/record/1707969/files/1812.03259.pdf
Reconstructing from random points
Goal : reduce