ORIGINAL ARTICLE: L . Filipe O . Costa, José Natário, F . Frutos - Alfaro, and Michael Soffel (2023). Reference frames in general relativity and the galactic rotation curves. Phys. Rev. D 108, 044056.
DOI: https://doi.org/10.1103/PhysRevD.108.044056
Abstract: This file contains material ancillary to Sections IV.E and IV.B of the paper Phys. Rev. D 108, 044056 (2023). In this paper we discuss the extension of the International Astronomical Union (IAU) reference system to the exact theory of general relativity. It is seen that such an extension, retaining some of its crucial properties, can be achieved in a special class of spacetimes, admitting non-shearing congruences of observers which, at infinity, have zero vorticity and acceleration. As applications, we consider the FLRW, Kerr and NUT spacetimes, the van Stockum rotating dust cylinder, spinning cosmic strings and, finally, we debunk the so-called Balasin-Grumiller (BG) model, and the claims that the galaxies’ rotation curves can be explained through gravitomagnetic effects without the need for Dark Matter. It is shown to be completely inappropriate as a galactic model: its dust is actually static with respect to the asymptotic inertial frame, its gravitomagnetic effects arise from unphysical singularities along the axis, and the rotation curves obtained are merely down to an invalid choice of reference frame (the congruence of zero angular momentum observers, which are being dragged by the singularities).

Balasin-Grumiller metric

This section contains an analysis of the so-called Balasin-Grumiller solution, ancillary to Section IV.E of the main text. The Balasin-Grumiller metric is an exact solution the Einstein field equations, claimed to describe a galactic model, and used in some literature as evidence that gravitomagnetic effects can partially, or even totally explain the galactic rotation curves without the need for dark matter. We show it to actually be totally unsuitable to model any galaxy; among other pathologies, its gravitomagnetic effects arise from unphysical singularities along the axis. Below we compute several quantities that help us identify and characterize the singularities present in this solution, namely, the Kretschmann curvature invariant (Kretschmann scalar), the mass-energy density, and the 2-forms needed for computing the Komar mass and angular momentum contained within cylinders along the symmetry axis.

Commands

In[]:=
MF=MatrixForm;​​S=Simplify;​​FS=FullSimplify;​​Inv=Inverse;​​Clear[coord,gmet,ginv,chrst,riemann,ricci,Kretschmann]​​n=4;

Coordinates:

In[]:=
coord={t,r,ϕ,z}
Out[]=
{t,r,ϕ,z}

The metric and its inverse

Here we define the metric tensor
g
αβ
, denoted by “gmet”, and compute its inverse
αβ
g
(“ginv”) and determinant. Note: W(r,z) denotes here the function N(r,z) in the main text.
In[]:=
​​MFgmet=FS
-1
0
W[r,z]
0
0
Exp[nu[r,z]]
0
0
W[r,z]
0
2
r
-
2
W[r,z]
0
0
0
0
Exp[nu[r,z]]

Out[]//MatrixForm=
-1
0
W[r,z]
0
0
nu[r,z]

0
0
W[r,z]
0
2
r
-
2
W[r,z]
0
0
0
0
nu[r,z]

In[]:=
TableForm[gmet]
Out[]//TableForm=
-1
0
W[r,z]
0
0
nu[r,z]

0
0
W[r,z]
0
2
r
-
2
W[r,z]
0
0
0
0
nu[r,z]

In[]:=
MF[ginv=FS[Inv[gmet]]]
Out[]//MatrixForm=
-1+
2
W[r,z]
2
r
0
W[r,z]
2
r
0
0
-nu[r,z]

0
0
W[r,z]
2
r
0
1
2
r
0
0
0
0
-nu[r,z]

In[]:=
MF[S[ginv.gmet]]
Out[]//MatrixForm=
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
In[]:=
Det[gmet]
Out[]=
-
2nu[r,z]

2
r
In[]:=
sqrtmg=r
nu[r,z]

Out[]=
nu[r,z]

r

Christoffel symbols

Here we compute the Christoffel symbols of the second kind
α
Γ
βγ
= (1/2)
αδ
g
(
g
αδ,β
+
g
βδ,α
-
g
αβ,δ
)
In[]:=
chrst:=chrst=Simplify[Table[(1/2)*Sum[(ginv[[i,s]])*​​(D[gmet[[s,j]],coord[[k]]]+​​D[gmet[[s,k]],coord[[j]]]-D[gmet[[j,k]],coord[[s]]]),{s,1,n}],​​{i,1,n},{j,1,n},{k,1,n}]]
In[]:=
listaffine:=Table[If[UnsameQ[chrst[[i,j,k]],0],{ToString[Γ[i,j,k]],"=",chrst[[i,j,k]]}],{i,1,n},{j,1,n},{k,1,j}]
In[]:=
TableForm[Partition[DeleteCases[Flatten[listaffine],Null],3],TableSpacing{2,3}]
Out[]//TableForm=
Γ[1, 2, 1]
=
W[r,z]
(1,0)
W
[r,z]
2
2
r
Γ[1, 3, 2]
=
-
-2rW[r,z]+
2
r
(1,0)
W
[r,z]+
2
W[r,z]
(1,0)
W
[r,z]
2
2
r
Γ[1, 4, 1]
=
W[r,z]
(0,1)
W
[r,z]
2
2
r
Γ[1, 4, 3]
=
-
(
2
r
+
2
W[r,z]
)
(0,1)
W
[r,z]
2
2
r
Γ[2, 2, 2]
=
1
2
(1,0)
nu
[r,z]
Γ[2, 3, 1]
=
-
1
2
-nu[r,z]

(1,0)
W
[r,z]
Γ[2, 3, 3]
=
-nu[r,z]

-r+W[r,z]
(1,0)
W
[r,z]
Γ[2, 4, 2]
=
1
2
(0,1)
nu
[r,z]
Γ[2, 4, 4]
=
-
1
2
(1,0)
nu
[r,z]
Γ[3, 2, 1]
=
(1,0)
W
[r,z]
2
2
r
Γ[3, 3, 2]
=
1
r
-
W[r,z]
(1,0)
W
[r,z]
2
2
r
Γ[3, 4, 1]
=
(0,1)
W
[r,z]
2
2
r
Γ[3, 4, 3]
=
-
W[r,z]
(0,1)
W
[r,z]
2
2
r
Γ[4, 2, 2]
=
-
1
2
(0,1)
nu
[r,z]
Γ[4, 3, 1]
=
-
1
2
-nu[r,z]

(0,1)
W
[r,z]
Γ[4, 3, 3]
=
-nu[r,z]

W[r,z]
(0,1)
W
[r,z]
Γ[4, 4, 2]
=
1
2
(1,0)
nu
[r,z]
Γ[4, 4, 4]
=
1
2
(0,1)
nu
[r,z]

Riemann Tensor

From the Christoffel symbols, we compute the Riemann curvature tensor using the convention
α
R
βμν
=
α
Γ
βν,μ
-
α
Γ
βμ,ν
+...
In[]:=
riemann:=riemann=S[Table[​​D[chrst[[i,j,l]],coord[[k]]]-D[chrst[[i,j,k]],coord[[l]]]+​​Sum[chrst[[s,j,l]]chrst[[i,k,s]]-chrst[[s,j,k]]chrst[[i,l,s]],{s,1,n}],{i,1,n},{j,1,n},{k,1,n},{l,1,n}]]​​listriemann:=Table[If[UnsameQ[riemann[[i,j,k,l]],0],{ToString[R[i,j,k,l]],"=",riemann[[i,j,k,l]]}],{i,1,n},{j,1,n},{k,1,n},{l,1,k-1}]​​TableForm[Partition[DeleteCases[Flatten[listriemann],Null],3],TableSpacing{2,3}]

Ricci Tensor

From the Riemann tensor, we compute here the covariant components of the Ricci tensor
R
αβ
=
μ
R
αμβ
In[]:=
ricci:=ricci=S[Table[Sum[riemann[[i,j,i,l]],{i,1,n}],{j,1,n},{l,1,n}]]​​listricci:=Table[If[UnsameQ[ricci[[j,l]],0],{ToString[R[j,l]],ricci[[j,l]]}],{j,1,n},{l,1,j}]​​MF[ricci]

Kretschmann Scalar (R.R)

Below we enter the semi-explicit expression for the gravitomagnetic potential W(r,z):
The functions left implicit (for convenience) here are:

Expression for R.R as given in the main text

Confirmation:

Energy density

This computes the energy density ρ, Eq. (68) in the main text:

Komar mass

2-form ★dξ needed for computing the Komar mass inside cylinders

Here we define the Levi-Civita symbol [αβγδ], with the orientation [rϕzt]=-1:

Transformation of ★dξ from cylindrical coordinates {t,r,ϕ,z} to spherical coordinates {t,ρ,θ,ϕ}

★dζ matrix in spherical coordinates

Komar angular momentum

2-form ★dζ needed for computing the Komar angular momentum inside cylinders

The NUT metrics

This section contains an analysis of two versions of the NUT metric in the literature, ancillary to Section IV.B of the main text.
The NUT metrics are exact solutions the Einstein field equations whose physical interpretation remains an open problem, namely the origin of their gravitomagnetic monopole. Below we obtain the 2-forms needed for computing the Komar angular momentum. The analysis of the angular momentum shows that the gravitomagnetic monopole present in version 1 of the metric can be consistently interpreted as arising from a semi-infinite cosmic string located at -∞<z<0; and, in the case of version 2, from a pair of counter-rotating cosmic strings, one located at -∞<z<0, and another one at 0<z<∞. In this analysis, it proves useful to consider the angular momentum contained within cylinders along the symmetry axis. In order to facilitate such computation, in addition to their expressions in the usual spheroidal-type coordinates, we compute also the corresponding 2-forms in cylindrical coordinates.

Coordinates:

The metric and its inverse:

Christoffel symbols:

2-form ★dζ needed for computing the Komar angular momentum

Covariant derivative:

Here we define the Levi-Civita symbol [αβγδ]:

Transformation of ★dζ from spherical coordinates {t,ρ,θ,ϕ} to cylindrical coordinates {t,r,ϕ,z}

Relevant components of ★dζ for computing the Komar angular momentum contained within cylinders along the z-axis

Evaluation of the Komar angular momentum J contained within cylinders along the z-axis

Here we compute the angular momentum contained within cylinders along the symmetry axis, through Eq. (46) of the main text. Up to a constant factor, it is given by the sum of the integration of the 2-form ★dζ along the base, top, and lateral surfaces of the cylinder.

Infinitely long cylinder (-∞<z<∞)

So, in this case, the integral of ★dζ along the base and top of the cylinder vanish; J reduces to the integral along the lateral surface:

Semi-infinite cylinder 2M<z<∞, of radius R

In this case, the integral of ★dζ along the top of the cylinder vanishes; J reduces to the sum of the integrals along the lateral and bottom surfaces. First we compute the integral along the lateral surface, leaving the upper z limit “a” arbitrary:
Now we take the limit a→ ∞ corresponding to the semi-infinite cylinder:

Semi-infinite cylinder -∞<z<-2M, of radius R

In this case, the integral of ★dζ along the bottom of the cylinder vanishes; J reduces to the sum of the integrals along the lateral and top surfaces. First, we compute the integral along the lateral surface, leaving the lower z limit “-a” arbitrary:
Now we take the limit a→ ∞ corresponding to the semi-infinite cylinder:
We now perform in version 2 of the NUT metric an analysis analogous to that done above for version 1. We refer to the text above for the detailed explanations of each step.

Coordinates:

The metric and its inverse:

Christoffel symbols:

2-form ★dζ needed for computing the Komar angular momentum

Covariant derivative:

Transformation of ★dζ from spherical coordinates {t,ρ,θ,ϕ} to cylindrical coordinates {t,r,ϕ,z}

★dζ matrix in cylindrical coordinates

Relevant components of ★dζ for computing the Komar angular momentum contained within cylinders along the z-axis

Infinitely long cylinder (-∞<z<∞)

Finite cylinder 2M<z<10M, R=10M, l=0.1M

Finite cylinder -10M<z<-2M, R=10M, l=0.1M

CITE THIS NOTEBOOK

Reference frames in general relativity and the galactic rotation curves​
by L. Filipe O. Costa, José Natário, F. Frutos-Alfaro, and Michael Soffel
Wolfram Community, STAFF PICKS, October 20, 2023
​https://community.wolfram.com/groups/-/m/t/3053345