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14
Cameron Beetar
[WSS20] A Path to Higher Order Corrections for Einstein's Equations
Cameron Beetar, University of Cape Town
Posted
8 months ago
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We consider the higher order corrections to geodesic ball volumes on Riemannian manifolds and compute an assumed form of the higher order corrections to the Einstein field equations. The form of these corrections is highly undetermined. As a potential method of resolving this indeterminacy, we note an analogy that exists between the derivation of the Einstein field equations in the context of Wolfram models and the derivation of hydrodynamic equations from discrete molecular dynamics. In particular, we note that one might be able to use relativistic Burnett and relativistic super-Burnett equations to find explicit values to constrain the undetermined values in the higher order corrections to the Einstein field equations in the context of Wolfram models.
Introduction
Where physics subfields meet - when there is only a
hint
of an underlying relationship - that is where the most exciting research can happen. With the development of Wolfram models and the parallel development of our understanding of what information they can convey (and how that relates to conventional physics) we are presented with many opportunities to extend our understanding.
The crux of the Wolfram physics project is to relate the
discretuum
to the
continuum
- a task that is both notoriously difficult and has exceptional promise. As physicists, we acknowledge the immense amount of time, effort and resources that have gone into finding a so-called
Theory of Everything
- some theory that might give a coherent description of physics at all scales; quantum and cosmological. While it is impossible to know whether the Wolfram physics project will take us
all
the way there, we would be foolish to sit idly by as others do all the ‘cool’ stuff. It is a part of the great
pleasure of finding things out.
Einstein’s field equations (EFE’s) were first published in 1915 and are a representation of the fundamental relationship between the geometry of spacetime and the bodies that exist within it. The equations are represented by a tensor equation of the form
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A geodesic describes the shortest path between two points on a manifold. There are various ways to visualise these but there is a particularly nice Wolfram repository function for these purposes. Most people are familiar with the geodesics on the surface of a sphere in 3-dimensional Euclidean space. The following code can be used for visualisation purposes [8].
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t
]
,
v
[
t
]
]
/
.
N
D
S
o
l
v
e
[
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
G
e
o
d
e
s
i
c
"
]
[
t
o
r
u
s
[
3
,
1
,
1
]
[
u
,
v
]
,
{
u
,
v
}
,
t
,
{
π
,
π
}
,
.
5
]
,
{
u
,
v
}
,
{
t
,
0
,
1
5
0
}
]
]
,
{
t
,
0
,
1
5
0
}
,
B
o
x
e
d
F
a
l
s
e
,
A
x
e
s
F
a
l
s
e
]
,
P
a
r
a
m
e
t
r
i
c
P
l
o
t
3
D
[
E
v
a
l
u
a
t
e
[
t
o
r
u
s
[
3
,
1
,
1
]
[
u
,
v
]
]
,
{
u
,
0
,
2
π
}
,
{
v
,
0
,
2
π
}
,
P
l
o
t
S
t
y
l
e
O
p
a
c
i
t
y
[
.
7
]
,
M
e
s
h
F
a
l
s
e
]
]
C
l
e
a
r
l
y
t
h
e
‘
v
o
l
u
m
e
’
o
f
a
2
-
s
p
h
e
r
e
o
n
t
h
e
s
u
r
f
a
c
e
o
f
t
h
e
t
o
r
u
s
i
s
u
n
l
i
k
e
l
y
t
o
b
e
t
h
e
s
a
m
e
a
s
o
f
t
h
o
s
e
o
n
t
h
e
s
u
r
f
a
c
e
o
f
a
s
p
h
e
r
e
.
A
l
f
r
e
d
G
r
a
y
p
u
b
l
i
s
h
e
d
a
p
a
p
e
r
i
n
1
9
7
4
e
n
t
i
t
l
e
d
T
h
e
v
o
l
u
m
e
o
f
a
s
m
a
l
l
g
e
o
d
e
s
i
c
b
a
l
l
o
f
a
R
i
e
m
a
n
n
i
a
n
m
a
n
i
f
o
l
d
[
2
]
.
I
n
i
t
,
h
e
p
r
o
v
i
d
e
s
t
h
e
g
e
n
e
r
a
l
f
o
r
m
o
f
t
h
e
v
o
l
u
m
e
o
f
g
e
o
d
e
s
i
c
b
a
l
l
u
p
t
o
O
(
6
r
)
.
W
e
d
o
n
o
t
r
e
-
d
e
r
i
v
e
t
h
e
r
e
s
u
l
t
,
b
u
t
s
i
m
p
l
y
s
t
a
t
e
i
t
h
e
r
e
.
T
h
e
o
r
e
m
:
L
e
t
M
b
e
a
n
n
-
d
i
m
e
n
s
i
o
n
a
l
a
n
a
l
y
t
i
c
R
i
e
m
a
n
n
i
a
n
m
a
n
i
f
o
l
d
,
a
n
d
l
e
t
r
>
0
b
e
s
m
a
l
l
e
n
o
u
g
h
s
o
t
h
a
t
e
x
p
m
i
s
d
e
f
i
n
e
d
o
n
a
b
a
l
l
o
f
r
a
d
i
u
s
r
i
n
t
h
e
t
a
n
g
e
n
t
s
p
a
c
e
M
m
.
W
e
p
u
t
V
m
(
r
)
=
{
e
x
p
m
(
x
)
|
|
|
x
|
|
≤
r
}
.
T
h
e
n
,
u
s
i
n
g
t
h
e
f
o
l
l
o
w
i
n
g
n
o
t
a
t
i
o
n
τ
(
R
)
=
n
∑
i
=
1
R
i
i
(
R
i
c
c
i
s
c
a
l
a
r
)
;
|
|
R
2
|
|
=
n
∑
i
,
j
,
k
,
l
=
1
2
R
i
j
k
l
(
N
o
r
m
o
f
R
i
e
m
a
n
n
t
e
n
s
o
r
)
;
|
|
ρ
(
R
)
2
|
|
=
n
∑
i
,
j
=
0
2
R
i
j
(
N
o
r
m
o
f
R
i
c
c
i
t
e
n
s
o
r
)
;
Δ
R
=
2
∇
R
=
n
∑
i
=
1
2
∇
i
i
τ
(
R
)
(
L
a
p
l
a
c
i
a
n
o
f
R
i
c
c
i
s
c
a
l
a
r
)
;
a
n
d
α
n
=
2
n
Γ
1
2
-
1
Γ
n
2
;
the higher order corrections to the volume of an
n
-dimensional geodesic ball are
V
m
(
r
)
=
α
n
n
r
n
1
-
τ
(
R
)
6
(
n
+
2
)
2
r
+
1
3
6
0
(
n
+
2
)
(
n
+
4
)
{
-
3
|
|
R
2
|
|
+
8
|
|
ρ
(
R
)
2
|
|
+
5
2
τ
(
R
)
-
1
8
Δ
R
}
4
r
+
O
(
6
r
)
.
(
2
)
A
s
a
n
a
s
i
d
e
,
i
n
n
-
d
i
m
e
n
s
i
o
n
a
l
E
u
c
l
i
d
e
a
n
s
p
a
c
e
V
m
(
r
)
=
α
n
n
r
n
.
W
h
a
t
w
e
n
o
w
h
a
v
e
a
r
e
t
h
e
h
i
g
h
e
r
o
r
d
e
r
c
o
r
r
e
c
t
i
o
n
s
t
o
a
g
e
o
d
e
s
i
c
b
a
l
l
o
n
a
n
a
r
b
i
t
r
a
r
y
R
i
e
m
a
n
n
i
a
n
m
a
n
i
f
o
l
d
.
O
u
r
p
r
i
m
a
r
y
g
o
a
l
,
t
h
e
n
,
i
s
t
o
e
x
t
e
n
d
t
h
e
a
n
a
l
o
g
y
p
r
o
p
o
s
e
d
b
y
G
o
r
a
r
d
i
n
[
1
]
t
o
h
i
g
h
e
r
o
r
d
e
r
i
n
r
.
G
e
o
d
e
s
i
c
b
a
l
l
s
c
a
n
b
e
t
a
k
e
n
i
n
t
h
e
d
i
s
c
r
e
t
e
c
a
s
e
b
y
c
o
n
s
i
d
e
r
i
n
g
d
i
s
c
r
e
t
e
s
t
e
p
s
o
f
s
o
m
e
c
h
a
r
a
c
t
e
r
i
s
t
i
c
l
e
n
g
t
h
(
c
o
n
v
e
n
t
i
o
n
a
l
l
y
o
n
e
e
d
g
e
)
o
n
a
h
y
p
e
r
g
r
a
p
h
o
r
c
a
u
s
a
l
n
e
t
w
o
r
k
.
T
o
b
e
m
o
r
e
p
r
e
c
i
s
e
,
a
g
e
o
d
e
s
i
c
o
n
a
h
y
p
e
r
g
r
a
p
h
b
e
t
w
e
e
n
v
e
r
t
i
c
e
s
A
a
n
d
B
i
s
t
h
e
s
o
l
u
t
i
o
n
(
s
)
t
o
t
h
e
s
h
o
r
t
e
s
t
p
a
t
h
p
r
o
b
l
e
m
.
T
h
e
d
i
s
c
r
e
t
i
z
a
t
i
o
n
p
r
o
c
e
s
s
o
f
g
e
o
m
e
t
r
i
c
o
b
j
e
c
t
s
a
n
d
p
r
o
p
e
r
t
i
e
s
i
s
d
e
s
c
r
i
b
e
d
m
o
r
e
c
o
m
p
l
e
t
e
l
y
i
n
[
1
]
.
Derivation of the Form of the Higher Order Corrections
Consider an n-dimensional analytic Riemannian manifold,
M
, and let
r
>
0
be small enough so that
e
x
p
m
is defined on a ball of radius
r
in the tangent space
M
m
. Then the ratio of the volume of a geodesic ball of radius
r
in the manifold to the volume of a geodesic ball of the same radius in Euclidean space is
V
m
(
r
)
V
E
(
r
)
=
1
-
τ
(
R
)
6
(
n
+
2
)
2
r
+
1
3
6
0
(
n
+
2
)
(
n
+
4
)
{
-
3
|
|
R
2
|
|
+
8
|
|
ρ
(
R
)
2
|
|
+
5
2
τ
(
R
)
-
1
8
Δ
R
}
4
r
+
O
(
6
r
)
.
(
3
)
While this relation might be sufficient when considering space alone, we must consider the spacetime form of this relation. To that end, Gray showed that the metric could be expanded to be of the form
g
p
q
=
δ
p
q
-
1
3
R
i
p
j
q
i
x
j
x
-
1
6
∇
i
R
j
p
k
q
i
x
j
x
k
x
+
1
1
2
0
-
6
2
∇
i
j
R
k
p
l
q
+
1
6
3
R
i
p
j
s
R
k
q
l
s
i
x
j
x
k
x
l
x
+
O
(
|
|
5
x
|
|
)
.
(
4
)
where
∇
is the Riemannian connection and we use the convention that the Ricci scalar curvature is the contraction of the Ricci curvature tensor which is, itself, a contraction of the Riemann tensor in the first and third indices. That is
R
=
a
b
g
R
a
b
;
a
n
d
R
a
b
=
c
R
a
c
b
.
Gray finds the higher order form of this procedure whose lower order terms agree with those found by Gorard. We can thus define the manifold’s volume element in terms of the standard Euclidean volume element (making use of the Einstein summation convention to neaten the results) by
d
μ
g
=
1
-
1
6
R
i
j
x
i
x
j
-
1
1
2
∇
i
R
j
k
x
i
x
j
x
k
+
1
2
4
-
3
5
2
∇
i
j
R
k
l
+
1
3
R
i
j
R
k
l
-
2
1
5
R
i
a
j
b
R
k
a
l
b
x
i
x
j
x
k
x
l
+
O
(
|
|
x
5
|
|
)
d
μ
E
(
5
)
which leads to [2]:
d
e
t
(
g
i
j
)
=
1
-
1
3
R
i
j
x
i
x
j
-
1
6
∇
i
R
j
k
x
i
x
j
x
k
+
1
2
4
-
6
5
2
∇
i
j
R
k
l
+
4
3
R
i
j
R
k
l
-
4
1
5
R
i
a
j
b
R
k
a
l
b
x
i
x
j
x
k
x
l
+
O
(
|
|
x
5
|
|
)
.
(
6
)
Now, introducing the conventional Einstein-Hilbert action [3]:
S
E
H
=
1
1
6
π
G
∫
R
-
g
4
x
=
1
1
6
π
G
R
-
1
-
1
3
R
i
j
x
i
x
j
-
1
6
∇
i
R
j
k
x
i
x
j
x
k
+
1
2
4
-
6
5
2
∇
i
j
R
k
l
+
4
3
R
i
j
R
k
l
-
4
1
5
R
i
a
j
b
R
k
a
l
b
x
i
x
j
x
k
x
l
+
O
(
|
|
x
5
|
|
)
4
x
(
7
)
where
g
=
d
e
t
(
g
a
b
)
and
G
is the gravitational constant (which, along with the speed of light in a vacuum
c
, we now set to be unity). Then our task is to extremize
S
E
H
(as this is the general procedure for producing the EFE’s in a vacuum). This is quite challenging, and I was unable to minimize this during the last few days of the Wolfram Summer School.
At this point, it would not make sense to end the derivation abruptly. Rather, I will make a rather flimsy assumption and will proceed naturally thereafter - the main point being to outline the
process
one would go through to achieve the desired results.
Under the variation
g
g
+
δ
g
such that
δ
g
0
as
|
|
x
|
|
0
, if we assume that the expansion of the metric behaves in approximately the same way as the unexpanded metric (this is the aforementioned flimsy assumption - it is almost certainly wrong) then with some tensor mathematics and several identities one can find [3]:
δ
R
-
g
=
-
μ
κ
g
λ
ν
g
δ
g
κ
λ
R
μ
ν
-
g
+
1
2
R
-
g
μ
ν
g
δ
g
μ
ν
(
8
)
such that
δ
S
E
H
=
1
1
6
π
-
μ
ν
R
+
1
2
R
μ
ν
g
δ
g
μ
ν
-
g
4
d
x
.
(
9
)
Requiring that the variation above be zero for any variation
δ
g
, we have the EFE’s in a vacuum
G
μ
ν
=
R
μ
ν
-
1
2
R
g
μ
ν
=
0
(
1
0
)
w
h
e
r
e
G
μ
ν
i
s
t
h
e
E
i
n
s
t
e
i
n
t
e
n
s
o
r
a
n
d
g
μ
ν
i
s
a
s
d
e
f
i
n
e
d
i
n
e
q
.
4
.
W
e
w
i
l
l
n
o
t
e
x
p
a
n
d
t
h
e
s
e
e
q
u
a
t
i
o
n
s
a
s
y
e
t
(
w
e
w
i
l
l
o
n
l
y
d
o
t
h
i
s
o
n
c
e
a
n
d
w
e
w
i
l
l
s
a
v
e
t
h
i
s
f
o
r
t
h
e
n
o
n
-
v
a
c
u
u
m
c
a
s
e
)
.
T
h
e
t
y
p
i
c
a
l
e
x
t
e
n
s
i
o
n
o
f
t
h
i
s
i
s
t
o
i
n
c
l
u
d
e
t
h
e
c
o
n
t
r
i
b
u
t
i
o
n
s
f
r
o
m
m
a
t
t
e
r
.
T
h
e
r
e
s
u
l
t
o
f
w
h
i
c
h
i
s
[
3
]
:
G
μ
ν
=
8
π
T
μ
ν
(
1
1
)
w
h
e
r
e
T
μ
ν
i
s
t
h
e
e
n
e
r
g
y
m
o
m
e
n
t
u
m
t
e
n
s
o
r
.
T
h
e
i
n
c
l
u
s
i
o
n
o
f
t
h
e
c
o
s
m
o
l
o
g
i
c
a
l
c
o
n
s
t
a
n
t
p
r
o
d
u
c
e
s
[
3
]
R
μ
ν
-
1
2
R
g
μ
ν
+
Λ
g
μ
ν
=
8
π
T
μ
ν
.
(
1
2
)
F
i
n
a
l
l
y
,
r
e
p
l
a
c
i
n
g
g
μ
ν
w
i
t
h
t
h
e
r
i
g
h
t
-
h
a
n
d
s
i
d
e
o
f
e
q
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(
1
3
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This equation is based on a weak assumption made in deriving eq. 8. Nevertheless, the only significant challenge that one might expect that was not addressed here was the extremisation of the expanded metric. This would be the obvious place to start in a more accurate derivation of the corrections to the EFE’s. While eq. 13 is quite devoid of things that might allow one to have an intuition of what is going on, we do have potential ideas for resolving this.
Proposed Method for Resolving Indeterminacy
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[
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.
A brief look at the analogy
We consider the case of a hypergraph and then move to the case of a causal graph. In all examples we run the same initialisation code:
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[
4
]
:
f
=
(
0
)
f
+
(
1
)
f
+
(
2
)
f
+
.
.
.
=
∞
∑
n
=
0
(
n
)
f
.
Developing the appropriate conservation laws and, thereafter, the appropriate constitutive relations, one has a path to solving for the
t
h
n
order distribution functions [4]. Investigating Chapman-Enskog expansion for the non-relativistic Boltzmann equation, one can arrive at the conventional hydrodynamic Boltzmann, Navier-Stokes, Burnett and super-Burnett equations (the latter two are notoriously long and challenging to solve in any circumstance). Applying the same techniques should allow one to attain their relativistic equivalent distributions. Furthermore, according to Gorard we have a candidate distribution function for
(
0
)
f
in the case of a causal graph [1]:
C
(
t
)
=
n
a
t
1
-
1
6
R
j
k
j
t
k
t
.
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F
u
r
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m
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,
t
h
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r
a
d
u
a
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d
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a
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o
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b
u
t
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o
n
.
M
a
n
i
p
u
l
a
t
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[
P
l
o
t
[
P
D
F
[
M
a
x
w
e
l
l
D
i
s
t
r
i
b
u
t
i
o
n
[
σ
]
,
x
]
,
{
x
,
0
,
1
0
}
,
P
l
o
t
R
a
n
g
e
{
{
0
,
1
0
}
,
{
0
,
1
}
}
]
,
{
σ
,
0
.
6
,
1
}
]
I
n
[
]
:
=
σ
Thus, while the distribution we found for the hypergraph may not be precisely Maxwellian, that it behaves like a distribution function is a good sanity check. It is also a good hint that the limiting distribution is ~0 for all radii,
r
, corresponding to an equal distribution of ‘matter’ (nodes in the case of the hypergraph) and being a good indication of ergodicity.
Of course, the above distributions are non-relativistic distributions only - they are taken on a hypergraph and
not
a causal graph.
Noticing that geodesic balls in a causal graph give a sort of distribution function and, therefore, recognising the link between the corrections to geodesic ball volumes in arbitrary Riemannian manifolds, distribution functions and the relativistic hydrodynamic equations, one can - at the very least - constrain the undetermined correction values in the higher order EFE' s by recognising analogous structures. This has the potential to influence the way we understand spacetime near singularities (such as black holes).
In summary, we know the corrections to the volumes of geodesic ball volumes from [2]. We have a means of calculating the EFE' s in the discrete hypergraph case (with certain constraints) from [1]. We have the form of the indeterminate higher order corrections to the EFE' s (this work). Finally, we have a means of removing the determinacy by using the Chapman - Enskog method to constrain the undetermined values (by analogy, as noted in [1]). As a result of constraining the values in the higher order EFE' s, one can obtain (to an as yet unknown determinacy) the higher order EFE' s.
A Brief Look at the Complexities to be Overcome
I
t
i
s
w
o
r
t
h
c
o
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d
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s
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)
b
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.
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h
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s
,
d
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-
t
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.
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h
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a
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d
b
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a
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m
a
g
e
f
r
o
m
[
4
]
:
B
e
g
i
n
n
i
n
g
f
r
o
m
t
h
e
(
n
o
n
-
r
e
l
a
t
i
v
i
s
t
i
c
)
B
o
l
t
z
m
a
n
n
e
q
u
a
t
i
o
n
:
∂
f
∂
t
+
c
k
∂
f
∂
x
k
+
F
k
.
∂
f