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Cameron Beetar

[WSS20] A Path to Higher Order Corrections for Einstein's Equations

Cameron Beetar, University of Cape Town

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Torus graphs with increasing/decreasing numbers of vertices.

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

μν

+Λ

μν

=8π

μν

(

)

where

μν

is the Ricci tensor,

is the Ricci scalar,

is the cosmological constant and

μν

is the energy-momentum tensor. The equation has remained unchanged for over a century and one should expect that finding corrections to these equations would be a challenge. There is, however, a potential method for finding higher order corrections. As Gorard found in his analysis of the relations between the Wolfram model of physics and general relativity [1], one may derive the EFE’s in the standard form (in the context of Wolfram models) by making use of geodesic ball volumes in the causal network of a hypergraph (this is a gross abbreviation of the process, but the method will be made clearer soon). In this essay, we discuss an extension of Gorard’s analysis and, in doing so, propose a method of finding an (indeterminate) form of the higher order corrections to the EFE’s.Of interest is how Gorard’s derivation is analogous to that of the derivation of the Navier-Stokes equations using the Chapman-Enskog method in the context of hydrodynamics. We conjecture that the indeterminate forms of the corrections obtained for the EFE’s may be determined by solving appropriate Burnett and super-Burnett equations.Below: A classic visualisation of general relativity (left) and results from a hydrodynamic simulation of the AREPO Code [6][7].

Higher Order Corrections to Geodesic Ball Volumes

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].

(*UseResourceFunction["Geodesic"]toaccessthegeodesicrepositoryfunction*)sphere[a_][u_,v_]:={aCos[v]Cos[u],aCos[v]Sin[u],aSin[v]}ge=ResourceFunction["Geodesic"][sphere[1][u,v],{u,v},t,{0,0},θ]sg=Flatten[(NDSolve[#1,{u,v},{t,0,4}]&)/@Table[ge,{θ,(2π)/30,2π,(2π)/30}],1];sphere[1][u[t],v[t]]/.sg[[1]]/.t.1Show[{Graphics3D[Table[Line[Append[#,First[#]]&[sphere[1][u[t],v[t]]/.sg]],{t,0,3,1/5}],BoxedFalse],ParametricPlot3D[Evaluate[sphere[1][u[t],v[t]]/.sg],{t,0,π},BoxedFalse]}]

Which produces:

A geodesic volume taken on the sphere (assuming we take the surface of the sphere to be some Riemannian manifold) is a 2-dimensional ‘bent’ disc. To see this, start from the upper pole and the move down until you hit a ‘latitudinal line’ - the enclosed area is a 2-dimensional ‘volume’.However, intuition begins to fail for less obvious structures. Consider a torus. What do its geodesics look like? We can visualise these again using the same repository function [8].

torus[a_,b_,c_][u_,v_]:={(a+bCos[v])Cos[u],(a+bCos[v])Sin[u],cSin[v]}ResourceFunction["Geodesic"][torus[3,1,1][u,v],{u,v},t,{2,3},θ]Show[ParametricPlot3D[Evaluate[torus[3,1,1][u[t],v[t]]/.NDSolve[ResourceFunction["Geodesic"][torus[3,1,1][u,v],{u,v},t,{π,π},.5],{u,v},{t,0,150}]],{t,0,150},BoxedFalse,AxesFalse],ParametricPlot3D[Evaluate[torus[3,1,1][u,v]],{u,0,2π},{v,0,2π},PlotStyleOpacity[.7],MeshFalse]]

Clearly the ‘volume’ of a 2-sphere on the surface of the torus is unlikely to be the same as of those on the surface of a sphere.Alfred Gray published a paper in 1974 entitled The volume of a small geodesic ball of a Riemannian manifold [2]. In it, he provides the general form of the volume of geodesic ball up to

). We do not re-derive the result, but simply state it here.Theorem: Let

be an

-dimensional analytic Riemannian manifold, and let

r>0

be small enough so that

exp

is defined on a ball of radius

in the tangent space

. We put

(r)={

exp

(x)|||x||≤r}

. Then, using the following notation

τ(R)=

∑

i=1

(Ricciscalar);||R

∑

i,j,k,l=1

ijkl

(NormofRiemanntensor);||ρ(R)

∑

i,j=0

(NormofRiccitensor);ΔR=

∇

∑

i=1

∇

τ(R)(LaplacianofRicciscalar);and

-1

;

the higher order corrections to the volume of an

-dimensional geodesic ball are

(r)=

τ(R)

6(n+2)

360(n+2)(n+4)

{-3||R

+8||ρ(R)

τ(R)

-18ΔR}

+O(

(

)

As an aside, in

-dimensional Euclidean space

(r)=

.What we now have are the higher order corrections to a geodesic ball on an arbitrary Riemannian manifold. Our primary goal, then, is to extend the analogy proposed by Gorard in [1] to higher order in

. Geodesic balls can be taken in the discrete case by considering discrete steps of some characteristic length (conventionally one edge) on a hypergraph or causal network. To be more precise, a geodesic on a hypergraph between vertices

and

is the solution(s) to the shortest path problem. The discretization process of geometric objects and properties is described more completely in [1].

Derivation of the Form of the Higher Order Corrections

Consider an n-dimensional analytic Riemannian manifold,

, and let

r>0

be small enough so that

exp

is defined on a ball of radius

in the tangent space

. Then the ratio of the volume of a geodesic ball of radius

in the manifold to the volume of a geodesic ball of the same radius in Euclidean space is

(r)

=1-

τ(R)

6(n+2)

360(n+2)(n+4)

{-3||R

+8||ρ(R)

τ(R)

-18ΔR}

+O(

(

)

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

ipjq

∇

jpkq

120

-6

∇

kplq

ipjs

kqls



+O(||

||).

(

)

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

;and

acb

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

dμ

=1-

∇

-

∇

iajb

kalb



+O(||x

)

dμ

(

)

which leads to [2]:

det(

)=1-

∇

-

∇

iajb

kalb



+O(||x

(

)

Now, introducing the conventional Einstein-Hilbert action [3]:

16πG

∫R

-g



16πG

R

-1-

∇

-

∇

iajb

kalb



+O(||x

)



(

)

where

g=det(

)

and

is the gravitational constant (which, along with the speed of light in a vacuum

, we now set to be unity). Then our task is to extremize

(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+δg

such that

δg0

||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

=-

μκ

λν

κλ

μν

-g

μν

(

)

such that

16π

-

μν

-g

(

)

Requiring that the variation above be zero for any variation

δg

, we have the EFE’s in a vacuum

μν

(

)

where

μν

is the Einstein tensor and

μν

is as defined in eq. 4. We will not expand these equations as yet (we will only do this once and we will save this for the non-vacuum case). The typical extension of this is to include the contributions from matter. The result of which is [3]:

μν

=8π

μν

(

)

where

μν

is the energy momentum tensor. The inclusion of the cosmological constant produces [3]

μν

+Λ

μν

=8π

μν

(

)

Finally, replacing

μν

with the right-hand side of equation 5, we get a form of the higher order corrections to the general Einstein field equations up to

O||x

μν

R+Λ

μν

iμjν

∇

jμkν

120

-6

∇

kμlν

iμjs

kνls



=8π

μν

(

)

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

Consider some causal invariant hypergraph whose causal graph demonstrates asymptotic dimensionality preservation (the dimension of the causal graph converges to some fixed finite value as the number of updating events becomes arbitrarily large [1]). Then Gorard has shown that there exist analogous discrete geometrical objects that can be used in place of the continuous geometrical objects we have used on Riemannian manifolds. In particular, Gorard was able to obtain discrete analogs of the Riemann curvature tensor and Ricci curvature tensor. Using these analogous objects, Gorard was able to show that, if there is some continuum limit in which the causal graph becomes a Riemannian manifold, then it must also satisfy the EFE’s. What is worth more consideration is the method by which the EFE’s that include matter are found. A coherent analog of the energy-momentum tensor needed to be defined. Since

μν

is the flux of the 4-momentum,

, across the hypersurface of constant

, an analogous definition in the discrete case comes from the flux of edges through a discrete hypersurface in the causal graph [1]. Using this interpretation, Gorard was able to derive the full EFE’s.This methodology of deriving continuum equations from a fundamentally discrete system is similar to the Chapman-Enskog method for deriving hydrodynamic equations from discrete molecular dynamics [1].

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:

WMData=ResourceFunction["WolframModelData"]WM=ResourceFunction["WolframModel"]WMPlot=ResourceFunction["WolframModelPlot"]HGNVol=ResourceFunction["HypergraphNeighborhoodVolumes"];HG2G=ResourceFunction["HypergraphToGraph"]GNVol=ResourceFunction["GraphNeighborhoodVolumes"]

Consider some hypergraph that meets the criterion specified (causal invariance, asymptotic dimensional stability). A good example of such a hypergraph that appears to have a good correspondence with a 2-dimensional Riemannian manifold is WM7714. The code shown will produce the image shown if the second ‘Automatic’ is changed to ‘2500’.

rules=WMData["wm7714"];WM[rules,Automatic,Automatic,"FinalStatePlot"]

The geodesic balls (or perhaps more appropriately ‘discs’) in this case, as stated before, are solutions to the shortest path problem. An example of geodesic balls having radii of 5, 10, 15 and 20 steps, from a particular vertex for the hypergraph shown above is presented below.

HG=WM[{rules},Automatic,2500,"FinalState"];GG=HG2G[HG];m=Median@VertexList[GG];HighlightGraph[GG,NeighborhoodGraph[GG,m,#]]&/@Range[5,20,5]

The choice of the median vertex was purely for aesthetic purposes (we can define more geodesic balls before boundary effects start appearing).Knowing the estimated dimension of such a manifold would allow you to define a distribution function of vertices as a function of the radius at a fixed time. However, this is a hypergraph and we are particularly concerned with causal graphs. Consider now WM1517 which is (heuristically) causal invariant.

The hypergraph (top left) has an almost identical causal graph (top right). The causal graph can be estimated as 2-dimensional based on it's dimension estimate data (bottom left) which predominantly estimates the causal graph to be 2-dimensional as demonstrated by the associated histogram (bottom right). Note that the falloff in the bottom left image is due to boundary effects. Clearly WM1517 is a good candidate for a Riemannian manifold in the continuum limit. Interpreting the flux of causal edges through some hypersurface as the energy-momentum tensor, one could perform a Chapman-Enskog expansion on the relativistic Boltzmann equation. This is a non-trivial process. The central idea of the Chapman-Enskog method is to expand the distribution function as an infinite series [4]:

(0)

(1)

(2)

+...=

∞

∑

n=0

(n)

Developing the appropriate conservation laws and, thereafter, the appropriate constitutive relations, one has a path to solving for the

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)

in the case of a causal graph [1]:

C(t)=

A reasonably satisfying look at how graphs can, indeed, lend themselves to being interpreted as distribution functions can be seen for the following hyperbolic hypergraph.

Taking geodesic balls on this distribution and normalizing by the assumed ~2-dimensional geodesic volume in Euclidean space we see the following (we do not include the code for this as it takes ~1H to run on conventional laptops).

These are not behaving as ‘perfect’ Maxwell-Boltzmann distributions, but they do share some undeniable structure with them. While it is indeed the case that the finiteness of the graph gives rise to the shape we see - this shape is being maintained for increasing time steps. Furthermore, the gradual flattening and spreading is characteristic of a Boltzmann distribution.

In[]:=

Manipulate[Plot[PDF[MaxwellDistribution[σ],x],{x,0,10},PlotRange{{0,10},{0,1}}],{σ,0.6,1}]

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,

, 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

It is worth considering the complexity of the method we are proposing for constraining the undetermined values in the higher order corrections to the EFE’s. This section is dedicated to providing an (exceptionally) brief overview of spatial Burnett equations. That is, despite their complexity, these are still not the relativistic forms of the equations - these are more complex. The overarching idea is summarised by an image from [4]:

Beginning from the (n