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

Posted 4 months ago

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 R μν 1 2 g μν g μν T μν ( 1 )where R μν R Λ T μν
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}],BoxedFalse],ParametricPlot3D[Evaluate[sphere[1][u[t],v[t]]/.sg],{t,0,π},BoxedFalse]}] Which produces: 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},BoxedFalse,AxesFalse],ParametricPlot3D[Evaluate[torus[3,1,1][u,v]],{u,0,2π},{v,0,2π},PlotStyleOpacity[.7],MeshFalse]] O( 6 r M n r>0 exp m r M m V m exp m τ(R)= n ∑ i=1 R ii 2 || n ∑ i,j,k,l=1 2 R ijkl 2 || n ∑ i,j=0 2 R ij 2 ∇ n ∑ i=1 2 ∇ ii α n n Γ 1 2 -1 Γ n 2 the higher order corrections to the volume of an n V m α n n r n τ(R) 6(n+2) 2 r 1 360(n+2)(n+4) 2 || 2 || 2 τ(R) 4 r 6 r ( 2 )As an aside, in n V m α n n r n r A B
Derivation of the Form of the Higher Order Corrections Consider an n-dimensional analytic Riemannian manifold, M r>0 exp m r M m r V m V E τ(R) 6(n+2) 2 r 1 360(n+2)(n+4) 2 || 2 || 2 τ(R) 4 r 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 pq δ pq 1 3 R ipjq i x j x 1 6 ∇ i R jpkq i x j x k x 1 120 2 ∇ ij R kplq 16 3 R ipjs R kqls i x j x k x l x 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= ab g R ab R ab c R 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 g 1 6 R ij x i x j 1 12 ∇ i R jk x i x j x k 1 24 3 5 2 ∇ ij R kl 1 3 R ij R kl 2 15 R iajb R kalb x i x j x k x l 5 || dμ E ( 5 )which leads to [2]: det( g ij 1 3 R ij x i x j 1 6 ∇ i R jk x i x j x k 1 24 6 5 2 ∇ ij R kl 4 3 R ij R kl 4 15 R iajb R kalb x i x j x k x l 5 || ( 6 )Now, introducing the conventional Einstein-Hilbert action [3]: S EH 1 16πG -g 4 1 16πG - 1 3 R ij x i x j 1 6 ∇ i R jk x i x j x k 1 24 6 5 2 ∇ ij R kl 4 3 R ij R kl 4 15 R iajb R kalb x i x j x k x l 5 || 4 ( 7 )where g=det( g ab G c S EH 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 gg+δg δg0 ||x||0 δR -g =-μκ g λν g g κλ R μν -g +1 2 -g μν g g μν ( 8 )such that δ S EH 1 16π μν R 1 2 μν g g μν -g 4 d ( 9 )Requiring that the variation above be zero for any variation δg G μν R μν 1 2 g μν ( 10 )where G μν g μν G μν T μν ( 11 )where T μν R μν 1 2 g μν g μν T μν ( 12 )Finally, replacing g μν O||x 4 || R μν 1 2 δ μν 1 3 R iμjν x i x j 1 6 ∇ i R jμkν x i x j x k 1 120 2 ∇ ij R kμlν 16 3 R iμjs R kνls x i x j x k x l T μν ( 13 )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 T μ p ν x
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"] HG=WM[{rules},Automatic,2500,"FinalState"];GG=HG2G[HG];m=Median@VertexList[GG];HighlightGraph[GG,NeighborhoodGraph[GG,m,#]]&/@Range[5,20,5] 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 th n (0) f C(t)= n at 1 6 R jk j t k 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. Manipulate[Plot[PDF[MaxwellDistribution[σ],x],{x,0,10},PlotRange{{0,10},{0,1}}],{σ,0.6,1}] In[]:=
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 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]: ∂f ∂t c k ∂f ∂ x k F k ∂f ∂ v |