I'm interested in describing physical systems with algebraic equations as opposed to differential equations. In an earlier Phys.SE question I gave an example - the Simple Harmonic Oscillator can be described by the functional equation:

$$f(n) = kf(n-1) - f(n-2)$$

where $n=2,3,4,5,..$ and $k$ is a constant.

But that's a simple system, what about a far more complex system? Can the EFE (Einstein Field Equations) be written as a set of functional equations?

  • $\begingroup$ Hi. That's an interesting question. I think the SHO works because it's essentially a simple ODE. However, the field equations are tensor equations. When you "solve" the Einstein equations, you are solving for a metric tensor, of which there are potentially 10 independent components in 4-dimensions. Then, there is the additional problem that in G.R., the governing equations are not just Einstein's equations, they are also the Killing equations as well. But, your question is interesting, and will require some more thought on my part! :) $\endgroup$ – Dr. Ikjyot Singh Kohli Sep 10 '16 at 14:55
  • $\begingroup$ @Dr.IkjyotSinghKohli You say, "in G.R., the governing equations are not just Einstein's equations, they are also the Killing equations as well". I have never heard of this. Could you provide a reference? $\endgroup$ – Prof. Legolasov Sep 10 '16 at 15:04
  • $\begingroup$ @Ken Abbott: Comment to the post (v5): What definition of functional equations do you use? Are you talking about a discretization of the EFE? Or perhaps algebraic equations? $\endgroup$ – Qmechanic Sep 10 '16 at 15:07
  • $\begingroup$ Well, the Ricci tensor is built up on the metric tensor. Assuming vacuum equations $R_{\mu\nu} = 0$ and substituting the components of the metric, you could say the metric is a matrix whose components satisfy the functional equation $R_{\mu\nu} = 0$? I'm no mathematician btw and this is just idle musing on my part. $\endgroup$ – Horus Sep 10 '16 at 16:27
  • $\begingroup$ @Qmechanic - By functional equation I mean a function define implicitly i.e. in terms of itself. In general this does not imply discretization, for example.. a body falling under (newtonian) gravity is ruled by the functional equation.. f(2t) = 4f(t) where f is the distance fallen in time t. However, functions of an integer variable (discretization) are clearly a subset of functional equations. $\endgroup$ – Ken Abbott Sep 12 '16 at 13:00

First, these are not functional equations, but discretized differential equations or difference equations.

What stops you from taking Einstein's equations and plugging a difference operator everywhere where a partial derivative occurs?

If you want to preserve manifest general covariance, I suggest looking in the direction of Regge calculus. It is based on splitting of the $n$-dimensional spacetime manifold in $n$-simplices and considering its triangulation. I suppose you are most interested in $n=4$ case.

A 4-simplex also known as pentachoron is a full graph with 5 vertices. Its sides are 3-simplices aka tetrahedrons - full graphs with 4 vertices. The sides of a tetrahedron are 2-simplices aka triangles, the sides of a triangle are 1-simplices aka segments and the sides of a segment are 0-simplices aka points.

The (Euclidean) Regge action is given by

$$ S(L_s) = \sum_h A_h(L_s) \, \delta_h(L_s). $$

Here, $s$ labels the segments (1-simplices) in the triangulation. $L_s$ is the length of the segment $s$. Note that these lengths determine the geometry of the simplicial complex, just as the metric determines the geometry of the Riemannian manifold. Thus, $L_s$ are analogous to the metric field $g_{\mu \nu}$.

Also, $h$ labels hinges or $(n-2)$-simplices. In our case $n=4$ and hinges are triangles. $A_h$ is the $(n-2)$-volume of the hinge $h$ (the area of the triangle) and $\delta_h$ is called the deficite angle and is analogous to the curvature of the Riemannian manifold.

The equations of motions (discretized Einstein's equations or Regge equations) are given by variating the action $S(L_s)$ with respect to the lengths of segments:

$$ \sum_h \frac{\partial A_h}{\partial L_s} \, \delta_h(L_s) = 0. $$

The term with the derivative of the deficite angle vanishes analogously to the vanishing of the term with the covariant derivative of the Ricci tensor when dealing with the Einstein-Hilbert action.

It was proven by Regge that this action converges to the Einstein-Hilbert action in the continuum limit.

This construction admits generalization to higher $n$ (straightforward) and Lorentz signatures (which I don't recall, but am absolutely certain that it exists). It plays a major role in GR-based computer simulations.

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  • $\begingroup$ Yes, the example I gave for the SHO was a difference equation. But still.. I was wondering if the EFE have ever be written as a set of functional equations? $\endgroup$ – Ken Abbott Sep 10 '16 at 16:08
  • $\begingroup$ @KenAbbott your terminology is unclear to me. What do you mean by a functional equation? $\endgroup$ – Prof. Legolasov Sep 10 '16 at 16:24
  • $\begingroup$ @Ken Abbott. If you mean functional programming, which is basically the ability to define functions on functions eg en.wikipedia.org/wiki/Functional_programming I don't see why not. Even C++ and C# have some of it, and it's getting into everything. Mathematica also. Is this what you mean? $\endgroup$ – Bob Bee Sep 10 '16 at 19:02
  • $\begingroup$ In mathematics a functional equation is one that's defined implicitly i.e. the function is defined in terms of itself. It's this "self referential" quality that interests me. My example for the SHO is a classic case. $\endgroup$ – Ken Abbott Sep 11 '16 at 0:42
  • $\begingroup$ Trivial example.. a body falling under gravity is ruled by the functional equation.. f(2t) = 4f(t) where f is the distance fallen in time t. $\endgroup$ – Ken Abbott Sep 11 '16 at 0:50

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