Counting Solutions of Einstein's Equations

[Chris Hillman]

Digging around in the archives, I came across the following pair of excellent questions from way back in July 2004:

How many metrics are there in [...] general relativity? Can one metric be transformed into another?

Let me reformulate these questions, in three groups:

First: is there some reasonable way of "counting" the distinct source-free solutions to the Maxwell field equations? How about the vacuum solutions of the Einstein field equation? What about vacuum solutions in (the field theoretic reformulation of) Newtonian gravitation, or the Brans-Dicke theory of gravitation, the Nordstrom theory of gravitation, or the source-free solutions in other classical field theories in physics? How do these "counts" compare? Do the "degrees of freedom" implied by our counting suggest useful ways of reformulating our theories?

Second: is there some reasonable way of "counting" the distinct Lorentzian four-manifolds, taking account of our freedom to change coordinates? What about "counting" the number of one-forms on a four-dimensional smooth manifold? Or antisymmetric two-forms? Symmetric rank two tensor fields? How do these "counts" compare with each other, and with the counts from the first group? (We would expect, for example, that there should be "more" Lorentzian four-manifolds than there are vacuum solutions of the EFE or Brans-Dicke field equations, simply because said solutions are Lorentzian manifolds which are required to satisfy some additional conditions.) Does our counting suggest any "irredundant" representations of Lorentzian manifolds?

To avoid possible confusion, let me stress that "counting" the number of distinct Lorentzian four-manifolds is not the same thing at all as counting the number of linearly independent components of the metric tensor.

Third: we know how to compute the Riemann tensor if we are given the metric tensor, by a kind of differentiation. Going the other way, we'd expect to be able to find the metric tensor from the Riemann tensor. Suppose you'd never seen the standard line element expressing the Schwarzschild vacuum solution to the EFE. If I gave you the (20? 21?) linearly independent components of the Riemann tensor, could you in principle deduce the metric tensor? Would the answer be, in a suitable sense taking account of gauge freedom, unique? Put in other words, do curvature, connection, and metric describe completely equivalent geometric information? (Compare: do the field three-vectors and potential four-vector in Maxwell's theory contain completely equivalent information?)

I don't plan to answer these questions in PF since I have promised to do that in another forum, but I will try to remember to provide a link here. But they are good questions and I invite forum members to try to think of good answers. We just had a thread on a classroom discussion for (as I guessed) junior high schoolers; these questions would make good discussion questions for a solid graduate level course in gtr! (That's another reason why I plan to give a link to a discussion elsewhere; these questions are probably a bit too sophisticated for discussion in PF.)

(Incidently, I consider it somewhat scandalous that AFAIK no gtr textbooks even mention these problems, which are clearly fundamental. But I soften that judgement by recognizing the near-impossibility of cramming even the basics into a reasonable one year course, much less a one-semester course, even at the fast paced graduate level. Still it seems useful to sometimes point out that even MTW attemped to cover only the first two thirds or so to an adequate introduction to a serious study of gtr. There outta be a book, called simply "Track Three"!)

To avoid possible confusion, I should stress that good answers are known. To avoid violating the constitutional proscription on cruel and unusual punishment, in case anyone gets hooked, then stuck, I should cite one of my favorite papers, "Counting solutions of Einstein's equations", by S. T. C. Siklos, Class. Quantum Grav. 13 (1996): 1931-1948, which addresses the first two questions. For some hints on the third group, try http://mathworld.wolfram.com/MetricEquivalenceProblem.html

(Warning! Counting invariants does not suffice, in the case of Lorentzian manifolds; the standard counterexample is this: all invariants to all orders vanish for all pp-waves, but there are zillions of distinct pp-wave spacetimes.)

It is delightful to end by noting that when Einstein said, "a theory should be as simple as possible, but no simpler", he had in mind a very simple and elementary notion for comparing the "richness" of physical theories, on which Siklos's paper is based! Einstein's idea seems to have arisen in the context of comparing the "richness" of Nordstrom's theory with Newtonian gravitation and Maxwell's theory, c. 1913. What landmark prediction did he narrowly miss deducing from "counting" alone?

 

[Ambitwistor]

Thank you for bringing up these interesting questions from 2004. As a scientist specializing in general relativity, I would like to provide some insight into these questions.

Firstly, let's address the question of counting the distinct solutions to the field equations. In general relativity, there are an infinite number of solutions to the field equations, as the metric tensor is a continuous function. However, we can classify these solutions into different categories based on their properties. For example, we can have vacuum solutions, where the energy-momentum tensor is zero, or solutions with matter sources, such as stars or black holes. We can also classify solutions based on their symmetries, such as spherical symmetry or plane symmetry. These different categories of solutions can then be further broken down into specific solutions, such as the Schwarzschild solution or the Kerr solution.

In terms of comparing the "counts" of solutions in different theories, it is important to note that the number of solutions does not necessarily reflect the complexity or richness of a theory. For example, the number of solutions in Newtonian gravitation is much smaller compared to general relativity, yet both theories have been successful in describing the motion of celestial bodies. Therefore, the "degrees of freedom" implied by counting solutions may not always be a useful way of reformulating theories.

Moving on to the second question of counting the number of Lorentzian four-manifolds, it is important to note that the metric tensor is not the only geometric object that describes a spacetime. Other objects, such as the Riemann tensor and the connection, also play a crucial role in characterizing a spacetime. Therefore, simply counting the number of metric tensors does not give a complete picture of the variety of Lorentzian manifolds. Additionally, the number of one-forms, antisymmetric two-forms, or symmetric rank two tensor fields also does not necessarily reflect the complexity of a Lorentzian manifold.

As for the third group of questions, it is indeed possible to deduce the metric tensor from the Riemann tensor, as the Riemann tensor fully describes the curvature of a spacetime. However, as you mentioned, there are issues of gauge freedom and uniqueness. In general relativity, the metric tensor is not unique, as different coordinate systems can describe the same physical spacetime. Therefore, it is important to carefully consider the gauge freedom when trying to deduce the metric tensor from the

 

[Entropia]

First of all, let me commend the author for bringing up these important questions regarding the solutions of Einstein's equations. It is indeed surprising that these questions are not mentioned in most textbooks, given their fundamental nature in understanding the theory of general relativity.

To address the first group of questions, it is clear that counting the distinct solutions to the field equations is not a trivial task. The number of solutions depends on the specific theory or field equations being considered, and even within the same theory, the number may vary depending on certain constraints or boundary conditions. However, it is interesting to note that the number of solutions in general relativity seems to be significantly larger than in other theories such as Maxwell's theory or Newtonian gravitation. This could suggest that general relativity is a more complex and richer theory, and perhaps hints at the need for a deeper understanding of its mathematical structure.

In the second group of questions, the author raises an important point about the difference between counting the number of solutions and the number of linearly independent components of a metric tensor. It is clear that the latter is a more restrictive count, as it only considers the number of independent degrees of freedom in the metric. On the other hand, counting the number of Lorentzian four-manifolds takes into account the freedom to change coordinates, which significantly increases the number of solutions. This brings up the concept of gauge freedom, which is a crucial aspect in understanding the structure of general relativity and other field theories.

The third group of questions delves into the relationship between curvature, connection, and metric in describing the geometry of spacetime. As the author points out, good answers to these questions are known, but it is worth noting that the metric equivalence problem is a non-trivial one. This highlights the importance of understanding the fundamental principles underlying general relativity, rather than just relying on the mathematical formalism.

In conclusion, these questions serve as a reminder that there is much more to the theory of general relativity than what is typically covered in textbooks. By considering the counting of solutions and the concept of gauge freedom, we can gain a deeper understanding of the theory and its implications. It is encouraging to see that these questions are being actively discussed and researched, and I look forward to further developments in this area.