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I was looking at the Chern-Gauss-Bonnett theorem in dimension 4. Here we can write the Euler characteristic of a compact 4-manifold as:

$$\chi(M)=\frac{1}{32\pi^{2}}\intop_{M}\left(|\mathrm{Riem}|^{2}-4|\mathrm{Ric}|^{2}+R^{2}\right)d\mu$$

Where the Riemann and Ricci tensors and Ricci scalar are respectively considered. Naturally, I'd like to apply this to spacetime. If I do this, It makes sense to say that topology is conserved in General Relativity. In other words that the variation of $\chi(M)$ vanishes under the dynamics of spacetime.

$\delta\chi(M)=0$

In this case, we might obtain an interesting “equation of motion” for spacetime dynamics that might help relate different quantities. For example, gravitational wave energy is carried by the Weyl tensor, so we can see how this changes with changes in the Ricci scalar and such.

However, I'm not entirely sure topology IS conserved. While looking for prior research I found an excellent paper (PDF) by Gibbons and Hawking that discusses topology changes to spacelike hypersurfaces, and restrictions thereof. Their conclusion is that the creation of wormholes can only be done in pairs. While I'm still digesting this paper, it appears that even then we obtain a restriction on the variation of the Euler characteristic such that it's quantized:

$$\delta\chi(M)=2n=\delta\frac{1}{32\pi^{2}}\intop_{M}\left(|\mathrm{Riem}|^{2}-4|\mathrm{Ric}|^{2}+R^{2}\right)d\mu$$

Where $n$ is an integer. In that case, we get severe limitations on the allowable structure of our curvature quantities. Can anyone weigh in here on any known utility for this??? While I haven't done the variation myself, does anyone know where this is addressed? Is my reasoning sound?

NOTE: I'm an enormous (understatement) fan of Wheeler's Geometrodynamics and his attempts to describe particles as curved spacetime, so naturally these selection rules on topology change pique my interest.

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  • $\begingroup$ Sadly I don't have the mathematical prowess to offer insight on your question, but your post makes me hopeful that you could be interested in this rough outline I've been working on, in which particles are spacetime solitons. It's how it explains the baryons that I feel is most compelling, but of course it's very half-baked, and I don't see a clear path forward to testing it. But it does invoke Chern-Gauss-Bonnet, albeit in a very loose, qualitative way at this point... $\endgroup$
    – Adam Herbst
    Commented Feb 10, 2021 at 18:13
  • $\begingroup$ I think there is a fundamental problem with your question: A compact, simply connected (which is [mostly][1] assumed in GR) manifold admits a Lorentzian metric if and only if its Euler Characteristic vanishes. See for example [this][2] mathoverflow answer for a proof. Therefore a variation of the Euler characteristic does not make sense. [1]: physics.stackexchange.com/a/12015/242169 [2]: mathoverflow.net/a/47446 $\endgroup$
    – hof_a
    Commented Feb 18, 2021 at 11:43
  • $\begingroup$ @hof_a That is true but there are ways around that such as those talked about by Gibbons here: arxiv.org/pdf/1110.0611.pdf How then is one to go about writing dynamics for topology change in spacetime if we can't take a variation? Surely there's a way? To be honest, I'm amazed that no one has mentioned that the Gauss-Bonnett term reduces to a surface term in 4D lol! $\endgroup$
    – R. Rankin
    Commented Feb 19, 2021 at 7:39
  • $\begingroup$ @R.Rankin are you talking about this connected sum idea on page 5 of the linked paper? Because this would still fix your manifold (upto diffeo.) so you still can't change the Euler characteristic. My whole problem with your question is that you can't add a Chern-Gauss-Bonnet term to your action in 3+1D gravity, because this term is trivial in this case. You can add other topological terms (Holst, Chern-Simons like for SO(1,3), ...) to your action, see for example arxiv.org/pdf/1604.07764.pdf. What do you mean with you last sentence? For CGB the manifold has to be closed afaik. $\endgroup$
    – hof_a
    Commented Feb 19, 2021 at 11:52
  • $\begingroup$ @hof_a Thank you, you're right of course. Since it has to vanish, I can't help but wonder if the term itself then can be written as the variation of another term? Thank you, I'm just trying to understand the dynamics of topology change in GR, for which general relativity isn't the biggest of help. $\endgroup$
    – R. Rankin
    Commented Feb 19, 2021 at 22:33

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