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This is based on a question by someone else on this site: Combining metric tensors/curvature tensors

They asked:

Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in the vicinity of the first particle, which causes a metric $h_{\mu\nu}$ on a spacetime which would have been Minkowski if not for these two particles.

Then, what would the metric in the vicinity of these two points be?

The answers given either stated that the metrics can't be added because GR is a non-linear theory, or that solutions for two particles can be found by solving the Einstein Field Equations, the latter strikes me as a bit odd.

Shouldn't there be a mathematical method for combining two spacetimes, or more generally two manifolds, that does not involve solving the Field Equations but rather relies on pure geometric arguments? How will that method depend on the type of geometry (for example, Riemannian geometry) of the manifolds?

I've tried to search throughout the internet and could barely find anything, some seemingly related topics I managed to find were the gluing construction and the pull-back/push-forward of manifolds.

Any further clarification on this subject or references to helpful sources would be greatly appreciated!

(EDIT) The Einstein Field Equations describe the relationship between spacetime curvature and matter, I don't see why this relationship should dictate how two spacetimes are to be combined.

Qmechanic
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Tachyon
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  • I can relate to your yearning for combining two spacetimes. Maybe taking both the red pill and the blue pill at the same time can get you everything everywhere all at once. – MadMax Mar 03 '23 at 18:09
  • Thanks will definitely try that – Tachyon Mar 03 '23 at 18:14
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    What makes you believe this --> "Shouldn't there be a mathematical method for combining two spacetimes, or more generally two manifolds, that does not involve solving the Field Equations but rather relies on pure geometric arguments?" I see no need for this to be true a priori. – Prahar Mar 03 '23 at 18:16
  • It it were possible to do this, it would be well-known and easily findable, because it would be very useful (such as when studying binary black hole systems). – Ghoster Mar 03 '23 at 18:24

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The answers given either stated that the metrics can't be added because GR is a non-linear theory, or that solutions for two particles can be found by solving the Einstein Field Equations, the latter strikes me as a bit odd.

Shouldn't there be a mathematical method for combining two spacetimes, or more generally two manifolds, that does not involve solving the Field Equations but rather relies on pure geometric arguments? how will that method depend on the type of geometry (for example, Riemannian geometry) of the manifolds?

It is possible to glue manifolds together, and this is sometimes done in General Relativity as well. However, this has nothing to do with the physical situation described in your linked question. When there are two particles in the same spacetime, the resulting spacetime is not a mere "combination of spacetimes". It behaves in a completely different, nonlinear manner.

One needs to solve the EFE again because the particles interact with each other and with spacetime itself. The resulting spacetime is not made of the two previous spacetimes glued together. It is actually something way more complicated.

The Einstein Field Equations describe the relationship between spacetime curvature and matter, I don't see why this relationship should dictate how two spacetimes are to be combined.

I agree. However, considering two particles on spacetime is deeply different from combining two spacetimes. One of the reasons for this is that gravity itself also gravitates (in other words, it is nonlinear—see also this post and links therein). Hence, having two particles on a spacetime leads to a different geometry due to the interaction between themselves, their gravitational fields, and so on. The problem is not a "gluing" problem, nor a geometry problem. It is a physics problem.

While it is possible to "patch" and combine manifolds, this is then an issue of mathematics that has nothing to do with the physical situation being considered here.

Níckolas Alves
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  • Thank you for your answer, There are still some things that are not quite clear to me.

    I understand that not only the gravity of both particles contributes to the spacetime curvature, but also the interaction of their gravitational fields. But say there was in fact a purely geometric way of defining the combining of two spacetimes, wouldn’t the contribution due to the interaction of the gravitational fields manifest itself as some correction term in the metric of the resulting spacetime?

     – Tachyon Mar 04 '23 at 14:49
  • Also, you agree that the EFE should not dictate how two spacetimes are to be combined, and yet you write that in order to solve this problem, one needs to solve the EFE. – Tachyon Mar 04 '23 at 14:49
  • "But say there was in fact a purely geometric way of defining the combining of two spacetimes" this is a speculative question. If there was a way, then it would somehow account for the interaction. But this is a really big "if", especially since the resulting spacetime must satisfy the EFE – Níckolas Alves Mar 04 '23 at 20:10
  • "Also, you agree that the EFE should not dictate how two spacetimes are to be combined" I do not agree. The EFE should not dictate how two manifolds are to be combined. Combining manifolds without the EFE (or some other equation of motion for the metric) in mind has no clear physical meaning. – Níckolas Alves Mar 04 '23 at 20:11
  • I can patch manifolds together in many different ways, but there is no reason to assume this will have an interesting physical meaning. Much less that this will represent two particles on the same spacetime – Níckolas Alves Mar 04 '23 at 20:12