Wormholes, Gravity & Interaction: Can a Star Pull a Planet?

[Ryan Reed]

Would objects interact via gravity through the wormhole? Let's say there's a star outside one wormhole, and a planet outside the other, would the star pull the planet through the wormhole?

 

[PeterDonis]

Would objects interact via gravity through the wormhole?

Putting a gravitating mass close to one mouth of a wormhole might make it impossible for the wormhole to remain open. The wormhole has to have a certain spacetime geometry, and the presence of a gravitating mass, heuristically, would be "trying" to make the spacetime geometry something different. I don't know that there are any known solutions of the Einstein Field Equation that cover this case; all the wormhole metrics that I've seen assume that the wormhole spacetime is asymptotically flat, which translates into English as "there aren't any gravitating masses close enough to the wormhole to affect its spacetime geometry".

 

[martinbn]

The translation into English doesn't seem right. You can have masses that affect the geometry as long as they are not too spread out, and far away the space-time can still be asymptotically flat.

 

[PeterDonis]

You can have masses that affect the geometry as long as they are not too spread out, and far away the space-time can still be asymptotically flat.

You can have other masses that affect the geometry, but then it isn't a wormhole geometry--at least not a wormhole geometry that appears in the literature. All of the wormhole solutions that have been studied assume that there is no stress-energy present except the "exotic matter" which is needed to hold the wormhole open. I should have clarified that. (Physicists often use the term "asymptotically flat" in the sloppy way I did--not just to mean what it means on its face, but in addition to mean that there is only a certain kind of stress-energy present in the non-asymptotic region.)

 

[pervect]

I'm fairly sure that "gravity" won't penetrate a wormhole. This is based mostly on some popular articles by Cramer in the science fact section of Analog on wormholes that one can find online, plus some recollections from Visser's book, "Lorentzian Wormholes", which I did read at one time but don't have handy to refer to to give an exact quote. Visser's book isn't terribly techical, if the OP can find it it might be good to order it from a library (interlibrary loan).

Anyway, the basic idea to sketch a proof would be to consider two separate asymptotically flat space-times, connected via a wormhole. Then at spatial infinity of each of in each asymptotically flat space-times, there is an ADM mass, which basically can't change as it's defined at spatial infinity. So if you move one end of the worhole around in it's own separate asymptotically flat space-time, there just isn't a way for the changes to propagate through to the other end, the continuity conditions prevent the ADM mass of the other end form changing.

It's also interesting to consider what happens when a mass passes through the wormhole, but this isn't strictly relevant to the OP's question.

[add]Perhaps it could be somewhat relevant, one can consider what hapens if a gravity wave, emitted by the changing configuration in one asymptotically flat space-time, propagates through the wormhole. Basically the total ADM mass of the wave + exit wormhole mouth doesn't change, but the distribution changes, so the exit end gets "lighter" and the gravity wave propagates normally.

Usually this effect will be negligible - gravity waves just don't carry that much energy under normal circumstances.