Does gravity "flow" through wormholes (if such can be constructed)?

Question

Let's start by assuming that a transversible wormhole can be constructed in the most "likely" way we can currently imagine... Say, we created two entangled black holes and use "negative matter" to make them transversible. Say they also have charge so you can separate them with a magnetic field and move the "mouths" around. Say you use a gas-giant worth of mass so that they don't evaporate in your face.

Now, I do completely realize some of it is very hypothetical and probably far from from being possible. But let's treat it as a thought experiment and assume all these things.

Now let's say we drive one "mouth" to an orbit around the Sun and the other to an orbit around Proxima Centauri (again, assume you got enough fuel to do that). In this situation, the Sun would "see" Proxima Centauri through the wormhole and the other way around.

My question is - would the gravitational effects of the stars flow through the wormhole (perturbing the stars as a result) the same way that light can travel through?

I'm assuming spacetime has to still be smooth so it's not clear to me how the different relative curvatures will (or are supposed to) align after the wormholes are made?

Answer 1

This post was originally supposed to be in two parts, first a static spherically symmetric proof that gravity does indeed go through wormholes, and then one for gravitational waves, but the second part sort of started to be a tad long and involved, so I will not include it for now, although it may come later.

Now then, proving the first part :

Consider a static Morris-Thorne wormhole,

\begin{equation} ds^2 = -f(l) dt + g(l) dl^2 + r^2(l) d\Omega^2 \end{equation}

This wormhole has a throat at $l = 0$ (ie, $\min_l r(l) = 0$). We will assume, furthermore, that the stress-energy tensor necessary to keep the wormhole open is compactly supported, so that outside of $l \in [-a, a]$, the stress-energy tensor is unrelated to the wormhole itself.

Now let's consider here that 1) for $l > a$, the stress-energy tensor vanishes and 2) for $l < -a$, we have some spherically symmetric static mass distribution, let's say something fairly ordinary, such as a spherical shell around the mouth or somesuch. If you're worried about specifics, I advise picking the thin-shell approximation of a wormhole connecting two copies of Minkowski space, which looks like

\begin{equation} ds^2 = -dt^2 + dl^2 + (|l| + R)^2 d\Omega^2 \end{equation}

Now what is the gravitational influence at $l > a$?

We can use Birkhoff's theorem here. The actual content of Birkhoff's theorem is fairly complex, but roughly we have that, given a vacuum spherically symmetric spacetime, the spacetime can be described by the vacuum Schwarzschild solution. Therefore, we can consider the $l > a$ part of our spacetime to have the Schwarzschild metric. The mass of this metric will be the Komar mass,

\begin{equation} M \approx \int_{l < a} R_{ab} u^a \xi^b d\mu[g] \end{equation}

Working out the integral, it's not too hard to show that this mass will be equal to the mass of the shell propping up the wormhole plus the mass of the matter outside of it,

\begin{eqnarray} M &=& M_T + M_{\odot}\\ &\approx& \int_{l \in [-a, a]} R_{ab} u^a \xi^b d\mu[g] + \int_{l < -a} R_{ab} u^a \xi^b d\mu[g] \end{eqnarray}

So yes, the gravity does propagate outside of it, although its effects will be somewhat masked by the matter of the throat itself.

Answer 2

I don't have a mathematical answer, but since no one knows this answer for sure, that is, insofar you suppose that gravity exists, that doesn't matter. Because no one knows for sure (and maybe even will never know if it indeed exists, in which case you can't know). I don't know exactly the (mathematical) solution in GR, but let me try to explain in words. I thought it was Feynman who said if you can't explain it simply (i.e. in words) you didn't understand it well enough.

In the first place, gravity doesn't flow in a static wormhole (you can say a condensate of gravitons exists, but then you assume quantum gravity). The spacetime is static. Though the question is (after reading a comment) if spacetime, in this or any case, can be static. And now I think, indeed, that it can't be static, except in highly specific cases like the Morris-Thorne wormhole with a very (almost impossible, practically seen) with a very specific spacetime- interval, as be read in the first answer.

So I think real gravitational waves can travel through the wormhole. Assuming QG doesn't exist. Look at my comment below to see some more.

By means of these gravitational waves, you can't see though what's on the other side of the mouth. This can only be seen with photons (which are quanta and how could quanta ever be able to flow through something quantized?) flowing through the wormhole (and why shouldn't they be able to travel the way through?). So you can see Proxima Centaury from the Sun (and vice-versa) though I don't think upside down. Although...I think it depends on how they both are oriented with respect to each other on both sides of the mouth.