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Let me try to break my question down by considering a hypothetical wormhole connecting the Solar system and the Alpha Centauri system:

  1. Would such a wormhole effectively decrease the distance between the Sun and Alpha Centauri? Would the light emitted by Alpha Centauri that happened to enter the wormhole take less than 4.37 years to travel to the Sun?
  2. If so, assume that in effect the distance between the stars shrank to a few light hours instead of a few light years. Given that the gravitational force is inversely proportional to the square of the distance, does it mean that the stars now experience much greater gravitational interaction?
  3. If so, is it possible that both star systems remain stable? Or would such a wormhole necessarily disrupt them? If two stars came that close to each other in normal space, they would form a binary system (or ternary in this case since Alpha Centauri is already a binary star). What would happen if they came close only because of a wormhole?
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Let's do an analogy. Imagine a very thin pancake shaped blob of water floating in space. Waves could travel along the pancake.

Now imagine a thin spherical shell of eater floating in space, small waves can travel all the way around.

Now imagine two very large very flat pancakes of water floating in space a few mm thick and a few cm from each other but each of them a few 100 km in circumference.

Now imagine you add some more water at the edges to connect the two pancakes together so in the connecting region waves can travel from one to the other because everything is still floating in space. So a wave can start on one pancake, heading towards the edge of that pancake go all the way to the edge, around the connecting part like water going around a basketball and then along the other pancake.

Wave waves travel at the speed of sound so if the pancake is large it takes a while for the wave to get from the center of one pancake to the other pancake.

Next imagine a different setup where everything looks the same except near the center of the pancakes. Near the place that used to be the pancakes it looks like a funnel on the top and an upside down funnel on the bottom, all connected up, and all made of water, and everything still floating in space.

In the new setup there isn't even a center of either pancake, there isn't water there. So a wave can't start out there. And if you start a wave on the top layer near where the center used to be and the wave heads to the edge of the former pancake then the wave has to travel at the speed of sound many many km before it gets to the edge and then to the other side and then many km again before it gets to the nearby part of the other pancake.

However the same wave if it were heading towards the former center will get to the other pancake quite quickly. If that funnel is small you'd have to aim well and if you miss you'd head to the opposite edge and take quite a while to get to the other pancake.

OK, so there is a shortcut but it only really effects those things near the shortcut, if you were far from a shortcut then an expanding wave would only have a very small portion take the shortcut. So the other pancake will see most of the waves come like the normal, no funnel setup, waiting quite a while for waves to get from one to the other. And a small portion will arrive much quicker.

The waves are similar to light traveling through the wormhole connected space, there is a shortcut but if you aren't near it you have to aim quite well to hit it and so most of your interaction ignores it. If you are nearby its like everything near it is near to each other, it doesn't really behave differently.

Now when we end the analogy there really bus something weird in the wormhole, at least if you want it to be traversable (we have to deal with curved spacetime not just curves space).

But for now let's add a funnel to the bottom and an upside down funnel to the top. Really small funnels. So the water is two flat layers a couple mm thick for many km out and they are a couple of cm apart. In the middle the water acts like it is avoiding a fireman's pole. The water is shaped like a funnel on the top and like an upside down funnel on the bottom connected together. Somewhere nearby on the top the water is shaped like an upside down funnel with a cap on it and somewhere nearby on the bottom the water is shaped like a funnel with a cap. The funnels in the middle just connect together.

Now was can still travel along the water, they just have the ability to get deflected if they pass near the two other funnels or to travel to the other side if they go towards the connected funnel.

We can noe end the analogy. If instead of water in a 3d space we had spacetime some things would be similar. Locally far from the wormhole or the stars spacetime would be pretty flat. Not totally flat like in the example, but pretty flat. So the wormhole and the stars have a pretty small effect far from the objects.

Every wave, including light, propagates through spacetime, just like that water wave had to go through the water and not the empty space in between.

So it can be deflected by going near the stars, and part of it can through the wormhole if part of it goes towards the wormhole. What is different is first there is no larger space, just a curved spacetime that acts as if it could have been in a larger space but there are actually lots of different spaces it could be in that would act the same, and it can be curved without a larger space even though that might be harder to visualize. With all those possibilities that make the sense predictions about what we see the point is to not take any of them too seriously, just focus on the predictions.

The effect of curving time is actually a primary effect not a small change to a curved space. But it still gets weaker farther from the star or the wormhole. And it basically just means that clocks and even brains work more slowly closer to those stars but since they both go more slowly to them things farther from the stars just seems to go a bit fast. And I say a bit because for star sized and star massed things none of the curvature is very large that's why it gets so very very small when you get a way from it.

OK but the wormhole is weird. And that is because it does need to focus and also defocus light and connecting up those two different kinds of curvature in time in a way that allows you to get through one and into the other is not something normal matter can do. But seeing that does require seeing the curvature of time. But it does mean that traversable wormholes require some exotic matter. The analogy might have had a funnel like any other but we want the wormhole to stay open and be traversable so it needs exotic matter.

Finally.

To your question. What about gravity. Gravity is not something that is transmitted from here to there. Back in the analogy the water was everything, the waves were changes in the water itself. In general relativity, gravity isn't a thing transmitted through space and time, it is a property of space and time.

In particular when a star forms there is a small amount of curvature already present outside a collapsing region, and what matter does is allow two different types of curvature to connect together. Imagine a large funnel and a small funnel, if you cut out the outside of one and the inside of the other you could connect them together along a seam of the same radius but the curved parts wouldn't line up the way a funnel normally does. Two different types can line up if there is matter along the seam. That is how spacetime works. So when the matter moves in to form the star the region where the two types of curvature line up moves in with it. And that curvature that was outside is a type that curves more near the center of that funnel so it is like you have a funnel with a cap that you keep replacing with a smaller and smaller cap.

So that lower surface was shaped like a funnel with a larger cap, just the cap and the part above it. Then latter a smaller cap so there was more of the funnel.

That is how spacetime gets more curved as stars form. The matter connects two different types of naturally curving regions of spacetime to connect up together. Spacetime can be curved even without matter, matter just connects two different types together.

So the spacetime was curved long ago way back when the star or wormhole formed. And spacetime evolves according to its own equation. Going back to the water wave, the eave of water evolves according to just how water itself evolves, you just had a surface that evolved the way bit evolved.

The same with spacetime it just evolves. And the kind of curvature outside a spherically symmetric non rotating star is a type of curvature that evolves into ... itself. It naturally evolves into itself. That is also how it filled itself in to the new region that was created as the star forming material collapsed.

But in this case there are electromagnetic waves in the spacetime that evolve as well ad the spacetime evolving, so there are two things.

But gravity doesn't become weaker or stronger just because of different distances. There is no gravity. There just is curved spacetime and it always evolves into its future self because that is what it does. The outside of the star isn't perfectly flat and the outside of the wormhole isn't perfectly flat but that just means things weren't totally flat back when these things formed, not that we know how to make wormholes, we do not.

The gravity is weaker farther from the stars because it was weak way out there way back when the star formed and it is the kind of curvature that evolves into more of itself. That is still basically true if other stars or wormholes are far away.

So now let's make it extreme, let's say that the stars are heading towards that wormhole, are they going to rip each other apart?

Yes. Eventually they would touch. And before that they will be close. But there isn't really a gravity of this thing acting on the other thing. The spacetime in between (including the wormhole) just evolves. And before the curvature of one starts to have a big impact on the other one it will have to impact the wormhole.

And the wormhole has exotic matter in it. So you need a theory about how your exotic matter moves in response to the evolving curvature.

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