Science fictiony questions about spacetime curvature

[StarkRG]

I'm writing a sci-fi story and I'd like to make it, at the very least, scientifically plausible (in the way that alcubirre warp drives are possible assuming we could get our hands on something with negative mass which, as far as we know, doesn't exist).

The basic assumption for these questions is that we have the ability to curve spacetime to arbitrary specifications at will. Presumably this is done using a machine of some kind which may or may not be turned on and off, but the how isn't important to my story since it's "cleverly" hidden behind a spacer guild/cult (the real reason is that I don't give a crap and want the reader to just accept that its possible).

The setup is this: a spaceship (which, for the sake of simplicity, is a sphere) has a double external hull and a double internal hull. For simplicity assume spacetime outside is perfectly flat. Between the two outer hulls spacetime is curved so that between the inner and outer hulls the slope of spacetime is linear (rather than actually curved) and roughly equivalent to that on the surface of Earth. Between the two inner hulls spacetime is again curved such that it is again flat on the inside of the inner sphere. If I understand the idea of curved spacetime this would mean that, for someone between the inner and outer hulls they'd feel a gravitational force towards the inner hulls, while those outside outer hulls or inside the inner would be in freefall (if this interpretation is wrong please me know as my questions are all based on this).

Now the questions.

As I understand it if the transition from flat to angled spacetime was discontinuous (pointy) it would act like a domain wall. If this is correct what would be the result of crossing it by way of a ladder? (I imagine something really bad involving some immense tidal forces)

Given that that's probably a bad idea what would the effect of passing through the transition if we made the gradient 1 cm in width? 1m? How wide would it have to be to make traversing the interface survivable? Comfortable? What would be the effect of something which lay permanently across the interface (such as power cables or support pylons keeping the hulls separated)?

If the gradient were caused by a machine located at the very centre which could be turned on and off (instead of something intrinsic to the double-walled hulls which permanently bent spacetime), what would be the effect to people and objects within fields of shutting it off immediately? Presumably spacetime would rebound to its flat equilibrium state, but how fast would this be? Would it oscillate or just return immediately to flat? If it was reduced slowly what would the effects be?

I would rather do this with an absolute bare minimum of equations, but if they're necessary I won't shy away from them (assuming I can understand them well enough to explain what they're doing).

 

[bahamagreen]

Not sure if it helps, but a similar gravitational boundary/transition happens at the shell of a hollow sphere...

If you have a hollow sphere of some thickness, all points in the interior are "weightless", but an individual standing on the surface of the sphere will be feeling a gravitational acceleration toward the center of the sphere.

If the sphere has a small hole in it and the individual walking on the outer surface sticks his hand through the hole, that portion of him inside the sphere will be "weightless", and if he passes through the hole, same thing, until he is completely inside. Likewise, once inside he will be "weightless", but any portion of himself that he sticks outside the hole will be subject to the gravitational acelleration of the sphere.

I'm just mentioning this as an example of a totally realizable localized surface on the sphere where the gravitational effect has a distinct geometric boundary to play with...

 

[StarkRG]

That's a good point, but shouldn't there be tidal forces within the boundary? Presumably the transition between curved spacetime outside the shell and flat spacetime inside would be on the same order as the width of the shell (give or take a little due to the existence of the hole).

 

[bahamagreen]

The interior of the shell of the sphere is flat space. For any position within the interior, the attraction to the mass of the shell is always perfectly balanced... the geometry of the distance of the parts of the shell from anyplace inside works out that getting nearer to one part of the shell presents a stronger pull from a closer but smaller mass balanced by the far side's weaker pull from more mass... the net result is null for all locations within.

See the Shell Theorem at wiki and go to 2. Inside a Shell

The transition width, I think, is 0, at the outer most boundary of the shell, because the thickness of the shell may be considered a series of layered shells, and each is yet subject to the shell theorem. That is, all points on any of the inner shell layers comprising a thick shell are interior to the outermost shell boundary.

 

[StarkRG]

As far as I can see that entry doesn't deal with the transition between curved spacetime outside and the flat spacetime inside. I'm assuming there would be some kind of tidal forces felt there, but how would they manifest? What would it feel like to climb through it?

Would the effects be the same if thou passed through the other interface?

 

[StarkRG]

Also, since there is a time aspect to the curvature how would straddling two reference frames (curved and flat spacetime) affect your body?

 

[bahamagreen]

You'll have to see if some GR folks want to address that...

 

[PeterDonis]

Between the two outer hulls spacetime is curved so that between the inner and outer hulls the slope of spacetime is linear (rather than actually curved) and roughly equivalent to that on the surface of Earth.

A better term for what I think you mean by "slope of spacetime" is "gradient of the gravitational potential". See further comments below.

Between the two inner hulls spacetime is again curved such that it is again flat on the inside of the inner sphere. If I understand the idea of curved spacetime this would mean that, for someone between the inner and outer hulls they'd feel a gravitational force towards the inner hulls, while those outside outer hulls or inside the inner would be in freefall

The inner hull part of this is easy (well, "easy" conceptually--not easy in practical terms, as we'll see in a moment). The inner hull just needs to be a thin spherical shell with sufficient mass to produce the same gravitational acceleration as the Earth does, given its radius. Just to give some idea of the numbers, if the inner hull has a radius of 1000 meters, its mass needs to be about 2 x 10^17 kg, or about 200 trillion metric tons. A person can then stand on the outer surface of the inner hull just like they would on Earth, and anyone inside the inner hull would be in free fall.

The outer hull part is more difficult; basically you want the outer hull to "shield" the surrounding space from the strong gravity created by the inner hull. The only way to do that is with what's called "exotic matter", which is generally believed not to be physically possible, because it will have a negative energy density as seen by at least some observers. The exotic matter basically creates a sort of "antigravity" that counteracts the inner hull's gravity.

This configuration would pretty much match what you are looking for: the gradient of the gravitational potential would be zero inside the inner hull and outside the outer hull, and would be nonzero between the hulls. The gradient would not be exactly linear between the hulls, but if the distance between the two hulls were small compared to the radius of the inner hull, the deviation from linearity would not be too great. (There's no way to get a perfectly linear gradient with the kind of configuration you're looking for; you would need an infinitely large flat plate, and it would create a linear field pointing towards itself on both sides of the plate.)

I'll comment on your further questions in a follow-up post.

 

[StarkRG]

Yes, "gradient of the gravitational potential" sounds like a reasonable term.

A hollow sphere is a reasonable approximation of what I want it to look like inside of the other hull, though, of course, I would like it to be a relatively manageable size. A planetary mass spacecraft is not particularly useful, you might as well just move a planet around.

As for the linear aspect, I'm not too fussed, as long as it feels like gravity without any apparent tidal forces around the surface. Tidal forces within the boundaries are acceptable long as they're not too uncomfortable to cross.

I'm really not concerned at all with HOW they can accomplish what I want, maybe it's exotic matter, maybe it's anti-gravitons, only the elders of the spacers guild know for sure and they're definitely not telling anyone (or, perhaps, the knowledge has been lost to time, either way, is something I have explain). However what I am concerned with is what effects would be observed if such a configuration were possible.

 

[PeterDonis]

Tidal forces within the boundaries are acceptable long as they're not too uncomfortable to cross.

This is the part I'm not quite sure about. For purposes of sci-fi you could probably finesse it as being a perceptible boundary, possibly uncomfortable to cross, but manageable. However, I'm not sure that would actually be the case, although I *think* it's close enough for sci-fi. I would have to look at the math in more detail to be sure.

 

[bahamagreen]

In the example of the man standing by the hole on the surface of the big hollow sphere, I don't see any discomfort passing through the hole, although he would feel it.

Passing into the hole, I imagine it would be like stepping into a swimming pool where the buoyancy makes the body parts below the water surface feel light while the head and arms still feel heavy; and likewise passing out the hole would be like getting out of the pool and feeling the heaviness again coming out of the water.

So passing into the interior of the sphere might just feel like moving into "completely buoyant" liquid without any hydrostatic pressure, sort of...?

 

[StarkRG]

This is the part I'm not quite sure about. For purposes of sci-fi you could probably finesse it as being a perceptible boundary, possibly uncomfortable to cross, but manageable. However, I'm not sure that would actually be the case, although I *think* it's close enough for sci-fi. I would have to look at the math in more detail to be sure.

Wouldn't it be highly dependant on the width of the boundary? The smaller the width the more drastic the curvature, and, thus, the greater the tidal force? We don't feel tidal force on (or near) Earth because the curvature is extremely slight (10% between the surface and the orbit of ISS), however the moon does feel it because it's competitively large, the gravitational force on the fat side is sufficiently smaller than the force on the near side.

I suppose it's possible that the difference between 0g and Earth's surface g is so slight as to make the transition over macroscopic scales negligible but it would be nice to know just how small I can make it before the effects are too intense.

I would also like to find out how spacetime would respond to the sudden disappearance of whatever it was that was causing it to be curved (the machine is turned off). Presumably it would rebound, but would it rebound like water or other physical substances, oscillating around the equilibrium? If this were the case would everything on the surface of the inner sphere suddenly be thrown outward (due to a positive curvature)? Or would the oscillations be so fast (speed of light, strong inclination to return to equilibrium) that they'd be all but imperceptible? Or, perhaps, instead would spacetime just return almost immediately to equilibrium?