Hyperspace and gravity

[Wyrm]

Heres the outline of how it might work.

Let us suppose that there is some quantity "stress" that the ship can only withstand a certain amount of, and this is what limits the closest approach to a star.

Now let us use the changing potential/force theory, and try and define this mysterious "stress" quantity as the rate of change of the (magnitude of the) gravitational force on the ship.
So Stress = dF/dt

I have a problem with this derivation already. Things break because the forces on an object vary with space, not time. A uniform force in space will just cause an acceleration, even if the force is nonconstant in time. A nonuniform force will cause stresses and failure, because in order for the object to keep a rigid shape, internal forces must even the forces out, as it were. In order for a bar to keep the same shape being pulled from one end, the bar must exert forces on its other end such that the average acceleration along the bar is the same — otherwise, one part of the bar goes at a different speed from other parts, and the bar flies apart.

Now, this is not to say that stress could not be an indirect function of time. It can. But when you start out with the basics wrong, no guarantee can be made that it will work.

As such, there is already a first-order approximation for tital forces:

F = 2GMm/R² ∆r/R

That is, the force on a ship at a particular point is 2∆r/R times the weight of the object in that gravity field. Since most objects are small compared to typical R's, this is usually insignificant, and the above equation good enough.

However, this is not true for a ship in hyperspace, and I am about to do something you should never, ever do at home: I am going to combine classical physics with relativistic physics, in a regime where relativistic physics produces silly answers without nonkosher modifications.

Since we are going very very fast, somewhere around 10 millon c, let's take a look at what would happen to the relative length of the ship. We're going to make the replacement γ → γi for v > c, so length "contraction" becomes

∆x' = ∆x/γi

At typical hyperspatial speeds of millions of c, this translates to a length expansion of a factor of ten million, and if we take a typical ship is on the order of a few kilometers, the above calculation yields lengths greater (though not that much greater) than a planetary radius, and forces on the order of the ship's own weight.

Of course, the tidal forces calculation I stated above is only good when ∆r/R is small, and ∆r/R here is not small. Adding in the other terms, it's clear that the maximum force on the really stretched out ship is on the order of its own weight in that gravity. This isn't a concern around planets, but around objects with significant gravity (such as stars), this could get significant.

[Ender]

Um-there's exactly ZERO evidence for the spiralling in on the destination and ALL of available evidence shows them straight-lining for the target.

And what evidence would that be?

None, as it turns out. While there STILL is no evidence for them spiralling in (and I completely fail to see why they should need to do that) neither the movies nor the EU as far as I can tell explicitly say they DON'T.

As we have repeatedly said in this thread, the traits of hyperdrives, much less the operations of it, are at no point defined. So since there is zero evidence period, your best option is to start from real science and work from there. Which is what we are attempting.


I recently learned something that I feel may be relevant - there is such a thing as hypernumbers, or as they are more commonly known hypercomplex numbers. I'm not real clear at all what these are - as near as I can tell it is for describing complex numbers on a coordinate plane. However, I think that using these in special relativity and general relativity will yield relevant answers to questions about how we could expect hyperdrives to behave. However I don't have the math education to do that - I'd be swinging in the dark. However, if one of our more math inclined people would like to dumb this down for my level (basic calculus) or to poke at it, I think we could get a few hints. I would also be rather obliged.

[Dooey Jo]

First, of course you can use the rubber sheet analogy in GR.

You can use it, but I don't think it gives a good view of what's going on beyond "look, some kind of curvature", and is much better suited to visualising gravitational potential energy of gravity wells (and so far, GR hasn't been used to explain anything in this thread; trajectories are much more conveniently explained by Newtonian mechanics here).

Now, this is not to say that stress could not be an indirect function of time. It can. But when you start out with the basics wrong, no guarantee can be made that it will work.

I suppose if you really want something like that, you could say that it has nothing to do with mechanical stress, and could instead be a consequence of the feedback system of inertial dampeners, which might have some limitation for how fast it can adapt to changes in acceleration. Maybe you could even get an approximation for this property by measuring how much shaking is going on inside the Millennium Falcon during the asteroid chase in ESB. On the other hand, such changes due to gravity would probably be small, even at lightspeed, or the acceleration gradient across the ship could become more significant than the average rate of change (I didn't check anything; no time for such maths right now). Also, if we do want to use GR for this, the idea might not be easily applicable to gravity at all, because gravity isn't a force in GR.


By the way, didn't Luke disengage some hyperdrive safeties in Heir to the Empire, to make a short jump to escape from an Interdictor?

[Mad]

By the way, didn't Luke disengage some hyperdrive safeties in Heir to the Empire, to make a short jump to escape from an Interdictor?

No. He used a trick with the inertial compensator to get free from a tractor beam long enough to get out of the Interdictor's projected mass shadow cone. The trick with the inertial compensator ended up damaging the hyperdrive just enough so that it failed 10 minutes into the hyperspace jump.

[Steel]

Heres the outline of how it might work.

Let us suppose that there is some quantity "stress" that the ship can only withstand a certain amount of, and this is what limits the closest approach to a star.

Now let us use the changing potential/force theory, and try and define this mysterious "stress" quantity as the rate of change of the (magnitude of the) gravitational force on the ship.
So Stress = dF/dt

I have a problem with this derivation already. Things break because the forces on an object vary with space, not time. A uniform force in space will just cause an acceleration, even if the force is nonconstant in time. A nonuniform force will cause stresses and failure, because in order for the object to keep a rigid shape, internal forces must even the forces out, as it were. In order for a bar to keep the same shape being pulled from one end, the bar must exert forces on its other end such that the average acceleration along the bar is the same — otherwise, one part of the bar goes at a different speed from other parts, and the bar flies apart.

Now, this is not to say that stress could not be an indirect function of time. It can. But when you start out with the basics wrong, no guarantee can be made that it will work.

I wasn't setting out to say here's something that is going to be "the answer", I was more going for the angle that there can be a logically consistent reasoning that explains why ships have trouble with stars, but are able to approach planets from hyperspace. I used the inverted commas around "stress" to show I wasn't talking about actual mechanical stress but rather some other quantity. There really isn't enough to go off to set about making a proper consistent theory for hyperspace travel, so just showing there are logically consistent systems that could describe this is just about all we can do. I could have used sqrt(|F|) instead of |F| and it would have worked better, but been a little bit more boring to chug through.

First, of course you can use the rubber sheet analogy in GR.

You can use it, but I don't think it gives a good view of what's going on beyond "look, some kind of curvature", and is much better suited to visualising gravitational potential energy of gravity wells.

I disagree. Putting some kind of object on a rubber sheet never produces an infinitely deep hole. In Newtonian potentials, all objects have their |grav potential| go to infinity as r->0, wheras in GR the curvature is not singular except in the case of special objects like black holes.

Now, this is not to say that stress could not be an indirect function of time. It can. But when you start out with the basics wrong, no guarantee can be made that it will work.

I suppose if you really want something like that, you could say that it has nothing to do with mechanical stress, and could instead be a consequence of *insert rationalisation of choice*

Bingo!

[Wyrm]

I wasn't setting out to say here's something that is going to be "the answer", I was more going for the angle that there can be a logically consistent reasoning that explains why ships have trouble with stars, but are able to approach planets from hyperspace. I used the inverted commas around "stress" to show I wasn't talking about actual mechanical stress but rather some other quantity.

Fine. But why is the quantity important? That is, why is exceeding a certain amount of this quantity such a concern that SW engineers build their hyperdrives with a cutout?

[Steel]

I wasn't setting out to say here's something that is going to be "the answer", I was more going for the angle that there can be a logically consistent reasoning that explains why ships have trouble with stars, but are able to approach planets from hyperspace. I used the inverted commas around "stress" to show I wasn't talking about actual mechanical stress but rather some other quantity.

Fine. But why is the quantity important? That is, why is exceeding a certain amount of this quantity such a concern that SW engineers build their hyperdrives with a cutout?

Well this quantity could be totally meaningless and unimportant, I initially just took something that would produce roughly the right behaviour, however if you are now asking me to retroactively justify my fudgy curve fitting example, I'll have a go

[begin hand waving bullshit]
This quantity is very important, as if a ship exceeds a certain level of "hyperstress" then it will be pulled apart and destroyed. Thus engineers build ships hyperdrives to cut out before this level is reached to avoid instant destruction.

We know that tidal forces are proportional to |local grav field strength|*(delta r)/R, and our hyperstress quantity is proportional to local gravitational field strength. Now because the ships tenser(-or?) fields (or whatever their structural integrity doodads they have been stated to have are called) can only react at a certain rate, if the hyperstress (I don't like making up technobabble...) is too large then the ship is destroyed by tidal forces.

So here the justification is that the ships active magic structural integrity field can only react at a certain rate and so you cant have the tidal forces change too quickly
[/end hand waving bullshit]

So had I started from the supposition "The thing that limits hyperdrive speeds is an aspect of active structural integrity technology." Then followed the chain of reasoning "What could this technology need to protect against? Perhaps tidal forces? Maybe it can only react so fast, and therefore the quantity of interest is the change in magnitude of the tidal forces over time, lets look at a bit of that." Would you have been happy?

All I was trying to do in my first post was show that there could be logically consistent ways to have the current apparent constraints on hyperspace travel, rather than just saying "lol writers fiat its impossible to reconcile". Given that nobody actually has knowledge of the innermost workings of hyperdrive a first principles approach was not terribly feasible, and I didn't want to get drawn into a big discussion about what the limiting factor really is when nobody can conclusively show what it is. Hence the curve fitting approach.

[Ender]

The cutout is key because if you hit a bardyonic object while tachyonic you release your equivalent mass-energy as you are accelerated to a infinite speed, becoming just a wave. When a billion tons of starship does this it tends to do things like break planets into little bits. This is usually considered to be a very bad thing.

We don't know much about hyperdrives, but we do know why the cut outs are important.

[Mad]

The cutout is key because if you hit a bardyonic object while tachyonic you release your equivalent mass-energy as you are accelerated to a infinite speed, becoming just a wave. When a billion tons of starship does this it tends to do things like break planets into little bits. This is usually considered to be a very bad thing.

We don't know much about hyperdrives, but we do know why the cut outs are important.

As a tachyon gains velocity, it loses energy and momentum. Is a tachyonic object going 10 million c really going to have that much momentum?

[Kuroneko]

Locally, the general expression measuring the Newtonian tidal force due to a potential V is -V_{,j,k} = -∂²V/∂x^j∂x^k when in rectangular coordinates. To see this, take the equation of motion d²x/dt = -∇V(x) for a test particle at x and another at y = x+z, separation z small. Expanding the latter in a Taylor series
[1] ∂V(y^i)/∂x^j = ∂V(x^i)/∂x^j + ∂_k[ ∂V(x^i)/∂x^j ] z^k + ...
and subtacting gives
[2] d²z^i/dt² = -δ^{ij} [ ∂²V/∂x^j∂x^k ] z^k.
For the spherically symmetric case V = -GM/r, the radial tidal force in the radial direction is proportional to -V_{,r,r} = 2GM/r³, since any particular radial direction can be declared to be one of the rectangular axes.

However, this is not true for a ship in hyperspace, and I am about to do something you should never, ever do at home: I am going to combine classical physics with relativistic physics, in a regime where relativistic physics produces silly answers without nonkosher modifications.

Not to worry--we can turn this into a dubious use of relativity rather than dubious use of both relativity and Newtonian physics. In the general-relativistic regime, pretty much the same thing happens when geodesic deviation is considered, with V_{,j,k} corresponding to the Riemann curvature tensor components R_{j0k0}. In the simplest nontrivial case of Schwarzschild spacetime, it is formally identical to the Newtonian case:
[3] (z^r)" = 2GM/r³ z^r, (z^θ)" = -GM/r³ z^θ, (z^φ)" = -GM/r³ z^φ,
in the orthonormal basis, except with differentiation with respect to proper time rather than coordinate time. Physically, this represents radial stretching and inward side squeezing.

Since we are going very very fast, somewhere around 10 millon c, let's take a look at what would happen to the relative length of the ship. We're going to make the replacement γ → γi for v > c, so length "contraction" becomes ∆x' = ∆x/γi

If we suppose that the superluminal ship has a physically meaningful reference frame in the first place, then we can vigorously handwave the transformation to be v→c²/v instead. A particle at the origin with velocity v has a wordline x = vt, which defines its time axis (i.e., duration as measured by the particle is the length of the projection onto this line); the line x = (c²/v)t is orthogonal to it and defines its spatial axis (in the same sense). From this point of view, as |v|→c, the axes get rotated "closer" to x = ct, and at |v| = c, they meet; this makes it natural to continue the rotation for |v|>c, with space and time switching their normal roles. Thus, there will be time contraction and length dilation, but for |v|≫c, γ≅1, making them irrelevant.

The transformation v→c²/v is interesting because it is preserves both Maxwell's equations and the relativistic velocity composition, i.e., u⊕v = (u+v)/(1+uv/c²) = (c²/u)⊕(c²/v), thus having a natural symmetry. However, whether it's more appropriate than γ→γi is anyone's guess; more likely, neither of them are, since hyperspace is implied to either modify the structure of spacetime or somehow be partially independent of it (via another spacetime?) altogether.

I recently learned something that I feel may be relevant - there is such a thing as hypernumbers, or as they are more commonly known hypercomplex numbers. I'm not real clear at all what these are - as near as I can tell it is for describing complex numbers on a coordinate plane.

Not quite--the identification of complex numbers with the coordinate plane is actually their canonical construction (even before it, Renaissance mathematicians suspected as much with cryptic planar constructions, interpreting i as the "sign of perpendicularity" (nowadays, it's well-known that multiplying by i rotates counterclockwise by 90°)). That construction generalizes the same. The next obvious step, quaternions, is actually seen frequently in physics, at least in the sense that a lot of things can be succinctly expressed in their terms. For example, the electromagnetic field can be seen as a combination of electric field E and magnetic field B. In the absence of monopoles (∇·B = 0), they can be written decomposed into electric (scalar) and magnetic (vector) potentials as {E = -∇φ - ∂A/∂t, B = ∇×A}. Thus, the "electromagnetic field" can be viewed as the scalar-and-vector quantity (φ,A), which is essentially what a quaternion is. Historically, quaternions came before vectors.

As for hypercomplex numbers in relativity, there are twistors. They have some formal advantages over the traditional tensor one in some contexts because they take the -+++ spacetime signature [time and space have opposite sign] and "balance" it by identifying spacetime events with subspaces of complex space of --++ signature.

However, I think that using these in special relativity and general relativity will yield relevant answers to questions about how we could expect hyperdrives to behave.

I don't yet see how, but are there any canonical (even vague) descriptions of hyperdrive operations?

[Ender]

The cutout is key because if you hit a bardyonic object while tachyonic you release your equivalent mass-energy as you are accelerated to a infinite speed, becoming just a wave. When a billion tons of starship does this it tends to do things like break planets into little bits. This is usually considered to be a very bad thing.

We don't know much about hyperdrives, but we do know why the cut outs are important.

As a tachyon gains velocity, it loses energy and momentum. Is a tachyonic object going 10 million c really going to have that much momentum?

Momentum? I'm not sure. Energy? Yeah. That's just about enough energy to blow off the earth's crust. For something bigger, like a battlecruiser the released energy will be enough to at least shatter a world, even if it doesn't mass-scatter it. Even something as small as a starfighter would hit with somewhere in the range of half a teraton. That will mess up the biosphere for a few generations at least, necessitating outside aid to restore it in a reasonable timeframe. This of course also assumes that it hits while still tachyonic. If the cutouts didn't activate and you left hyperspace too close to the planet you would still be decanting at relativistic velocities when the ship hit. This could conceivably be far worse.

Kuroneko, I have no doubt that, as usual, your post was dead on accurate in explaining the situation. However I didn't understand it - looks like I will be hitting the books harder in my of time when school resumes. As to the operations of hyperdrive, the AOTC ICS tech entries makes mention of hypermatter being complex matter and necessary to uphold conservation of mass and energy (and I presume, momentum). It is used as a ballast in some kind of bound-state with the rest of the ship so the whole thing flies. I figured this was just technobabble until I recently became aware of papers dealing with hypernumbers and special relativity, which leads me to believe that it is less technobabble than re-appropriating a mental exercise for sci-fi purposes. So I am chipping at it to try and break it down to a more abstract concept than its current state of "something that makes me press the 'I Believe' button"

[Ender]

here is a HTML version of one of those papers I was talking about. WAY out of my league, my general plan was go back and talk to people in Math Club and the physics profs I'm in good with to try and get an explanation.

[Kuroneko]

I'd recommend backing off that paper for quite a while. If you're interested in working with quaternions and octonions, there's a nice book by Conway (On Quaternions and Octonions) you could use (although I've never made that far into it); for a quicker introduction, look for John Baez's website (I'm sure you'll find it quickly enough on google). But whatever you do, I hope you're not doing just for Star Wars analysis.

If you're interested in the constructions and how they connect to spacetime, it's actually not all that complicated because it's really just the same trick repeated over an over again. Let's take a pedestrian approach and just say we got some set R with some sort of arithmetic of sufficient complexity defined on it and an operation *:R→R. In fact, to start with, let's make it even simpler and assume that * does absolutely nothing at all (x* = x for all x). Pick some number α in your set R.

The Cayley-Dickson construction takes the set of ordered pairs R² = { (a,b): a,b in R } and defines new arithmetic on it:
[1] (a,b)+(c,d) = (a+c,b+d)
[2] (a,b)·(c,d) = (ac+αdb*,a*d+cb)
Each x in R is naturally identified with (x,0) in R².

Suppose that R is the set of natural numbers {0,1,2,...} and α = 1. What happens? There's an interesting element (0,1), which has (0,1)·(0,1) = (α,0) = 1. In the naturals, there was only one square root of 1 (itself), but we've just constructed another. Let's call it -1, and write
(a,b) = a(1,0) + b(0,1) = a + b(-1).
That almost looks useful; if we define equality not by (a,b) = (c,d) iff a=c and b=d, but rather by a+d=b+c, then we've just constructed the integers {...,-2,-1,0,1,2,...} from the naturals, with the pair (a,b) identified with the integer a-b. (1)

Now, let's assume that R has additive inverses, so that the following definition of a new operation * makes sense:
[3] (a,b)* = (a*,-b)
Let R be the set of real numbers and α = -1. Then, as before, (0,1)·(0,1) = (α,0) = -1, and so we've just constructed a square root of -1. Calling it i and writing
(a,b) = a(1,0) + b(0,1) = a+bi,
we have the complex numbers, in which * is the usual complex conjugate, (a,b)* = (a,-b).

If on R, * is also an involution, i.e., a self-reversing operation: (x*)* = x for all x, then we can define
[4] |(a,b)|² = (a,b)*·(a,b) = (a,b)·(a,b)* = aa* - αbb* = |a|²-α|b|².
In the case of the complex numbers, this is just the modulus, |a+bi|² = a²+b². And here's where it gets interesting.
[5] For complex x,y, let <x,y> = Re[ x*y ]
Where "Re" just picks out the first component, i.e., Re(a,b) = a. Hence <(a,b),(c,d)> = ac+bd, which is exactly the Euclidean inner product of vectors [a;b] and [c;d]. We'll call this a ++ signature.

The split-complex numbers are constructed from the reals using α = 1, introducing a new element j = (0,1) with j² = (α,0) = 1. The corresponding modulus will be |(a,b)|² = a²-b² and inner product <(a,b),(c,d)> = ac-bd, a +- signature.

If you start with the complex numbers instead and take α = -1, you get a set of ordered pairs {(w,z)} in which w and z are both complex, which you can expand back into real components, treating them as ordered quadruples {(a,b,c,d)} of real numbers instead. The modulus will be |(a,b,c,d)|² = a²+b²+c²+d² with a similar inner product matching the four-dimensional Euclidean inner product, with ++++ signature.

Doing it from quaternions with α = +1 gives something isomorphic to the "hyperbolic octonions" in the paper you've linked, with an inner product having signature ++++----. That's more than necessary, since spacetime has signature +--- (or -+++, depending on convention--the only important part is that time and space dimensions have opposite sign), which is why Lemma 2 picks out a subalgebra.

The significance of the Minkowski inner product is if two observers have four-velocities u = [t;x;y;z] and v = [t';x';y';z'], then <u,v> = tt' - xx' - yy' - zz' is the Lorentz gamma corresponding to their relative velocity. It is also the generalization of the Euclidean <u,v> = |u||v|cos θ, except now the cosine is the hyperbolic (γ = cosh θ). Note: four-velocities have unit norms by definition, |u| = |v| = 1 in units of c = 1.

(1) Defining equality in this manner is completely optional. Usually, R is taken to be an algebra (which the naturals are not), and this special step is avoided. For example, α = 0 gives an idempotent element ε = (0,1) with ε² = (α,0) = 0.

[Dooey Jo]

You can use it, but I don't think it gives a good view of what's going on beyond "look, some kind of curvature", and is much better suited to visualising gravitational potential energy of gravity wells.

I disagree. Putting some kind of object on a rubber sheet never produces an infinitely deep hole. In Newtonian potentials, all objects have their |grav potential| go to infinity as r->0, wheras in GR the curvature is not singular except in the case of special objects like black holes.

That's just for point masses, though. The potential inside a solid sphere would go to some constant as r goes to 0, and "smooth out" much like the rubber sheet does. But anyway, my main objection is that it can easily give the impression that it's the space-part of the curved space-time that's causing things to move (because that's how it looks), when it's actually the time-coordinate that's most important under normal circumstances. You could say the curvature represents gravitational time dilation, but that could be confusing (and proportional to the gravitational potential anyhow).