Wormholes, Warp Drives and Time Machines - Part III

[You have read the Disclaimer, haven't you?]

The assiduous reader (that's you, O Sister Mine) who has been following the theme of Wormholes, Warp Drives and Time Machines has of course read Parts I and II - everyone else may wish to catch up...

So, dear Sister, take nieces Scarlet & Em on your knee and read on... I'm sure they'll enjoy this!

The story so far: many who probably know better than I am claiming to, say that one can construct a time machine by moving one mouth of a wormhole at very high velocity relative to the other (the closer the velocity is to the speed of light the better if we want a significant time machine). Time dilation - as per Einstein's Special Theory of Relativity - will then cause a time difference to occur between the two ends, but because the two ends are linked this makes a time machine (see Part II for how this is supposed to work).

My naive objection to this is that a separation in time is in fact a separation in spacetime - and the key words here are in and spacetime. The implied rhetorical question is: can one apply things like the Lorentz transformations to spacetime itself rather than to the separations of things and events in it. To which the unequivocal (but possibly wrong) answer has to be: No!

But' it's time for a diversion. Let's get practical - another little experiment will help illustrate the concepts. For this you will need a lump of some malleable substance, such as dough (e.g. pizza dough - in which case, make plenty in anticipation of pat IV) ; some superglue, and a finger.

• Take lump of dough and make a hole in it, slightly larger than your chosen finger
  
  Look at it for a moment - you've got a torus, a thing with a hole in in it. Now, flatten and stretch it slightly to make a fat pancake - but don't close the hole (it's still a torus - they don't have to be doughnut shaped: anything with one hole in it is topologically a torus) and then imagine that the dough is spacetime.
  
  You can go from one side of the pancake to the other one of two ways: you can either avoid the hole and go to the edge and round, or you can go through the hole. The hole is our experimental wormhole because, if you happen to be closer to the hole than to the edge, the distance you need to travel to get to the "same point" on the other side is less if you go through the hole. Note however (for future reference) that whether you go around or through, if you keep going you can get back to where you started - and are still facing the same way.
  
  
• Now, apply superglue to your chosen finger in a narrow ring around the first joint
  
• Insert finger into the pre-pared hole and squeeze the dough onto the ring of superglue

You should now be superglued to the dough. And since the dough represents spacetime and the hole in the dough the wormhole, we can simulate moving the wormhole by moving our finger. Try it - gently.

As you move your finger, dough piles up against the forward edge of your finger and gets stretched everywhere else. If a real wormhole is going to move, it will do something similar - or we will have find a way to stop dough piling up in front and stretching thin behind.

The reason for saying that spacetime "piles up" or "stretches out" and that a moving wormhole has to deal with such things is that one cannot create and destroy spacetime - there's no such thing as "Cut and Paste" spacetime", no "delete" let alone "insert" (unless perhaps you are a theoretical cosmological topologist. Don't be shy in coming forward). However, one can stretch and squeeze space(time), thereby changing the separations between things, which, strangely is all that gravity is about really.

So a small digression is in order, and without reproducing and trying to explain the Einstein field equations... Oh what they hell, there are are, courtesy of Wikipedia (see Einstein Field Equations) - because it is one of the most beautiful equations in physics.


What this this says is this. The left hand side of the equation, the "G" thing, is The Einstein Tensor - it is a (nonlinear) function of the metric, which is that thing that tells us how bendy and twisty spacetime is at every point. The right hand side of the equation, the "T" thing, is the Stress-Energy Tensor, which describes the density and flux of energy due to everything at a point except gravity. So the bendiness and twistiness of space(time) is determined by the matter and energy in it, or as I believe John Wheeler so beautifully put it "Space tells matter how to move, matter tells space how to curve".

Without matter etc. spacetime would be flat, but put in some ordinary mass and it bends in just the right way to produce what we previously thought of as the mysterious force of gravity. The usual analogy for this is a bowling ball on a trampoline: the big heavy ball causes a dent towards which any peas, marbles, or other small round objects therefore move (don't overthink the analogy, it will just make your head hurt). A nice little animation from YouTube shows the Sun denting spacetime, earth circling around and the way that light also gets bent by the gravitational dent - until the sun "disappears", at which point, there being no "T" thing, all the curvature disappears...



The key point here is that without the "T" thing to keep spacetime bent and/or twisted it will flatten out, thus, returning finally to our wormhole, we need some T in the wormhole to squeeze stuff up on the leading edge and some effectively "anti-T" to stretch it out behind.

Now, I have previously drawn an analogy between this stretching and squeezing with that of the Alcubierre "warp drive", which also squeezes up the space in front (so you have less to travel across) and stretches it out behind (so that space overall doesn't get an overall pinch in it.

At that time and rather unfortunately for the eternal optimist, it occurred to me that the Alcubierre drive couldn't work, because I had assumed that the whole point of this Faster Than Light warp drive was that you could point somewhere, intone "Engage!", and then be off, and that therefore the warp bubble was somehow propagated from its initial position just like a wave. This clearly wouldn't work, I thought, because when the metric does change the effects propagate at the speed of light (speed of gravity = speed of light) and what would be the point of an "FTL" drive when you had to wait four years to sail across the four light years that had then been reduced to a mere, say, centifurlong.

And indeed it seems I was not the first to raise an eyebrow in Alcubierre's direction - but since they know better, it seems the experts never for a moment thought the metric change would be propagated... so they said it wouldn't work because you'd have to traverse the entire route in advance in order to deposit suitable amounts of T and anti-T along the way. It turns out that the Wikipedia article on the Alcubierre drive makes this point (see "Difficulties), quoting Krasnikov and Dr David Coule's work to the effect that "there's no warp drive without warp drive" (and thanks to DHC for providing a copy of his original paper).

Or, to spoil a rather nice line, if we do ever get any sort of warp drive we will only be able "To boldly go where at least one other has been before."

However, at least my intuition is not clearly leading me astray, so maybe the dénouement (when it arrives) will be worth the wait, though if I have not in fact drifted a long way off track the geodesics of Mr Stuff's idea space have clearly been severely bent bythe abnormal density and flux of... well, I suppose, Stuff, so I should probably let you digest this properly before going on...

Spacetime - Bendy, Twisty, Wavy Stuff