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On the Wikipedia article on the Alcubierre drive, it says:

Since the ship is not moving within this bubble, but carried along as the region itself moves, conventional relativistic effects such as time dilation do not apply in the way they would in the case of a ship moving at high velocity through flat spacetime relative to other objects.

And...

Also, this method of travel does not actually involve moving faster than light in a local sense, since a light beam within the bubble would still always move faster than the ship; it is only "faster than light" in the sense that, thanks to the contraction of the space in front of it, the ship could reach its destination faster than a light beam restricted to travelling outside the warp bubble.

I'm confused about the statement "conventional relativistic effects such as time dilation do not apply".

Say Bob lives on Earth, and Jill lives on a planet in Andromeda, and we'll say for the sake of argument that they're stationary. If I were to travel from Bob to Jill using an Alcubierre drive such that the journey would take me, say, 1 week from my reference frame... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?

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Funny this should come back up now: translate.google.com/… –  Emilio Pisanty Jun 22 '12 at 9:54

3 Answers 3

up vote 9 down vote accepted

Spacetime can dynamically evolve in a way which apparently violates special relativity. A good example is how galaxies move out with a velocity v = Hd, the Hubble rule, where v = c = Hr_h at the de Sitter horizon (approximately) and the red shift is z = 1. For z > 1 galaxies are frame dragged outwards at a speed greater than light. Similarly an observer entering a black hole passes through the horizon and proceeds inwards at v > c by the frame dragging by radial Killing vectors.

The Alcubierre warp drive is a little spacetime gadget which compresses distances between points of space in a region ahead of the direction of motion and correspondingly expands the distance between points in a leeward region. If distances between points in a forwards region are compressed by a factor of 10 this serves as a “warp factor” which as I remember is $w~=~1~+~ln(c)$, so a compression of 10 is a warp factor 3.3. The effect of this compression is to reduce the effective distance traveled on a frame which is commoved with the so called warp bubble. This compression of space is given by $g_{tt}$ $=~1~-~vf(r)$.

Of course as it turns out this requires exotic matter with $T^{00}~<~0$, which makes it problematic. Universe is also a sort of warp drive, but this is not due to a violation of the weak energy condition $T^{00}~\ge~0$. Inflationary pressure is due to positive energy. The gravity field is due to the quantum vacuum, and this defines an effective stress-energy tensor $T^{ab}$ with components $T^{00}~=~const*\rho$, for $\rho$ energy density, and $T^{ij}~=~const*pu^iu^j$, for $i$ and $j$ running over spatial coordinates $u^i$ velocity and $p$ pressure density. For the de Sitter spacetime the energy density and pressure satisfies a state $p~=~w*\rho$ where $w~=~-1$. So the pressure in effect is what is stretching out space and frame dragging galaxies with it. There is no need for a negative energy density or exotic matter.

Negative energy density or negative mass fields have serious pathologies. Principally since they are due to quantum mechanics the negative eigen-energy states have no lower bound. This then means the vacuum for these fields is unstable and would descend to ever lower energy levels and produce a vast amount of quanta or radiation. I don’t believe this happens. The Alcubierre warp drive then has a serious departure between local laws of physics and global ones, which is not apparent in the universe or de Sitter spacetime. The Alcubierre warp drive is then important as a gadget, along with wormholes as related things, to understand how nature prevents closed timelike curves and related processes.

Addendum:

The question was asked about the redshift factor and the cosmological horizon. This requires a bit more than a comment post. On a stationary coordinate region of the de Sitter spacetime $g_{tt}~=~1~-~\Lambda r^2/3$. This metric term is zero for $r~=~\sqrt{3/\Lambda}$, which is the distance to the cosmological horizon.

The red shift factor can be considered as the expansion of a local volume of space, where photons that enter and leave this “box” can be thought of as a standing wave of photons. The expansion factor is then given by the scale factor for the expansion of the box $$ z~=~\frac{a(t_0)}{a(t)}~-~1 $$ The dynamics for the scale factor is given by the FLRW metric $$ \Big(\frac{\dot a}{a}\Big)^2~=~\frac{8\pi G\rho}{3} $$ for $k~=~0$. The left hand side is the Hubble factor, which is constant in space but not time. Writing the $\Lambda g_{ab}~=~8\pi GT_{ab}$ as a vacuum energy and $\rho~=~T_{00}$ we get $$ \Big(\frac{\dot a}{a}\Big)^2~=~H^2~=~\frac{\Lambda}{3} $$ the evolution of the scale factor with time is then $$ a(t)~=~\sqrt{3/\Lambda}e^{\sqrt{\Lambda /3}t}. $$ Hence the ratio is $a(t)/a(t_0)~=~ e^{\sqrt{\Lambda /3}(t-t_0)}$.The expansion is this exponential function, which is Taylor expanded to give to first order the ratio above $$ a(t)/a(t_0)~\simeq~1~+~H(t_0)(t_0-t)~=~1~+~H(t_0)(d-d_0)/c $$ which gives the Hubble rule. $z~=~a(t)/a(t_0)~-~1$. It is clear that from the general expression that $a(t)$ can grow to an arbitrarily large value, and so can $z$. On the cosmological horizon for $d~-~d_0~=~r_h~=~\sqrt{3/\Lambda}$ we have $z~=~1$.

Looking beyond the cosmological horizon $r_h~\simeq~10^{10}$ly is similar to an observer in a black hole looking outside to the exterior world outside the black hole horizon. People get confused into thinking the cosmological horizon is a black membrane similar to that on a black hole. Anything which we do observe beyond the horizon we can never send a signal to, just as a person in a black hole can see the exterior world and can never send a message out.

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Isn't $z=\infty$ the redshift of the horizon? $1+z = \frac{\lambda_{now}}{\lambda_{then}}$ –  Jerry Schirmer Apr 19 '11 at 13:18
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May I ask what you do for a living Lawrence B. Crowell? –  Mr. Manager Jun 16 at 14:21
    
So, with all this mathematically gobbilty-gook you've written (that I barely comprehend) are you saying in terms of outer space, gravity is such a weak force that it wouldn't take "much" energy for the warp drive to expand/contract space? Similar to the way cleaning vacuums create very high suction by only generating a very small amount of negative pressure? –  Mr. Manager Jun 16 at 14:26

Relativistic effects happen when you travel near light speed through space-time. With the Alcubierre drive, you don't travel through space-time, but remain stationary and the space-time around you is warped in a way that brings you closer to your destination. So if it only took a week for you, it would only take a week for your observers.

It'd be like if you put two objects on a sheet, and instead of moving them across the sheet, you scrunched up the part of the sheet between them--moving them closer together even though they both remained stationary relative to the sheet.

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2 things about your 2nd paragraph: 2nd, if you move objects around on a sheet, there's no reason time dilation would enter into the situation unless we're talking about a very large sheet, and very large speeds. 2nd, I'm not really able to visualise the "scrunching". Assuming the sheet is elastic, you could move a region of the sheet closer to a point. In that case, there would be compressed space in between, and expanded behind, but the only way the region can stay there is if the distortions stay. Only solution I can visualise would be to cut the region, and place it at the destination? –  Ozzah Apr 19 '11 at 22:51
    
its not a particularily enlightening analogy in this case.. usually sci-fi movies about wormholes try to use the "folded paper with a pen through it"-trick when they explain how to go from A to B in space by topology though :) –  BjornW Apr 20 '11 at 0:01
    
I was just excited to see a question on this site that I could at least partially answer :) –  Carson Myers Apr 21 '11 at 2:31

String theory (and all other theories involving hidden dimensions) predict that gravity and electromagnetism unify in hidden dimensions and that the hidden dimensions are indetectible because of their small size. It does also predict that sufficiently short-waved photons, with wavelengths shorter than the size of the hidden dimensions, can enter them. Producing ultra-short photons can thus manipulate gravity, with revolutionizing space travel applications such as cheap anti-gravity launches. The problem that it would require high energy can be practically solved by concentrating several laser beams on a nanoparticle, heating it to locally extreme temperatures. An Alcubierre metric can be created by ejecting multiple nanoparticles from the craft and then beam perfectly timed laser beams on them (fire at the most distant first so that they are hit simultaneously), so each nanoparticle contributes a slower than light effect but together add up to faster than light, creating no discrete event horizon and thus no Hawking radiation.

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@ Martin J Sallberg - Can you provide any citations supporting your answer? –  Richard Terrett Aug 1 '11 at 10:48

protected by Qmechanic Jan 8 '13 at 16:28

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