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In most literature I have found, research into the Alcubierre Metric, and Warp Drives seems to be highly focused on the plausibility of superluminal travel.

Today I was thinking to myself that maybe we're asking hard questions before easier ones. After all we built sub-sonic planes before we built supersonic ones.

So are there any solutions to the Alcubierre Metric that are focused on sub-luminal travel. Like in the realm of 0.1x-0.5x the speed of light?

RudyJD
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    Could such a sub-luminal drive be made without a negative energy density? – mmesser314 Mar 10 '23 at 01:27
  • @mmesser314 I'll admit I'm not exactly a physicist but this area has always interested me. However, my understanding of Alexey Bobrey's recent paper showed that negative mass/energies are not necessary. https://arxiv.org/abs/2102.06824 – RudyJD Mar 10 '23 at 02:46
  • I would think that one really does not need a warp drive to travel at subluminal velocities. How do you even define a “subluminal warp drive”? – A.V.S. Mar 13 '23 at 04:07
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    @A.V.S. A sub-luminal warp drive would just be a solution to the Einstein Field Equations, (similar to the Alcubierre Metric) where the warping of space allows for the 'riding' of a warped spacetime. But where v<c. This is a 'Class 1" warp drive in the paper I linked above. Clearly warp drives are not necessary for subluminal travel, but I don't see how that's relevant to my question. – RudyJD Mar 13 '23 at 18:19
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    the warping of space allows for the 'riding' of a warped spacetime… Use of vague, Sci-Fi language does not help, and the paper you cited is not much better in its definitions. Would you consider the metric of a rotating hollow shell written in a frame where it is moving with velocity 0.1c a “warp drive”? (“Rotating” is in order to have nontrivial frame-dragging since Bobrick&Martire seem to be big on that). – A.V.S. Mar 14 '23 at 05:07
  • @A.V.S. I'm not sure. I don't have a lot of experience with general relativity, nor with warp drives. But my point in asking this question is to understand how feasibility of warp drives changes when you deal with super-luminal vs sub-luminal travel. You are right that this is very 'sci-fi-ey" in its language, maybe thats due to my limited education. – RudyJD Mar 14 '23 at 05:39
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    A.V.S. is right - a warp drive spacetime would normally be defined as one where you can outrun a light beam in a certain sense. If you can't do that then it's unclear what counts as a warp drive and what doesn't. You can construct a subluminal "warp" geometry with no energy at all – that's the point of the principle of relativity. It doesn't generalize to the superluminal case. – benrg Mar 14 '23 at 07:35
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    Interesting... What do you mean that a warp geometry could be constructed without any energy at all? My understanding was that mass (and therefore energy?) is responsible for the warping of spacetime. – RudyJD Mar 14 '23 at 16:42
  • If we set $f(r_{s})=1;;$in the Alcubierre metric, we find the Gullstrand–Painlevé metric with $\beta=-\sqrt{\frac{2M}{r}}$ https://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates – The Tiler Mar 18 '23 at 21:57
  • with the change $;t -->t_{r};$ – The Tiler Mar 18 '23 at 22:07
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    v in Alcubierre is generic, it can be also subluminal. – Rexcirus Apr 17 '23 at 10:19

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I will expand my comment in an answer:

$v$ in Alcubierre is an arbitrary function of time, which can be interpreted as the velocity of the warp drive. In particularly it can lead to arbitrary high acceleration, resulting in arbitrary low round trip times in proper time without experiencing time dilation. The latter is a truly coordinate independent statement, which to me is the core feature of a warp. See this answer for more details: Is observing (superluminal) velocity independent of choice of coordinates for asymptotically flat spacetimes?

In any case yes, you can build a completely subluminal Alcubierre drive by always taking longer proper time than a light ray to perform a round trip, but you will still need negative energy.

Rexcirus
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