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Alcubierre / GR Warp drive and Entering/Exiting a Light Cone

  1. Jun 8, 2014 #1
    So, this has really stirred my interest. To be clear, I'm not citing these as sources, simply linking for discussion;
    An article on the subject,
    And the abstract.

    Talking about this elsewhere I seem to find no shortage of objections. But to me it seems fundamentally pretty sound.

    One thing in particular has me asking, though;

    One of the objections is that it could violate causality by allowing for even apparent FTL travel. I'm not so sure.

    Special Relativity shows us how FTL = time travel, true, but that's only in the context of flat spacetime. The thing is, though, with this "warp drive," although you achieve apparent FTL, it comes off to me as a way to expand your light cone, not break out of it.

    But that's the thing - can you expand your light cone and reach areas that were previously inaccessible? It seems to be a big claim, but it just seems to be a natural result of inflationary cosmology.

    As I understand it, Inflationary Cosmology states that the universe didn't start off expanding, but in the earliest moments its rate of expansion rapidly shot up, then came down to a more reasonable level, until large volumes of empty space caused it to increase again.

    If the universe expands at rate [itex]H[/itex] per distance [itex]D[/itex], then your cosmic horizon will simply be

    [itex] D_{Horizon} = \frac{C}{H} [/itex]

    If [itex]H[/itex] decreases, then, the distance [itex]D_{Horizon}[/itex] will increase, thus your "light cone," with the expanding universe taken into account, will include regions of spacetime that were previously inaccessible to it. That is, your light cone now includes regions of spacetime you would have had to exceed [itex]c[/itex] in order to reach at an earlier point in time.

    So, isn't it possible, then, in the context of GR, to expand a light cone to include previously inaccessible regions of space, thus, spacetime?
     
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  3. Jun 8, 2014 #2

    Simon Bridge

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    Sadly Nature does not care about whether an idea seems fundamentally sound to you or me or not.

    The warp drive is supposed to work in flat space-times.
    Supposing that one could enter and leave the warp-bubble, you've have to be able to enter or leave in a flat spacetime so the causality violation still holds.

    Do you know how to show such a violation in SR without talking about light-cones.

    What does "expand your light cone" mean?
     
  4. Jun 8, 2014 #3
    Well, let's start with the worldline of a ray of light in a normal lightcone:

    [itex]ct = x[/itex]

    [itex]c = \frac{dx}{dt}[/itex]

    Now, let's consider that the universe is expanding at rate [itex]H[/itex] per distance [itex]x[/itex]. This expansion gives rise to an apparent velocity [itex]v[/itex] of stationary observers distance [itex]x[/itex] apart. ie, two galaxies that would otherwise be stationary to eachother, appear to move away from eachother at velocity [itex]v[/itex] due to the expansion of space.

    [itex]v = Hx = \frac{dx}{dt}[/itex]

    Taking this into consideration, we get a worldline like this:

    [itex]\frac{dx}{dt} = c - Hx[/itex]

    Because as a light ray travels outwards, it will have to "catch up" to any object that is "moving away" (ie, riding the expansion of spacetime away from the observer). If [itex]Hx \geq c[/itex] , then the light ray cannot reach the object at [itex]x[/itex] distance away, because it lies outside the cosmic horizon of the origin.

    Which leads to

    [itex]ct = \frac{x}{1-\frac{Hx}{c}}[/itex]
    (eq. 1)

    The important feature is the asymptote. The cosmic horizon is the furthest you could ever reach; the distance at which space is expanding outwards at the speed of light;

    [itex]\frac{c}{H} = D_{horizon}[/itex]

    When

    [itex]x = D_{horizon}[/itex]

    As expected, this adjusted light cone given by eq. 1 has an asymptote at the cosmic horizon. I'm calling this the light-cone because this is the only region of spacetime accessible to any observer at the origin, to which there can be any causal ties. This is the light cone, taking the expansion of the universe into account.

    But, inflationary cosmology tells us that the expansion of the universe, [itex]H[/itex] changes, and at the earliest moments of the universe it took on an extremely large value that rapidly fell. Notice, that if [itex]H[/itex] decreases over time, then the asymptote of eq. 1 exists at a greater distance away from the origin, thus the accessible light-cone of an observer at the origin comes to include a larger region of spacetime.

    That is, eq. 1 with [itex]h[/itex] substituted for [itex]H[/itex], where [itex]h < H[/itex], will have a greater amount of spacetime accessible to the observer at the origin. So if an event lies outside of the lightcone given by eq. 1, by a very small amount, than a sufficiently decreasing value of [itex]H[/itex] will allow that event to enter the lightcone.

    If, as inflationary cosmology states, the value of [itex]H[/itex] ever dropped, then larger regions of spacetime became accessible to any observer.

    Therefore, one cannot use the fact that larger regions of spacetime become accessible due to the Alcubierre drive as an objection, without also objecting to inflationary cosmology.
     
    Last edited: Jun 8, 2014
  5. Jun 9, 2014 #4
    I almost wish I could discuss this with colleagues and possibly even write a paper on it, but unfortunately, I am only an undergrad Sophmore and I'm home for the summer, so that's not really an option for me at the moment :/
     
  6. Jun 9, 2014 #5

    PAllen

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    One reason you are not getting more responses is that this topic has been discussed to death on these forums, with many links collected to review papers on (in)feasibility, and wild impracticalities if constructed. People are tired of rehashing the same topic, especially one with no connection to experiment or any known phenomena in the universe. There is also a paper linked on one of those threads that rigorously proves that if an Alcubierre drive exists, you can build closed timelike curves using it and produce causality violations. So please search these forums a bit.
     
  7. Jun 9, 2014 #6

    Simon Bridge

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    GR is usually taught after the sophmore college year isn't it?
    It may be difficult to explain how a closed time-like curve happens in GR without the maths.

    The possibility of causality violations occur for FTL travel of any kind - unless there is some, hitherto undiscovered, mechanism to prevent it.

    You already know that maths can describe things that are not real. Students discover this when equations give them two answers (as in some ballistics problems) when Nature only gives one. Some extra criteria has to be used to pick one of the solutions as the "physical" one.

    Another place where we would usually discard a solution as "non-physical" is when the answer is imaginary (a multiple of the square-root of minus one). This is the case for the Alcubierre drive - it requires imaginary mass.
    Unfortunately there is some vagueness about what imaginary mass may mean which is pretty much what the current "warp drive" research is actually investigating.

    ... do you have a credible reference for anyone using the resulting availability of otherwise forbidden regions as an objection?

    The main objection is the causality violation. That could be interpreted as a consequence from larger regions of spacetime becoming accessible... in which case, the argument reads that we cannot use causality violations as an objection to warp drives, without also objecting to cosmological expansion, because cosmological expansion makes more spacetime available.

    That would be a valid argument if cosmological expansion could result in a causality violation.

    Do you have a reference showing that inflationary cosmology makes "larger regions of spacetime become accessible" in such a way as to produce causality violations?

    At the bottom of this page is a "related discussions" list - have you had a read through them in context with your speculations?
     
  8. Jun 10, 2014 #7

    Ah, yes, I am familiar with mathematical solutions that have no physical meaning - the ballistics example and working the relevant equations for [itex]n[/itex]-slit interference patterns comes to mind (particularly where we go from [itex]e^{i\phi}[/itex] to [itex] cos(\phi)+isin(\phi)[/itex] and "throw out" the imaginary term).

    Oh, and yes. I think it's not even taught at the undergraduate level at all, unfortunately. So as much as I'm absolutely enthralled with the idea of mathematics describing spacetime, it'll probably take at least two more years to build the mathematical knowledge to work with the Differential Geometry necessary (BYU offers classes in GR to undergraduates as 600-level courses, from what I've gleaned it seems to be a new or unusual kind of thing to offer).

    Haha, I like to think in the meantime I might discover some way to take advantage of mathematica to do the maths for me, so I can poke around with GR a bit even without the mathematical knowledge. Of course I fully intend to learn the math, but waiting to play with various models in GR is a bit like waiting to open presents on Christmas morning - and to fit this analogy perfectly, my family has a tradition of opening one present on Christmas Eve - that'll be using mathematica to poke around with GR before learning Differential Geometry.

    EDIT:

    That's the thing - I'm wondering if GR would somehow come in and "save the day," so to speak, and prevent the drive from being used to violate causality. Heh, as you've stated, I, too, would anticipate that any FTL drive would somehow maintain causality, and I was thinking perhaps GR would pick up the slack with the Alcubierre drive - that a sufficient analysis of how it behaves might reveal that it does preserve causality.

    But there again, I don't know enough about the mathematics of GR to really do any more than speculate on such a possibility. Though from what I've read, I don't think anyone has fully explored a true time-variant model of it, both in formation and collapse? I think if GR were to somehow preserve causality, it would be something that happens during formation and collapse. But there again, speculation, but it's what I would anticipate, so I'd love to see some models on how it behaves over time, during creation and collapse.

    Another edit: I think my hunch comes from this - as I understand it, relativity of simultaneity is the mechanism that would be exploited in a FTL drive to allow causal violations. Relativity of simultaneity, though, and all Lorentzian transformations, have [itex]x[/itex] and [itex]t[/itex] as variables in their equations.

    It seems, then, that whatever mechanism that acts to preserve causality would also have to incorporate [itex]x[/itex] and [itex]t[/itex] into its equations in a similar manner. However, the drive in a constant state - ie, non time-variant - does not describe its displacement from its original position where it was formed ([itex]x[/itex]), nor does it describe the elapsed time it took from the formation of the warp bubble to its collapse ([itex]t[/itex]), so by that virtue alone, it would seem impossible for a constant-state warp bubble description to incorporate any mechanism to preserve causality, even if a full description of its formation to its collapse, does.

    (note, although the alcubierre metric does describe [itex]dx[/itex], [itex]dy[/itex], [itex]dz[/itex], and [itex]dt[/itex], it does not describe [itex]x[/itex] to mean displacement from original position (where the warp bubble was formed to where it collapses), or [itex]t[/itex] to mean time from formation to collapse, as would be necessary to incorporate into a specific worldline, and thus have a mechanism that prohibits violation of causality. That is - not that the drive is doomed - but that a constant-state description of it is doomed to not incorporate a causality-preserving mechanism. To discover such a mechanism, the equations would need to describe the warp bubble from formation to collapse)

    /another edit

    That being said, it would certainly be very interesting if it turns out that inflationary cosmology implies that causality violation is possible, thus the Alcubierre drive could also be a time machine, of sorts, perhaps maintaining consistency through something like the Novikov self-consistency principle, or by generating alternate timelines.

    Rather unlikely, but would be fascinating.

    /Edit

    I've been scouring for every bit I could read about the Alcubierre drive for the last seven or so months, and that right there is something of great significance I haven't seen mentioned anywhere.

    Okay, that puts the whole thing in a new light. Negative mass is one thing. The Casimir effect shows that virtual negative mass isn't anything completely unheard of. But imaginary is certainly a bit different. I guess if there's vagueness about it, though, that makes it a lot more interesting, in terms of possible meanings.

    Though on second thought, that does also bring to mind how black holes create imaginary solutions within the Schwarzchild Radius, though I suppose while the meaning of imaginary time and space is somewhat apparent - swapping roles - the meaning of imaginary mass is a bit less obvious.

    Thanks for the insight!

    Ah, thanks for clarifying that for me. Yes, that is what I meant, "we cannot use causality violations as an objection to warp drives, without also objecting to cosmological expansion, because cosmological expansion makes more spacetime available." so it does rely on inflationary cosmology making larger regions of spacetime accessible in such a way as to produce causality violations, for it to be a valid argument. I guess I'd just as much assumed that the two were inseparable, but thanks for pointing out that key difference.

    And I can't say I do have a reference. It seems like something pretty major, though, if it were the case. I mean, wow, that'd be a pretty big kicker for an argument for causality violations/time travel being possible if it's implied by inflationary cosmology!

    I'll have to search around and see if I can find anything on it and then I'll come back with another reply.

    In the meantime, though: With the equation I derived above about the light-cone in the context of an expanding universe; if I were to plot the lorentzian transformations for events in this model, would I still use the equations

    [itex]x' = \gamma (x-vt)[/itex]
    [itex]t' = \gamma (t-\frac{v}{c^{2}}x )[/itex]
    where
    [itex] \gamma = \frac{c}{\sqrt{c^{2}-v^{2}}} [/itex]

    Or would my modification of the worldline of a light ray from
    [itex] \frac{dx}{dt}[/itex] = c
    to
    [itex] \frac{dx}{dt}[/itex] = c - Hx

    change how the transformations are done?

    Intuitively I'd want to say no, but perhaps as the value of [itex]H[/itex] changes that might make an effect on the Lorentzian transformations, but I'm not sure exactly what the rules would be for working in this situation, though I'll try to hash it out and come back with the results.

    EDIT: So perhaps because the expansion of space is a phenomenon that takes place over changing time (imparting an apparent velocity on objects; [itex]\frac{dx}{dt} = Hx[/itex]), then it has no effect on the positions and times of instantaneous events, since being instantaneous, the events have no change in time, and no change in space?

    As of reading that, no, but as of now, yes.

    (And as an aside, your note on a warp drive leaving a plasma stream akin to an interstellar contrail is probably one of the most amazing, epic mental images I've had in quite awhile!)

    I'll also see what reading I can find on causal relations and what exactly forms the mathematical basis for that so I can dive into this in a more informed way.
     
    Last edited: Jun 10, 2014
  9. Jun 10, 2014 #8

    PAllen

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    Alcubierre drive does not require imaginary mass. I think the confusion is with tachyons which (if they existed) would have imaginary rest mass but positive energy. What Alcubierre drive needs is negative mass, which is required to exist in some forms, in quantum theory.

    In case your researches haven't brought you to these:

    1) Two review papers:
    http://arxiv.org/abs/1001.4960
    http://arxiv.org/abs/0710.4474

    2) This paper demonstrates the obliteration of the destination by such a drive:

    http://arxiv.org/abs/1202.5708

    3) This paper shows that the ability to construct Alcubierre drive implies the ability to construct causal violations:

    http://exvacuo.free.fr/div/Sciences...tt - Warp drive and causality - prd950914.pdf
     
  10. Jun 12, 2014 #9

    Simon Bridge

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    I have it down, off the last time I canvassed the lit and discussions on this topic, as needing negative energy - so I guess I may have mixed up the sources.
    That's odd since I recall being careful to distinguish this situation from tachyons ... obviously not careful enough.
    It must have been a very common mistake.

    I got bored with it ages ago and lost track. I see from the paper list that the state of the art has moved on and lightspeed survives even weird topologies.
    That makes for a seriously robust postulate.
     
  11. Jun 12, 2014 #10

    Simon Bridge

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    Doesn't work that way - you can get computers to crunch the numbers for you but you have to know what the mathematical constructs mean.

    If if if ... interesting it may be but...
    http://ned.ipac.caltech.edu/level5/Liddle/Liddle6_1.html
    http://arxiv.org/pdf/gr-qc/9811037.pdf?origin=publication_detail
    1st is lecture notes and the second a paper - afaik causality is usually treated as a limiting factor in inflationary models - not a feature.
    ... we can always have fun with "if"s, but there's no end of them.
    "If" is not generally a useful direction to take your questions.

    I may have got caught up in a general confusion there - my apologies.
    Alcubierre drive requires negative energy.

    If we count the energy of the vacuum as zero, then the Casimir effect is an example of negative energy. But we could say that the zero is the theoretical minimum energy in the absence of quantum fluctuations. Which do we use in GR?

    iirc some experiments are being done (have been) where light is shone through a casimir gap and timed (via interference methods) to see if there is something funny happening to space in there. Is the speed of light faster in the gap? So far results have been "inconclusive".

    One of the ways to get negative energy is supposed to be inside the ergosphere of rapidly rotating black holes - I don't know enough about that to comment.

    One of the things to be very careful of in anything involving relativity is "who is doing the observing". It may not be very meaningful to talk about what happens "within the Schwarzchild Radius" from the reference frame of an observer outside it.

    Just so we're clear.
    One of the arguments is that FTL may be possible in principle, so long as you don't try to make a time machine.

    More speculating on "if".
    There is a science fiction section.

    That is pretty much how the Hubble expansion is defined - yes.

    It sounds like you may benefit from an introduction to general relativity.

    http://preposterousuniverse.com/grnotes/grtinypdf.pdf [Broken]
    ... this is a crash course, but I don't know how good it is.
    It's attached to Sean Carrol's blog - it skims OK but I've been caught out before.
    MIT has online course notes for GR and tensor calculus - which assume you've done the earlier courses, but those are provided too.

    Make sure you are familiar with SR first.
    http://www.physicsguy.com/ftl/html/FTL_part1.html
    ... you'll like this since the last part deals with FTL in the SR framework.

    Both links are on the "fun" side and should not be taken as a substitute for acquiring a deep understanding through sustained study.

    Enjoy.
     
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