How does FTL = time-travel?

[skotos]

You can have such a frame, yes - but it renders portions of relativity false.

Which portions of relativity are rendered false? The laws of physics are still the same in all frames, we still have time dialation, length contraction, etc. Different observers still disagree on the temporal order of events seperated by a spacelike interval.

The one thing I can think of is that one can establish a cause and effect relationship between two events that are seperated by a spacelike interval, which is not possible under special relativity as we understand it today. So if this new possibility is considered a contradiction of relativity, then I concede the point.

Yes it would, because it determines a frame that experiences, for certain, the most proper time

I don't believe that this is correct. All I can do is to appeal to authority on this point, and quote Kuroneko from this thread:

In the standard Friedmann-Robertson-Walker family of models, for example, there is indeed a "privileged frame" [1] in which the "cosmic dust" is maximally isotropic. This frame also experiences the most proper time, which is the resolution to an interesting version of twin paradox for closed universes

If I understand Kuroneko's post correctly, there already is a frame of reference that experiences, for certain, the most proper time (assuming the Friedmann-Robertson-Walker family of models applies to our universe). This frame would be unrelated to the FTL frame (although as I've said before, it is my prefered frame for doing FTL calculations, but that's just a coincidence).

[Ariphaos]

Which portions of relativity are rendered false? The laws of physics are still the same in all frames, we still have time dialation, length contraction, etc. Different observers still disagree on the temporal order of events seperated by a spacelike interval.

It violates the name. There is a true ordering. Observers may disagree, but one or both of them is wrong. Thus, it's not relative any longer - it's just an illusion. The equations don't work for going faster than light - you can't use them for your calculations to come up with a result.

Technically, in your example, a postulate is also violated - the constancy of the speed of light. Because in our frame, it's measured to be 1/100.84 of your constant value. At .84c, it's measured to be 1/50.42. This is mostly just a flaw of your example, though.

The one thing I can think of is that one can establish a cause and effect relationship between two events that are seperated by a spacelike interval, which is not possible under special relativity as we understand it today. So if this new possibility is considered a contradiction of relativity, then I concede the point.

Not that I'm aware of. Any form of FTL which travels within its own light-cone still permits causality and relativity both - it's just considered unphysical by other means (requiring a negative energy density).

I don't believe that this is correct. All I can do is to appeal to authority on this point, and quote Kuroneko from

If I understand Kuroneko's post correctly, there already is a frame of reference that experiences, for certain, the most proper time (assuming the Friedmann-Robertson-Walker family of models applies to our universe). This frame would be unrelated to the FTL frame (although as I've said before, it is my prefered frame for doing FTL calculations, but that's just a coincidence).

Note 1 from Kuroneko:

[1] Not in regards to the laws of physics themselves! It may be useful to use "preferred" for that sort of phenomenon and reserve "privileged" for frames in which symmetry occurs.

Intergalactic dust experiences the most proper time because it is outside of the massive gravitational wells produced by galaxies and their dark matter halos. This is significant - the sun's gravitational well 'slows' us far more than Earth's gravity does, to the tune of nearly 20 billionths compared to Earth's 1.3 billionths. The gravitational well of Andromeda causes eight times the dilation of our own sun on Earth. The Milky Way is half again as massive as Andromeda.

This doesn't apply when you consider the expansion of the Universe - an observer at rest in the Sculptor void would disagree that they were sharing a common frame with an observer in the Bootes void, for example., because they are moving apart at some 5-10% of c.

[Loner]

How does the FTL drive of the Planet Express ship from Futurama fit in all of this?

[Kuroneko]

The way I deal with faster than light is I assume that everything, irrespective of the speed or accelleration it's going through has a definitive point in space and time that's the universal moment of the present. If you froze the universe at one point in time and rearranged your location to anywhere else in it upon resumption, I don't really see what problems with causality that would engender.

What is "the universe at one point in time"? According to you, it's some set of events simultaneous to (Rye,right now); call it X. According to your neighbor, the set of events simultaneous to (Rye,right now) is some set of events Y. And if your neighbor runs away from you, X is not the same as Y. Simultaneity is relative, which is why what is instantaneous teleportation to you may be time travel to the past to others.

I would've thought something like this: At any point in time from any observer, everything else must correspond to a specific time/space position relative to it, work out the average for everything in the universe and you have the "true" position of everything relative to everything else and have everything truly plottable.

However, for any two events A and B with spacelike separation, there is an inertial frame in which A precedes B and another inertial frame in which B precedes A. Unless there is already an absolute frame to which one can compare inertial frames and "scale" their contribution to this "average", there is no way that this can meaningfully come out to be anything but zero.

... after all, there's all the confusing stuff with virtual particles, entanglement and the origin of the modern universe itself. Or there is the idea that relativity is not the whole story since on the tiniest scales, the physical universe is connected to the warp and chaos or whatever.

Such things are counterintuitive, but they do not carry information superluminally.

---

How does picking a common frame of reference in order to determine the times and locations of FTL events (say, the entry and exit into "hyperspace") violate relativity? We pick common frames of reference in order to do physics all the time. If I'm solving a mechanics problem on Earth, I pick the frame of reference in which the Earth is at rest.

The issue is not whether you can pick a reference frame in which you can make sense of the situation, but that you are now forbidden from picking certain ones. The relativistic motto is that all observers are equivalent (for STR, inertial; for GTR, all) in the sense that the laws of physics are the same for all--in other words, you could in principle do your mechanics problem in coordinates which treat the galactic center at rest, or some such. The situation described prior would have causality preserved in some reference frames but not others, so that it will not be proper to use the same laws of physics

The one thing I can think of is that one can establish a cause and effect relationship between two events that are seperated by a spacelike interval, which is not possible under special relativity as we understand it today. So if this new possibility is considered a contradiction of relativity, then I concede the point.

In special relativity, any a signal is faster than light if and only if it causally connects two events with spacelike separation. In general relativity, this goes for "locally faster than light" (in STR, locally FTL is equivalent to FTL).

Intergalactic dust experiences the most proper time because it is outside of the massive gravitational wells produced by galaxies and their dark matter halos.

That's certainly an important factor, but more directly to how this frame was defined: if the universe is isotropic in this frame, then "gravitational effects cancel out" by this symmetry. Our universe is only approximately uniform, but this uniformity is apparently rather impressive on the very large scale.

This is significant - the sun's gravitational well 'slows' us far more than Earth's gravity does, to the tune of nearly 20 billionths compared to Earth's 1.3 billionths.

Not that I have any reason to dispute these figures, but I'm somewhat curious as to how they were calculated. If it was by comparing to an idealized observer at infinity assuming an asymptotically flat spacetime, this comes out to be
[1-(1-GM/(c²AU))^{1/2}]:[1-(1-Gm/(c²r))^{1/2}] ≅ [M/(2AU)]:[m/(2r)] ≅ 14:1
where M is the mass of the Sun, m that of earth, and AU and r are Earth's orbital and physical radii, and the nonlinearity of GTR was blithely ignored (that should be fine on this scale). The ratio is extremely close (14:1 vs. 15:1 = 20:1.3), although the numbers are a bit more off (4.9e-9 vs. 0.35e-9), so there's probably one more factor affecting both in the same manner. Was it the mass of the Milky Way or was the calculation simply done with a different method of comparison?

[The Prime Necromancer]

This is some pretty interesting stuff, and after reading both of these threads I believe I have at least a dim understanding of what's going on.

But then I thought about Dr. Saxton's work to bring Star Wars at least somewhat more in line with known physics (albeit rather theoretical physics), and wondered what, if anything, he had to say about the "pick two" problem. After all, Star Wars clearly has causality and FTL travel, and with his being an astrophysicist, it seemed unlikely to me that he would throw out Relativity.

This is his take on the matter:

Causality paradoxes.

A naive interpretation of Special Relativity would suggest that under some circumstances superluminal travel results in time reversal. The trouble arises when observers at the starship's destination are moving at some significant velocity relative to the origin. If v is the velocity of the ship in hyperspace, u is the difference of velocity between the origin and destination points, then a kind of reverse time-dilation occurs when v u > c² (where c is lightspeed). This is considered a paradoxical problem for spaceflight and communications because it seemingly allows someone at the destination to send out a superluminal signal to the origin system, affecting the circumstance of the ship's launch. For instance we might suppose that the Grand Moff Tarkin would send a message to Tatooine ordering the impoundment of the Millennium Falcon, preventing the ship from launching in the first place. Thus (on a superficial reading) it would seem as if hyperdrive travel would lead to confusing violations of causality and historical paradoxes.

The galaxy rotates, with star systems and interstellar material orbiting the common centre of mass. The nett rotation and other internal motions give rise to velocity differences of dozens of maybe a hundred kilometres per second between distant regions of the disk. The drift speeds are much smaller between neighbouring systems. For a jump between systems with velocity separation of 50km/s observers at the destination system will witness the time-reversal effect if the vessel travels faster than about 6000c. For jumps of more typical length (only a few hundreds or thousands of light years) the velocity differences will be smaller, but the threshold for time reversal rises to only a few tens of thousands of times lightspeed. The range of velocities involved in hyperdrive travel are very much greater, far into the realm of possible causality violation.

Fortunately it turns out that causality paradoxes actually cannot arise. Although nobody will see starships travelling backwards in time, other exotic effects will be observed. A particle moving backwards in time is equivalent to the motion of some antiparticle forwards in time. In relativistic quantum mechanics that's exactly what antiparticles are: time-reversed versions of ordinary particles. In those situations where naive relativistic considerations would suggest the observation of a starship travelling backwards in time what the bystanders will really witness is an antimatter starship travelling forward in time. This mirror ship will be seen to head in the opposite direction.

When the reentry jump occurs the real ship would emerge from a point in space and the antimatter mirror ship appears simultaneously at the same location and then accelerates off at superluminal velocity in the opposite direction. At the jump point, where both the vessel and the mirage are indistinguishably close to lightspeed, they would each appear to be compressed into zero length. This is due to a length-dilation which is similar to the relativistic time dilation effect. (All objects appear to be shortened along their direction of motion by a factor which depends on speed. This contraction becomes indefinite at lightspeed.)

Thankfully, even this disconcerting sight is unlikely to be witnessed under ordinary circumstances. In preparation for the jump back to realspace the ship must reduce velocity to very near lightspeed, well below the time-reversal / antimatter threshold speed. It will cover great distances while doing so, and even more distance while it decelerates from lightspeed to rest. Even if it is possible to see a ship in hyperdrive, the transition back to realspace will take place far from inhabited space, on the outer reaches of the destination system or even further away. Similarly a ship jumping into hyperspace covers a great distance during the pre-jump acceleration phase.

Thoughts?

[Winston Blake]

How does the FTL drive of the Planet Express ship from Futurama fit in all of this?

It gets lumped in with everything else, since there's no difference between moving and moving the universe around you. You would still need to make the universe go FTL, although in the Futurama-verse they admittedly decided to change the speed of light in 2208. Really all you've got are FTL in special relativity (e.g. tachyonic 'Saxton drive') and FTL in general relativity (e.g. spacetime shortcuts like wormholes and warp-based systems).

Thoughts?

I'll just make a note that I recently found that the 'antimatter ship' solution to tachyon causality violation has a short name - the Feinberg reinterpretation principle. Interestingly, I think it means the acausal triangle shown in my earlier link is prevented from ever happening, kinda like how the Chronological Protection Conjecture prevents making time-machine wormholes.

The property of causality is a fundamental principle of theoretical particle physics; tachyons, if they existed, would not violate causality, even if they interacted with ordinary (time-like) matter[3]. Causality would be violated if a particle could send information into its own past, forming a so-called causal loop, leading to logical paradoxes such as the grandfather paradox. Tachyons are prevented from violating causality by the Feinberg reinterpretation principle[3] which states that a negative-energy tachyon sent back in time in an attempt to violate causality can always be reinterpreted as a positive-energy tachyon travelling forward in time. This is because observers cannot distinguish between the emission and absorption of tachyons. For a tachyon there is no distinction between the processes of emission and absorption, since there always exists a sub-light velocity reference frame shift that alters the temporal direction of the tachyon's world-line, which is not true for bradyons or photons.

The attempt to detect a tachyon from the future (and violate causality) actually creates the same tachyon and sends it forward in time (which is causal). A tachyon detector will seem to register tachyons in every possible detection model; in reality the tachyon "detector" is spontaneously emitting tachyons. The effect of the reinterpretation principle on any tachyon "detector" is that any incoming tachyonic message would be lost against the tachyon background noise, which is an inevitable accompaniment of the uncontrollable emission. The counter intuitive conclusion is that tachyons (if they existed) could be used to transmit energy-momentum, but they can't be used for communication. Thus there is no need to fall back on some quantum field theory form of the Novikov self-consistency principle to preserve causality.

As for the OP, here's a succinct explanation:

Consider a duel with tachyon pistols. Two duelists, A and B, are to stand back to back, then start out at 0.866 lightspeed for 8 seconds, turn, and fire. Tachyon pistol rounds move so fast, they are instantaneous for all practical purposes.

So, the duelists both set out --- at 0.866 lightspeed each relative to the other, so that the time dilation factor is 2 between them. Duelist A counts off 8 lightseconds, turns, and fires. Now, according to A (since in relativity all inertial frames are equally valid) B's the one who's moving, so B's clock is ticking at half-speed. Thus, the tachyon round hits B in the back as B's clock ticks 4 seconds.

Now B (according to relativity) has every right to consider A as moving, and thus, A is the one with the slowed clock. So, as B is hit in the back at tick 4, in outrage at A's firing before 8 seconds are up, B manages to turn and fire before being overcome by his fatal wound. And since in B's frame of reference it's A's clock that ticks slow, B's round hits A, striking A dead instantly, at A's second tick; a full six seconds before A fired the original round. A classic grandfather paradox.

Note, this is NOT a matter of when light gets to an observer, it is NOT an optical illusion. It is due to the fact that, in SR, the question of what occurs at the "same time as" something else is observer dependent.

[Ariphaos]

That's certainly an important factor, but more directly to how this frame was defined: if the universe is isotropic in this frame, then "gravitational effects cancel out" by this symmetry. Our universe is only approximately uniform, but this uniformity is apparently rather impressive on the very large scale.

It's annoying how long I've been aware of this but did not end up making that connection when I saw you mention it.

Coincidentally, in note [2] in the linked thread - I assume the twin that doesn't undergo acceleration with respect to the 'medium' ages faster, but what happens if they both split, heading towards eachother?

Not that I have any reason to dispute these figures, but I'm somewhat curious as to how they were calculated. If it was by comparing to an idealized observer at infinity assuming an asymptotically flat spacetime, this comes out to be
[1-(1-GM/(c²AU))^{1/2}]:[1-(1-Gm/(c²r))^{1/2}] ≅ [M/(2AU)]:[m/(2r)] ≅ 14:1
where M is the mass of the Sun, m that of earth, and AU and r are Earth's orbital and physical radii, and the nonlinearity of GTR was blithely ignored (that should be fine on this scale). The ratio is extremely close (14:1 vs. 15:1 = 20:1.3), although the numbers are a bit more off (4.9e-9 vs. 0.35e-9), so there's probably one more factor affecting both in the same manner. Was it the mass of the Milky Way or was the calculation simply done with a different method of comparison?

Hmm, my constants may be off. It was awhile ago so I need to reconstruct - I used the Schwarszchild metric: 2GM/r, basically (discounting the change in radius)

The mass of the sun in 'meters' as I ended up calculating it out was:
132712440018000000000 meters
based on the heliocentric gravitational constant (so it incorporates G)

So M/c² = 1476.62503825

Earth's Time-Average Orbital Distance is about 149618773700 meters

Leaving 2M/r at

0.00000001973849941

(about)

I would have posted asking, but I see so many quotes about a solar mass black hole having a radius of 3 km I figured I was correct.

For reference, I got
0.000000001390697013

For Earth's (I guess I looked at my notes too hastily - the ratio is about 14.2:1 with my math - I did this months ago).

I ignored delta r because I figured it would be trivial. (Granted, this whole thing is - but I was making a sort of Helios 'Day' calendar for a sci-fi setting I've been designing)

Doing the mass of the Milky Way would require me running through the interior Schwarzschild metric... which is not a particularly simple equation for me.

[Nyrath]

Yes, the duel with tachyon pistols is gone into with more detail here:
http://sheol.org/throopw/tachyon-pistols.html

From
http://www.projectrho.com/rocket/rocket ... #causality

The aphorism at rec.arts.sf.written goes "Causality, Relativity, FTL travel: chose any two."

Your average physicist holds Relativity quite strongly. It has been tested again and again with an accuracy of many decimal places. They hold onto Causality even tighter. Without Causality the entire structure of physics crumbles. Causes must preceed effects, or it becomes impossible to make predictions. If it is impossible to make predictions, it would be best to give up physics for a more profitable line of work.

Therefore, they chose to jettison FTL travel.

Why only two? Relativity proves that FTL travel is identical to Time travel (to help your research, the technical term for time travel is "Closed timelike curve"). Time travel makes Causality impossible, since it can be used to create paradoxes. So if you have Relativity and FTL, Causality is impossible. If you do not have Relativity, FTL is not Time travel, so you can have Causality. Or more mundanely you can have Relativity and Causality, but no FTL/Time travel (the latter is the opinion of physicist Stephen Hawking, he calls it the chronology protection conjecture).

Clever readers will have already spotted a possible loop-hole. What if there was some law of physics that prevented Time travel from creating paradoxes?

Hinson shows there are four ways of enforcing a "no-paradox" rule for time travel. Parallel Universes, Consistency Protection, Restricted Space-Time Areas, and Special Frames. In some ways Special Frames is the best, though it directly contradicts part of Relativity. Oh well. For details, you'd best read the Hinson article.

The latter three are examples of the Novikov self-consistency principle.

In some late-breaking news, physicists Daniel Greenberger and Karl Svozil have shown that the laws of quantum mechanics enforces Consistency Protection. You can read their paper here, but it makes my brain hurt. Translated into English, they maintain that time travellers going back into the past cannot alter the past (i.e., the past is deterministic). This is because quantum objects can act sometimes as a wave. When they go back in time, the various probabilities interfere destructively, thus preventing anything from happening differently from that which has already taken place.

[His Divine Shadow]

If you're operating with the premise that physics is a part of it, then you've got to either assume one thing or the other, as I stated above. You can't have FTL and not have time-travel in one case, for instance. So every time the Millennium Falcon jumps, it's fucking up time lines or something else fundamental to physics.

How did saxton get around that now again?

[Surlethe]

If you're operating with the premise that physics is a part of it, then you've got to either assume one thing or the other, as I stated above. You can't have FTL and not have time-travel in one case, for instance. So every time the Millennium Falcon jumps, it's fucking up time lines or something else fundamental to physics.

How did saxton get around that now again?

He's got Star Wars ships with some sort of temporal shielding or whatnot -- obviously since they're not in the business of fucking up time lines, they've got some way to retard or speed up time while they're traveling FTL (this may also explain why the trip to Bespin from Anoat didn't seem to take as long as it must have).

[Kuroneko]

I'll just make a note that I recently found that the 'antimatter ship' solution to tachyon causality violation has a short name - the Feinberg reinterpretation principle. Interestingly, I think it means the acausal triangle shown in my earlier link is prevented from ever happening, kinda like how the Chronological Protection Conjecture prevents making time-machine wormholes.

It's certainly true that one can embrace the ambiguity between sender and receiver, it's a bit silly to apply it to this context. Since the ship must come out of its supposed 'tachyonic state', there is genuine information that has traveled superluminally--just ask the crew as to whether they're beginning or ending their journey. That's certainly still a problem in plain STR.

[Saxton]'s got Star Wars ships with some sort of temporal shielding or whatnot...

Well, that makes a bit more sense, at least in-universe.

Coincidentally, in note [2] in the linked thread - I assume the twin that doesn't undergo acceleration with respect to the 'medium' ages faster, but what happens if they both split, heading towards eachother?

If they have the same velocity relative to the 'privileged' frame, then their proper times will be equal when they meet up again. Note that in a closed universe that's not expanding too much, one can in principle determine speed relative to this frame without reference to the background matter per se (which for some closed universes might not even exist). Simply send out light signals in directly opposite directions and measure the discrepancy between their arrival times when they meet up with you again. Only in the privileged frame will they arrive simultaneously. This can be easily seen from suppressing all spatial dimensions except one, so the universe is basically a cylinder R×S.

Hmm, my constants may be off. It was awhile ago so I need to reconstruct - I used the Schwarszchild metric: 2GM/r, basically (discounting the change in radius)

Aa, that explains the factor of four--yours is twice too big; mine is twice too small. A two disappeared on me for no good reason.

So M/c² = 1476.62503825 Earth's Time-Average Orbital Distance is about 149618773700 meters Leaving 2M/r at 0.00000001973849941.

From the Schwarschild metric, dτ² = (1-2R)dt², R = GM/c²r, where dr,dθ,dφ are ignored because they are extremely small in the case of the Earth. This means that dτ/dt = [1-2R]^{1/2} = 1 - R + O(R²) from the Maclaurin series. So the relevant quantity is R = GM/(c²r) = 9.87e-9, which is half of your figure and twice my previous.

Doing the mass of the Milky Way would require me running through the interior Schwarzschild metric... which is not a particularly simple equation for me.

Not quite. The interior Schwarzschild metric is inapplicable to this situation; it treats t as spacelike at r as timelike. We are not inside a black hole. Modeling the Milky Way as anything more complicated than the crude exterior-Schwarzschild (or Kerr) is a rather thorny issue. A particularly dramatic attempt was done by Cooperstock and Tieu, whose model actually has no dark matter.

[Surlethe]

[Saxton]'s got Star Wars ships with some sort of temporal shielding or whatnot...

Well, that makes a bit more sense, at least in-universe.

He goes into it a bit more here. Apparently, vehicles traveling faster than light would, for distance travelled x, experienced travel time would limit to x/c (in the frame of the travellers), so there has to be some sort of time-retardation in effect anyway. It makes sense that they'd use it to shield from causality.

[Ariphaos]

If they have the same velocity relative to the 'privileged' frame, then their proper times will be equal when they meet up again. Note that in a closed universe that's not expanding too much, one can in principle determine speed relative to this frame without reference to the background matter per se (which for some closed universes might not even exist). Simply send out light signals in directly opposite directions and measure the discrepancy between their arrival times when they meet up with you again. Only in the privileged frame will they arrive simultaneously. This can be easily seen from suppressing all spatial dimensions except one, so the universe is basically a cylinder R×S.

Nifty, if somewhat time consuming. In such a Universe wouldn't you be able to see that sort of disparity on smaller scales? Assuming you had a large enough region to test over, it seems there shouldn't be any difference between firing them in opposite directions and firing them directly towards each other.

Aa, that explains the factor of four--yours is twice too big; mine is twice too small. A two disappeared on me for no good reason.

From the Schwarschild metric, dτ² = (1-2R)dt², R = GM/c²r, where dr,dθ,dφ are ignored because they are extremely small in the case of the Earth. This means that dτ/dt = [1-2R]^{1/2} = 1 - R + O(R²) from the Maclaurin series. So the relevant quantity is R = GM/(c²r) = 9.87e-9, which is half of your figure and twice my previous.

So the 2 vanishes... I need to pick up some refresher calculus material.

Not quite. The interior Schwarzschild metric is inapplicable to this situation; it treats t as spacelike at r as timelike. We are not inside a black hole. Modeling the Milky Way as anything more complicated than the crude exterior-Schwarzschild (or Kerr) is a rather thorny issue. A particularly dramatic attempt was done by Cooperstock and Tieu, whose model actually has no dark matter.

Ah, thanks. That saves a lot of wasted effort.

Given the size and seeming incompressibility of the dark matter halo, is discounting it really worse?

[Kuroneko]

In such a Universe wouldn't you be able to see that sort of disparity on smaller scales? Assuming you had a large enough region to test over, it seems there shouldn't be any difference between firing them in opposite directions and firing them directly towards each other.

Are you thinking of having an observer in the middle of two comoving emitters, which send a signal toward the observer? In order to properly measure the velocity relative to the privileged frame, the experiment should not be localized to a spatial region smaller than the universe. To see this, first imagine this performed in plain Minkowski spacetime, which definitely should not yield anything peculiar. Now, take a closed static universe. One can treat the spacetime of this universe as a quotient space of the Minkowski spacetime by simply identifying (t,x) with (t,x+α) for some constant α representing the 'size' of the universe [1]. Since within each "strip", the quotient space is identical to the corresponding region of Minkowski spacetime, if this experiment is localized there, it will not work. The bottom line is that in general, one needs some signal to travel "around" the universe, as in the previous version. Of course, one might simply look at the background radiation if there is any.

So the 2 vanishes... I need to pick up some refresher calculus material.

Maclaurin series: sqrt[1-a] = 1 - a/2 - a²/8 - a³/16 - ... . In our case, a = 2R is very small, so the O(R²) terms can be ignored with impunity.

Given the size and seeming incompressibility of the dark matter halo, is discounting it really worse?

You misunderstand. It's not that their ignored the dark matter halo, but that their model was intended to reproduce the observed rotation curves of stars without any appreciable amount of dark matter, which is something that is not tenable with Newtonian gravitation. In other words, according to their model, the dark matter halo doesn't exist.


[1] Like so:

Code: Select all

^t     ^t     ^t  The quotient space with equivalence
|      |      |   relation (t,x) ~ (t,x+α) folds Minkowski
*      *      *   spacetime into a cylinder. For example,
|      |      |   the points marked * are now the _same_ point.
+------+------+-->x
0      α     2α

[skotos]

Well, given that one can preserve causality and have FTL travel, it seems to me that there is still another problem: Conservation of energy and conservation of momentum.

Imagine a ship that "jumps" from one location to another instantaneously (in some frame). In a different frame it departs at T=10 and arrives at T=5. From T=0 to T=5 there is only one ship, and then from T=5 to T=10 there are two, and after T=10, only one again. Mass and (possibly) momentum are created and then lost.

In some other frame the ship might depart at T=10 and arrive at T=11. Between T=10 and 11 there are no ships, and so mass and momentum are lost.

Does any explanation of FTL travel explain where this mass and momentum come from and/or go? Can any explanation account for this problem or do we have to discard conservation of mass and momentum in order to have FTL?

[Ariphaos]

Are you thinking of having an observer in the middle of two comoving emitters, which send a signal toward the observer? In order to properly measure the velocity relative to the privileged frame, the experiment should not be localized to a spatial region smaller than the universe. To see this, first imagine this performed in plain Minkowski spacetime, which definitely should not yield anything peculiar. Now, take a closed static universe. One can treat the spacetime of this universe as a quotient space of the Minkowski spacetime by simply identifying (t,x) with (t,x+α) for some constant α representing the 'size' of the universe [1]. Since within each "strip", the quotient space is identical to the corresponding region of Minkowski spacetime, if this experiment is localized there, it will not work. The bottom line is that in general, one needs some signal to travel "around" the universe, as in the previous version. Of course, one might simply look at the background radiation if there is any.

I understand in intuition but not in reality :-/

We have a pair of light beams that, over the course of equidistant journeys, will appear to have undergone a shift if sent to meet eachother the long way round.

Where did this shift occur? How are we prevented from slicing it up, for instance? Place observers and retransmitters (or simply splitters that slice an insignificantly small portion of the signal off for measure) in a 'ring' around the Universe to measure what should be a gradual change?

You misunderstand. It's not that their ignored the dark matter halo, but that their model was intended to reproduce the observed rotation curves of stars without any appreciable amount of dark matter, which is something that is not tenable with Newtonian gravitation. In other words, according to their model, the dark matter halo doesn't exist.

Ah, something like MOND? After the image of the bullet cluster and friend a few months ago, are there still serious arguments for such things?

I get ~.000011 when I plug in 1.9 trillion Solar Masses for the galaxy. Is this a 'decent' estimate, at all? Obviously, it gets quite flawed near the center of the galaxy...

[Kuroneko]

[img=left]http://i4.photobucket.com/albums/y121/v ... -sig03.png[/img]

Well, given that one can preserve causality and have FTL travel, it seems to me that there is still another problem: Conservation of energy and conservation of momentum. Imagine a ship that "jumps" from one location to another instantaneously (in some frame). In a different frame it departs at T=10 and arrives at T=5. From T=0 to T=5 there is only one ship, and then from T=5 to T=10 there are two, and after T=10, only one again. Mass and (possibly) momentum are created and then lost.

In STR, that is definitely a problem. In GTR, one would still have local energy-momentum conservation, regardless of whether or not there are closed timelike curves--it's a consequence of the Einstein field equation itself. The proper question to ask in regards to conservation is no longer "is it the same amount as before?" but rather "for every small piece of spacetime, is there any created or destroyed there?" If you're familiar with electromagnetism, think of ∇·B = 0 (except that here it's a covariant derivative of the stress-energy tensor).

We have a pair of light beams that, over the course of equidistant journeys, will appear to have undergone a shift if sent to meet eachother the long way round.

There is no shift. A static cylindrical universe R×S³ has no intrinsic curvature--the Riemann curvature tensor vanishes everywhere. In other words, the cylindrical universe is flat, which can be made obvious by the fact that a flat piece of paper can be rolled into a cylinder without changing any of the metrical relationships between the points on it, which is exactly what we did to Minkowski spacetime above.

For reference, on the left (top), there's a spacetime diagram of the opposite-ray experiment. The vertical axis is time, and the horizontal axis is space, scaled so that the entire universe has diameter 1. There are two light signals (imagine wrapping this into a cylinder), of slope 1 and -1, in blue. A stationary observer at the origin would receive both of them simultaneously, at (t,x) = (1,0). There is another observed, in red, that travels at half the speed of light; he or she will receive the left-bound signal at (t,x) = (2/3,1/3) and the right-bound signal much later, at (t,x) = (2,0).

Where did this shift occur? How are we prevented from slicing it up, for instance? Place observers and retransmitters (or simply splitters that slice an insignificantly small portion of the signal off for measure) in a 'ring' around the Universe to measure what should be a gradual change?

Once again, the problem is the ambiguity of simultaneity. Take a look at the bottom half of the diagram on the left. It has an observer (magenta) with velocity of half the speed of light flanked by two equidistant comoving probes (red), which simultaneously send a light signal toward the observer. The green consists of all events simultaneous according to the moving observer; note that these events are not simultaneous in the stationary frame. Note that the signals arrive simultaneously at (t,x) = (2/5,2/5). If they are allowed to travel onwards, we have essentially the same experiment as above--they will not be simultaneous when they intersect the observer's worldline next time.

A geometrically and topologically important distinction is that only in the stationary frame are the surfaces (here, lines) of simultaneity compact. For any other observer, they are not (green line).

Ah, something like MOND? After the image of the bullet cluster and friend a few months ago, are there still serious arguments for such things?

The model uses only classic GTR. There are no new physics--that's why it's such a dramatic model. It's unfortunate that it has some features that make it ambiguous as to whether it's physically reasonable.

I get ~.000011 when I plug in 1.9 trillion Solar Masses for the galaxy. Is this a 'decent' estimate, at all? Obviously, it gets quite flawed near the center of the galaxy...

I'm not certain, since I don't know how you arrived at this figure. we take Newtonian gravitation seriously for this scale, the gravitational parameter of the mass inside the Sun's orbit is μ = 4π²r³/T². Roughly, r = 2.5e20m and T = 7.5e15s, giving μ/(c²r) = [2πr/cT]² = 4.9e-7. But this method employs the Schwarzschild metric, which is not reasonable for the galaxy.

[Ariphaos]

*snip explanation*

Alright, my mind was just being stubborn and I had to sleep on it a bit before I got it. Thanks.

The model uses only classic GTR. There are no new physics--that's why it's such a dramatic model. It's unfortunate that it has some features that make it ambiguous as to whether it's physically reasonable.

How does that work without fudging (to a significant extent, observed and verified) masses?

I'm not certain, since I don't know how you arrived at this figure. we take Newtonian gravitation seriously for this scale, the gravitational parameter of the mass inside the Sun's orbit is μ = 4π²r³/T². Roughly, r = 2.5e20m and T = 7.5e15s, giving μ/(c²r) = [2πr/cT]² = 4.9e-7. But this method employs the Schwarzschild metric, which is not reasonable for the galaxy.

I used the full mass as described here and get a 'galactic gravitational constant' of ~2.8e15 (just multiplying solar result above by 1.9e12). Divide by r=2.5e20 for 1.1e-5.

The orbital period equation is nifty, though... If all I'm seriously looking for is a means to measure the difference in 'temporal drift' between two points in the Galaxy, would that be a more reasonable estimate?

[Kuroneko]

How does that work without fudging (to a significant extent, observed and verified) masses?

Their thesis is that the only thing that's fudged is the mass of the dark matter, for which there is no measurement besides the stellar rotational curves run through Newtonian gravitation; if GTR reproduces the rotational curves without dark matter, it would follow only that there is no dark matter, or at least that there is only a negligible amount compared to the Newtonian prediction. How seriously one should take their model depends, in part, on the significance of the naked singularity in their model--I haven't followed up on it recently, so I'm not sure of what the overall physicists' opinion is on this matter.

I used the full mass as described here and get a 'galactic gravitational constant' of ~2.8e15 (just multiplying solar result above by 1.9e12). Divide by r=2.5e20 for 1.1e-5.

Ah. The 1.9e12M_☉ figure is more than I expected. In any case, that would treat all of the Milky Way as a point-mass at the center, which is a dubious method given that the Sun is inside the galaxy. For Andromeda, that's fine, but the 'effective mass' acting on the Sun is closer to 8.3e10M_☉ by Kepler's third law.

The orbital period equation is nifty, though... If all I'm seriously looking for is a means to measure the difference in 'temporal drift' between two points in the Galaxy, would that be a more reasonable estimate?

If you need a qualitative measurement for storyline reasons, that should be sufficient. As for quantitative measurements, I would not trust either approach because not only is Schwarzschild geometry inapplicable to something like a galaxy except at very large distances, but GTR is also non-linear, meaning one can't simply decompose the overall 'drift' into a sum of separate contributions by considering them in isolation.

[Ariphaos]

Their thesis is that the only thing that's fudged is the mass of the dark matter, for which there is no measurement besides the stellar rotational curves run through Newtonian gravitation; if GTR reproduces the rotational curves without dark matter, it would follow only that there is no dark matter, or at least that there is only a negligible amount compared to the Newtonian prediction. How seriously one should take their model depends, in part, on the significance of the naked singularity in their model--I haven't followed up on it recently, so I'm not sure of what the overall physicists' opinion is on this matter.

O_o That a naked singularity actually needs to be present or just the resulting model of one?

Ah. The 1.9e12M_☉ figure is more than I expected. In any case, that would treat all of the Milky Way as a point-mass at the center, which is a dubious method given that the Sun is inside the galaxy. For Andromeda, that's fine, but the 'effective mass' acting on the Sun is closer to 8.3e10M_☉ by Kepler's third law.

I get that - as much as 83 billion solar masses also seems frightfully low, given the seeming mass of the galaxy (there are later estimates but they specifically constrain the size of the halo).

If you need a qualitative measurement for storyline reasons, that should be sufficient. As for quantitative measurements, I would not trust either approach because not only is Schwarzschild geometry inapplicable to something like a galaxy except at very large distances, but GTR is also non-linear, meaning one can't simply decompose the overall 'drift' into a sum of separate contributions by considering them in isolation.

I don't really need to get too specific outside of much simpler situations like moving close to a neutron star or black hole. The setting takes place over timescales where small relativistic adjustments need to be accounted for even with 'human' observers, so it should at least be mentioned.

[Kuroneko]

That a naked singularity actually needs to be present or just the resulting model of one?

I'm not sure; I've not been keeping up with this development. If I recall correctly, C&T revised the model to exclude the any singularities, but I don't know how successful that was.

I don't really need to get too specific outside of much simpler situations like moving close to a neutron star or black hole. The setting takes place over timescales where small relativistic adjustments need to be accounted for even with 'human' observers, so it should at least be mentioned.

Ah. Well, in that case, this should be more than adequate.