Particles traveling back in time

[michael879]

*disclaimer* I am not suggesting any crazy theories here, I am merely examining some of the more extreme situations allowed in relativity!

ok so first off, let's allow particles to travel in timelike trajectories backwards through time relative to us. Although this does have some strange causal effects, relativity remains perfectly consistent as it is symmetric under time reversal. If we assume these particles have positive rest mass, then in our reference frame they would appear to have negative mass. I stress the word "appear" because the invariant mass remains positive. However, dt/dτ = -γ so you end up with negative signs every place "relativistic mass" appears.

Now, my question is if there is any conservation law (or any other mechanism to prevent this) broken by the spontaneous "creation" of two particles of equal mass, where one is traveling back in time and one is traveling forward in time. The "creation" point is the space-time point where the particle's worldlines intersect. Before this point space was empty, and afterwards we see two particles moving away from each other, one with positive energy and the other with negative energy. It is pretty trivial to show that both energy and momentum should be conserved in this process, and similarly in the time-reversed process (annihilation).

However, there is one thing I find very bothersome about this process, and I can't shake the feeling that it shouldn't be allowed on the grounds of conservation laws. If you draw the spacetime diagram of the process, you will see one particle going backwards in time from infinity, and then spontaneously start moving forwards in time back to infinity. How is it possible that momentum is conserved in a process where there is a clear change in direction of the particle!

The only explanation I can think of is that special relativity is actually degenerate in the sense that for any 4-momentum and initial conditions there are 2 possible worldlines that are "solutions" (the normal one and the time reversal of it). So what we're actually seeing here is a "spontaneous" switch between two intersecting worldlines with identical 4-momenta.

 

[HallsofIvy]

I'm not sure what you mean by "negative mass" or how you conclude that a particle, moving "backward in time" will have negative mass.

Richard Feynman once proposed (I'm not sure how seriously. I'm never sure how seriously to take Feynman!) that positron be considered an electron "moving back in time". But he then asserted that its energy, not its mass, would be negative.

By the way, Newtonian physics also has no "conservation" laws forbidding backward movement in time. There is, in thermal physics, a law about "increasing entropy" but that's a statistical law.

 

[michael879]

I think you should read my post more carefully, because I addressed everything you commented on already..

I thought I made it clear I wasn't talking about particles with negative rest mass. However, their relativistic mass (i.e. their energy) would be negative. So they would behave similarly (but not necessarily identically) to a particle traveling forward in time with negative mass. The derivation of why a particle with positive rest mass traveling backwards in time would have negative energy is very straightforward from basic SR.

I'm not asking for a law to forbid backward movement in time. In fact I'm making the assumption that no such law exists! I'm looking for a law that would forbid spontaneous creation of mass/negative "mass" pairs from nothing, if particles were allowed to travel back in time. Basically, if there is no mechanism to stop this spontaneous particle production (which we have never observed), then clearly there must be something preventing backwards time travel! Since relativity is perfectly symmetric under time reversal, it must come from some other theory/domain

 

[nitsuj]

I have no idea the consequence/meaning of what is said , but the use of dimensions to explain the stuff I don't understand is neat, imo time/length should explain all the mechanics as they apply to matter, and perhaps they do. An interesting read in other words

I'm looking for a law that would forbid spontaneous creation of mass/negative "mass" pairs from nothing, if particles were allowed to travel back in time.

It's a continuum, easily spoiled by semantics, wiki uses the term "quantitative transition", SR usually uses just c.

so what's backwards in time besides physically meaningless? Not so much a law, and probably circumnavigates your point.

Geometrically I think math shows the relationship between time & length. so maybe equally; what would it mean to go "backward" in length?

I wonder if "Foward / Backward" time are used the same as "up / down" in length, which is merely orientation. so forward in time must mean towards me, Backward in time is away from me lol. Or is it orientation of the matter itself is backward/forward in time and ONLY from a casual perspective the WHOLE universe can agree on, less spacetime expansions ect., yea i like that one

Oh and comparative geometric "backward" time is just that; a comparative & in 2D no less, where's the invariance of this metric?(invent can't go at or less than c, which the only thing I can see as being "backwards time/length")?

 

[HallsofIvy]

I think you should read my post more carefully, because I addressed everything you commented on already..

I thought I made it clear I wasn't talking about particles with negative rest mass. However, their relativistic mass (i.e. their energy) would be negative. So they would behave similarly (but not necessarily identically) to a particle traveling forward in time with negative mass. The derivation of why a particle with positive rest mass traveling backwards in time would have negative energy is very straightforward from basic SR.

Okay, do you have any reason to believe that there are particles with negative mass?

I'm not asking for a law to forbid backward movement in time. In fact I'm making the assumption that no such law exists! I'm looking for a law that would forbid spontaneous creation of mass/negative "mass" pairs from nothing, if particles were allowed to travel back in time. Basically, if there is no mechanism to stop this spontaneous particle production (which we have never observed), then clearly there must be something preventing backwards time travel! Since relativity is perfectly symmetric under time reversal, it must come from some other theory/domain

 

[michael879]

Again, see the FIRST thing I said in my thread. I'm not talking about reality here, I'm talking about the world governed by general relativity, classical E&M, and relativistic dynamics. It doesn't matter if negative mass particles DO exist, I'm examining what special relativity predicts IF they existed

 

[nitsuj]

Again, see the FIRST thing I said in my thread. I'm not talking about reality here, I'm talking about the world governed by general relativity, classical E&M, and relativistic dynamics. It doesn't matter if negative mass particles DO exist, I'm examining what special relativity predicts IF they existed

SR says there is no backwards in time, less you define "backwards in time" in some not backwards in time way . SR isn't independent of reality.

orientation of the matter itself is backward/forward in time and ONLY from a casual perspective the WHOLE universe can agree on, less spacetime expansions (spoiling the assumed metric) ect.

 

[michael879]

Well yes if you want to get technical there is no such thing as "backwards" or "forwards" in time in the absolute sense. I implicitly meant with respect to the "laboratory" frame though..

Ignoring that technicality, SR IS independent of reality. It's a mathematical model, which as far as we can tell accurately depicts reality in all domains. It is the ONLY theory that can make this claim though (arguments can be made for statistical mechanics and E&M but its not as clear cut), which is why I said I wanted to stay away from "reality" in this discussion of macroscopic physics.

My question is simply about a thought experiment conducted in a universe that follows our macroscopic laws of physics. Let me try to condense it further:

Is there any mechanism within the general relativistic framework preventing the spontaneous appearance of two identical particles with equal and opposite 4-momenta?
i.e. is there anything preventing a particle from spontaneously reversing its propagation through time?

 

[nitsuj]

Well yes if you want to get technical there is no such thing as "backwards" or "forwards" in time in the absolute sense. I implicitly meant with respect to the "laboratory" frame though..

Ignoring that technicality, SR IS independent of reality. It's a mathematical model, which as far as we can tell accurately depicts reality in all domains. It is the ONLY theory that can make this claim though (arguments can be made for statistical mechanics and E&M but its not as clear cut), which is why I said I wanted to stay away from "reality" in this discussion of macroscopic physics.

My question is simply about a thought experiment conducted in a universe that follows our macroscopic laws of physics. Let me try to condense it further:

Is there any mechanism within the general relativistic framework preventing the spontaneous appearance of two identical particles with equal and opposite 4-momenta?
i.e. is there anything preventing a particle from spontaneously reversing its propagation through time?

it's true as far as has been tested the rest is theoretical, what in SR hasn't been validated?

Its a physical technicality and not semantics and see it being as simple as a geometrical consequence of spacetime as we have measured it.

I don't know 4-momenta let alone the equal and opposite of it, I am of no use to this thread now , spectating...

 

[WannabeNewton]

http://en.wikipedia.org/wiki/Tachyon

You cannot spontaneously reverse time orientation along a time-like curve if space-time is time orientable. The notion of "future directed" along time-like curves must vary smoothly.

 

[PeterDonis]

Now, my question is if there is any conservation law (or any other mechanism to prevent this) broken by the spontaneous "creation" of two particles of equal mass, where one is traveling back in time and one is traveling forward in time.

I'm not sure this is actually distinguishable from a single particle existing from t = minus infinity to t = plus infinity. Or, more precisely, I think that for what you are suggesting to be consistent with conservation of energy-momentum, it could not be distinguishable from a single particle.

The "creation" point is the space-time point where the particle's worldlines intersect.

This part is OK, but what will the "two" worldlines look like? I don't think they'll look like you think they will.

Before this point space was empty, and afterwards we see two particles moving away from each other, one with positive energy and the other with negative energy.

I don't think this is how things will look, because as you describe it, to the observer, both particles are moving forward in time, so both of them will have positive energy. Energy is relative to the observer, not the particle. So this process does not conserve energy. (I'm assuming it conserves momentum by hypothesis, with the particles having equal rest mass and velocities of equal magnitude and opposite directions.)

For the second particle to have negative energy while still having positive rest mass, the time component of its 4-velocity, in the observer's frame, must be negative. But this means that the direction of increasing proper time for the particle is the *negative* time direction. Which means that we really have a single worldline, going from t = minus infinity to t = plus infinity; and the "spontaneous creation" event is just a particular event on the worldline, with the first particle's worldline being the portion to the future of this event, and the second particle's worldline being the portion to the past of the event. (The worldline can't "change direction" at the event because momentum has to be conserved.)

 

[michael879]

it's true as far as has been tested the rest is theoretical, what in SR hasn't been validated?

Well, SR and GR allow for tachyons, closed timelike curves, time travel, and plenty of other things people rule out as "unphysical". However, there is nothing in the theories themselves that prevents these phenomenon. Also, just because we've validated a theory in all the domains accessible to us doesn't mean its valid in all domains. You can't claim SR and reality are completely compatible because that can never be known for sure. This is exactly why I don't want to bring "reality" into this discussion at all. It's a purely theoretical question about a physical model we have.

 

[michael879]

I'm not sure this is actually distinguishable from a single particle existing from t = minus infinity to t = plus infinity. Or, more precisely, I think that for what you are suggesting to be consistent with conservation of energy-momentum, it could not be distinguishable from a single particle.

This part is OK, but what will the "two" worldlines look like? I don't think they'll look like you think they will.

Sorry, I know exactly what you're thinking and where the confusion is. I just wish I could draw a picture on here more easily to clarify. Hopefully the attached picture will help a little bit.

The particle to the left is traveling through time in the opposite direction as we are, so in our frame it has negative energy.

 

[nitsuj]

Well, SR and GR allow for tachyons, closed timelike curves, time travel, and plenty of other things people rule out as "unphysical". However, there is nothing in the theories themselves that prevents these phenomenon. Also, just because we've validated a theory in all the domains accessible to us doesn't mean its valid in all domains. You can't claim SR and reality are completely compatible because that can never be known for sure. This is exactly why I don't want to bring "reality" into this discussion at all. It's a purely theoretical question about a physical model we have.

Good point

 

[DrGreg]

Sorry, I know exactly what you're thinking and where the confusion is. I just wish I could draw a picture on here more easily to clarify. Hopefully the attached picture will help a little bit.

The particle to the left is traveling through time in the opposite direction as we are, so in our frame it has negative energy.

But on the diagram the particle on the right has reversed energy but not reversed momentum, i.e. you've drawn it in the wrong direction.

 

[michael879]

But on the diagram the particle on the right has reversed energy but not reversed momentum, i.e. you've drawn it in the wrong direction.

No its drawn correctly, I think the arrows are just misleading you. The y-axis is the lab time, and the arrows just show the worldlines of the particles. In the lab frame the left particle is moving to the left, and the right particle is moving to the right.

 

[PeterDonis]

Hopefully the attached picture will help a little bit.

This picture matches what I imagined you to be describing, but I see now that I didn't fully understand how you were imagining the signs of the energy and momentum of the two particles. Basically, you're exploiting the fact that the dispersion relation $E^2 - p^2 = m^2$ only has squares of the energy and momentum in it, so we can switch the signs of either $E$ or $p$ without violating it.

At first blush, there is nothing in this that violates conservation laws, because the net energy and momentum is zero everywhere! Before the "spontaneous creation" event, there are no particles, for a net energy and momentum of zero; after that event, there are two particles with net energy $E + (- E ) = 0$ and net momentum $p + ( - p ) = 0$. Classically, nothing at all has changed. But see below for a wrinkle you might not have considered.

(Quantum mechanically, of course, the Feynman picture of the creation of particle-antiparticle pairs out of the vacuum is similar to what you're describing, as HallsOfIvy mentioned. The difference there is that the particle and antiparticle have to come back together and annihilate each other within the time/space limits of the uncertainty principle, which means that they can't be on the mass shell, i.e., they don't satisfy the dispersion relation I wrote down above. You are describing a classical process where both particles *are* on the mass shell, which guarantees, as I said above, that the net energy and momentum is zero everywhere in the spacetime.)

The wrinkle, though, is this: what happens if we change frames? Suppose we boost into the rest frame of the positive energy particle. Its 4-momentum in this frame is $(m, 0)$. What is the 4-momentum of the negative energy particle in this frame? We don't even have to calculate it explicitly; it should be obvious that it's going to be $m (- \gamma', - \gamma' w)$, where $- w$ is the negative energy particle's ordinary velocity in the new frame and $\gamma'$ is the gamma factor associated with $w$. This violates conservation of *both* energy and momentum in the new frame, since before the spontaneous creation event the net energy and momentum is zero, and after it the net energy is $m (1 - \gamma')$ and the net momentum is $- m \gamma' w$, neither of which are zero (in fact, both are negative).

So it looks like the key issue with your proposal is that it violates 4-momentum conservation in every frame except the center of mass frame.

 

[harrylin]

Well, SR and GR allow for tachyons, closed timelike curves, time travel, and plenty of other things people rule out as "unphysical". However, there is nothing in the theories themselves that prevents these phenomenon. Also, just because we've validated a theory in all the domains accessible to us doesn't mean its valid in all domains. You can't claim SR and reality are completely compatible because that can never be known for sure. This is exactly why I don't want to bring "reality" into this discussion at all. It's a purely theoretical question about a physical model we have.

There is nothing in the mathematical toolbox that SR uses to prevent it; however, as a physical theory it is based on such things as effects following causes, and with time defined by means of clock readings that increment only.

 

[OCR]

I'm not sure this is actually distinguishable from a single particle existing from t = minus infinity to t = plus infinity. Or, more precisely, I think that for what you are suggesting to be consistent with conservation of energy-momentum, it could not be distinguishable from a single particle.

Yup... another guy said that, and about the same way.

http://en.wikipedia.org/wiki/One-electron_universe

http://en.wikipedia.org/wiki/Identical_particles

http://en.wikipedia.org/wiki/John_Archibald_Wheeler




OCR

 

[michael879]

There is nothing in the mathematical toolbox that SR uses to prevent it; however, as a physical theory it is based on such things as effects following causes, and with time defined by means of clock readings that increment only.

Causality is really an outdated notion. There are theories that exploit and allow retrocausality. You might just call these mathematical "tricks", but since they are just different interpretations of the same theory, whose to say which is "correct"?

Examples:
Feynman and Wheeler's absorber theory
General Relativity (CTCs)
Retrocausal interpretation of QM
QFT! (some off-shell virtual particles, and any anti-matter can be viewed as traveling backwards in time)

 

[michael879]

This picture matches what I imagined you to be describing, but I see now that I didn't fully understand how you were imagining the signs of the energy and momentum of the two particles. Basically, you're exploiting the fact that the dispersion relation $E^2 - p^2 = m^2$ only has squares of the energy and momentum in it, so we can switch the signs of either $E$ or $p$ without violating it.

At first blush, there is nothing in this that violates conservation laws, because the net energy and momentum is zero everywhere! Before the "spontaneous creation" event, there are no particles, for a net energy and momentum of zero; after that event, there are two particles with net energy $E + (- E ) = 0$ and net momentum $p + ( - p ) = 0$. Classically, nothing at all has changed. But see below for a wrinkle you might not have considered.

I mean yes, you could ignore the whole time travel aspect of this problem and just ask about negative energy particles, nothing would change. However the energy and momentum of the negative particle in this case weren't just chosen to be negative! I started with a 4-vector of a particle traveling backwards in time and calculated its energy and momentum in our frame (p = mdv/dτ=md²x/dτ²). It's a trivial calculation, but its a lot more rigorous than just looking at the dispersion relation and saying "hey, I could change the sign of E and p if I wanted to!" :P

(Quantum mechanically, of course, the Feynman picture of the creation of particle-antiparticle pairs out of the vacuum is similar to what you're describing, as HallsOfIvy mentioned. The difference there is that the particle and antiparticle have to come back together and annihilate each other within the time/space limits of the uncertainty principle, which means that they can't be on the mass shell, i.e., they don't satisfy the dispersion relation I wrote down above. You are describing a classical process where both particles *are* on the mass shell, which guarantees, as I said above, that the net energy and momentum is zero everywhere in the spacetime.)

The wrinkle, though, is this: what happens if we change frames? Suppose we boost into the rest frame of the positive energy particle. Its 4-momentum in this frame is $(m, 0)$. What is the 4-momentum of the negative energy particle in this frame? We don't even have to calculate it explicitly; it should be obvious that it's going to be $m (- \gamma', - \gamma' w)$, where $- w$ is the negative energy particle's ordinary velocity in the new frame and $\gamma'$ is the gamma factor associated with $w$. This violates conservation of *both* energy and momentum in the new frame, since before the spontaneous creation event the net energy and momentum is zero, and after it the net energy is $m (1 - \gamma')$ and the net momentum is $- m \gamma' w$, neither of which are zero (in fact, both are negative).

So it looks like the key issue with your proposal is that it violates 4-momentum conservation in every frame except the center of mass frame.

Thank you Peter, this was exactly the kind of answer I was looking for! I'm a little confused by it though and I need to think about it a little more.. Isn't the conservation of 4-momentum Lorentz invariant?? If what you're saying is true, then this would of course prevent the spontaneous creation of particles. However I find it a little shocking that 4-momentum can be conserved in 1 frame but not another..

 

[michael879]

Yea I think you're wrong Peter. Let's say the initial 4 momenta are (γm,γmv) and (-γm,-γmv). They only differ by an overall sign so any Lorentz transformation will affect them identically! Therefore 4-momentum would be conserved in ALL frames

 

[harrylin]

Causality is really an outdated notion. There are theories that exploit and allow retrocausality.

I was talking about how "time" is defined in SR and GR, which are theories about observables based on operational definitions. The result is as some others here already mentioned: particle speeds over a particle's trajectory are positive by definition - positive displacement during positive time interval. Other theories may have it differently, but that is irrelevant for this thread and this forum.

 

[michael879]

I was talking about how "time" is defined in SR and GR, which are theories about observables based on operational definitions. The result is as some others here already mentioned: particle speeds over a particle's trajectory are positive by definition - positive displacement during positive time interval. Other theories may have it differently, but that is irrelevant for this thread and this forum.

I'm not really sure what point you're trying to make here. Yes, a particle has a positive displacement over a positive proper time interval. Nowhere did I suggest that isn't the case.. And like I said the only theory I'm directly addressing here is Special Relativity. GR and E&M can enter into the discussion as they form a self-consistent set of theories with SR, but the question itself lives entirely in the SR domain.

 

[nitsuj]

Causality is really an outdated notion. There are theories that exploit and allow retrocausality. You might just call these mathematical "tricks", but since they are just different interpretations of the same theory, whose to say which is "correct"?

Examples:
Feynman and Wheeler's absorber theory
General Relativity (CTCs)
Retrocausal interpretation of QM
QFT! (some off-shell virtual particles, and any anti-matter can be viewed as traveling backwards in time)

Certainly not from the perspective of reality, anything else is insane.

CTCs is not "traveling back" in time, a light trick of sorts.

In what sense does QM reflect GR/SR? Certainly not dimensionally. I understand GR/SR to be "geometric" theories. The one that's "Correct" is the one even I can understand, the simply one.

I'll have to check out absorber theory, but still is merely theory.

the orientation of the dimensions limits a metric to a speed when those dimensions are properly defined. How then can any computation of length/time intervals be invariant if one object magically doesn't follow this structure, this orientation of the dimensions? It ruins the c postulate, which then ruins the rest of SR, the thing you say supports these ideas.

 

[michael879]

Certainly not from the perspective of reality, anything else is insane.

Classically, yes. It is clear that any retrocausal effects at the classical level are entirely hidden from us. However, retrocausality is perfectly capable of explaining the strangeness of QM, and its entirely possible that QM effects are actually cases of time travel.

On a side note, retrocausality in general can cause paradoxes (I'm not sure of this but I'd assume so). However, there are many situations in which retrocausality works fine. For example in SR, retrocausality is allowed and doesn't break the theory. No real paradoxes will form when you allow it.

CTCs is not "traveling back" in time, a light trick of sorts.

I'm not sure what you mean by a light trick. You can interpret the predictions of GR however you like, but its a fact that there are geometries allowed within GR that contain CTCs, and you can manipulate these geometries to travel through time (within the context of GR, not reality).

This statement reminds me of the claim people tend to make about falling into a black hole. An observer falling in crosses the event horizon and reaches the center in a finite time. An observer at infinity sees the other get closer and redder to the event horizon and never pass through it. The common interpretation of this result is that only the light emitted from the falling object gets "stuck" at the event horizon and the observer is merely seeing a shadow of something that has already fallen in.

However, this interpretation is completely wrong according to the principles of GR! The falling object never crosses the event horizon in the reference frame of the observer at infinity. For example, imagine the falling object can get a sudden boost away from the black hole and meet up with the observer at "infinity". For the observer at infinity this will always be a possibility, so he can never say for certain whether or not the object has fallen in (although the math says that in his frame it hasn't).

In what sense does QM reflect GR/SR? Certainly not dimensionally. I understand GR/SR to be "geometric" theories. The one that's "Correct" is the one even I can understand, the simply one.

I never said QM reflects GR/SR I was just giving examples of retrocausality in modern physics

I'll have to check out absorber theory, but still is merely theory.

It's more of an interpretation than a theory. Its actually very interesting, and removes many of the unsettling parts in classical E&M. The retrocausal effects are entirely hidden though, so its not like you can purposefully send information through time.

the orientation of the dimensions limits a metric to a speed when those dimensions are properly defined. How then can any computation of length/time intervals be invariant if one object magically doesn't follow this structure, this orientation of the dimensions? It ruins the c postulate, which then ruins the rest of SR, the thing you say supports these ideas.

I have no idea what you mean by this?


This is really off topic, but an interesting discussion. So if you want to talk about it more can you move it to a new thread or PM?

 

[harrylin]

I'm not really sure what point you're trying to make here. Yes, a particle has a positive displacement over a positive proper time interval. Nowhere did I suggest that isn't the case.. And like I said the only theory I'm directly addressing here is Special Relativity. GR and E&M can enter into the discussion as they form a self-consistent set of theories with SR, but the question itself lives entirely in the SR domain.

The same points as nitsuj and others. A particle that has coordinates (0,0,0) at t=0 and (1,0,0) at t=1, moved according to SR with speed 1 along x from x=0 to x=1. According to SR it did not move from x=1 to x=0, simply because the clocks were constructed to accumulate time and the synchronization is done following convention.

 

[HallsofIvy]

Again, see the FIRST thing I said in my thread. I'm not talking about reality here, I'm talking about the world governed by general relativity, classical E&M, and relativistic dynamics.

So you are sayjng that you don't believe the "real" world is governed by "general relativity, classical E&M, and relativistic dynamics"?

It doesn't matter if negative mass particles DO exist, I'm examining what special relativity predicts IF they existed

 

[PeterDonis]

Yea I think you're wrong Peter. Let's say the initial 4 momenta are (γm,γmv) and (-γm,-γmv). They only differ by an overall sign so any Lorentz transformation will affect them identically!

Yes, I see the argument mathematically, but it doesn't make sense physically, because the two particles are moving in opposite directions, so they can't possibly both be at rest in the same frame, which is what would be required to conserve both energy and momentum. Just look at the picture you drew: it's impossible for both particle's worldlines to be the time axis of the same inertial frame, since they have different slopes. So either the picture is wrong, or the argument you've made based on the math of the Lorentz transformation is wrong.

My previous post assumed that the LT math argument was wrong, that a "correct" transformation of the negative energy particle's 4-momentum, as drawn in your picture, would be as I said. However, there is another way out: perhaps your picture is wrong, and *both* worldlines slope up and to the right, with the same slope! This would match the LT math and would allow 4-momentum to be conserved in all frames; but it also means that, as I said before, the situation you describe is classically equivalent to nothing being there at all--but now in *all* frames. We basically would have a negative energy worldline and a positive energy worldline overlapping each other and canceling each other out.

 

[PeterDonis]

For example, imagine the falling object can get a sudden boost away from the black hole and meet up with the observer at "infinity". For the observer at infinity this will always be a possibility

Somewhat off topic for this thread, but no, it won't, because if the falling object stops falling at some point, the light emitted outward from it will look different: its redshift will stop increasing. That difference is observable at a distance.

 

[nitsuj]

I'm not sure what you mean by a light trick. You can interpret the predictions of GR however you like, but its a fact that there are geometries allowed within GR that contain CTCs, and you can manipulate these geometries to travel through time (within the context of GR, not reality).This statement reminds me of the claim people tend to make about falling into a black hole. An observer falling in crosses the event horizon and reaches the center in a finite time.

By light trick I mean in a CTC I (me) cannot go back in time in any sense. I could "see" an earlier version of myself that I should not be able to see, unlike my reflection in a mirror.

I've said it in another thread, there is only one of me no matter which metric you chose. It's these types of "technicalities" that seems to spoil backward time travel, logically however.

For that last part, in other words c is a very important part of spacetime/geometry (length/time). It also means within SR there is no faster then c. So even if we idealize away the physics and allow a particle to go faster then c, motion is still relative. Faster then c is then not invariant, and negative time makes no sense as a measurement. In turn this new invented speed (metric and mechanical physics as a whole) is frame dependent, so ruins invariance. You could postulate a minimum speed though, I think that's what tachyons things have.

Woo hoo I finally get to "use" this equivalence lol, "falling" into a black hole reminds me of discussions about what would be like to travel at c.

 

[michael879]

Somewhat off topic for this thread, but no, it won't, because if the falling object stops falling at some point, the light emitted outward from it will look different: its redshift will stop increasing. That difference is observable at a distance.

Yes but the "decision" to turn around and escape can be made at any point, meaning until t=infinity the observer won't know if the object fell in or not. Anyway this is off topic so let's not digress

 

[michael879]

I've said it in another thread, there is only one of me no matter which metric you chose. It's these types of "technicalities" that seems to spoil backward time travel, logically however.

Yes there is only one of you (classically), but its not the 3 dimensional object you are thinking of. Rather its a very complicated 4 dimensional object that CAN (in theory) come back and "interact" with itself (interact = intersect). You think photons are the only things that can fall into CTCs? You can send fermions into CTCs just as easily. Now while a single particle might not be able to interact with its past self (I have no view on whether or not it can), surely each particle in your body can interact with every other past and future particle!

 

[michael879]

So you are sayjng that you don't believe the "real" world is governed by "general relativity, classical E&M, and relativistic dynamics"?

I'm saying whether or not it is is completely irrelevant to this entire thread... I don't care what is "realistic" I'm only asking about what a world perfectly described by SR would allow.

To answer your question though, the real world is NOT governed by these classical theories, as nearly every person in the world remotely interested in physics can tell you. Classical E&M is flat out wrong (without modifying it to be quantized), and general relativity has so many problems with it most physicists believe it has to be wrong (CTCs, singularities, non-quantized fields, etc.).

 

[michael879]

Yes, I see the argument mathematically, but it doesn't make sense physically, because the two particles are moving in opposite directions, so they can't possibly both be at rest in the same frame, which is what would be required to conserve both energy and momentum. Just look at the picture you drew: it's impossible for both particle's worldlines to be the time axis of the same inertial frame, since they have different slopes. So either the picture is wrong, or the argument you've made based on the math of the Lorentz transformation is wrong.

My previous post assumed that the LT math argument was wrong, that a "correct" transformation of the negative energy particle's 4-momentum, as drawn in your picture, would be as I said. However, there is another way out: perhaps your picture is wrong, and *both* worldlines slope up and to the right, with the same slope! This would match the LT math and would allow 4-momentum to be conserved in all frames; but it also means that, as I said before, the situation you describe is classically equivalent to nothing being there at all--but now in *all* frames. We basically would have a negative energy worldline and a positive energy worldline overlapping each other and canceling each other out.

Yes I agree this is bizarre, but I can't argue with the very simple math involved.. My only thought is that maybe the derivation of the Lorentz transformations assumes everything is moving in the same direction through time, and some ± was dropped somewhere? It's really simple to see though, if you boost a particle (-E, -p) by v (the particle is moving in the -v direction) you find that the resulting 4-momentum is (-m,0)! Basically if a particle has negative energy and you perform a Lorentz boost in 1 direction, it will speed up in that direction instead of slow down!

I might try rederiving the Lorentz boost to see if I can find some badly made assumption on energies always being positive...

However, as strange as this result is, wouldn't it be much stranger for an event to conserve 4-momentum in 1 frame but not another? Tha completely breaks Lorentz invariance!

 

[nitsuj]

Yes there is only one of you (classically), but its not the 3 dimensional object you are thinking of. Rather its a very complicated 4 dimensional object that CAN (in theory) come back and "interact" with itself (interact = intersect).

Weird, so the matter is duplicated. I can't for the life of me see this as conceivable. To your point the dimensions cannot be separated (4D), taking me from light path a in space and place me in a earlier place along that time path in space. Time cannot circumnavigate a length & vice versa.

I don't find 4D complicated and am not sure which 3D object you have me thinking of, or how the alternative (4D) differs. If you can see so clearly the 3D -4D distinction how can the absurdity of backward time travel even be considered. It can intersect at the same instant. interact, no. Go "before" the instant, no not even close.

There is only one of me no matter the metric, this is up to and including GR. Another way to say it there is only one "present moment" for any object in spacetime.

 

[WannabeNewton]

Any Lorentz boost can be decomposed into consecutive infinitesimal transformations belonging to the connected component of $O(3,1)$ which contains the identity transformation (the decomposition will start with the identity transformation). The time reversal transformation (which is non-orthochronous by definition) belongs to a connected component which does not contain the identity. Lorentz boosts are orthochronous transformations in the above sense i.e. they preserve the notion of "future-directed" or "past-directed" of vectors.

 

[PeterDonis]

if you boost a particle (-E, -p) by v (the particle is moving in the -v direction) you find that the resulting 4-momentum is (-m,0)!

Yes, and that indicates to me that the alternate solution in my previous post is the correct one: your picture was wrong. A particle with negative energy and negative x-momentum must be moving in the *positive* x-direction, not the negative x-direction. So *both* particles are moving in the positive x-direction from the spontaneous creation event--i.e., both worldlines slope up and to the *right* in your picture.

Basically if a particle has negative energy and you perform a Lorentz boost in 1 direction, it will speed up in that direction instead of slow down!

That's not quite right; what happens is that the meaning of spatial momentum is reversed for a particle with negative energy, as above.

Here's another way of seeing it: look at the 4-velocity of such a particle (which, with a constant rest mass, is just the normalized 4-momentum). The 4-velocity components in a given frame are $(dt / d\tau, dx / d\tau)$. For a particle with negative energy, $dt / d\tau$ is negative; and for a particle with negative x-momentum, $dx / d\tau$ is negative. But that means that $dx / dt = (dx / d\tau) / (dt / d\tau)$ is *positive*; which means, as I said above, that a negative energy particle with negative x-momentum is moving in the positive x-direction, just like a positive energy particle with positive x-momentum. So both worldlines overlap completely; that allows the two particles' energy and momentum to cancel each other out in all frames.

This is indeed somewhat bizarre, as you say, but I agree it's much less bizarre than violating the invariance of 4-momentum conservation.

 

[michael879]

Yes, and that indicates to me that the alternate solution in my previous post is the correct one: your picture was wrong. A particle with negative energy and negative x-momentum must be moving in the *positive* x-direction, not the negative x-direction. So *both* particles are moving in the positive x-direction from the spontaneous creation event--i.e., both worldlines slope up and to the *right* in your picture.

Ok I see what your saying, it makes perfect sense, but I can't get it to come out. If you're right can you explain where the flaw in my logic is?

left particle:
x' = (t',x')
u' = dx'/dτ = -γ(1, dx'/dt')
p' = mu' = -γm(1, dx'/dt')

right particle:
x = (t,x')
u = dx/dτ = γ(1, dx/dt)
p = mu = γm(1, dx/dt)

dx/dt = dx'/dt' = v

Therefore the 3-velocity of the left particle would be -v and it should be moving to the left...

*edit* Similarly: dx'/dt = dx'/dt' * dt'/dτ * dτ/dt = v * (-γ) * (1/γ) = -v

 

[PeterDonis]

can you explain where the flaw in my logic is?

I don't think there's any flaw in what you wrote up to your final paragraph (see below), except that your variable naming seems a little confusing; the variable "v" is usually used to mean dx/dt, not dx/dtau, but that's a detail. You're just agreeing with me that both particles have positive dx/dt. But that means both of their worldlines will slope up and to the right in a spacetime diagram; your diagram had the negative energy particle's worldline sloping up and to the left.

Therefore the 3-velocity of the left particle would be -γv and it should be moving to the left...

No, its 3-velocity is its dx/dt, not its dx/dtau. Both particles have a positive 3-velocity.

 

[michael879]

Sorry, you commented while I was in the middle of editing my post, please reread it? v is defined as dx/dt of the right particle

*edit* I just noticed I used v for both 4-velocity and 3-velocity. I replaced it with "u" for 4-velocity

 

[michael879]

No, its 3-velocity is its dx/dt, not its dx/dtau. Both particles have a positive 3-velocity.

The 3-velocity would be dx'/dt not dx'/dt', and since dt = -dt' you end up with a negative velocity..

 

[PAllen]

The 3-velocity would be dx'/dt not dx'/dt', and since dt = -dt' you end up with a negative velocity..

You don't use a separate reference frame for each particle. You can represent both particles in anyone reference frame, but it is nonsensical to combine statements from two different frames. Thus, Peter is right.

 

[PeterDonis]

The 3-velocity would be dx'/dt not dx'/dt', and since dt = -dt' you end up with a negative velocity..

I'm getting confused with your variables; I'm not sure what the primed and unprimed ones are supposed to mean. But I think dt and dt' are just the coordinate time differentials for the two particles; and those are both positive, because we're looking at both particles' worldlines going into the future, not the past. In other words, dt and dt' refer to two different differentials of the *same* coordinate time, t; they do not refer to differentials of two different time coordinates. (I see PAllen has made this point too.)

To make this clearer, let me go back to the two 4-momentum vectors, which are the fixed points we know we agree on. They are $(E, p)$ for the positive energy particle and $(- E, - p)$ for the negative energy particle. Factoring out the rest mass $m$, which is constant, we have the two 4-velocities $(\gamma, \gamma v)$ and $(- \gamma, - \gamma v)$. So far, so good.

But what do the 4-velocity components mean, physically? They are rates of change of the $t$ and $x$ coordinates with respect to *proper* time, right? So if we label the positive energy particle's coordinates as $(t_p, x_p)$, and the negative energy particle's coordinates as $(t_n, p_n)$, we have

$$
\frac{dt_p}{d\tau_p} = \gamma
$$
$$
\frac{dx_p}{d\tau_p} = \gamma v
$$

for the positive energy particle, and

$$
\frac{dt_n}{d\tau_n} = - \gamma
$$
$$
\frac{dx_n}{d\tau_n} = - \gamma v
$$

for the negative energy particle. These are easily integrated to obtain parametrizations of the two worldlines:

$$
(t_p, x_p) = \tau_p ( \gamma, \gamma v)
$$
$$
(t_n, x_n) = \tau_n ( - \gamma, - \gamma v)
$$

But we still have two $\tau$ variables; how are they related? We'll assume they are scaled the same, so the only question is the sign. Suppose the signs are the same. Then we have

$$
(t_p, x_p) = \tau ( \gamma, \gamma v)
$$
$$
(t_n, x_n) = \tau ( - \gamma, - \gamma v)
$$

If we assume that $\tau = 0$ is the spontaneous creation event, and that $\tau > 0$ describes the two worldlines after that event, then we have a problem: the two worldlines don't describe the "V" shape you drew. Instead, they describe two segments of an infinite single worldline from t = minus infinity to t = plus infinity, with the "spontaneous creation" event simply being whichever event on this single worldline we pick as the origin of $\tau$.

So now suppose the signs are opposite; then we have
$$
(t_p, x_p) = \tau ( \gamma, \gamma v)
$$
$$
(t_n, x_n) = \tau ( \gamma, \gamma v)
$$

In other words, the two worldlines both go into the future (increasing $t$) from the spontaneous creation event, but they overlap--they are identical, so they cancel each other out.

The upshot of all this is that the picture you drew can't be right, no matter which assumption we make about $\tau$. This "negative energy particle" business can't possibly describe a "V" shaped pair of worldlines; it can only describe either a single fully infinite worldline, or two half-infinite worldlines that overlap and cancel.

 

[michael879]

If we assume that $\tau = 0$ is the spontaneous creation event, and that $\tau > 0$ describes the two worldlines after that event, then we have a problem: the two worldlines don't describe the "V" shape you drew. Instead, they describe two segments of an infinite single worldline from t = minus infinity to t = plus infinity, with the "spontaneous creation" event simply being whichever event on this single worldline we pick as the origin of $\tau$.

In other words, the two worldlines both go into the future (increasing $t$) from the spontaneous creation event, but they overlap--they are identical, so they cancel each other out.

The upshot of all this is that the picture you drew can't be right, no matter which assumption we make about $\tau$. This "negative energy particle" business can't possibly describe a "V" shaped pair of worldlines; it can only describe either a single fully infinite worldline, or two half-infinite worldlines that overlap and cancel.

Again, I understand what you're saying and I really appreciate the detailed, well thought out response. However, why would the intersecting wordlines necessarily need to go from -∞ to ∞? Aren't all conservation laws obeyed if both particles simply cease to exist at the intersection point? This would be the time-reversal of my original question.

Now I understand that so far we've assumed the particles are non-interacting, and an annihilation of both would be a type of interaction. So what about instead of annihilation we allow the particles to have some type of elastic interaction with each other? At the "collision" point there would be an infinite number of possible outcomes that preserve both energy and momentum (which one would depend on the details of the interaction)! One of these possible outcomes would be both particle coming to rest, where they would overlap and effectively "annihilate" leaving space-time empty again! I don't see any law preventing this scenario...

 

[PeterDonis]

However, why would the intersecting wordlines necessarily need to go from -∞ to ∞?

I didn't assume they would; I derived that conclusion for the case where $\tau_p = \tau_n = \tau$. The only assumption I made, other than the intial 4-momentum vectors, was that each worldline is "half-infinite", which is what you postulated in your original scenario. The rest pops out of the math, no assumptions required.

Aren't all conservation laws obeyed if both particles simply cease to exist at the intersection point? This would be the time-reversal of my original question.

Sure, in my notation that would be the case where $\tau_p = - \tau_n = \tau$ and $\tau <= 0$ for both worldlines ($\tau = 0$ would be the "spontaneous destruction" event in this case). And in that case, we would have two overlapping worldlines coming from t = minus infinity and cancelling each other out. We still wouldn't have a "V" shaped pair of worldlines.

At the "collision" point there would be an infinite number of possible outcomes that preserve both energy and momentum (which one would depend on the details of the interaction)!

Sure, this just corresponds to the infinite number of possible choices for $v$ in the equations I wrote down. But the conclusions remain the same; what I wrote down is valid for any $-1 < v < 1$. (I put the choice in terms of $v$ rather than $\gamma$ because the sign of $v$ can be positive or negative, as I just showed, so the mapping from $v$ to $\gamma$ is not invertible--a given $v$ and $- v$ map to the same $\gamma$.)

Once again, the only assumptions I made were the initial 4-momentum vectors, which have to sum to zero net energy and momentum by the conservation laws, and that each particle's worldline was "half-infinite". Everything else follows from that. You appear to be looking for extra degrees of freedom in the problem that aren't actually there.

 

[michael879]

I didn't assume they would; I derived that conclusion for the case where $\tau_p = \tau_n = \tau$. The only assumption I made, other than the intial 4-momentum vectors, was that each worldline is "half-infinite", which is what you postulated in your original scenario. The rest pops out of the math, no assumptions required.

You couldn't have assumed that they were half-infinite because your conclusion was that they were both infinite!

 

[PeterDonis]

You couldn't have assumed that they were half-infinite because your conclusion was that they were both infinite!

No, that wasn't my conclusion. Look at the parametrizations I wrote down, carefully. For the case $\tau_p = \tau_n = \tau$, we have

$$
(t_p, x_p) = \tau ( \gamma, \gamma v)
$$
$$
(t_n, x_n) = \tau ( - \gamma, - \gamma v)
$$

where $\tau >= 0$ for both worldlines (and $\tau = 0$ is the point where they meet). So the positive energy particle's worldline starts at $(t, x) = (0, 0)$ and goes to t = plus infinity, and the negative energy particle's worldline starts at $(t, x) = (0, 0)$ and goes to t = minus infinity. Both are half-infinite.

(Of course this could equally well be interpreted as describing a single positive energy particle going from t = minus infinity to t = plus infinity. That was part of my point. But if you insist on calling the part with negative t the worldline of a "negative energy particle", then we have two "particles" each with a half-infinite worldline.)

 

[michael879]

ooo ok I think I realized my mistake. I was confusing proper time with the time coordinate (i.e. I was using the "lab time" coordinate as τ and two different time coordinates t and t'). Thank you, this answer makes far more sense and makes the concept of particles traveling backwards in time much more sensible. Like you said, to conserve energy and momentum the "spontaneous" creation of positive/negative particles must combine into a single worldline that is either (E,p) or (0,0). This essentially forbids the process, which means you can allow particles to travel back in time and remain consistent with both theory and experiment!

 

[remmeler]

How do you reconcile the problems of going backward in time

How can you reconcile the problems with time travel
Everyone know about killing you grandfather but how about -

Traveling 5 days back in time and for some reason not being able to travel forward except in the normal way of 1 day at a time.

There would be two of you existing in that time, but in 5 days your original self would travel back again and now there would be two of you who have traveled back and so forth.

The only way it could happen would be that you traveled back in time and ended up in that part of the multi universe where you traveled back in time.

Also there is the problem that no one has ever met someone that had information that would prove that they traveled back in time.

Just a little stupid thought I had, just to see if anyone has any thoughts on it.