How does the twin paradox work with wormholes?

Question

Wormholes are connections of different points in space-time which occur due to bending of the space time geometry. Like a piece of paper can be folded and a hole can be created to connect two points on the paper, which causes the distance between those two points to reduce to zero by creating a pathway directly from one point to another point. There are no experimental evidences to prove the existence of wormholes, however there are some theoretical solutions for wormholes.

I will describe a thought experiment which questions the existence of wormholes. I haven’t yet found out the flaw in the thought experiment or proper explanation other than that wormholes can’t exist. It would be great if I can get suggestions regarding the matter.

The idea goes thus:

Suppose two twins Howard and Raj are at the same point A in space-time. One day Howard gets the chance to travel in space (Basically the same setup as in the twin paradox). Howard boards a spaceship and starts moving away from point A in a straight line at a speed comparable to the speed of light. When he reaches point B in his journey (still travelling) the statements recorded by both of them are:

Raj: Howard has reached point B and time on my clock is 5 hours and time on Howard’s clock must be 3 hours (assume that Howard is moving with velocity corresponding to the above time intervals). So he is younger than me. Howard: Raj is moving away from me and hence he is younger than me. Howard is still moving at constant speed and hence both the frames are inertial. This is a paradoxical situation that who is correct and who is wrong. But on closer observation we find that there is no paradox. The two statements are not simultaneous according to relativity of simultaneity. Suppose event P is the event when Raj’s clock ticks 5 hours and event Q be the event in which Howard reaches point B. These two events are simultaneous in reference frame of Raj, but they are not simultaneous in Howard’s reference frame i.e. when Raj’s clock ticks 5 hours according to Howard, he is not at point B.

Now comes the main part of this thought experiment. Suppose if there is a wormhole at point B which connects directly to point A. Now Howard reaches the same point in space-time where Raj is, and still moving with constant velocity so that his frame is still inertial. Now what happens? Both of them again give the same statement that the other one is younger than me. As they have reached the same point in space-time the two statements must be simultaneous!

If we consider that wormhole connects two points only through space and not through time, than who is younger? If wormholes connects two points through both space & time, does that mean they have time travelled with respect to one another?

Answer 1

This question has nothing to do with wormholes; it's actually about Lorentz symmetry.

In special relativity, there is global Lorentz symmetry, in the sense that global inertial frames exist and physics is independent of which of those frames you work in. As a result, there's no notion of who's 'really' moving in the twin paradox.

General relativity breaks this principle in just about every case. For example, the expanding universe has a preferred frame of reference, the comoving frame, and you can detect motion relative to it. A toroidal universe also breaks global Lorentz symmetry, since the torus cells look different in different frames.

Your situation is like the torus. To see directly why it breaks Lorentz symmetry, note that you implicitly used the fact that the wormhole, which transports Howard from spatial point B to spatial point A, keeps the time coordinate constant in Raj's frame. That is, according to Raj, Howard enters the wormhole at about the same time he exits it on the other side. This is not a Lorentz invariant statement! It picks out Raj's frame as special, and as a result we can detect absolute motion relative to this frame.

Answer 2

This depends upon how you define a wormhole. A traversable wormhole of the sort common in science fiction narratives probably does not exist. These require negative energy quantum states with $\langle T^{00}\rangle~<~0$. The stress-energy tensor that is negative means that $\hbar\sum_n\omega(n) a^\dagger(n) a(n)$ or Hamiltonian that defines the number of particles is negative. This has some strange physical implications. It means that the frequency $\omega_n$ for some values of $n$ is negative, which is physically odd. The other is that there is no minimum energy state; the spectrum is not bounded below. This means an infinite amount of radiation could be emitted as this exotic quantum state falls to lower energy levels.

There is however a form of wormhole associated with black holes, called the Einstein-Rosen bridge. This is seen in the conformal diagram for the Schwarzschild black hole.

This diagram is an idealization and it may pertain to a pair of quantum entangled black holes. We may think of generating these black holes from EPR pairs of spins, and two sets of these are then brought together in separate regions to form the entangled black holes. The observer entering from this universe then can observe the world in this distant region, here called the parallel universe. The formation of a black hole may involve aspects of how elementary particles are entangled through the multiverse.

This is of course not traversable. There is no causal path that connects these two regions, or universes. So a black hole can't serve as a type of wormhole, such as in Joe Haldeman's novel “The Forever War,” where travel is facilitated by jumping through “collapsars.” The formation of the black hole is with states in entanglements $$ |\psi\rangle~=~\sum_ne^{-E_n\beta/2}|n_1\rangle\otimes|n_2\rangle $$ which in this Euclideanized form is boltzmann. The formation of the entangled black holes is from these entangled states. This is a standard case, and Hawking radiation produces particles in both universes, shifting entanglement from the black hole to these worlds. The Hawking radiation is blackbody in its statistics.

We may then consider the case with squeezed states or the squeezed vacuum. The Boltzmann case with $\rho~\sim~e^{-E\beta}$ is replaced with Poisson statistics $$ \rho_{nn}~=~\frac{\langle n\rangle^n e^{\langle n\rangle}}{n!}. $$ We may think of the density matrix describing the statistical distribution of points in a spacetime volume $V$, with $V\beta~\simeq~\langle n\rangle$. The parameter $\beta$ is the Planck volume $\beta~=~\ell_p^4$. The Poisson distribution assumes the form $$ \rho_{nn}~=~P_n(V)~=~\frac{B\beta^ne^{-\beta V}}{n!}. $$ One thing that is apparent is that the exponent $e^{-\beta V}$ is very small for large volume. This will not contribute much to the quantum gravity of connected regions except on very small regions close to the size of the spacetime volume $\beta$. We may improve this situation of course by considering this on event horizons of black holes. In this case we can let $\beta~=~\ell_p^2$ as Planck areas on a black hole horizon. This gives a small bit of traversible properties to a black hole. The squeezed vacuum lowers the energy, and the statistics on the occurrence of these pixels on the horizon by holography imprints the same on the spatial bulk outside. This is however very small compared to a stellar mass black hole. To devise a wormhole this way requires quantization on the massively large scale.

We have no observational evidence from astronomy of wormholes in the universe. Work has been done to understand what a wormhole would appear as in the universe. So far nothing has suggested the existence of a traversable wormhole.

Answer 3

Your thought experiment was discussed at length in Chapter 14 of Kip Thorne's 1994 book, Black Holes & Time Warps -- Einstein's Outrageous Legacy. It was considered more recently in Chapter 10 of the 2010 book, The Physics of Stargates -- Parallel Universes, Time Travel, and the Enigma of Wormhole Physics by Enrico Rodrigo.

In short, the apparent paradox is resolved by Raj necessarily traveling forward in time, so that the wormhole between Raj and Howard becomes a time machine.

[I believe that both authors obtained their Ph.D.s under John Wheeler (who introduced the terms "black hole" and "wormhole" to physics) a few decades apart.]

Answer 4

If there is one future connected to the present an event and its' negation cannot exist in that same future. There must be multiple futures connected to the present for any device to give exact information of any of the futures or some other noise (jamming) in the device. If not, people with such device would change the undesirable for them changeable events. Samuel Lewis Reich

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Answer 5

I do not believe your argument can work, because in a spacetime with wormholes there are no global inertial frames. Before Raj catches back up with Howard, he enters a region to which Howard's frame cannot be extended, which means that Howard has no clear opinion about how fast Raj's clock is running.

What is true is that when they come back together, each will have aged by the length of his own worldline, and both can agree on what these ages should be.

Another way to see the same thing is that your argument has nothing to do with wormholes: If it were correct it would prove that there can be no spacetime with closed timelike curves. But there are spacetimes with closed timelike curves (e.g. Godel's); therefore your argument cannot be correct.