# Does a Weak Energy Condition Violation Typically Lead to Causality Violation?

In the answer to this question: ergosphere treadmills Lubos Motl suggested a straightforward argument, based on the special theory of relativity, to argue that light passing through a strong gravitational region which reaches infinity cannot reach infinity faster than a parallel light ray which does not pass through the strong gravitational region.

The argument is just the standard special relativistic one, that if you can go faster than light, you can build a time-machine. To do this, you just have to boost the configuration that allows you to go faster than light, and traverse it in one direction, then boost it in the other direction, and go back, and you have a closed timelike curve.

In the condition of the question, where the strong gravitational region is localized, you can do this using pre-boosted versions of the solution. Preboost two far away copies of the solution in opposite direction, one boosted to go very fast from point B to point A (at infinity) and the other boosted to go from point A to point B.

Then if you traverse a path from A to B, crossing through the first boosted solution, then cross the second on the way back, you can make a CTC.

I was skeptical of this argument, because the argument I gave required the Weak energy condition, not a no CTC condition. The argument from null energy just says that a lightfront focuses only, it doesn't defocus, the area doesn't ever spread out. Any violation of weak energy can be used to spread out a light front a little bit, and this corresponds to the geodesics passing through the violating point going a little bit faster than light, relative to the parallel geodesics nearby.

So if both arguments are correct (and although I was skeptical of Lubos's argument at first, I now believe it is correct), this means that a generic violation of the weak energy condition which is not hidden behind a horizon could be turned into a time machine. Is this true?

Question: Can you generically turn a naked violation of weak energy condition into a closed timelike curve?

A perhaps more easily answerable version: If you have an asymptotically flat solution of GR where there is a special light ray going from point A to point B which altogether outruns the asymptotic neighbors, can you use this to build a time machine, using boosted versions of the solution?

• i'm not sure i understand this argument; if two light rays can follow different paths to reach the same point, does that mean that you can make a closed timelike curve? – lurscher Nov 18 '11 at 15:05
• @lurscher--- No--- that is allowed. What seems to be not allowed is for one member of an infinite parallel sheet of light rays to outrun it's partners at infinity. The sheet represents light from a far away point in some direction headed to another far away point in some other direction. If one of the light rays on the sheet passes by the origin, and gets sped up a little bit so that it outruns its asymptotic neighbors, then a massive body can outrun the asymptotic light ray, and you seem to get a causality violation. I think this argument is correct, but it is a links null energy/causality. – Ron Maimon Nov 18 '11 at 21:14
• so the argument holds as long as the family of rays is continuously connected? – lurscher Nov 18 '11 at 21:19
• I am not sure why you would say continuously connected is important--- the important thing is that the light-front is all emerging from a really distant point, so that it is a planar sheet moving to the right at the speed of light. If the area of this sheet is growing, so that area is being shoved outward, this means some little patch is outrunning light, so maybe you have a causality violation. Or maybe you need just need a region of positive energy later to fix the growth and turn the area back down. I don't know, that's why I'm asking. – Ron Maimon Nov 19 '11 at 22:46
• This paper--phys.uconn.edu/~mallett/main/research_activities.htm--claims to be ale to generate closed timelike curves from ordinary matter. I have not examined its details, so I cannot vouch for it beyond the fact that it was at least credible enough to make the PhysRevLetters – Jerry Schirmer Jul 29 '12 at 8:13

There are no obvious causality violations in spacetimes containing Hawking radiation, which must violate the weak energy condition because they also violate the area theorem. But to my knowledge, there hasn't been a lot of study of spacetimes involving black holes that completely evaporate.

• The Hawking radiation is not in a consistent classical theory, it's semiclassical (classical gravity and quantum fields) and the back-reaction is inferred, and is not added in a self-consistent way to make a new theory. The answer does not answer the question which is about pure classical GR. If you mocked up Hawking radiation as some sort of classical negative-energy flux into the black hole accompanied by positive energy emissions, it is not clear you would not get causality violations at the same time. – Ron Maimon Aug 26 '14 at 4:47
• @RonMaimon: I understand that, but you can model shrinking black holes clasically, certainly. And I don't see any obvious CTCs in Vaidya spacetime with $\dot M < 0$ – Jerry Schirmer Aug 26 '14 at 12:56
• I don't see how to model shrinking black holes classically in any way that doesn't have negative focusing somewhere. Regarding M<0 Vaidya spacetime, there is no CTC in one of these, but if you put together two boosted copies of these moving relative to each other very fast, you get CTCs. – Ron Maimon Aug 26 '14 at 13:31
• @RonMaimon: so? you asked for an example of a spacetime that violates weak energy and doesn't have closed timelike curves. I provided one. That another model has one is immaterial. – Jerry Schirmer Aug 26 '14 at 15:10
• No, I didn't ask for an example of a spacetime that violates weak energy and doesn't have a CTC (reread the question). The negative mass Schwartschild solution is such an example. I asked for a spacetime which violates weak energy where you can't use the violation to produce a different spacetime, which is necessarily present in the theory if the first is, which does contain a CTC. – Ron Maimon Aug 27 '14 at 5:03

Although I have no insight into your interesting approach based on the defocussing of light rays, your question does seem to have a simple answer. If the WEC violation implies a negative energy density, then the associated particles have negative mass. This means that the causal direction of their world lines is negated. Assuming that it is possible to interact with such ghost particles, they could therefore be used by an observer to send signals into his own past -- a violation of causality.

• Negative local energy density does not imply negative mass particles. – Ron Maimon Mar 31 '12 at 6:53

The question gives two examples of weak energy condition violations in supergravity which do not lead to causality violations.

• Neither is an example. The second example is flawed because the domain wall energy is always bigger than the bulk energy, because the surface/volume relation is altered. The first example is also flawed because the "tachyonic" perturbations don't violate null energy condition at all, even though they superficially seem to. – Ron Maimon Aug 28 '12 at 10:13