Fifteen hours ago, I thought that the null energy condition was enough to forbid superluminal signals. Well, that was wrong.
As you know, special relativity strengthens the notion of causality. Newton thought that causality meant that the time "t" when a cause occurs must be smaller than time "T" when its effect takes place, and that's it. Causes precede their effects. This modest condition is everything you need to protect the life of your grandmother during her encounter with your grandfather and to protect the Universe from logical absurdities. Einstein showed that a spacelike vector directed to the future is equivalent, via Lorentz transformations, to a spacelike vector directed to the past.
This means that if "T > t" holds for all observers, then we must require that the signals can't propagate faster than light. You are unable to influence not only your past, but even the future at very distant places that would require signals faster than light. This is what the principle of relativity implies in combination with the previous, weaker version of the causality requirement and my extensive experience with various types of discussions taught me to use the word "crackpot" for everyone who thinks that this principle is routinely violated. In some cases, these "crackpots" can also be geniuses whose path to knowledge differed from ours. :-)
Now imagine that you construct a classical non-gravitational field theory - for example, with a scalar field and complicated functions of its gradient - and you ask the question whether the signals can propagate faster than light on any background. In other words, whether you can create an environment that allows you to send the information faster than light. This can't be possible according to the relativistic causality.
Tachyons, particles that classically move superluminally, have a greater momentum than the energy. So you suspect that an inequality between the energy density and momentum density could be the appropriate local counterpart of the causality conditions for field theory.
But which inequality? The simplest inequality you can think about is
- "T_{00} >= 0"
i.e. the energy density is positive. The vacuum is stable. If you start with a Lorentz-invariant theory, it holds in any frame if it holds in one of them for any configuration. In a general frame, you can write it as
- "T_{mn} v^m v^n >= 0"
which must hold for any time-like vector "v^m". This is the weak energy condition (WEC). A nearly equivalent but slightly weaker is the null energy condition (NEC) - the weakest condition among all
- "T_{mn} n^m n^n >= 0"
for any null vector "n^m". You can get it, up to a normalization, by taking a light-like limit of the previous inequality. The WEC is clearly violated by the vacuum with a negative cosmological constant (the NEC marginally holds), and you may choose how to respond to this fact. My response is that the negative cosmological constant is allowed to have a waiver. Another possible reaction is that only the NEC is a legitimate requirement. There exists a stronger condition, the dominant energy condition (DEC):
- "T_{mn} v^m = u_m is never space-like"
This condition is stronger than NEC and WEC: DEC implies NEC and WEC. People say that DEC is equivalent to the requirement that the signals can't be superluminal - something that seems to violated by an example but I must be doing a silly error. (Finally I found a violation of DEC in the superluminal theory I looked at: it seems that a Lorentz-invariant classical theory satisfying the DEC can never propagate superluminal signals?) Chris Hillman who discussed with John Baez argues that the DEC is as essential and fundamental for general relativity as the field equations themselves. Well, he probably exaggerates.
Also, for the cosmological evolution, the DEC seems equivalent to the holographic principle. Note that for isotropic geometries, the DEC implies that the pressure "p" must be between "-rho" and "+rho" where "rho" is the energy density. Note that "-rho" is the pressure of the (positive) cosmological constant while "+rho" is the "black hole gas" that saturates the holographic bounds.
Are there other energy conditions? Yes, another energy condition is the strong energy condition (SEC) that essentially says that in every reference frame,
- "R_{00} >= 0"
for the time-time component of the Ricci tensor. According to Einstein's equations, the Ricci tensor is proportional to the stress-energy tensor minus 1/2 times the metric times the trace of the stress energy tensor. In four dimensions and some intuitive conventions, the SEC means that
- "T_{00} + T_{11} + T_{22} + T_{33} >= 0"
where the signs are such that the sum is not proportional to the trace, and it reduces to (plus) the energy density if the spatial components vanish. Actually, the SEC is the only energy condition that seems to be violated by a certain example of a theory that admits superluminal signals above a certain background.
Because I deeply respect special-relativistic causality and because of the previous sentence, it means that I have to like the SEC. Don't be confused by the confusing terminology: SEC is not terribly "strong" and it implies neither WEC nor DEC. It's independent.
What do you gain if you believe SEC? For example, some wormholes become unphysical. Some rates of tunnelling are affected. And so forth, and so forth. I am being told that DEC and SEC are routinely violated by "regular" physical systems. It is not easy to believe this statement as long as we really mean "regular" systems.
