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Energy conditions

Energy conditions may be an example of a topic that the classical general relativists probably understand better than most of us in particle physics and string theory.

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.

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reader Robert said...

If you leave flat backgrounds you should define what exactly you mean by "faster than light" as this can have different meanings locally and globally. Take for example the Alcubierre warp drive (which of course violates some energy conditions, the strong?): There you have an asymptotically flat space-time and take two points in the sufficiently flat region.

Then you can pretend that space-time everywhere is flat and compute the proper time it takes to travel between the two points. Then you turn on the warp drive and now there exists a path with shorter proper time.

If the points are space-like in flat space but become causally related in the warp drive geometry, you could say that that geometry allows you to travel faster than light. But of course this is only a global effect and locally you can never overtake a light ray.

Of course of great interest is not only which energy condition ensures causality but also the opposite: Nobody would claim that AdS violates causality even if it violates energy conditions.

Even with tachyons the analysis is subtle, at least if you treat them as fields and not as particles: A scalar with Mexican hat potential looks tachyonic when expanded around phi=0 but of course has no causality issues.

Even in the free case, you could argue the following: You give initial data for your scalar field on a Cauchy surface. You might want to exclude data for which phi grows exponentially for t-> +/- infinity. But that implies that the fourier transform of the data has compact support which again implies that the data itself does not fall off rapidly and you can question if such data can describe a localised particle.

Alternatively you can ask what happens to your classical solution if you change data in a bounded region on your Cauchy surface. If I am not mistaken, the solution to phi only changes in a region which is causal to that bounded region. So again no way to send information to your ancestors, even with tachyons.

For more information, check out L. Robinett ``{\it Do tachyons
travel more slowly than light?}'', PRD 18(10)1978


reader Quantoken said...

Lubos:

Your whole discussion is completely classical and not a single notion of quantum mechanics is entered. Therefore it's all wrong.

Remember uncertainty principle? The exactly value of T and t can not be measured perfectly precisely, They observe a Gauss distribution, extending from -infinity to + infinity. Time, as most other physics quantities, is statistical in nature. They can not be determined exactly and precisely. They are fuzzy.

If T and t and NOT known precisely, how could you be absolutely sure T > t, not T < t? How could you be 100.00000000000% sure, not 99.9999999999999% sure? And of course, whether T > t or T < t tells you the causal relationship between the two. The best you can say is t is most likely the cause of T, but there could also be a chance, however small that chance is, that the opposite could be true.

The whole logic of causality becomes fuzzy logic. Remember in the quantum world, things do get fuzzy as everything is statistical. It's almost absolutely certain that your parents gave birth to you, but there is an almost infinitesimal chance, but NOT a zero chance, that you gave birth to your parents. Notions like this completely defy our knowledge in the classical world. But they are true in a quantum world. And the world we live in is a quantum one, not a classical one.

The textbook often argues that air molecules in a room could simutaneously gather in one corner of the room, leaving most part of the room in vacuum. There is a very small but none zero probability that that could happen.

On the same token, all the atoms that form your body, Lubos, could have a very small but none zero probability that they simultaneously fly away, and you no longer exist. So even the question whether you exist or not is a statistical one where both possibilities exist. That is the strange quantum mechanical view. But that's the correct view.

Quantoken


reader JimG said...

So Lubos, what caused you to change your mind about the null energy condition 15 hours before posting?
Is there a reference?
I looked for a new paper on the ArXiv, but didn't find anything.
Jim G.


reader Lumo said...

I agree it's a classical discussion and therefore incomplete. The relation of these conditions with the underlying quantum physics is one of the interesting questions in this context.

20 hours ago I just simply started to look at a particular system discussed by some of our colleagues here, and realized that they are right and it has superluminal signals even if the Hamiltonian density is positive definite.

The DEC and SEC are both violated, as I checked during the midterm exam (Waterloo?) right now. ;-) Sorry, little time for a better answer.


reader Physics Commenter said...

Do you mind sharing with us what the particular model you have in mind is?

There was some controversy a while back in nonlinear electrodynamics. Even with a Lorentz invariant nonlinear action, it's possible for the group velocity of light to exceed the speed of light (the standard speed of light, that is) at certain frequencies. This controversy was settled by defining yet another velocity (after the phase velocity and the group velocity) called the front velocity, which is the velocity at which the initial wavefront travels. In the cases of interest, the front velocity turned out not to travel faster than light.


reader Quantoken said...

Why should there be any controversy? Only the signal progagation speed can not exceed the light speed. In this particular case where group velocity exceeds light speed, the wave dissipates and so no signal or information is propagated. I do not think it is right to invent another concept of front velocity. Since the wave dissipates, you really can not define what the wave front is.

You can show in experiments that group velocity can exceed the light speed exactly because of quantum mechanics played out in the condensed matter state. In a pure classical system where there is no QM, it's not possible for such thing to happen.

Quantoken


reader Quantoken said...

My point is not that there is zero information progagated. My point is in such particular case, when the leading edge of the wave arrives, you can not decipher the information yet. Only when the later part of the wave also arrives, you can combine the two and actually decipher the information, like doing a Flourier Transform or such. But by that time, it's already no faster than light speed.

Quantoken


reader Fyodor Uckoff said...

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.

Err...that's an interesting way of dealing with the problem. :-) Maybe a better way is just to accept that string theory is compatible with violations of the WEC and that we just have to learn to live with this.





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"

Actually that's NOT what the DEC says, but your version is actually an improvement! The usual version says that u_m has to be either timelike or null AND FUTURE-POINTING. But your version is better because it allows negative energy density and so it allows AdS. In AdS u_m is never spacelike but it will be past-pointing. That doesn't mean that anything is actually going backwards in time. In short, your version of the DEC allows AdS, and that is sensible.



People say that DEC is equivalent to the requirement that the signals can't be superluminal -

That's correct [see http://arxiv.org/abs/gr-qc/0205010]

but I'm pretty sure that it is still true with your version of the DEC. By the way you can safely ignore Carter's comments, just look at his proof.

John Baez argues that the DEC is as essential and fundamental for general relativity as the field equations themselves. Well, he probably exaggerates.

JB didn't say this, C. Hillman did. It's nonsense.


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"


Because I deeply respect special-relativistic causality and because of the previous sentence, it means that I have to like the SEC.

Maybe, but Nature does not share your enthusiasm. It is now known with about 10 sigma confidence that our Universe is accelerating, and that means that the SEC is dead. That's right, "universe accelerates" = "SEC is dead".


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

Well, if the whole universe is "regular", they are right about the SEC. The strict DEC is violated by Casimir effects, but your version of it is not. Summary: you can forget the SEC. But your version of the DEC may be ok. [See however http://arxiv.org/abs/hep-th/0312009 where every version of the DEC is violated!] The strict version [disallowing negative energy density] is incompatible with AdS so I would say it conflicts with string theory.


reader Fyodor Uckoff said...

Oh yeah, I meant to direct you to this reference too:
http://arxiv.org/abs/gr-qc/0205066


reader Lumo said...

By saying that the negative cosmological constant gets a waiver, I mean that the positive energy condition definitely has to be refined in such a way that it agrees with manifestly consistent anti de Sitter vacua of string theory. The same thing holds for the positive cosmological constant that violates the SEC. I don't think that the positive cosmological constant is inconsistent, and it is more reasonable to separate the C.C. term as an extra term in Einstein's equations that is not included in T_{mn} - which is how it was originally written. This treatment is not "fundamental", but the energy conditions aren't either.

Even if it is a good idea, its details in the presence of gravity and cosmological constant have to be reconsidered.

String theory or AdS/CFT or N=4 gauge theory can't be modified, and they're consistent. Even though I consider energy conditions morally right, they're not quite fundamental.

I think that the requirements that macroscopic amounts of physical signals can't propagate outside the light cone is more fundamental than a particular energy condition. And I leave to Nature - i.e. string theory - tel control what exactly is a allowed and what is not.

Finally. The stable vacua of string theory do not allow macroscopic violations of the speed of light for physical signals and does not exhibit any further inconsistencies of this type either.


reader Physics Commenter said...

The vacuum stability theorem only applies to fluctuations propagating across a vacuum (assuming the dominant energy condition). It doesn't apply to signals propagating over some nonvacuum background with a nonzero stress-energy tensor.

There are models of nonlinear electrodynamics and "Klein-Gordon" theories with nonlinear kinetic terms which admit superluminal signals propagating over a nontrivial background despite satisfying the weak energy condition.

There are also models admiting superluminal propagation which don't satisfy the weak energy condition, like the ghost cosmology.

The theorems limiting the speed of propagation of signals in hyperbolic PDEs only apply if the highest derivative term in the PDE is linear. These theorems don't apply to models with nonlinear kinetic terms.