# Time dilation and entropy

1. Oct 13, 2015

### newjerseyrunner

I was watching a Youtube video and the narrator mentioned that sometimes antimatter can be thought of matter traveling backwards in time and that confused me. I started thinking about what time is and it doesn't make any sense that antimatter acts in any way like it's traveling backwards in time. Wouldn't that imply gravity would be reversed and even weirder, it'd entropy backwards?

That got me thinking that I can define the direction of time by which way entropy points and it's possible that everything we observer seems to only travel in one chronological direction simply because we can only observe things that experience entropy.

That got me wondering if the rate of entropy changes, so I googled entropy and relativity together and was surprised to find that I don't see any formulations of entropy that includes relativity. Could time dilation be thought of as changes in the rate of entropy? I tried to continue my thought process to see other manifestations of such things and I also realized that faster entropy in one location over another would cause objects to drift, just like gravity.

So then I googled entropy gravity and found a few articles. Is this an active area of research in physics or has the idea since been thrown out? And why does the equation of entropy not have any reference to the curvature of space? It seems that the more space is curved, the slower the rate of entropy.

Last edited: Oct 13, 2015
2. Oct 13, 2015

### SlowThinker

Antimatter will fall down, just like photons or matter does.
If antimatter's gravity was repulsive, it would be easy to construct a perpetuum mobile, so while we haven't yet (to my knowledge) measured antimatter's gravity field, it's quite unlikely that it would repel things.
That's an interesing question, maybe someone more knowledgeable can elaborate?
The light clock's entropy does not change over time, yet it measures time, and dilates with movement.
I believe that's a fairly high-level, behind-the-scenes concept, not something that would be directly observable.

3. Oct 13, 2015

### Staff: Mentor

This is a common heuristic way of describing antimatter in ordinary language (it was a favorite of Feynman, for example), but it has to be used carefully. The precise mathematical description of antimatter does not have all of the properties you might guess from this ordinary language description. If you want more detail about this, you should post a separate thread in the Quantum Physics forum.

This is one way of defining an "arrow of time", yes.

This is a bit garbled. Here is a more precise version: there is a "thermodynamic arrow of time" (the direction of time in which entropy increases), and there is a "subjective arrow of time" (the direction of time in which we anticipate events rather than remember them). If we then look at how the process of memory actually works, physically, we find that it involves an increase in entropy: in other words, the state of the universe in which we have a memory of some past event must have higher entropy than the state of the universe in which the past event has occurred, but we have not yet formed a memory of it. This means that the thermodynamic arrow of time and the subjective arrow of time must point in the same direction--the "future" in the thermodynamic sense must also be the "future" in the subjective sense.

I assume you mean "the rate at which entropy increases changes". Yes, that can certainly happen; there is no requirement that entropy always has to increase at the same rate; the only requirement is that it increases.

There are, but they're probably not going to be easily accessible to non-experts. There are a lot of dragons lurking in them there mountains...

If you want a very simple heuristic to use, though, you can observe that along any timelike worldline in relativity, a "proper time" is defined--it's just the time that a clock traveling along that worldline measures. The Second Law in relativistic form then simply says that entropy must increase with proper time along any timelike worldline. This trick works for a lot of cases where the word "time" appears in non-relativistic physics and you're wondering how to translate it into relativistic terms, where "time" in the coordinate sense is frame-dependent.

There is a sense in which this sort of works, but I don't think it's a generally fruitful line of thought--at least not unless you've already digested all the expert-level material on how entropy works in relativity.

How would this work? Do you have a reference?

Do you mean "entropic gravity", as described here? I don't know that this idea has been thrown out, but it's highly speculative and not really appropriate for discussion here (except perhaps in the Beyond the Standard Model forum), since there is not an accepted model that is backed up by experiments.

Which "equation of entropy" do you mean? Do you have a reference? (Bear in mind, as I said above, that relativistic treatments of entropy are not likely to be accessible to non-experts, so it may simply be that you're looking at the wrong equation.)

Again, how would this work? Do you have a reference?

4. Oct 13, 2015

### newjerseyrunner

Thanks, that's all exactly what I was not understanding. I had a few more questions.

I guess I gotta find some good relativity textbooks. What kind of dragons? I'm scared.

If the increase in entropy occurs faster in one part of the universe than an area right next to it, there would be a continuous gradient. Would the statistical probabilities of particle locations be skewed? How does increasing entropy affect Schrodinger's equations if there is a gradient?

Gravity slows time, which slows the increase in entropy. Sorry, this wasn't meant to be a scientific statement or imply any correlation, just of an observation.

5. Oct 13, 2015

### Staff: Mentor

You probably won't find much discussion of relativistic entropy in them. MTW has some discussion of it, but only to basically assume that it can be defined as a scalar function on spacetime, like temperature. They don't go into any of the stuff lurking underneath. (Not that this is necessarily a bad thing--see below.)

It's nothing to be scared of, necessarily; just that the whole concept of "entropy" is somewhat of a contentious one in physics, because different experts have different ideas about what it's supposed to mean. When you throw in all the complications of relativity on top of that, it's not a recipe for easy understanding.

I personally favor the MTW approach, described above, where you assume just enough to allow you to work on whatever practical problem you want to analyze, without trying to delve any deeper. This is usually a good strategy in areas we don't understand very well. For example, it is how many physicists approach quantum mechanics in general: they don't try to answer the "philosophical" questions about what it all means, they just make reasonable assumptions and get on with their work.

Why would that happen? See further comments below.

This is getting close to one of the dragons.

If you're viewing "entropy" as derivable from the statistics of microstates of the system, and you also want to have a relativistic theory, then you have to figure out how to define the microstates of the system in a way that is relativistically invariant. You certainly can't just say that "the positions and velocities of all the particles at an instant of time" is the microstate (or its quantum analogue, "the wave function at an instant of time"), since that's not relativistically invariant (because "an instant of time" is frame-dependent). Once you start digging into this, you will find that you need to tackle all the complexity of quantum field theory (and possibly quantum gravity, if you want to include gravity in your definition of the microstates), plus all the complexity of statistical mechanics. It can be done, but it's not easy, and there's no good place to stop and think in between the simple MTW view ("entropy is just a scalar function on spacetime, never mind how it's derived") and the full-blown, all the complexities at once view.

(Btw, Schrodinger's Equation itself is non-relativistic, so if you want to do this sort of thing in a relativistically invariant way, you can't use that equation. Exploring the reasons why not, and what happens when you try to fix them, led to quantum field theory.)

But that assumes that the "rate of increase of entropy" is the same as the "rate of flow of time". There's no reason why that has to be true--i.e., there's no reason why entropy has to always increase at the same rate, as measured by a clock traveling on the worldline along which we are measuring the entropy increase.

6. Oct 17, 2015

### vanhees71

Well, although Feynman had this way explaining anti-particles as flowing "backwards in time", it's to be taken with a grain of salt. It is just a pop way to express a quite formal and not so simple mathematical idea to solve the apparent problem with negative-energy solutions of all possible relativistic field equations with the fields to be interpreted as wave functions as in the SchrÃ¶dinger-wave mechanics formulation of non-relativistic quantum mechanics of a system of point particles with strictly conserved particle numbers.

The solution is to use many-body theory, i.e., quantum field theory to formulate a consistent relativistic quantum theory. This comes also handy, because you can use the concept of local interactions in a very natural way. The negative-frequency modes of the free fields are simply rewritten as creation operators of the corresponding field operators instead of annihilation operators. With both the positive-frequency annihilation and negative-frequency creation parts of the fields, you can formulate local ineracting QFTs, which have a Hamiltonian that has a ground state, i.e., a state of minimum energy (usually "the vacuum", because it describes the state where no particles are present). In this way all asymptotic free particles have positive energy and run forward in time. The theory is also causal, in the sense that there cannot be faster-than-light information propagation about the states of particles.

The question with the entropy is more complicated. Here you have to derive macrosocpic laws from quantum field theory. This is done in some way of "coarse graining", i.e., you look at the collective motion of many particles, antiparticles, and fields. You end up with something like the Boltzmann-Uehling-Uhlenbeck transport equation, which obeys the famous H theorem, according to which the entropy is increasing with time. There is no difference in principle to the non-relativistic transport theory, which is derived in an analogous way from many-body quantum theory also in the non-relativistic case.