T-symmetry (What is it? An easy one for the physicists of the appropriate stripe)

Factual Questions
1

Every now and then, I try to buff up my woefully inadequate scientific knowledge. Today, that eventually brought me to reading about CPT symmetry, a topic which I’ve read about before, and I now find myself with the same questions I have every time I read about it. Embarrassingly basic questions. Luckily, I now have the SDMB to help me out.

My basic question is… what is T-symmetry? Give me a definition. I was thinking it must be that a law satisfies T-symmetry if it is unchanged under inversion of time, but that cannot be it; the law of gravity is a simple violation of this (it goes from “Objects get closer as time increases” to “Objects get further apart as time increases”), and I gather that real violations of T-symmetry are not so simple. So then some reading made it appear as though a law satisfies T-symmetry if it is unchanged under inversion of time and spatial coordinates. That seems reasonable enough, but now what of P-symmetry? It seems that P-symmetry must be symmetry under inversion of handedness and spatial coordinates, by similar reasoning (laws that violate handedness alone are too simple). Fine.

But what’s PT symmetry? It can’t be that laws are unchanged under inversion of time, spatial coordinates, and handedness; this combined with the notion of P-symmetry above would give a symmetry under simple inversion of time, which brings us back to the problem with the law of gravity. And it can’t be that laws are simply unchanged under inversion of time and handedness, because that also is simply violated by the law of gravity.

So, clearly, as my rambling thoughts show, my understanding of what these symmetries are is flawed. So, could someone explain to me what all these symmetries really are? A definition of what it means for a law to satisfy them?

2

It might prove instructive to start here, until the real experts show up. :slight_smile:

3

Believe me, that’s the first place I went. It gave a definition of symmetry under mere inversion of time, which seems violated by the law of gravity, as mentioned above. Stumbling around Wikipedia found me some references which made it seem like T-symmetry involved inversion of time and spatial positions, but as I explained above, I can’t quite make sense of this either, at least not when combined with P-symmetry.

4

On re-reading my OP, my rambling reasoning was bunk from almost the get-go; making T-symmetry into symmetry under simultaneous inversion of both time and spatial coordinates doesn’t salvage the law of gravity from T-symmetry violation at all; it will still go from saying “Objects get closer as time increases” to “Objects get further apart as time increases”.

So, I’m totally lost as to what T-symmetry would be.

5

A definition of T-symmetry might be: If you watch a video of a process and you cannot determine whether the video is being shown in reverse or not, then the process is T-symmetric.

Gravity works fine. Your definition of gravity as “objects get closer as time increases” is only true when, well, objects are getting closer. A rock starting from rest at eye level and dropping to the ground could just be a T-reversed rock projected upward from the ground and coming to rest at eye level. The rock and the earth did not “get closer as time increased” in the second case.

A cleaner gravity example: orbits. If you flip all velocities in the solar system (note: x --> x and t --> -t implies v = dx/dt --> -v), you’ve got a valid solar system going in reverse. (The rock example is just another orbit example, really – it’s just that the rock’s orbit intersects the earth.)

6

Right: Gravity’s not about how close or far apart objects get to each other, but rather, how their velocities change with respect to each other over time.

If you reverse time, then things that were “getting closer” under time-forward gravity will indeed “get farther away” under time-reversed gravity, but in the latter case, (assuming no other objects around,) the objects will separate at slower and slower speeds (this deceleration corresponding to the acceleration that occured time-forward) in a way reflecting exactly the law of gravitation we all know and love.

-FrL-

7

Hm. Alright. So then an example a T-symmetry violation would be if there were something like, say, a hypothetical physical law “Wherever there is fire, there is smoke there five seconds later” but no physical law saying “Wherever there is fire, there is smoke there five seconds earlier”. So if you saw a video that went smoke, then, five seconds later, fire, then five seconds later, no smoke, you’d be able to say “Aha! This isn’t valid running this way, but it would be valid running the other way.” Validity of videos, so to speak, would not be symmetric under time reversal.

So that’s kinda like what I thought originally. I guess the gravity thing was screwing me up, but your explanation seems to have cleared that away. I was thinking “The Law of Gravity says that objects get closer to each other as time increases”, but, of course, that’s not it at all. It just says that objects will undergo an attractive gravitational force, and force will of course be invariant under time reversal for the same reasons that acceleration would be while velocity, momentum, and time (obviously) would be flipped. Man, that all makes sense now. Thanks.

8

Heh. Just to clarify, the above post was in response to Pasta, but Frylock’s explanation is good too.

9

I have a related question.

To my understanding, all the forces directly involved in chemical reactions are time-symmetric. Also to my understanding, combustion is a chemical process. Also to my understanding, certain statistical laws about the motion of chemical units (i.e. atoms, molecules) are not time-symmetric.

Assuming the above to be correct, the question is, Do people consider it a puzzle at all that a process composed, for all intents and purposes, entirely of time-symmetric sub-processes, can nevertheless itself fail to be a time-symmetric process? I have in mind specifically the case of fire, but it’s a rough example. Fuel-then-fire-then-smoke is not time-reversible in the sense that if you had a closed system with a fire in it, and view it time-reversed, you will see an increase in entropy (smoke-then-fire-then-unburned-fuel) rather than a decrease in entropy. Yet, all the particle motions involved in constituting the process of burning are, in fact, governed by time symmetric laws. This seems puzzling to me, at least when I put it that way.

The fire thing may be confusing, but just think about air in a box (with no outside forces affecting it). Imagine all the air crowded to one half of the box. The air, of course, rapidly expands to fill the vacuum in time-forward mode, and so rapidly shrinks to leave behind a vacuium in time-reverse mode. The latter process breaks the (apparently, at least to me) non-time-symmetric second law of thermodynamics. Yet the actual individual motions of the molecules constituting the air are all governed by purely time-symmetric forces of inertia, gravity, electromagneticism, and so on.

Am I making the puzzle clear at all?

-FrL-

10

Yeah, this is a classical problem, Loschmidt’s paradox. As I understand it, the solution is that entropy need not necessarily increase over time, air need not necessarily rapidly expand to fill a box starting in only half the box; certain starting states lead to this, but, as demanded by T-symmetry, particular corresponding starting states with the air filling the box lead to the air contracting to fill only half the box. That we generally observe the former and not the latter, that in general we generally observe the Second Law of Thermodynamics and not entropy decreases over time, then goes back to particular details of the state of the world back at the Big Bang.

Note, if you try to prove that air will fill the box starting in only half of it, you will probably use an assumption of molecular chaos in the initial state; however, you will find that by the end, when the box is filled, there is no longer molecular chaos. This breaks the time-symmetry in a suitable way.

11

Also, relatedly, the more I reflect on it, the more embarrassed I am that I never, in all those years of sporadically re-thinking about it, realized that the law of gravity actually was symmetric under time-reversal. I guess I got too caught up in everyday examples of how I’d know a process was running in reverse: if I saw a broken glass coming together and shooting up from the floor, I’d say I was watching a reversed video; that’s like the canonical example. But I guess that if you consider all the energy jolted into the ground by the glass falling to shards on the floor, when put in reverse, you have a physically acceptable scenario in which the particles of the earth happen to move in such a coordinated manner as to jolt all the glass shards back into a unified whole and up onto the table.

12

The apparent asymmetry comes about because we consider very few states of the system special. Consider a box with N=2 molecules bouncing around. If every configuration of the system is equally likely, then we can have

L=1, R=1
L=1, R=1 (swapping molecules)
L=2, R=0
L=0, R=2

Where L and R indicate how many molecules are on the left side or right side of the box. If we consider only “L=0,R=2” to be special, then a random configuration has a 1/4 chance of being special. Not too unlikely. But, If N=(Avogadro’s number) or so, 10[sup]24[/sup], then:

number of special states = 1
number of possible states = 2[sup]10[sup]24[/sup][/sup]

==> probability of seeing our human-defined “special” state = mind-blowingly small.

If we drop a rock and it hits the ground with a thud and a poof of dust, we’ve picked out a particular state of the universe. If we are to see the reverse happen spontaneously, we’d need to have the positions and velocities of some enormous number of dust/air/earth molecules be just right, but that particular configuration is very unlikely (an understatement) if the configurations are chosen “randomly” from all possible configurations.

Hence, the statistical mechanics (and underlying) definition of entropy as the (logarithm of) the number of available states for a given set of macroscopic quantities. A system moves toward macroscopic configurations which have more available states (higher entropy) simply because those configurations are more likely… often mind-blowingly more likely.

13

This brings me to another question that’s been on my mind for years. What exactly is entropy? Can it be physically measured? Could I buy or devise an entropometer? I gather that the entropy of a situation is something like the logarithm of the number of microscopic states consistent with the macroscopic properties of the situation (or, I guess, more accurately, given that microstates may not be uniformly distributed, we can use Shannon’s formula: the sum of -p_i * log(p_i) where p_i is the probability of the i-th microstate given the macrostate). But what exactly makes a property a microscopic one vs. a macroscopic one?

14

It occurs to me that this is entirely lacking as an explanation for the Second Law of Thermodynamics. Sure, states of higher entropy are more likely than states of lower entropy. But there’s no way to infer from this that a system is more likely to be at low entropy at early times and high entropy at later times than it is to be at high entropy at early times and low entropy at later times. If the high probability of high entropy states pushes for them as likely futures, it also pushes for them as likely pasts, and so it can’t explain directionality to changes in entropy.

It’s simply not possible to derive the Second Law of Thermodynamics by any logical reasoning which assumes nothing more than the T-symmetric laws of physics, because you can’t derive a non T-symmetric fact from purely T-symmetric facts. Given only the T-symmetric laws of physics, every possible history of the universe in which entropy rises is equiprobable to its time-flipped dual, a history of the universe in which entropy falls. These laws can’t on their own account for the directionality in the Second Law.

If we take the necessary truths of the universe to be just those T-symmetric laws of physics, then the Second Law would have to be a contingent truth of the universe. Thus, as I said before, we could account for it by noting that it follows as a result of some contingent fact about our universe; e.g., facts about the particular details of the start of our universe during the Big Bang. Different starting conditions would then give different results (for example, starting conditions corresponding to a time-flipped slice of our own universe would have to lead to a universe in which entropy generally decreased over time).

One might object that proposing that details of the Big Bang “explain” the existence of the Second Law is improper, as it suggests a time-asymmetric point of view wherein facts about early events are to be used to explain facts about later events. But, one need not actually view this as a time-asymmetric process of explanation. After all, the only reason we think there was a Big Bang is because of facts about the present; modern day observations. So just as well as we can say “Contingent facts about the beginning of the universe lead us to knowledge that ours is a universe in which entropy rises over time”, we can also say “Contingent facts about the current state of the universe lead us to knowledge that ours is a universe in which entropy rises over time”. Indeed, what’s really going on is always going to be “Contingent facts about our observational state lead us to knowledge that ours is a universe in which entropy rises over time”, since, in the end, after all, everything comes down to our observations; all knowledge of past, present, future, etc., springs from this. (One might even say that we define the past, present, future, etc., in accord with this knowledge, but, enough drifting from physics into philosophy).

15

Although, of course, as has been well-noted, in the time-flipped dual of our own universe, when people use the word “past” it refers to events at later times, and when people use the word “future” it refers to events at earlier times, what people remember are the events from later times, etc., and the whole thing would be precisely the same as our own universe in terms of observables. So it could be that if there were any directionality to entropy change, then the people of that universe would feel it was a case of entropy rising over time, as they would orient their notion of increases in time accordingly. And it’s certainly possible for T-symmetric laws to demand that there be some directionality to entropy change, as long as they allow the directionality to go either way with equal probability. So this serves as a sort of resolution to the paradox as well.

16

Except that a random configuration is going to evolve into the more likely higher entropy states. If you T-reverse the universe and then pick a random state, that T-reversed universe will also obey the 2nd Law. This is quite distinct from:

(1) Let a T-forward universe evolve from state A to B, with entropy being higher in B.
(2) Then, T-reverse the universe and start with state B, watching it evolve to A.

In this case, you’ve picked out a very particular state, one that is unlikely(*) to occur naturally. It’s equivalent to the glass shards reassembling in the backwards video.

(*) where “unlikely” is the mind-blowing kind, P~1/(10^(10^(10)…))

17

If there’s an expectation value in its definition, or relatedly, if it can be expressed in terms of the partition function, it’s macroscopic.

You can’t measure entropy, but you can measure things (pressure, temperature) which have known relations to it. And, like temperature, entropy is a statistical quantity itself. If one is dealing with microscopic quantities (individual atoms and their positions, etc.) instead of with an ensemble, then thermodynamic entropy isn’t defined. Nor is the 2nd Law.

18

Why will a random configuration almost certainly evolve forward into higher entropy states but not almost certainly evolve backwards into higher entropy states, given that the laws governing its particles’ behaviors are time-symmetrical?

19

Rather, let me reword:

Why will a random configuration almost certainly evolve forward into higher entropy states but not almost certainly evolve backwards into higher entropy states, despite the laws governing its particles’ behaviors being time-symmetrical?

20

Point taken, and thank you – I had not had opportunity before to think through to this contradiction in my understanding. And upon some quick reading, it does seem (as you point out) that one must invoke particular initial conditions to get the 2nd law we observe. Whaddya know? I’ll have to add this to my list of Things-I-want-to-think-through-when-time-permits…