# Why does time reversibility imply that there is no arrow of time fundamentally?

The fundamental laws of physics (classical and quantum) are time-reversal invariant. This symmetry is often taken to imply that, at the microscopic level, there is no preferred direction of time, no fundamental distinction between past, present and future. The emergence of an "arrow of time" is solely attributable to the second law of thermodynamics.

But there is something I just don't understand. I get that the behaviour of particles can be equally well described either moving forward or backward in time, but clearly, once the initial conditions of the universe are set, the movement of particles either go in one direction or another, not both. Just because their motions are reversible doesn't mean that they oscillate forwards and backwards in time. So even if there is no obvious qualitative difference between past and future on the microscopic level*, as opposed to macroscopic phenomena (a broken cup never reassembles), this doesn't mean that time is not always ticking forward.

Honestly, is it that surprising that the laws of physics work backwards? I don't expect that when I rewind a recording from back to front, somehow I might end up with a different beginning.

*Just because there isn't an "obvious" difference with microscopic phenomena doesn't mean there isn't any distinction at all. If shown a video of a particle travelling from A to B, you wouldn't be able to tell if it was played forwards or backwards. But still, you can tell they are 2 distinct videos: either A→B or B→A. It doesn't really matter WHICH direction they're going, just that they're going in ONE direction.

Help?

• Ignore quantum stuff for a moment: if I show you a microscopic "video" of a particle and it looks like it is travelling from a point A to a point B at a later time. How do you know I'm not playing back the video in reverse, and that it was actually travelling from point B to point A? Time reversibility tells us we can't know: those two scenaria would play out identically. Hence, there is no correct choose for which one is forward in time and which one is backward in time. Commented May 8, 2023 at 5:09
• Time reversibility does not imply a lack of arrow of time, if you argue technically by the argument you gave. The problem, however, is how would you know beforehand that the arrow of time is in the direction that we observed it to be? Because of the time reversal symmetry, you cannot deduce the arrow of time a priori, and instead have to have something else tell you. That is the point of the analysis. Your argument will not have anything to bear upon that, and that is why we do not speak of it in the analysis of this problem. Commented May 8, 2023 at 5:59
• IMO, the arrow direction of time is the same as the error direction we mark on a vector in space. Space reversal has only a meaning in the mathematics we use to model a phenomenon. The same is true for time. Commented May 8, 2023 at 6:54
• The arrow of time is a trivial consequence of the fact that there is more "there" than "here" and that clocks are nothing other than devices that are slowly releasing energy from a local energy reservoir ("here") to the rest of the universe ("there"). Of course that energy flow is irreversible. The only reason why you believe that there are reversible systems is because you have been taught in school to treat all systems as reversible. That, however, is nothing but an idealization. In reality all systems are irreversible, which makes them really hard to calculate, so we ignore that. Commented May 8, 2023 at 7:13
• What prevents me from labelling these events by t=2 at A, t=1 at B, and t=0 at C? How do you know, from just looking at the particles, whether the Big Bang was at the start of, or at the end of, the process? Commented May 8, 2023 at 9:37

Math is a powerful tool, and exploring math leads to valuable discoveries about physics, but in the end, for a statement to be physics, it has to predict the results of an experiment (even if you have no idea how you could ever actually set up such an experiment). Time reversal is a mathematical operation that we do to a mathematical model to make the mathematical model do what models are for, which is to predict future experimental results on the basis of data collected in the past.

We can measure a system's present state, load that state into a model that applies the laws of physics, apply the relevant time reversal operator to that state in that model, and run the model to make a prediction about the system's state at some time in the past. A model is time-reversible if, when you do that and take some measurement that enables you to infer facts about the system's past, your measurements reliably match your predictions.

[...] once the initial conditions of the universe are set, the movement of particles either go in one direction or another, not both. Just because their motions are reversible doesn't mean that they oscillate forwards and backwards in time. So even if there is no obvious qualitative difference between past and future on the microscopic level*, as opposed to macroscopic phenomena (a broken cup never reassembles), this doesn't mean that time is not always ticking forward.

Of course, the symmetry in respect to time direction (T-symmetry) is broken since the birth of the Universe. How it came around to be broken is a problem in itself. However, there are more everyday difficulties with the lack of time reversal symmetries. If we take an example of particles in the OP, coming from some symmetric velocity distribution, we definitely expect the particle with positive velocity to move to the right, while a particle with negative velocity goes to the left, and all the particles split in 50/50 proportion. This is however not what we observe in reality: a block sliding along a surface comes to a stop, energy being transferred to heat, but we rarely see a block at rest suddenly accelerate by cooling itself down. Water fills a void, but void suddenly appearing in a sea would be considered a miracle.

The reason for this behavior is that

• we do not have (and arguably cannot have) the full information about the initial conditions

AND

• even if we knew the initial conditions, we would not be able to perform computation to predict the behavior of the system. Hence the reversibility.

Loschmidt, Zermelo, etc.
The reversibility also may show in other unexpected ways. The objection that by reversing all the velocity one could made the system to return to its initial state is known as Loschmidt's paradox. Another well-known paradox is Zermelo's recurrent paradox - that any closed system (e.g., performing Hamiltonian evolution) would eventually return to its initial state. There are also some other difficulties/paradoxes.

In the end it boils down to the fact that physics/physicists encounter mathematical difficulties in describing the real world, which manifests irreversible behavior, using the fundamental laws that are reversible.

• Although not strictly relevant to the original question, can it be argued that the symmetry in respect to time direction (T-symmetry) is not broken since the birth of the Universe in a CPT Symmetric Universe, as discussed by Boyle, Finn and Turok? arxiv.org/abs/1803.08928 Commented May 9, 2023 at 18:45
• @iSeeker I can't really say - my angle is more statistical physics. I suggest asking a specific question - there are some people here who know a lot more than I about the origin of the universe. Commented May 9, 2023 at 18:51

This is a difficult question that continues to draw a lot of debate.

In a sense your statement that

the emergence of an "arrow of time" is solely attributable to the second law of thermodynamics

is a tautology - it is not clear what we would mean by an 'arrow of time' if the second law was not true. It is true that we would still expect e.g. General Relativity to hold, and that would still contain a time parameter in the metric, but that is distinct from our experience of e.g. eggs not spontaneously unscrambling themselves.

Viewed this way, the second law of thermodynamics is a statement of the fact that the universe started in a low entropy (high order) state, and the arrow of time is merely a corollary of that. Why the universe started in such a state (the initial conditions you refer to) is an open question.

Beyond that I am not sure what you asking. When you state that

this doesn't mean that time is not always ticking forward

your language brings to mind a clock; but let's imagine there was no free energy available anywhere (the so-called heat death scenario). While we conceptually could imagine that time still 'ticks' forward, there in fact wouldn't be a way to wind a clock, so there would be no ticking.

I agree that it isn't necessarily surprising that physics works backwards; but it is surprising that despite this we don't see certain macroscopic processes running in reverse.