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From time to time I encounter things on the internet about physics mysteries concerning the "arrow of time". It is held that the laws of physics at a microscopic level are the same regardless of which of two directions time runs, but at a macroscopic level, if you drop a glass onto a concrete pavement it shatters, and yet we never see the shards leap off the pavement and reassemble themselves into an intact glass.

The contrast between the symmetry of the laws and the asymmetric behavior of the glass (and other things like it) is sometimes held to be a paradox.

However, it appers to me that there is no conflict between the two, if one has asymmetric initial conditions --- a Big Bang or the like: \begin{align} & \Big( \text{past-future-symmetric physical laws}\Big) \\ & {} + \Big( \text{temporally }\textbf{a}\text{symmetric initial conditions} \Big) \\[10pt] \Longrightarrow {} & \Big( \text{glasses shattering and not reassembling} \Big). \end{align} The observed asymmetry of the glass shattering and never reassembling seems adequately explained by asymmetry of the initial conditions without any need for an additional asymmetry, which would be in the laws governing the microscopic particles.

Perhaps this leaves unexplained the initial asymmetry in the initial conditions, but it leaves us with no unexplained mystery of where the arrow of time is in the laws governing the microscopic particles: those laws are not where the "arrow" is located; the "arrow" is located only in the asymmetry of the initial conditions.

So: Do physicists pose the question of the arrow of time as a paradox whose resolution they don't know, or do they pose it only as an exercise for freshmen, whose solution I present above? If the former, why is the solution I present above inadequate?

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Nobody really understands the flow of time, presumably because it has something to do with how we perceive time. There are some plausible explanations and models out there, but if I had to guess, I'd guess that the flow of time has little to do with fundamental physical laws and a lot to do with human memory and consciousness. I assume most of physicists would agree.

But to answer your question, we know how to resolve the paradox and it's exactly what you said, temporally asymmetric boundary conditions. In other words, in moments after the big bang, the universe was in a state of extremely low entropy which has been increasing ever since.

The paradox has morphed into another problem: Why did the universe start out in a state which such low entropy?

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If you look at the shattering glass from the point of view of naive physics, you consider it as a series of spatial configurations that are somehow abstractly located on an "eternal" timeline. That is the reason why you think you can imagine the process in reverse order: because you already know the end result as one slice of the eternal reality of the process, which you just need to trace back to its origin.

However, a vast majority of people (physicists and non-physicists alike) can confirm to you, that they can remember their past, but they cannot remember their future. So what you construct in your brain, i.e. the movie of the breaking glass ending in a bunch of shards on a concrete floor, does not yet exist at any point in time. Although you think you know how it all ends, this imagination is just a product of your simplified, abstract thinking. As is the concept of remembering the pile of shards (i.e. remembering the future) and then being surprised that they unforeseenly reassemble to a glass (not knowing the past).

Or do you know where exactly all the shards lie around on the floor, what their sizes and shapes are, what colors the refracted light from the sun casts on the walls of the room where the glass broke, what kind of carpet lies next to the broken glass, and so on? No, you didn't even think about all these aspects. You didn't even notice that a machine which was coupled to a random number generator unexpectedly nudged the glass beyond the edge of the table that it was standing on. You thought about the breaking glass just like you think about a tree when the word "tree" pops up in your mind: as a very careless abstraction that does not take into account the fact that you don't know what will happen before it has happened. And the glass actually never broke. There simply is no glass.

So the real question is why the laws of physics allow the past to be encoded in our brains, while they don't allow the future to be encoded. It is also not a question about whether there is time symmetry or not because even if the laws of physics were asymmetric in time, this would not preclude that past as well as future could be encoded in our brains at every point in time. So if we accept the hypothesis that physics is responsible for our asymmetric memory (and not that everything is just an illusion), then the laws of physics must be temporally asymmetric in a very specific way.

That is the "mystery about the arrow of time". You can't just explain this away by choosing specific initial conditions, because by saying "initial conditions" you already talk about the past, which presumes that there is a physical reality called "past" which has the special property of influencing what comes next, but not what came before it. Are you sure that what you have called "temporally asymmetric initial conditions" isn't actually the end of a process, like the shattered glass? Of course, we all know that there are initial conditions, but we can't define them without already surrendering to the subjective reality of the asymmetry between past and future, between potential memories and the unknown.

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The problem you ultimately get bogged down into is the Boltzmann brain problem. As pointed out in Schlomo's answer, you have to address the low entropy initial conditions. But as you can read in section 6, page 18 of this paper, you have to consider that Poincaré recurrences will happen, it's then always far more likely that you owe your existence to a small local downward fluctuation of the entropy, rather than a large one that would have given rise to the Big Bang.

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  • $\begingroup$ Would the sun be an example of the sort of "small local downward fluctuation of the entropy" that you have in mind? $\qquad$ $\endgroup$ – Michael Hardy Oct 11 '16 at 22:42
  • $\begingroup$ @MichaelHardy The entropy of the Sun and its surroundings is increasing. A truly downward fluctuation in the entropy would be exceptionally rare, this is considered in detail in this paper. $\endgroup$ – Count Iblis Oct 11 '16 at 22:47
  • $\begingroup$ But I meant it's a low-entropy area. That's why it's radiating energy. However, isn't the size of the "Bang" inessential? $\endgroup$ – Michael Hardy Oct 11 '16 at 23:07
  • $\begingroup$ @MichaelHardy Yes, but then most of the universe is at low entropy relative to where it would be had it been at thermal equilibrium. The size of the low entropy area is huge, at least the size of the visible universe. $\endgroup$ – Count Iblis Oct 12 '16 at 18:30
  • $\begingroup$ The breaking of time reversal symmetry and the implied low entropy initial conditions doesn't resolve the problem, it leads to the Boltzmann brain problem in a rather generic way. $\endgroup$ – Count Iblis Oct 14 '16 at 23:14
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I am a bit confused about your concept of "temporally asymetric initial conditions".

The laws of physics as we understand them are encoded in a set of second order differential equations. That means that we only need to know the positions and velocities of all particles at a given time (in fact, in a Cauchy hypersurface) in order to predict both their past and future trajectories. Any time, it doesn't have to be the initial or final time.

In you comment to @knzhou ("A Big Bang in the past and not in the future"), you say that the temporal asymmetry means that we consider "initial" conditions with a big bang and "final" conditions without it. But our cosmological models don't work that way: we use as boundary conditions (of a Cauchy problem) the positions and velocities (Hubble's law) of galaxies as seen today. Then we trace them back, and we obtain a big bang. On the other hand, the future evolution of the Universe seems a little harder to predict. An idea that was popular for some time is that it would end in a time-reversed big bang, or big crunch. The big crunch theory was dismissed by the experimental observation of accelerated expansion, but as far as I known (maybe I'm wrong about this), it wasn't incompatible with an arrow of time.

To sum up, I think that your premise is flawed.

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    $\begingroup$ I don't believe the Big Bang resulted from physicists deducing that it happened. $\qquad$ $\endgroup$ – Michael Hardy Oct 11 '16 at 22:40
  • $\begingroup$ That's not what I'm saying. I'm saying that, for this discussion (the arrow of time), the big bang is no more special than the present or 1 billion years in the future, since they are all connected by causality relations $\endgroup$ – Bosoneando Oct 11 '16 at 22:46
  • $\begingroup$ You said we use positions and velocities that we see today and then obtain a Big Bang. That is how physicist deduce that it happened, but that seems off topic. But the glass falls on the pavement and shatters and we never see the shards leap from the pavement and reassemble themselves. Even if that's a result of a local and ephemeral low-entropy region rather than a Really Big Bang, it's perfectly consistent with temporally symmetric laws if one has temporally asymmetric conditions in that local and ephemeral region. $\qquad$ $\endgroup$ – Michael Hardy Oct 11 '16 at 23:03
  • $\begingroup$ If light were flowing in toward the sun rather than radiating out from it, might we not reconcile this with temporally symmetric laws by saying it's caused by some future low-entropy conditions whose cause we don't know? $\endgroup$ – Michael Hardy Oct 11 '16 at 23:05

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