# Time symmetry and general relativity

It's often mentioned that all (?) basic physical laws are time reversible.

For example if I imagine a planet sized perfect metal ball, and then a smaller ball falls onto it from far away (starting with zero velocity), it will reach the surface of the planet at (almost) the escape velocity, then bounce back (a perfect elastic collision, i.e. ignoring all energy dissipation) and go back to where it started, and the whole thing will repeat again and again (it's an oscillator). All the equations involved are time reversible, meaning that if I take a video of this, playing the video forward or backward is indistinguishable.

But what if a black-hole is involved? Is it conceivable for an object to reach just a tiny distance into the event horizon and then hit something massive enough (inside the event horizon) to reverse its speed and make it bounce back? For an outside observer, if the falling object seems frozen forever at the horizon, it seems that time reversibility is broken. So the Einstein equations of general relativity are non-time reversible? If that's the case, can't we then define an absolute arrow of time by the way objects fall into a black-hole? i.e. they can never "bounce back", so that this system acts a bit as a "one way" function?

(I had initially a more convoluted scenario where the small object was falling through a narrow tunnel drilled into the planet, and coming out at the other side in perfect symmetry. And same idea with a rotating black-hole, where the singularity is a torus).

• I don't think the event horizon of black holes of stellar origin is known to work only on objects travelling in one direction: My impression, mostly from work by Nikodem J. Poplawski visible on Arxiv, is that the gravitational time distorsion within the BH would keep anything from crossing inward until all matter within the BH would've decohered into extremely small, faint, weak versions of its smallest subatomic particulate components, while all matter that might've been heading outward toward the horizon would've fallen back because of the intensity of the star's gravitational field. Commented Aug 31, 2020 at 18:18
• I'm not sure whether the EH of very large BHs formed by the "direct collapse" of dust would work in the same way: The common descriptions of BHs as "gobbling" or "eating" other astronomical objects may apply to them. ("Direct collapse" is much less common than stellar collapse, but an instance of it was observed a couple of years ago, and discussed on the Astronomy Stack Exchange.) Commented Aug 31, 2020 at 18:23

Classical general relativity is time-symmetric: flip the $$t$$ direction on your space-time manifold, and you get another valid solution to the field equations.

The difference is that in this picture the black hole is replaced with a white hole occasionally spewing our particles... or metal balls. Nothing can get into it.

The weird arbitrariness of white holes spewing out non-random things hints that something is not right. In normal life things are not time-symmetric, and we typically attribute this to thermodynamics: entropy typically increases and we cannot undo past events without spending considerable (entropy increasing) resources.

This similarity between black holes always absorbing things and entropy always increasing led Hawking and others to suggest black hole thermodynamics: black holes behave like thermodynamic systems with temperature, and this is why the time-reversed case does not happen. But black hole thermodynamics is not classical physics: extra assumptions have been introduced, and many think that quantum gravity will make it clearer what is going on.

• “Classical general relativity is time-symmetric: flip the 𝑡 direction on your space-time manifold, and you get another valid solution to the field equations.“ I cannot understand. What Notion of time? Maybe you are speaking of the time orientation which does not need a definition of a preferred time coordinate. Commented Aug 25, 2020 at 21:26
• @Anders Sandberg --I'm wondering why you feel things are not time-symmetric in normal life. In what I'd thought to be the relativistic view of time (as being a dimension thru which passage may occur, analogous to the spatial dimensions), I might, in my "present", be standing in a field, looking at stars (which would be in my past), and then turn around to look at other stars, also in my past, but in the future of some of those I'd been looking at before turning around, so that all the stars would appear to be in a past and future identifiable as such, and symmetrical around my own present. Commented Aug 31, 2020 at 17:22
• I hope it's clear that I'm not talking about "my past" in any personal sense, and that I am aware that any reversed passage thru time might well require energy balancing at least half the gravity in a universe perhaps both infinite and eternal: My own nervous system would nevertheless have been in possession of some of the photons concerned. Commented Aug 31, 2020 at 17:47
• As I've posted elsewhere on this site, it is, in fact, in regard to versions of the cosmos that underestimate (by half) the force of gravity concerned that I'm effectively making an objection. Commented Aug 31, 2020 at 17:56

I am far from an expert but I think I can help.

The arrow of time is a macroscopic phenomenon built on top of completely symmetric dynamical laws of physics held through CPT-symmetry. Where does the arrow of time come from then, why is there observed T-assymetry? First note CPT together is held, witnessing T-assymetry alone is not a problem, and we see it all the time, usually as the second law. T-symmetry is only violated in the dynamical laws (in the dynamical laws themselves, not talking about the macroscopic probability of second law which is another t-asymmetry from the laws themselves) in two narrow cases, "one through the mixing of different flavours of quarks in their weak decays, the second through a direct CP violation in strong interactions. The first is seen in experiments, the second is strongly constrained by the non-observation of the EDM of a neutron." [1] But neither of these two dynamics is important to your question, and this dynamical T-assymetry is NOT why we have the macroscopic second law anyway. In fact, we have a second law because "the constant increase of entropy we observe happens only because of the initial state of our universe." [1] The low entropy big bang prior condition is taken as a given in modern physics, and the second law is a consequence of that fact plus the dynamical laws we know and love acting upon those conditions.

Onto GR specifically. "But because the equations of general relativity are time-reversible – they exhibit Time reversal symmetry – general relativity must also allow the time-reverse of this type of "realistic" black hole that forms from collapsing matter. The time-reversed case would be a white hole that has existed since the beginning of the universe, and which emits matter until it finally "explodes" and disappears.[10] Despite the fact that such objects are permitted theoretically, they are not taken as seriously as black holes by physicists, since there would be no processes that would naturally lead to their formation; they could exist only if they were built into the initial conditions of the Big Bang." [2]

So just because a white whole is a T reversible solution in the equations of GR does not mean it has to or can occur. In fact we think they don't occur because they'd have to be around at the big bang, and be stable since. We can't and shouldn't expect to see every naively allowable solution to the equations of physics without taking initial conditions/cosmology into account. And we assume a hot big bang prior condition.

Lastly, about playing a physics video in reverse. Your setup would still betray an arrow of time. The smaller ball will have greater velocity and acceleration in one direction due to the planet's own gravity. And knowing what I know about physics, I bet I can pick out which direction is the "natural" one. I would see the ball accelerating away from the planet in one version and instantly know due to the second law and fluctuation theorem (exponentially decreasing odds to witness reversing of entropy further a system is from non-equilibrium). Thus I am almost positively seeing the reversed video. The only way this works is for an extremely small setup, where the second law (which is describes the macroscopic average) doesn't have a large role. Like a single photon in an idealized box.

So in summary, you can't expect the exact time reversed macroscopic behavior in a world with the second law, even outside of the dynamical t-assymetry in the strong and weak force.

• A local version of the hypothetical process mentioned by Hawking and J Kusin has been elaborated since 2010 by Nikodem J. Poplawski in papers found free by his name in Arxiv, but uses Einstein-Cartan Theory (developed by E.&C. in 1929), not GR. Assigning a tiny mass to fermions, it provides for materialization of new fermions from the (locally) intensifying gravitational field of a collapsing star, with the interaction of their spin with that of the star's own fermions forming a white hole. The arrow of time's inherited from a parent itself formed thru the same past-eternal process. Commented Nov 13, 2020 at 20:48
• As all stars rotate, the proof of Poplawski's theory might lie in a universal direction of rotation of stellar groupings: However, recent observations suggest that the addition of relatively small amounts of mass to a neutron star might artificially change one into a BH establishing a new temporal sequence of such "local universes" (white holes), and the establishment of such new sequences (by advanced civilizations) would tend to mask any more generalized rotation. Commented Nov 13, 2020 at 21:00