Over the holidays, I read a book written back in 2005 titled "A World Without Time."
It's a good read for over the holidays because the first half of the book is largely biographical chapters on Albert Einstein and Kurt Gödel, and the second half is a step-by-step presentation of Godel's argument that, if the General Theory of Relativity is true, then our perceived "flow of time" is not real.
This is an interesting argument, so I'd like to discuss it further in this post. It's actually quite similar to the argument that has been made by Julian Barbour for the last couple of decades. (In this prior post, I discuss Dr. Barbour's latest addition to his last-standing argument that there is no such thing as time because 'time' in General Relativity is nothing more than another spatial dimension.)
So, let's look at Kurt Gödel's argument in more detail:
What Kurt Gödel did was to build a hypothetical universe that was consistent with GR. (While the universe was nothing like our universe, it was entirely consistent with the laws of GR.) In this hypothetical universe, there were closed space-time paths. In other words, there were closed space-time paths in the same way that the Earth has a 'essentially' closed space path around the Sun. In other words, on this closed space-time path you would wind up right back where you started. Meaning that you could revisit the past and it would be exactly the same as before,
Kurt Gödel then argued that in this hypothetical universe, there can be no such thing as 'flow of time' because you could easily go back in time or forward in time, just as easily as you can go left or go right at a T-intersection.
Kurt Gödel then argued that, since the 'flow of time' does not exist in this universe and since this universe is entirely consistent with General Relativity, then the 'flow of time' does not exist in our universe because our universe is governed by the laws of General Relativity.
So, I think that this is a valid argument, except for the last step. The problem with this last step is that there are four laws of physics in our universe (gravity, E&M, weak nuclear, and strong nuclear.) I would agree with Kurt Gödel if the only law of physics had been gravity, but the weak nuclear force just doesn't cooperate so easily.
It's well known that the weak nuclear force violates both CP and T symmetry. In other words, there is an arrow of time associated with the weak nuclear force, and this arrow of time does not exist in the other forces of nature. So, what keeps us from ever being able to make a closed space-time path is the weak nuclear force, because we are ultimately surrounded by particles that interact via the weak nuclear force, i.e. neutrinos (and perhaps dark matter particles interact via the weak nuclear force.) While the interaction of space-ships with neutrinos is negligible at normal velocities, the interactions is extreme when the space-ship starts moving at relativistic velocities...i.e. having a directed energy per nucleon on the ship of around ~10 GeV.) For example, I calculated that, when traveling at a speed where your kinetic energy is equal to your rest mass energy, then the protons in your start converting into neutrons at a rate of 1 proton every 2 milliseconds. (While this isn't particularly fast given the number of protons in your body, you can hopefully see that there's no way for you to travel anywhere near the speed of light without having neutrinos significantly destroy the structure of your body.)
So, there's no way to get a space-ship up to the required velocity/energy to create a closed space-time loop without bumping into neutrinos who will create a drag force on the space-ship as they bump into electrons. The irreversibly of drag (due to collisions with particles that interact via the weak nuclear force) is what prevents us from creating a closed space-time path.
But what if there were no neutrinos for use to run into? Would time travel be possible? (i.e. would a closed space-time path be possible?)
I would answer that that time travel would be possible in a world without the weak nuclear force. However, we live in a world with the weak nuclear force, and there is no way to get around it. In fact, the real question is: are the background neutrinos a requirement of our time asymmetric world?
I don't think that it's coincidental that we are surrounded by neutrinos (the very particles that prevent us from traveling back in time.) It's the neutrinos and dark matter particles that carry most of the entropy in the universe. It's the high-entropy, diffuse nature of neutrinos that pushes back against attempts to create closed space-time loops (just as it's impossible/difficult to create vortexes in extremely viscous liquids like honey.)
So, in summary, I think that Kurt Gödel has a valid point that there is no flow in General Relativity (alone.) But when you combine GR with the weak nuclear force, then space-time travel is not possible.
What I have argued in prior posts is that we live on the wrinkled, 3D surface of a 4D space-time sphere. The amount of mass at any point tells us the local curvature of space-time (i.e. the distance away from the center of the space-time sphere.) Large amount of mass mean small distance (i.e. radius.) So, at the Big Bang, everything in the universe was at a small radius. Now that mass is less densely packed, we are further from the center of the 4D sphere. But GR doesn't tell us about how space-time grows.
It's possible that the weak nuclear force determines how space-time grows. Via the weak nuclear force, heavy particles decay into lighter particles in such a way that entropy increases. Heavy quarks convert into light quarks, and heavy leptons convert into lighter leptons. (Also, perhaps the non-Maxwell-Boltzmann velocity distribution in Dark Matter slowly turns into a Maxwell-Boltzmann distribution due to weak force interactions between dark matter particles.)
Rest mass slowly converts into into kinematic energy, and the total entropy of the universe increases. Likewise, non-equilibrium energy distributions slowly get thermalized. Perhaps dark matter slowly decays into light neutrinos. My point here is that the number of bits of information required to describe the universe is increasing with time due to the weak nuclear force.
As this is happening, the 4D space-time volume within the 4D space-time sphere increases. The 3D surface area of the universe (at any given point of time) is in one-to-one correlation with the entropy on the 3D surface area of universe (at any given point of time.)