[raattgift]

Thanks, I enjoyed your reply to my reply too.

The sequence of discoveries or formalisms weighs heavily on how we teach students; it's not just because earlier formalisms are necessarily easier or more intuitive, although they certainly appear to be when it comes young people who have grown up very close to the surface of the earth when it comes to classical mechanics and Newtonian gravity versus the post-Newtonian extensions.

Certainly lots of physicists took varying amounts of time coming to terms with Special Relativity; few today are au fait with General Relativity. Indeed, even relativists who are will tend to prefer to cast problems as Special Relativity ones, using (or even deliberately abusing) the approximately flat spacetime close by the strict definition of "local", because even when they are comfortable with General Relativity, it is faster to use SR where one can, even in cases where one has to manually put in corrections arising from slight curvature.

In an SR setting one usually teaches Lorentz transformations by trying to impart understanding about three things: firstly, the constancy of the speed of light for all observers in uniform motion everywhere, and secondly, thinking of a "clock" that is a pair of parallel mirrors with a pulse of light continuously bouncing back and forth between them. An observer moving with the parallel mirrors will see the pulse "forever" moving perpendicularly back and forth at the same frequency. An observer in any other uniform motion will see the pulse follow a non-perpendicular path (try it with your thumb and forefinger on one hand held parallel and representing the mirrors, with your index finger on your other hand pretending to be the front of the pulse of light -- hold your hands at a fixed distance in front of your face and watch, then try moving your arms left and right, or towards and away from you, or up and down.). The third thing to understand is that the zig-zagging of your finger between your moving thumb-and-finger appears to be a longer path because it is a longer path (think of a set of coordinates on a wall you see past your hands -- bricks or a wallpaper pattern may help). Moving-with-mirrors twin sums up the length traversed by the pulse of light and arrives at something shorter than not-moving-with-mirrors twin's sum, since the latter sees the pulse travelling along a zigzag between the moving mirrors. Since light always travels at a fixed speed, the longer zig-zagging path must take more time than the shorter always-parallel path. That is, each zig-zag "bounce" takes longer, i.e., the zig-zag bounce frequency is lower, or equivalently, the moving-with-mirrors twin's time is passing more slowly.

Einstein wrote about light bouncing between parallel mirrors, but unfortunately almost always in technical settings. I wonder if that would have helped people like Bergson.

'map is not the territory' -- funnily that's exactly what General Relativity is about; diffeomorphism invariance means that you can have arbitrarily many maps of the same matter, all exactly equivalent, and that you can apply arbitrary coordinates over the configuration of matter.

'Relativity allows for a space-like time that can be 'run in reverse'

Well, sorta. Flat spacetime is time-symmetric; since the symmetry group of flat spacetime is fundamental to the Standard Model, all Standard Model interactions are time-reversible.

BUT... the Hubble volume is extremely curved and in an expanding universe, time-reversibility is far from clear. Indeed, there is a pretty clear thermodynamic arrow-of-time, since the earlier universe, being hotter and denser, had less entropy than the later universe (which has lots of almost wholly empty space, and space with a tiny tiny tiny energy-density can be arranged in all sorts of ways and look the same macroscopically). As the metric expansion of space continues, entropy increase because of all that extra new practically empty space. The empty space can pop up all over the place and in almost any sort of configuration, and we get the same overall picture of the cosmos (in particular everything on Earth looks fundamentally, if not absolutely exactly, the same). Reversing the "movie" of the expanding universe with lots of galactic clusters in it requires very careful positioning of all the "almosts" in the empty space as it disappears, otherwise the overall picture of the cosmos diverges dramatically from our history of it. So at that scale, time-symmetry appears to vanish.

(You could also think of it this way: if you blow up the earth you get a cloud of dust and rocks and stuff. Following Boltzmann's definition of entropy as above, one cloud of dust and rocks and stuff can be pretty indistinguishable from another. But if you reverse the explosion (say, via gravitational collapse), you aren't going to get the dust and rocks coalescing into cities and coral reefs and the Himalayas as we know them unless you are very very precise. So even at that scale, time-symmetry vanishes.)

I am not certain that our brains are actually sensitive to those sorts of time-symmetry violations. Maybe we don't reconstruct the future as well as we reconstruct the past because some part of our brain was lost during the evolutionary periods in which we lost various features found in our common ancestors with birds (e.g., the ability to synthesize vitamin C in our own bodies; tails; nictitaing membranes on our eyes; ...). It'd be interesting to have a conversation with a corvid or a grey parrot or something, or a cetacean. Maybe they have a more symmetrical view of "past" and "future", in that they can remember both. Maybe we are good at playing catch because our brains actually "remember" where the ball will be, rather than doing some sort of calculative prediction.

General Relativity is not quite silent on these points; the theory is a "block world" one in which the whole of spacetime is fully determined. Formalisms that do a 3+1 foliation to look more like pre-Einsteinean physics can produce surprisingly bogus results, even though the "block world" suggests that if we know the entire configuration of the universe at any "slice", we know the configuration of the whole "block" history of the universe. (Why is the subject of a substantial amount of current research).

'actual time isn't space-like, in the sense that it can be traversed in one direction only'

So, above, I said that in an expanding universe, or in the presence of curvature near planetary masses, time-reversal fails, but it fails globally. The individual local interactions within atoms and within molecules are all fully time-symmetric (and we can more-or-less show this in labs).

Again, this is a hot topic in physical cosmology. However, I think everyone agrees that no humans are known to have travelled backwards in time, even if subatomic parts of humans may have (due to e.g. the presence of positrons from radionuclide decays within their bodies, or the uncertainty principle).

[comment too long, so dividing it here]