[math-fun] Simple description of general relativity. Physics tries not to waste time :)
You've heard of the "principle of least time" in optics... light follows a path which minimizes the time it takes to get there... in quantum field theory there is a more general action principle, which is that the integral, over all paths followed by all particles, of exp(i*T*m*c^2/hbar) dT is stationarized, where T is the proper time consumed by that particle (of rest mass m) in following its trajectory. (Actually things a bit trickier for spinor particles, and there are certain constant factors inserted whenever two particles merge or bifurcate, but I will ignore that here.) So my claim (which I haven't seen before but it isn't really new in the sense it is just a physical rephrasing of the mathematical Hilbert action) is: general relativity is also viewable as a stationarized-time principle. Sit someplace. (And by "you" I mean, a particle of infinitesimal mass & diameter, equipped with clock, free will, observation and communication capabilities, etc.) Distribute your pals (similar particles) around you uniformly on the surface of a infinitesimal-radius sphere centered at you. Now wait. The clock readings of you and your pals will gradually diverge if you are not in "flat space." There will be a beginning and an end of this experiment. At the end, the total score is the difference between your clock time, and the mean among your pals' clock readings ("wasted time"). So what is general relativity? It is just this. At the beginning, imagine such experiments are happening everywhere, uniformly on the spatial metric. (The "beginning" is a spatial slice of a 3+1 dimensional metric. So is the "end.") In between the beginning and the end, the 3+1 metric evolves in some manner. There are time-slices, each a spatial section (3-metric) of the 3+1 metric, which are infinitesimally close together in time. Each adjacent time slice should be regarded as the end of an old, and beginning of a new, such experiment. My claim is: the 3+1 metric is the one which stationarizes the total score of all the experiments. Physics is weird and "paradoxical" in the sense time is not the same for me and for you. But the Einstein vacuum is sort of trying to keep everything as consistent as possible given the situation at the beginning. Trying to be the least paradoxical it can be, sort of. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
On May 19, 2016, at 10:27 AM, Warren D Smith <warren.wds@gmail.com> wrote:
You've heard of the "principle of least time" in optics... light follows a path which minimizes the time it takes to get there... in quantum field theory there is a more general action principle, which is that the integral, over all paths followed by all particles, of exp(i*T*m*c^2/hbar) dT is stationarized, where T is the proper time consumed by that particle (of rest mass m) in following its trajectory. (Actually things a bit trickier for spinor particles, and there are certain constant factors inserted whenever two particles merge or bifurcate, but I will ignore that here.)
Taking seriously -- in quantum mechanics -- the sum over all paths, one has to include particle world-lines parts of which are not time-like and over which T is pure imaginary. With the right choice of sign, “tachyonic” propagation is thereby suppressed in a direct way. If the Hilbert action formulation is the right way to think about classical GR, i.e. the Einstein equations are just a consequence, then it seems reasonable that action-stationarity in that setting is also the hbar->0 asymptotics of a sum-over-whatever (e.g. metrics) principle. Nobody has yet made sense of that sum. Are there real-exponentially suppressed configurations? -Veit
participants (2)
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Veit Elser -
Warren D Smith