Ever since special relativity we've had this equation that puts time and space on an equal footing:
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.$$
But they're obviously not equivalent, because there's a sign difference between space and time.
Question: how does a relative sign difference lead to a situation where time only flows forward and never backward? We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time? It sounds like something I should know, yet I don't - the only thing I can see is, $dt$ could be positive or negative (corresponding to forwards and backwards in time), but after being squared that sign difference disappears so nothing changes.
Related questions: What grounds the difference between space and time?, What is time, does it flow, and if so what defines its direction? However I'm phrasing this question from a relativity viewpoint, not thermodynamics.
Answer
We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time?
As illustrated in the answer by Ben Crowell and acknowledged in other answers, that relative sign doesn't by itself determine which is future and which is past. But as the answer by Dale explains, it does mean that we can't "move back and forth in time," assuming that the spacetime is globally hyperbolic (which excludes examples like the one in Ben Crowell's answer). A spacetime is called globally hyperbolic if it has a spacelike hypersurface through which every timelike curve passes exactly once (a Cauchy surface) [1][2]. This ensures that we can choose which half of every light-cone is "future" and which is "past," in a way that is consistent and smooth throughout the spacetime.
For an explicit proof that "turning around in time" is impossible, in the special case of ordinary flat spacetime, see the appendix of this post: https://physics.stackexchange.com/a/442841.
References:
[1] Pages 39, 44, and 48 in Penrose (1972), "Techniques of Differential Topology in Relativity," Society for Industrial and Applied Mathematics, http://www.kfki.hu/~iracz/sgimp/cikkek/cenzor/Penrose_todtir.pdf
[2] Page 4 in Sanchez (2005), "Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision," http://arxiv.org/abs/gr-qc/0411143v2
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