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In Witten's article, 'Light Rays, Singularities, and All That', he wrote

"A spacelike geodesic in a spacetime of Lorentz signature is never a minimum or a maximum of the length function, since oscillations in spatial directions tend to increase the length and oscillations in the time direction tend to reduce it. Two points at spacelike separation can be separated by an everywhere spacelike path that is arbitrarily short or arbitrarily long. However, what we have said does have a close analog for timelike geodesics. Here we should discuss the elapsed proper time of a geodesic (not the length) and spatial fluctuations tend to reduce it. So a sufficiently short segment of any timelike geodesic maximizes the elapsed proper time."

I am confused.

  1. Isn't space-like geodesic also a critical point of some length function, with proper sign chosen?
  2. I can imagine for two space-like separated points, space-like path very close to null direction give small length while those have lots of zig-zags in spacelike direction give large lengths. But how does these examples fail in time-like separation? In other words, why zig-zaging in time-like direction does not lead to arbitrarily large proper time?

Any hints or comments are appreciated!

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2 Answers 2

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Hint:

  1. Be aware that when discussing timelike separation, say that a spacetime event $Q$ lies in the future of a spacetime event $P$, then it is implicitly implied that we only consider paths $\gamma$ from $P$ to $Q$ whose derivative $\dot{\gamma}$ everywhere lie within the future light cone. In particular, traversing the same timelike path back and forth in both directions is not allowed.

  2. A similar restriction$^1$ does not exist/make sense for spacelike separations. We can in principle add a spacelike self-loop to any spacelike path.

  3. Any separation (in a connected spacetime) can be connected by a piecewise lightlike path.

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$^1$ Here we assume that spacetime has more than 1 spatial dimension.

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  • $\begingroup$ Thank you! I think this is indeed the key that I missed. $\endgroup$
    – Bowen Zhao
    Commented Jan 13, 2023 at 1:25
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Isn't space-like geodesic also a critical point of some length function, with proper sign chosen?

Hint: Critical points can be minima, maxima, or inflection/saddle points. We're told it's not a minimum or maximum. So what is it?

In other words, why zig-zaging in time-like direction does not lead to arbitrarily large proper time?

Hint: In the 'Twin Paradox' of special relativity, does the zig-zagging twin age faster or slower?

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