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I know this may appear to be a duplicate question but the other question Straight lines and longest distance doesn't seem to explain in laymans terms. So... I'm trying to understand this but have read loads of stuff about Minkowski space etc but it's still beyond my tiny brain to grasp.

So, would anyone like to tackle this in layman's terms for me please.

The quote I've seen and would like explaining is this..

In space, a straight line describes the shortest distance between two points. In spacetime, by contrast, a straight path yields the longest elapsed time between two events.

I just can't get my head around why the straight line is the longest time in spacetime!

Qmechanic
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ScaryAardvark
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  • Possible duplicates: https://physics.stackexchange.com/q/303461/2451 and links therein. – Qmechanic Nov 22 '22 at 11:07
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    It may not be obvious to OP why the linked question is (functionally) a duplicate. Only one of the answers really gets to the heart of the issue, which is that space and time have opposite signs in the (pseudo-)metric of spacetime $ds^2=-dt^2+dx^22+dy^2+dz^2$. Using that sign convention it becomes clear that one minimizes the length (value of the metric) by maximizing dt, which makes the length become more negative (hence smaller). – Eric Smith Nov 22 '22 at 13:17

2 Answers2

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Because of time dilation. If I have to go from A to B in a given coordinate time and I do so by zigzagging all over the place, on average my speed will be pretty high. As seen from an inertial frame, this means that my clock will run slow, and I will measure a shorter time. By going in a straight line at constant speed from A to B, I keep my average speed as low as possible, thus making my clock run as fast as possible, which will make me measure a longer time.

Javier
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As I mentioned in a comment: space and time have opposite signs in the spacetime (pseudo)metric. That means that in Minkowski space the "distance squared" $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$ can become negative. So the "shortest distance" corresponds to the "longest time" in the same way that, say, -20 is smaller than +10 or even -10.

There are some subtleties in the math about what a geodesic is, but for flat spacetime straight (timelike) lines representing massive objects are ones which maximize dt^2 and minimize the other terms.

Eric Smith
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  • Not sure I agree with this answer. If you use the opposite signature, the longest time maximizes the squared interval instead of minimizing it. Not to mention that proper time is the square root of your expression. – Javier Nov 25 '22 at 20:16
  • Yes, if you use the opposite signature then the longest time maximizes the squared interval; but note that then a spacelike straight line will be the longest distance rather than the shortest distance (because spacelike intervals will be negative). That's why I included the disclaimer about geodesics. My general point was to give an intuitive overview for OP rather than rigorous definition. – Eric Smith Nov 26 '22 at 11:00