Straight line motion with uniform velocity in spacetime according to General Relativity

Question

“The analog of a straight line in space is for space-time a motion at uniform velocity in a constant direction. The curve of shortest distance in space corresponds in space- time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the t-terms in relativity. “Straight-line” motion—the analog of “uniform velocity along a straight line”—is then that motion which takes a watch from one place at one time to another place at another time in the way that gives the longest time reading for the watch. This will be our definition for the analog of a straight line in space-time.”

This is from Richard Feynman’s Six Not So Easy Pieces

Please explain “ The curve of shortest distance in space corresponds in space- time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the t-terms in relativity.”

What will happen if the object does not travel with uniform velocity? Will he be unable to travel the straight line or will it just cause a change in its time taken? Why does this happen? If this does happen what is the conclusion?

Answer 1

The distance, $ds$, between two points in Euclidean space is $ds^2 = dx^2 + dy^2$ (according to Pythagoras). The "distance" $d\tau$, (proper time) between two events in Special Relativity is $d\tau^2 = dt^2 - dx^2 - dy^2$. The presence and the sign of $dt$ in Special Relativity, together with the signs of $dx$ and $dy$, are the source of the "funny things that happen to signs of the t-terms in relativity" which means we look for a maximum of proper time rather than a minimum.

There are many subtleties beyond what I have said here, but I hope this is enough of a pointer to enable you to get to the bottom of it. See Spacetime Interval for more details.

If the object travels with non-uniform velocity, then you just have to split the journey up into tiny pieces and add up the times for each piece to give you a total.