Yes.Is the twin paradox settled by saying that any non-straight path between two events (points) in space-time has less proper time that a straight path between the two events?
No. Be careful not to mix in your Euclidean thinking into the argumentation. ”Longer” in spacetime equates to larger proper time and so the ”longer” path is the straight path. The only geometry you should be referring to is the Minkowski geometry of spacetime.So the twin in the frame which has a longer trajectory between the two pints(curved) will have less elapsed time?
I implicitly assume flat spacetime, so it's right in this context right?No. Be careful not to mix in your Euclidean thinking into the argumentation. ”Longer” in spacetime equates to larger proper time and so the ”longer” path is the straight path. The only geometry you should be referring to is the Minkowski geometry of spacetime.
No. You are implicitly assuming Euclidean spacetime rather than Minkowski spacetime. Both are flat but only the latter is the spacetime relevant to special relativity.I implicitly assume flat spacetime, so it's right in this context right?
but in Minkowski spacetime, the interval is equal to the proper time only when the observer moving in a straight line between the events. If it deviates from the straight line, it's not enough to measure the coordinate time for that frame to say it's equal to the distance of its spacetime trajectory and thus shorter than the straight trajectoryNo. You are implicitly assuming Euclidean spacetime rather than Minkowski spacetime. Both are flat but only the latter is the spacetime relevant to special relativity.
Ah but I could measure its proper time between infinitesimally close points along its own trajectory and they're going to add up to less than that of a straight line. Got the point thanksNo. You are implicitly assuming Euclidean spacetime rather than Minkowski spacetime. Both are flat but only the latter is the spacetime relevant to special relativity.
This can be some (IMO) bad terminology by certain authors who define the spacetime interval or the proper time between events. They should both be defined along a path rather than between events.the interval is equal to the proper time only when the observer moving in a straight line between the events
Yeah, I was confused at first because I thought proper time had to be only defined for straight paths. Nevertheless, the strights path traveller ages absolutely more than the one who moves on a curved path, am I correct?This can be some (IMO) bad terminology by certain authors who define the spacetime interval or the proper time between events. They should both be defined along a path rather than between events.
The interval should be ##ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2## and the proper time should be ##d\tau^2=-ds^2/c^2##. Both of these are then integrated along a specified path to get the interval, ##s##, or proper time, ##\tau##.