• I
• Ahmed1029
In summary, the twin paradox can be settled by saying that any non-straight path between two events in spacetime has less proper time than a straight path between the two events. However, it is important to note that this is only applicable to Minkowski spacetime, and not Euclidean spacetime. The spacetime interval, ##ds^2##, and proper time, ##d\tau^2##, should be defined along a path rather than between events. Therefore, the twin traveling on a straight path will age more than the one moving on a curved path in Minkowski spacetime.f

#### Ahmed1029

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? So the twin in the frame which has a longer trajectory between the two pints(curved) will have less elapsed time?

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?
Yes.

So the twin in the frame which has a longer trajectory between the two pints(curved) will have less elapsed time?
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.

• vanhees71 and Ahmed1029
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.
I implicitly assume flat spacetime, so it's right in this context right?

I implicitly assume flat spacetime, so it's right in this context right?
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.

• vanhees71, Ahmed1029 and Dale
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.
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 trajectory

• PeroK
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.
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 thanks

the interval is equal to the proper time only when the observer moving in a straight line between the events
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##.

• vanhees71 and Ahmed1029
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##.
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?

• Dale
the strights path traveller ages absolutely more than the one who moves on a curved path, am I correct?
Yes. With @Orodruin ’s caution that we are talking about paths in spacetime, not paths in space.

• vanhees71 and Ahmed1029