Yes.Ahmed1029 said: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.Ahmed1029 said: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?Orodruin said: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.Ahmed1029 said: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 trajectoryOrodruin said: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 thanksOrodruin said: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.
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.Ahmed1029 said: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?Dale said: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##.
The Twin Paradox is a thought experiment in special relativity that explores the concept of time dilation. It involves two identical twins, one of whom stays on Earth while the other travels through space at high speeds. When the traveling twin returns to Earth, they will have aged less than their twin who stayed on Earth.
The Twin Paradox can be resolved by understanding that time is relative and depends on the observer's frame of reference. The traveling twin experiences time dilation due to their high speed, while the twin on Earth experiences time at a normal rate. This results in the traveling twin aging less than their Earth-bound twin.
Non-straight paths, such as those involving acceleration and deceleration, can affect the resolution of the Twin Paradox. These paths can cause changes in velocity, which can impact the amount of time dilation experienced by the traveling twin.
Proper time is the time experienced by an observer in their own frame of reference. In the Twin Paradox, the proper time of the traveling twin is less than that of the Earth-bound twin due to their different frames of reference and the effects of time dilation.
While the Twin Paradox is a thought experiment, it has been observed in experiments with atomic clocks on airplanes and in space. These experiments have confirmed the principles of time dilation and the resolution of the Twin Paradox.