# Time like spacetime interval, proper time, and time dilation

• B
Homework Helper
So this notion of "at rest" is actually w.r.t. a coordinate system.
Yes.
What about the notion of "being at rest" w.r.t. another body ?
In flat space-time, there is relative velocity -- the velocity of the one body in the [momentary, inertial] rest frame of the other. If this is zero then the objects are at rest relative to each other.

In curved space-time, the notion of relative velocity gets slippery. Non-local comparison of velocities becomes ambiguous and one needs to "parallel transport" the velocity of the one into the local frame of the other. The result can depend on the path over which this "parallel transport" is performed.

There is a different notion which might be used. Have you Googled "born rigidity"?

cianfa72 and Ibix
italicus
Hello to every body. I’ve found the attached image among my old draft papers, which I used in a conference to illustrate the so-called twin paradox:

Do you have difficulties with Italian ? In short , the test says that , in the flat spacetime of SR (no mass- energy that causes curvature) , one can go from event O to event Q in two (or more...) ways :

1) staying at home, sitting in a very comfortable arm chair, (but let’s ignore for the moment that the Earth is not an inertial RF , from the point of view of relativity) , and just letting proper time (wristwatch time) flow. So, OQ is a piece of geodesics. Spatial coordinate x doesn’t change.
2) moving from O to Q along a curved universe line OSQ , that can be approximated by a succession of short segments , during which the speed maintains constant , and varies from piece to piece. Axes t’ , t” ...are the instantaneous time axes of the MCRFs which are tangent to the curved universe line.

Then, mathematics and physics formulae don’t need to be translated. Fortunately , their language is (or should be) universal and well understood by everybody.
In the end, the result is that the integral of proper time along the curved universe line is shorter than the coordinate time from O to Q : the geodesics OQ , on the time axis of the stationary twin, is the "longest time line” . No paradox, then .