Question

Alice is driving a race car around an essentially circular track.  Brian, who is sitting at a fixed position at the edge of the track, measures the time that Alice takes to complete a lap by starting his watch when Alice passes by his position (call this event E) and stopping it when Alice passes his positin again (call this event F).  This situation is also observed by Cara and Dave, who are passengers in a train that passes very close to Brian.  Cara happens to be passing Brian just as Alice passes Brian for the first time, and Dave happens to pass Brian just as Alice passes Brian the second time.  Assume that the clocks used by Alice, Brian, and Cara are close enough together so that we can consider them all to be "present" at event E, and similarily that those used by Alice, Brian, and Dave are "present" at event F.  Assume that the ground frame is an inertial reference frame.

Who measures:

a) the proper time,

b) the coordinate time,

c) and the spacetime interval between events E and F?

Explain why.

Answer 1

Brian measures the proper time. In relativity, proper time along a timelike (or lightlike) world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and a Lorentz scalar.^[1] The proper time interval between two events on a world line is the change in proper time. This is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. Hence.

Cara and Dave measure coordinate time. In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time. Hence

The difference in times do prove that time slows in faster references.