[Physics] Physics Digest, Vol 2, Issue 11

[Ilja Schmelzer]

2016-11-12 16:13 GMT+01:00, cj at mb-soft.com <cj at mb-soft.com>:
> You certainly all absolutely are convinced that General Relativity causes
> Time Dilation. I need each of you to teach me WHY you believe that. The MATH
> would be best, but even the LOGIC that you feel applies could help me
> learn.

Let's start with the point that "time dilation" is about the behavior
of clocks.  A clock is some peace of matter which allows, by observing
how its state changes between two events, to extract some number - the
clock time between these two events.

In GR, but in SR too, this clock time depends not only on the initial
and the final event, but on the trajectory of the clock.  In SR, it
depends only on the velocity of the clock, in GR also on the
gravitational field at the place of the clock.

The gravitational field is defined by ten functions
g_{mn}(x)=g_{nm}(x), and the clock time of a clock moving along a
trajectory x(t) has nothing to do with some time coordinate t and is
computed by

tau = \int \sqrt{ g_{mn}(x) dx^m(t)/dt dx^n(t)/dt}  dt

Here the four coordinates x = (x^0,x^1,x^2,x^3) are arbitrary
coordinates of this four-dimensional animal named spacetime, and t is
simply some parametrization of the trajectory of the clock. If you use
other coordinates, there are formulas to recompute the g_{mn}(x) into
some other g'_{mn}(y), same for other parametrizations s(t) along the
trajectory. The point of these rules is that anyway, if you apply the
same formula to the g'_{mn}(y) and the differently described but the
same trajectory y(s), you obtain anyway the same result for clock
time.

If you compare two clocks at two events, but moving at different
trajectories between these events, the formula gives different
results.  That there is such a difference (also called "twin effect")
is what is named "time dilation".  Of course, once two clocks show
different time differences, one is slower than the other, and the
slower clock can be named "time dilated".
But the other is faster, so one could have named it "time enhancement"
with essentially the same right.  There is no rule that this formula
"always resulted in a value which is greater
than one".  You can compare clocks only with other clocks. Once they
show something different, one is slower, one is faster.

Ok, this is not the whole truth.  Because there is a trajectory of the
clock where the clock is the fastest one.  And this trajectory is a
very special one, namely, it is the trajectory predicted for the clock
by GR if no force (except gravity) acts on the clock.  So, if you act
on a clock with some non-gravitational force, the clock will go
slower.

All this has, of course, not much to do with logic, it is simply a
description what GR tells us about the behavior of clocks.