Re: HELP! - Constant acceleration in SR and event horizon


"Dean D" <Dean@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message news:A9jYh.39479$E02.15813@xxxxxxxxxxxxxxxxxx
Hello everybody



I'm looking at my lecture notes on four-vectors and there are some things I have trouble understanding.
I remember the lecturer used invariance of s^2 to prove that an observer in constant acceleration (hyperbolic motion) always have constant distance to some point behind him (in his instantaneous inertial frame). The lecturer then went on and proved that there is an event horizon at some distance behind and that a photon sent to chase the observer can never catch up with him. I can see that this is so by inspecting the space-time diagram but unfortunately I don't have the proof written down ad I need the mathematical proof (with focus on physics, not the rigorous mathematics).
Could someone help me out with this?

See for instance at the end of
http://users.telenet.be/vdmoortel/dirk/Physics/Acceleration.html

Using x and t as coordinate distance and ditto time, the covered distance
as seen in the (permanent) inertial frame is
x(t) = c^2/a ( sqrt( 1 + (a t/c)^2 ) -1 ) ,
provided x(0) = 0
where a is the constant proper acceleration felt by the traveler.

You can easily verify that
limit { t -> inf ; x(t) / t } = c
and that
limit { t -> inf ; x(t) - c t } = -c^2/a
so the worldline x(t) has an oblique asymptote with equation
x = c t - c^2/a
which is the 'worldline' of a light signal sent ot at event
( t, x ) = ( 0, -c^2/a )
Since it is on the assymptote, the signal will not reach the traveller.

Dirk Vdm
.

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