Re: Two astronauts in 2 frames...

On 4 Jun 2005 "Curious" wrote:
> Al's acceleration actually brings him to rest, whereas Bert reaches
> a speed close to 0.8c; before they decelerate back to 0.4c after 1
> space-station hour... from the Earth's frame of reference, Bert's
> clock should be behind Al's, because Bert spent that hour at 0.8c
> whereas Al spent it at rest, is that right?

Of course.

> Can you show me how the Lorentz transformations would work in
> this case?

Sure, it's trivial. With respect to inertial rest frame coordinates of
the space station, set their departure point at the origin (t,x) =
(0,0), with t and x in units of years and light years. Then let them
accelerate abruptly to +-0.4, so they follow inertial paths to the
points (1, -0.4) and (1, 0.4), where they abruptly decelerate to rest
(relative to the station). The elapsed proper time on their clocks is
sqrt(1^2 - 0.4^2) = 0.916 years at the points where they abruptly
decelerate.

Now, in terms of inertial rest frame coordinates (t', x') of the
Earth, we can still place the origin at their departure event (0,0),
but now all other points are related according to the Lorentz
transformation t' = (t - vx)g and x' = (x - vt)g where g =
1/sqrt(1-v^2). In your scenario, v = 0.4, because that's the relative
velocity between your two inertial coordinate systems (the rest
frames of the Earth and the space station).

So, the events at which Al and Bert abruptly decelerate have the
coordinates

Al:
t' = (1 - 0.4(0.4))/sqrt(1-0.4^2) = 0.9165...
x' = (0.4 - 0.4(1))/sqrt(1-0.4^2) = 0.0000...

Bert:
t' = (1 - 0.4(-0.4))/sqrt(1-0.4^2) = 1.2656...
x' = (-0.4 - 0.4(1))/sqrt(1-0.4^2) = -0.8728...

So the time from their departures to the events where Al and Bert
decelerate are different with respect to the Earth's inertial rest
frame. In other words, although they decelerate simultaneously with
respect to the space station's inertial rest frame, they do not
decelerate simultaneously with respect to the Earth's inertial rest
frame. (This is why people have been laboring to get you to take the
concept of simultaneity seriously.) Of course, the absolute lapse of
proper time for each of their paths is the same, because that's an
invariant. In terms of the Earth's rest frame coordinates it is

Al: sqrt(0.9165^2 - 0.0000^2) = 0.916...
Bert: sqrt(1.1256^2 - (-0.8728)^2) = 0.916...

just as we computed in terms of the space station rest coordinate
system. So everyone agrees that Al and Bert's clocks read the same at
their deceleration points, but those events occur at different times
with respect to any system of inertial coordinates in terms of which
the Earth is at rest.

But I doubt that any of this is getting you any closer to actually
understanding special relativity. To actually understand it, you need
to understand the full physical significance of inertial coordinate
systems, and the physical meaning of the invariant spacetime interval.
You can find some good explanations on the web if you look hard
enough.
.

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