Re: Accelerating train paradox

From: sal (believer_at_nospam.org)
Date: 06/14/04 

Date: Mon, 14 Jun 2004 20:18:21 GMT

On Mon, 14 Jun 2004 21:37:03 +0200, Henri wrote:

>> > What about GR? In the train frame proper time must slow down even more
>> > than predicted by SR cause acceleration is alike gravity. What are the
>> > formulas to compute this?
>>
>> Doesn't make any difference -- GR and SR are the same in this case
>> because there isn't any gravity.
>
> I think it does make a difference.
>
> If a train, or a car accelerates at, let's say 10 m/s2 along the
> x-axis, the passengers feel a force like gravity.
>
> Acceleration along the z-axis can cancel out the 'force' of gravity in a
> spaceship frame.
>
> I write 'force' because gravity is the curvature of spacetime, at least
> that is what the textbooks want to let me believe.
>
> So acceleration also should lead to a curvature of spacetime, and maybe
> a warping of time.

It does. Acceleration produces exactly the same effect a uniform G-field
would produce.

It is possible to use a metric, and associated coordinates, that describes
a uniform gravitational field, and analyze the problem in that way (or so
I understand; I haven't done it). That's the "GR solution" -- it uses a
non-Lorentz metric, and non-flat coordinates.

But it's not necessary. You can obtain exactly the same results by
analyzing the whole problem using SR from the point of view of the tracks,
which are not accelerating.

To do it the SR way, at each moment we find the instantaneous velocity of
each part of the train, as seen by the tracks. From the instantaneous
velocity, we can find an instantaneous value for gamma, and the Lorentz
transform to an inertial frame which happens, at this exact moment, to be
moving at the same speed as the part of the train we're examining. That's
the MCRF for that object at that moment.

In the MCRF, at a particular time, we can find the coordinates of
everything else, and from that we can find the apparent speed _and_ the
apparent time (clock reading) of everything else, from the point of view
of an accelerating object.

Once we've done that, we can differentiate the clock rate we found with
respect to time in the object whose MCRF it was, and hence see "time
dilation" or "time contraction" as an apparent consequence of
acceleration.

The result of that analysis will be identical with the result of
pretending there's a "uniform gravitational field" filling all space.
And that's what I mean when I say the SR and GR approaches yield the same
result.

Either way, it's a mess.

One simple consequence is that, while you are accelerating _toward_
something, its clock appears to run fast -- exactly the way clocks high up
in a gravitational potential well run faster than clocks lower down in the
well. And when you're accelerating _away_ from something, its clock
appears to run slow -- just as the clock lower down in a gravitational
field runs slower.

> The only thing I want to understand is a mathematical connection between
> the 'simple' uniformly moving train in SR and the accelerating train in
> GR.

Acceleration is just like a gravitational field, _but_ because space is
still "flat" in this case it isn't necessary to use GR techniques to
analyze it.

The hallmark of gravity is that space is (almost) always curved when it's
present, and that curvature makes it impossible to find the momentarily
comoving inertial reference frames which are needed in order to analyze it
using SR and Lorentz transforms.

> If time slows down in the train frame, due to 'gravity along the
> x-axis', then what do ground observers measure?

No, it doesn't slow down, not in that sense.

Time slows down on the train only as a result of the train speeding up --
time relative to time on the tracks goes as 1/gamma where
gamma=1/sqrt(1-v^2) at each moment, just as you would expect(?).

>From the point of view of someone on the train, however, time of a point
FAR AHEAD of them on the tracks seems to _speed_ _up_ while the train is
accelerating, and time at a point FAR BEHIND them appears to _slow_
_down_. But note: This "acceleration" effect can't be observed! It can
only be calculated!

Anyway, I think I've contributed enough confusion to the topic for now,
and I've got to run anyway.

I hope something I said helped at least a little.

Cheers...

>
> The problem might appear easy, but I'm sure it is pretty difficult.

Yes, it is. 

-- 
To email me directly, take out nospam and put back physicsinsights.

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