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Re: [Phys-l] geometry of spacetime (was: relativisitic mass ...)



John M wrote:
| The fact that that object *appears* to another observer to
| asymptotically approach the speed of light is the *observer's*
| problem!

Bob Sciamanda wrote:

Whatever this means, is it not in the same way the "object's" problem that the left behind observer "appears" to asymptotically approach the speed of light?

We understand that speed is a relative quantity and that the idea of an "intrinsic" speed is without meaning or use. So what's the point?


1) I understood what John M. meant, and I agree with it in spirit.

2) However Bob raises a valid objection; the original statement is
perhaps a bit glib and/or open to misinterpretation.

First of all, let's get some facts on the table. I hope we all agree that:

A) As JM said, a particle can undergo uniform acceleration forever
(uniform in the particle's own frame). WLoG let this be an acceleration
in the X direction.

B) The X-component (in the lab frame) of the particle's 4-momentum (p)
increases uniformly, and increases without bound.

C) The X-component (in the lab frame) of the particle's 4-velocity (u)
increases uniformly, and increases without bound.

D) In all generality, frame-independently, we can say p = m u.

E) The 3-velocity is !!not!! the spatial part of the 4-velocity:
4-velocity is d(position)/d(tau), whereas 3-velocity is d(position)/d(t).
They differ by a factor of gamma.

THEREFORE, perhaps it would be better to say something like:
The fact that that object's 3-velocity *appears* to another observer
to asymptotically approach the speed of light is not a property of
the object, and it's not a property of the observer, either. It is
a property of the projective geometry of spacetime ... and of the
definition of 3-velocity.

To repeat, it's not a problem with the object, it's not a problem
at all, and it's not even a _property_ of the object or observer.
It's just geometry.

The geometrical situation is discussed on page 10 of the odometer paper:
http://www.av8n.com/draft/odometer.pdf