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# Re: [Phys-L] relativistic acceleration of an extended object

The equation for the first part should read x1(t) = x1+a1 t^2, that is "+" instead of "=" on the right. Sorry for the typo.

Moses Fayngold,
NJIT

On Saturday, November 1, 2014 12:51 PM, Moses Fayngold <moshfarlan@yahoo.com> wrote:

I was forewarned against opening the given links by the message:

"This Connection is Untrusted.
You have asked Firefox to connect securely to www.av8n.com, but this site's identity can't be verified."

But I have some comment on the basic statement in the text itself: "...in order for the object to maintain its shape, different parts will need to accelerate at different rates".

This statement is ambiguous. Its truth value depends on the chosen reference frame (RF) and on definition of "shape". Let us define the shape as an instantaneous configuration of the object in a given RF. Then it is easy to see that in the initial rest frame of the object (frame A), the statement is wrong. Different acceleration rates for different parts will surely distort the shape. In order to conserve it in A, we need the same acceleration a for all parts. Consider two small parts of the object - one initially at point x1 and the other at x2 along the direction of the boost. If the boost starts at t =0, then at t >0 we will have x1(t)=x1=a1 t^2, x2(t)= x2+a2 t^2, and the instantaneous distance between the parts will be

D(1, 2, t) = x2-x1 + (a2-a1) t^2

It reduces to the initial distance D(1, 2, 0) = x2-x1 only if a1=a2 .
You can find more details in the "Dynamics of the relativistic length contraction and the Ehrenfest paradox", arXiv:0712.3891 or in my book "Special Relativity and How It Works", Wiley-VCH, 2008

On Sunday, October 26, 2014 6:19 AM, John Denker <jsd@av8n.com> wrote:

Hi --

I posted some notes and diagrams on the subject:
https://www.av8n.com/physics/hyperbolic-motion.htm

Suppose we have an object with some large length (L) undergoing a
large constant acceleration (a). We consider the relativistic case,
where La/c^2 is not small compared to unity.

It is well known that the world-line of pointlike object undergoing
constant acceleration is a hyperbola. When we generalize to a larger
object, we find a few features that may seem non-obvious at first,
but can be given a simple interpretation in terms of the geometry of
spacetime. We call particular attention to the geometrical /center/ of
the hyperbolas.

We shall see that in order for the object to maintain its shape,
different parts will need to accelerate at different rates. This can
be considered a generalization of the notion of centrifugal force, as
applied to the case of a rotation in the xt plane.

Contents

1 Introduction

2 A Numerical Example : Hyperbolic Motion
2.1 Cluster of Pointlike Objects
2.2 Rigid Extended Object

3 Discussion
3.1
Center of the Parabolas
3.2 Centrifugal Stress
3.3 The Relativistic Criteria

4 References

For the rest of the story, see
https://www.av8n.com/physics/hyperbolic-motion.htm

This is new and probably full of typos. Comments, questions, and
suggestions are welcome.

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