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

Ken,

Thank you for correcting my elementary blunder (omitting the factor (1/2) in x(t) = x0+(1/2)at^2). As to the whole expression itself (with this factor included!), it is definitely correct under conditions that I have specified - the fixed reference frame (RF) and fixed time-independent acceleration a.

I was not discussing the motion of a real physical body, but the motion of a point under given acceleration. This was only to show in the simplest possible way that in the specified RF two separate points starting from rest and undergoing the same acceleration program along the initial separation line, will remain at the same distance from each other. This is general, purely mathematical argument having nothing to do with relativity as long as we consider the process in the same inertial RF (regarding integrations, see also Appendix F in my book indicated before).
As the one who worked on and taught relativity, including its effects as observed from accelerated RF, I am not against using a set of instantly co-moving RF. But I am also free to use the RF I have used - according to relativity, it enjoys equal rights with all others, and it is also quite natural to use in the given problem. It is natural to observe from Earth the motion and behavior of a spacecraft launched from Earth, regardless of how large its final speed might be. All my initial comment was that in such RF, conserving the shape of the spacecraft requires equal, not different, accelerations of its parts. And of course, one should not confuse the shape as observed from given RF with the proper shape. The latter will definitely change under conditions specified in my comments.

If you do have a chance to look through my book and find any errors, please, let me know. I already found a lot (so far, purely technical), which is quite natural for the first edition of a book of this size. If it comes to the next edition, I want to make it more correct and less verbose.

Best,
Moses Fayngold
NJIT

On Sunday, November 2, 2014 2:34 PM, Ken Caviness <caviness@southern.edu> wrote:

I hope this thread doesn't degenerate into a shouting match, because it interests me a great deal, having played with acceleration within the SR formalism in graduate school, eventually leading up to my dissertation:

http://www.researchgate.net/publication/234390233_Considerations_of_acceleration_effects_in_relativistic_kinematics

MF: Your brief analysis in this thread appears to me to contain a couple of conceptual mistakes, which I presume you avoided in your published articles and books on the subject. Trivially, no one believes x(t) = x0 + a t^2, even in the non-relativistic domain that's incorrect, lacking a factor of (1/2) in the second term. Secondly, remember that the formula is only valid for cases involving non-relativistic speeds and constant acceleration according to some observer, being found by twice integrating (with respect to t) the definition dv/dt = a, for constant a. Thirdly, with these integrations and corresponding differentiations w.r.t. time t, one must specify _whose_ time is being used. The logical way to deal with constant acceleration under SR is to consider constant _proper_ acceleration (the acceleration of the accelerating observer). This yields the hyperbolic motion that JD refers to, although I think he should more carefully include the
adjective "proper" in order to avoid commonly occurring confusion. [*] It has been studied frequently over the years, I'd have to check my files at the office to verify, but I expect the Wolfgang Rindler treats it his classic text, "Essential Relativity", as do Christian Møller, Sears & Brehme, and others in various books and articles. JD seems to be laying out a nice reiteration of parts of this treatment and extending it in a logical way, although other possible ways come to mind as well.

Fun stuff! I will definitely read the things you both (JD & MF) have shared, although it may have to wait for Thanksgiving break, much to do before then!

Enjoy,

Ken Caviness
Physics

[* Constant proper acceleration is quite different from constant acceleration according to an inertial observer. All my relativity materials are at work, but if memory serves, for the 1-d case, (proper acceleration) = (gamma^3)(acceleration), where gamma = 1/sqrt(1-v^2/c^2) and both velocity and acceleration on the RHS are as measured by an inertial observer, proper acceleration (by definition) as measured by the accelerating observer, or by a momentarily co-moving inertial observer.]

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of Moses Fayngold
Sent: Sunday, 2 November, 2014 12:13 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] relativistic acceleration of an extended object

I reread carefully the previous correspondence. UNDER THE CONDITIONS THAT I TOOK CARE TO EXPLICITLY FORMULATE, I do not see any errors in my conclusions. If you admit that they may be correct "...perhaps in the non-relativistic limit... ", this is already killing for your statements since the non-relativistic limit is merely a part of relativistic domain. The high school relationship between given acceleration and resulting displacement (starting from rest) which I used, is universal, mass-independent, and applies at any, even superluminal, speed. The latter may be realized, e.g., by an appropriate light spot zipping across a screen (the mathematical structure of Relativity treats superluminal and subluminal motions on the same footing).

As to your article, I had explained why I could not read it. Reciprocally, you could read my article with much more detailed analysis of the whole situation than presented here, and under different possible conditions. As to your statement "Shape is determined by proper length..." , I can also refer to another relevant article, "Two Permanently Congruent Rods May Have Different Proper Lengths" in the arXiv.

I would appreciate if anyone could show where and why my arguments are wrong.

Moses Fayngold,

NJIT

On Saturday, November 1, 2014 1:19 PM, John Denker <jsd@av8n.com> wrote:

On 11/01/2014 09:51 AM, Moses Fayngold wrote:
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".

If you would read the article, you would find the answers to those questions. Acceleration means proper acceleration at each point. Shape is determined by proper length, measured along a contour of constant time. All observers agree that the contour in question *is* a contour of constant time, so there is no ambiguity whatsoever.

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.

It's not 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.

False (except perhaps in the non-relativistic limit, which
is not what we are talking about here).

Would it kill ya to read the article?

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