Acceleration in Special Relativity

Pmb

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Apr 2009
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Hi :) I'm working on how to resolve the twin paradox in special relativity for my 4th-year dissertation. I've found several different equations which supposedly allow for the calculation of the time experienced by each twin at set distance intervals from Earth, but they don't seem to work during the decelerations.

The travelling twin's flight consists of 4 stages; acceleration towards a distant planet for 3 lightyears, deceleration towards the planet for 3 lightyears, acceleration back to Earth for 3 lightyears, and finally, decelerating towards Earth for 3 lightyears. The absolute acceleration experienced by the travelling twin is 1 ly/yr^2 (approximately 1g).

The equations are as follows;

t = (c/a) sinh(aT/c) = sqrt[(d/c)2 + 2d/a] (time experienced by stationary twin),

d = (c2/a) [cosh(aT/c) − 1] = (c2/a) (sqrt[1 + (at/c)2] − 1) (distance travelled),

v = c tanh(aT/c) = at / sqrt[1 + (at/c)2] (velocity),

T = (c/a) asinh(at/c) = (c/a) acosh[ad/c2 + 1] (time experienced by travelling twin),

γ = cosh(aT/c) = sqrt[1 + (at/c)2] = ad/c2 + 1 (Lorentz factor).

Specifically, I'm interested in the 't' and 'T' equations. They seem to work fine for the 1st phase (acceleration towards the distant planet), but break during the 2nd phase (deceleration towards the distant planet).

I'm unsure as to whether I should plug in distance travelled or displacement from Earth for 'd', and when 'a' becomes negative (during stages 2 and 3), the acosh functions don't seem to work. Does anyone have any ideas? Any help would be appreciated!
You're using the wrong equations. I.e. you're using the time dilation formula for a clock accelerating relative to an inertial frame when in fact you want the expression for a non-accelerating clock in a non-inertial frame. In that case you have to use general relativity.
 
Dec 2012
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Boulder, Colorado
You're using the wrong equations. I.e. you're using the time dilation formula for a clock accelerating relative to an inertial frame when in fact you want the expression for a non-accelerating clock in a non-inertial frame. In that case you have to use general relativity.
It's not necessary to use general relativity for accelerating observers. In fact, doing it in GR is a very contorted and artificial way to do it.

By far the easiest and least error-prone way to do it is explained here:

https://sites.google.com/site/cadoequation/cado-reference-frame

and here:

"Accelerated Observers in Special Relativity", PHYSICS ESSAYS, December 1999, p629.
 

Pmb

PHF Hall of Fame
Apr 2009
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Boston's North Shore
It's not necessary to use general relativity for accelerating observers. In fact, doing it in GR is a very contorted and artificial way to do it.

By far the easiest and least error-prone way to do it is explained here:

https://sites.google.com/site/cadoequation/cado-reference-frame

and here:

"Accelerated Observers in Special Relativity", PHYSICS ESSAYS, December 1999, p629.
I strongly disagree. By definition, physics in an accelerated frame is general relativity. It's simply a matter of definition. In order to determine the rate of clocks in an accelerated observers frame their position must be used and that position determines the gravitational potential which effects the rates of clocks. Recall the equivalence principle: A uniformly accelerating frame of reference is equivalent to a uniform gravitational field.

If you read Einstein's 1911 derivation of gravitational time dilation you'll see that he uses what appears to be special relativity. But in reality its a step to GR which in fact is what relativity of accelerated frames is called.

Also, if you read Peacocks text on cosmology (i.e. the text used at MIT) you'll see the author explain the twin paradox from the accelerated observers FOR using GR.

So while one need not be required to use GR to analyze this it most certainly can be used. And in fact it is being used by definition when one is using accelerated frames.

Otherwise I'm curious to hear what you think GR is?
 

Pmb

PHF Hall of Fame
Apr 2009
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Boston's North Shore
Hi :) I'm working on how to resolve the twin paradox in special relativity for my 4th-year dissertation. I've found several different equations which supposedly allow for the calculation of the time experienced by each twin at set distance intervals from Earth, but they don't seem to work during the decelerations.
See

Cosmological Physics by John Peacock, pages 7-8
Cosmological physics | J. A. Peacock | download