# Acceleration in Special Relativity

#### Astro

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!

#### topsquark

Forum Staff
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!
Special Relativity normally does not work with accelerations; accelerations imply a force and forces belong to General Relativity. If you don't mind my asking, where did you get/derive the equations from?

-Dan

#### Astro

Special Relativity normally does not work with accelerations; accelerations imply a force and forces belong to General Relativity. If you don't mind my asking, where did you get/derive the equations from?

-Dan
From what I understand, Special Relativity can work with acceleration, provided that the phases in which acceleration occurs are split up into several instantaneous reference frames. General Relativity can be used to explain the acceleration phases in the proposed scenario, but is not necessary.

I was unsure of where to even start to derive the equations required for my research, but I found the equations here; http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

I've also found some of these equations (particularly 'v' and 'd' equations) elsewhere.

#### topsquark

Forum Staff
Thanks for that!

Without seeing the text I can only speculate, but the idea of instantaneous reference frames is a concept from General Relativity. (Or from Differential Geometry, upon which GR is based Mathematically.) So the text is actually doing GR rather than SR...something they apparently didn't tell you.

Also, what the link you sent me claims is the "Equivalence Principle" is not what Einstein called the Equivalence Principle. But it's a good enough name for what is being described anyway.

-Dan

#### topsquark

Forum Staff
The simplist way to do it is with the CADO equation, described here:
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and here:

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

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Thanks for the page. I'll look at it later.

Why are you posting so small? Is there an intruder creeping around your home and you're trying to make sure not to be heard?

-Dan

#### MikeFontenot

Why are you posting so small?
When the compose window came up, and I copy/pasted in those two links, they printed extremely large, so I reduced them to something sized about like your sentence above. And when I posted that, it printed on the forum (as displayed on my mac) in a reasonable-sized font. I don't understand why it was so small on your display.

#### kiwiheretic

The simplist way to do it is with the CADO equation, described here:

and here:

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

Oh, the CADO equations, I remember them well. Well almost. This is the thread that on CADO I remember from time ago.

As I recall, this only works for flat Minkowski space. Is that correct?

I am thinking I might take lessons learned from this and put on my physics website.

#### MikeFontenot

As I recall, this only works for flat Minkowski space. Is that correct?
It works wherever and whenever special relativity works ... no significant masses involved.