# proper acceleration

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What is the magnitude of the proper acceleration of an object moving in a circle at a given (unchanging) relativistic speed?

For an object moving in a straight line proper acceleration = acceleration*gamma^3

My question has to do with the relationship between transverse mass and longitudinal Mass

Edited by granpa
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I'm not exactly sure what your question has to do with transverse and longitudinal mass. Proper acceleration is the magnitude of four-acceleration. So if you have a path (in this case a circle), you can parametrize it, find four-acceleration, then take its norm to find proper acceleration:

$x^\mu (t)= (ct,Rcos \, \omega t,Rsin \, \omega t)$

$u^\mu (t)= \frac{dx^\mu}{d \tau} = \frac{dt}{d \tau} \frac{dx^\mu}{dt} = \frac{dt}{d \tau} (c,-R\omega sin \, \omega t,R\omega cos \, \omega t)$

we know that:

$d\tau^2 = dt^2 -c^{-2}( dx^2 + dy^2)$

or:

$\left (\frac{d\tau}{dt} \right )^2 = 1 -c^{-2} \left ( \left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right )^2 \right ) = 1 -c^{-2} \left ( R^2 \omega^2 sin^2 \, \omega t + R^2 \omega^2 cos^2 \, \omega t \right ) = 1-\frac{R^2 \omega^2}{c^2}$

So:

$u^\mu (t) = \frac{1}{\sqrt{1-R^2 \omega^2/c^2}} (c,-R\omega sin \, \omega t,R\omega cos \, \omega t)$

Then we can compute four-acceleration:

$a^\mu (t) = \frac{dt}{d \tau} \frac{du^\mu}{dt} = \frac{1}{1-R^2 \omega^2/c^2} (0,-R\omega^2 cos \, \omega t,-R\omega^2 sin \, \omega t)$

And from there we can compute proper acceleration:

$a_{prop}= \sqrt{a^\mu a_\mu} = \sqrt{\frac{R^2\omega^4 cos^2 \, \omega t+R^2\omega^4 sin^2 \, \omega t}{(1-R^2 \omega^2/c^2)^2}} = \frac{R\omega^2}{1-R^2 \omega^2/c^2}$

So, given the circular path's radius and the object's angular velocity, the proper acceleration felt by the object is $R\omega^2/ (1-R^2 \omega^2/c^2)$. Note that the object's tangential speed $v=R \omega$ can also be plugged into this equation to get:

$a_{prop} = \frac{v^2/R}{1-v^2/c^2} = \frac{\gamma^2 v^2}{R}$

EDIT: Whoops, forgot to square the demoninator .

Edited by elfmotat
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Thank you Elfmotat

So for circular motion proper accelleration = (proper velocity)^2/r works for all velocities!

F = m*(proper acceleration)

So centripetal force should equal m*(proper velocity)^2/r

This result makes perfect sense though I don't understand why for an object moving in a straight line proper acceleration = acceleration*gamma^3

And I don't see any way of reconciling it with the concept of transverse mass

This is what I am trying to understand

https://en.wikipedia.org/wiki/Relativistic_mechanics#Force

I'll keep working on it

I was expecting everything to reduce to F = m*(proper acceleration)

https://en.wikipedia.org/wiki/Proper_acceleration#Acceleration_in_.281.2B1.29D

F=dp/dt

p = m*gamma*v

So I assume

F=m*d(gamma*v)/dt

It then says that the result is that

F = gamma^3*ma for straight line acceleration

F = gamma*ma for acceleration at a right angle (like a circle)

a= v^2/r for a circle

????

Therefore F = gamma^1*m*v^2/r for a circle

Edited by granpa
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What is the magnitude of the proper acceleration of an object moving in a circle at a given (unchanging) relativistic speed?

Let's consider forces affecting an object swung around using a rope.

If the Newtonian centripetal force measured on the rope is F, then the relativistic centripetal force is gamma * F, because the transverse mass of the object is gamma*rest mass. This is the force at the other end of the rope. I mean at the end that is not moving relativistically.

At the end that is moving relativistically forces are measured to be gamma times bigger. I like to think that's because of time dilation.

The proper acceleration is measured by an accelerometer, which is a force meter.

So a force meter attached to the object measures this force: Newtonian centripetal force times gamma squared.

So when used as an accelerometer the force meter measures this acceleration: Newtonian centripetal acceleration times gamma squared.

What is the magnitude of the proper acceleration of an object moving in a circle at a given (unchanging) relativistic speed?

For an object moving in a straight line proper acceleration = acceleration*gamma^3

My question has to do with the relationship between transverse mass and longitudinal Mass

Longitudinal acceleration and transverse acceleration are quite different:

Longitudinal acceleration changes the longitudinal mass and the transverse mass.

While transverse acceleration does not change the longitudinal mass or the transverse mass.

So it makes sense that longitudinal mass and transverse mass are quite different.

Edited by Toffo
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I'm trying to get away from the idea of relativistic mass and instead use proper velocity since momentum equals mass times proper velocity and force equals the rate of change of momentum

Edited by granpa
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What is this proper velocity thing? If a driver measures his speed relative to road to be 0.5 c, then a cop standing on the road agrees that the car moves at speed 0.5 relative to the road.

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Proper velocity = gamma * v

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So the rate of change of momentum equals force but does not equal proper acceleration

Seems like proper acceleration is badly named

Edited by granpa

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