Relativistic Spring-Mass Oscillator: A Paradox?

[Alfred Cann]

Consider a spring-mass oscillator on a train moving at relativistic speed.
According to SR, to a stationary observer, both the mass and the period will appear to have increased by a factor of γ.
But the period is supposed to be proportional to the square root of the mass. Something is wrong.
Don't talk about longitudinal and transverse masses; I can orient the oscillator any way I want.
Don't talk about the mass oscillating at relativistic speed; I can keep the oscillation slow.

 

[Ibix]

Have you looked at the relativistic force transformation rules?

 

[PeterDonis]

the period is supposed to be proportional to the square root of the mass.

It's proportional to the square root of the mass, and inversely proportional to the square root of the spring constant. So how does the spring constant transform when you change frames?

I can orient the oscillator any way I want.

So which way do you want to orient it?

 

[Dale]

Something is wrong.

Yes, the formula for the period of a spring is non-relativistic.

 

[Alfred Cann]

I had assumed the spring constant is invariant; you guys are implying that it changes.

 

[Dale]

I had assumed the spring constant is invariant; you guys are implying that it changes.

I suspect that not just the spring constant, but also Hookes law is not invariant.

 

[phyzguy]

Yes, the formula for the period of a spring is non-relativistic.

This shouldn't be an issue. As the OP said, the oscillations in the frame of the moving train can be slow, so Hooke's law should apply in the frame of the train to whatever accuracy you want. The question is how to reconcile this with what the stationary observer sees as the train speeds by. I think the right answer, as others have implied, is that the spring constant as perceived by the stationary observer is different from that perceived by the observer on the train. @Alfred Cann, imagine the train speeding by multiple times (or multiple trains) with different weights hanging from the spring. What would he see?

 

[Dale]

This shouldn't be an issue.

How could it not be? The derivation by which the OP expects the frequency to depend on the square root of the mass is based on Hooke’s law and Newton’s laws. If those are non relativistic then the relationship should not be expected to hold relativistically.

 

[PeterDonis]

I had assumed the spring constant is invariant

You should not assume. You should try to calculate in order to either demonstrate that it's correct or realize that it's not correct. In order to do that calculation you will first need to find a relativistic version of Hooke's law, since the ordinary version is written in terms of 3-vectors, not 4-vectors, and is therefore not relativistically invariant, as @Dale has pointed out.

 

[Mister T]

According to SR, to a stationary observer, both the mass and the period will appear to have increased by a factor of γ.

The period will increase, that's a consequence of the postulates. The same cannot be said for the mass. Saying that the mass increases is a choice about the meaning of a word, it's not a consequence of the postulates.