Period of a pendulum on a train at relativistic speeds as observed by 2 different observers

Question

My friend asked me this question

"Two observers, A and B:

QUESTION : "A is on a train travelling at a constant speed of 0.9c and B is standing on the ground outside. Both observe the motion of a 1m long pendulum swinging (in the direction of the motion of the train), compare the observations made by A and B with regards to the motion of the pendulum. Observer A measures the period of the pendulum to be 2.01 seconds."

I'm not too familiar with Special relativity but, A is stationary relative to the equilibrium position of the pendulum so he will observe the period of the pendulum according to T = 2π√(l/g)

The pendulum is moving relative to B and so B will observe the effects of time dilation, length contraction etc and the period of the pendulum will be greater than that observed by A. But the length of the pendulum is unchanged (I think), but the angle it makes with the vertical is changed and so is the period.

However, according to Special Relativity, the Laws of Physics are the same in inertial frames, so

T = 2π√(l/g) should hold for B's frame of reference as well so the gravitational field observed by A and B should be different.

From what I've read, the period measured by A shouldn't be 2.01 seconds, as the Earth is moving relative to the pendulum, and A on the train and so we can not take g = 9.8 ms^-2, but it is true that time dilation effects will take place and B will observe a longer period? What exactly are the effects at play here that make the period observed by B longer?

Thanks in advance, and apologies I'm not very familiar with Special Relativity at all

Answer 1

Well firstly I think you've got the equivalence principle a bit misunderstood. It is that the laws of physics are the same for an observer in their frame of reference (whatever it is). In order to achieve this there are some effects, distortions of relative observations, and consequences of changing frame of reference (accelerating). Principally, no-one will observe that the speed of light is exceeded, as this is our yardstick, it is the maximum speed of interaction between any particles on which we rely for our measurement and very perception of time. This in turn drives our measurement and perception of distance.

So it's consistent with the theory for the period of the pendulum to appear different to observer B.

Secondly you've asked about special relativity, effects due to relative speeds. In general relativity similar but different and independent effects occur due to acceleration, including acceleration from a gravitational mass. Some of the existing answers cover both sets of effects.

Working with relative velocities alone (special relativity) there is often a paradox as to who experiences time dilation. This can often be resolved by considering who has accelerated to achieve the particular scenario. It is the party that accelerated that experiences time dilation.

In this case A is on a train, it must be them that accelerated. Time dilates for them, and observer B sees their every action slow down, including the motion of pendulum and anything else.

Time is perceived as scalar. If it is running at a particular rate when measurements are made in one direction, we assume/perceive this is true in all directions. Unfortunatly, along the direction of motion light cannot travel as far in the same time, or it would exceed c. In order to preserve this rule we perceive/measure that instead of time running differently, the local distances in the observed frame of reference are shorter. This is length contraction.

That being the case, on top of the initial time dilation effect your pendulum will also appear length contracted in the direction of motion. This ensures the time dilation you observe is not weirdly directional in nature, which would just be a really hard and unnatural thing for us to make sense of.

So that's the "no equations" version. I went through the introductory maths myself not so long ago for the famous "light clock" derivation of the lorentz transform, here Relativity tangential light clock

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Note1 : that during an oscillation the pendulum moves repeatedly in two opposite directions. These alter it's relative velocity. In this answer I assume that the pendulum velocity in its frame of reference is not relativistic, as that would complicate matters somewhat.