Special relativity and acceleration of an object falling in a direction perpendicular to the direction of motion

Question

So I'm redoing special relativity because I want to start GR and I thought it might help make the learning somewhat easier, I was doing some SR problems and this thought came to me.

Imagine a train moving along a smooth surface with a velocity 0.98c along the x axis with respect to some observer on the surface. Imagine in the train's frame of reference (let's call it S2), an object falls from top of the train to the bottom and it takes 1 second to reach the bottom. In the frame of the observer outside (let's call it S1), the time taken by the object to reach the bottom should be approximately 5 seconds.

In S2, the distance traveled by the object is some short length $L_0$, in S1, the distance traveled is a lot more since the train is moving. But the y component of this displacement is the same in both frames since the Lorentz transformation along the y axis in this scenario just says that $y_2 = y_1$. Now if we assume that gravity acts along y axis with respect to both observers, wouldn't the observers disagree about the strength of gravitational field inside the train?

Or to be slightly more precise, won't S1 conclude that gravitational field is weaker inside the train since the time taken to fall is more? I'm not sure how we would apply velocity transformations here, from what I understand, it should say that approximately $\dot{y_1} = (0.2) \dot{y_2} $, I'm not sure if this should hold at every instant of time however. If it does hold, doesn't it again imply that acceleration in S1 appears weaker? Could this be resolved by S1 assuming that acceleration of the object in gravitational field is dependent on mass of the object (as opposed to S2) and doesn't that violate the first postulate of special relativity? Do we need to invoke relativistic momentum here?

Answer 1

The disagreement is caused by combining Newtonian gravity with special relativity, which is not general relativity (GR). In weak GR (and 1g is weak) there is a graviomagnetic field that has a $\vec v/c \times \vec g_M$ Lorentz type force, where $\vec g_M$ is a solenoidal (like magnetism) gravitational field caused by masses moving at high speed. That mass is the source of the "usual" gravitational field moving at $0.98c$ under the train.

Answer 2

Imagine a train moving along a smooth surface with a velocity 0.98c along the x axis with respect to some observer on the surface. Imagine in the train's frame of reference (let's call it S2), an object falls from top of the train to the bottom and it takes 1 second to reach the bottom. In the frame of the observer outside (let's call it S1), the time taken by the object to reach the bottom should be approximately 5 seconds.

From the times taken by the object to reach the bottom in S1 and S2 we can easily deduce these additional things:

1. In S1 y-speed reached is 1/5 times the y-speed reached in S2.

2. In S1 y-acceleration is 1/25 times y-acceleration in S2.

Now if we assume that gravity acts along y axis with respect to both observers, wouldn't the observers disagree about the strength of gravitational field inside the train?

Or to be slightly more precise, won't S1 conclude that gravitational field is weaker inside the train since the time taken to fall is more? I'm not sure how we would apply velocity transformations here, from what I understand, it should say that approximately y1˙=(0.2)y2˙ , I'm not sure if this should hold at every instant of time however. If it does hold, doesn't it again imply that acceleration in S1 appears weaker? Could this be resolved by S1 assuming that acceleration of the object in gravitational field is dependent on mass of the object (as opposed to S2) and doesn't that violate the first postulate of special relativity? Do we need to invoke relativistic momentum here?

S2 says that the strength of the gravity field of the earth is the same huge number inside and outside the train. Because "inside the train" is a normal place on earth. (But earth is not normal, it's contracted and has a huge gravity field.)

S1 says that the strength of the gravity field of the earth is the same number (9.8 m/s^2) inside and outside the train. Again because "inside the train" is a normal place on earth. (Objects inside the train are not normal, they are contracted and have a hugely increased gravity fields.)

For some reason I can't explain why in S1 objects falling inside the train almost float in the air. ... Oh yes they don't almost float in the air. They fall quite normally. The falling motion causes a slowing down of the x-motion, which causes the falling motion to not be very time-dilated. But in S2 the falling is still 5 times quicker than this quite normal falling ... so acceleration in S2 must be 25 times higher, as noted earlier.