# Effect of Acceleration on (Special) Relativity of Simultaneity

We often refer to the train/station analogy of inertial frames to determine the relativity of simultaneity. So, if the experiment was done on Earth, the train is one inertial frame, and the station is another inertial frame. But in reality the Earth is accelerating in a number of ways - revolving once a day, orbiting the sun once a year, and orbiting the galaxy centre once every 250mil(?) years. So both the train and the station are not in reality inertial frames. So my question is - can we treat them as inertial frames, because they are both subject to the same acceleration, or do we in reality need to invoke some other correction or indeed General Relativity?

Note, I'm not a physics student, so if it's possible to give a "for dummies" answer, that would be great.

## 1 Answer

The answer depends on how long your train is. In one second, a photon falls the same as a feather or a bowling ball: 16 feet, but it travels over 186,000 miles. That is not very much curvature to worry about.

A train car is 50 feet long, and the speed of light is 1 foot per nanosecond. That mean it takes 50 ns to traverse it, so it falls about 40 femtometers (1 fm $=10^{-15}$ m)--that is roughly the width of 40 protons, or several thousand times smaller than the diameter of an atom. These corrections are insignificant.

If your train station were on the surface of a neutron star, then perhaps General Relativity would complicate things.

• So do we conclude that even though they aren't technically inertial frames, "nearly inertial frames" is more or less equivalent? – erv Mar 27 '18 at 4:06
• Basically is ok for the purpose, at least. I never done such experiments, usually we draw train and station as suspended in a space and are inertial for the discussion. – Alchimista Mar 27 '18 at 9:29
• @erv Yes. That's the thing about Gedanken experiments. You can just unthink your systematic problems, unlike actual experimental physics. – JEB Mar 27 '18 at 13:32
• Frankly you can't build a light projector inside a physically sized train car with opening angle less than $3 \times 10^{-15}$ radians which is what would be needed for this "systematic" to affect your experiment. – dmckee --- ex-moderator kitten Mar 27 '18 at 13:45
• @erv Yes, we can generally pretend that the Earth is an inertial frame, although sometimes we do need to account for its acceleration due to its axial rotation; the other accelerations are relatively miniscule. Formally, we use a co-moving inertial frame, i.e., a frame that has zero velocity wrt Earth at the start of the experiment, and (hopefully) by the end of the experiment the comoving frame and the true Earth frame will still be in close agreement. – PM 2Ring Mar 28 '18 at 9:33