Question

I'm lay about relativity and I want to understand how does c does not change between frames of reference.

Imagine a train of length L0 at a relativistic speed and a light beam inside it. For an inside frame, the time taken for light to travel across the train would be Δt0=L0c.

Now imagine an outside still frame of reference. To him, both time and length should receive a Lorentz transformation, thus ΔtR=γΔt0 and LR=L0γ. As velocity is ΔsΔt, the velocity to him would be

LRΔtR=L0γ2Δt0=cγ2≠c

Where is the mistake?

Answer 1

The mistake is in assuming that the velocity, Δs/Δt, stays the same in both frames of reference. The correct velocity transformation between frames is given by the Lorentz transformation equations.

Step-by-step explanation:

The mistake in your calculation is in assuming that the velocity, Δs/Δt, stays the same in both frames of reference. In reality, the velocity does not add up in a simple way as in classical mechanics. According to the Lorentz transformation equations, the correct velocity transformation between frames is given by:

v' = (v - u) / (1 - v*u/c^2)

where v is the velocity of the train as observed from the outside frame, u is the velocity of the train as observed from the inside frame, v' is the velocity of the light beam as observed from the outside frame, and c is the speed of light.

By plugging in the values from your example, where v = L0γ/T0 and u = cγ^2, you will find that v' equals c, confirming that the speed of light is the same in both frames of reference.