Einstein's derivation of the time dilation and length contraction equations in his 1905 paper on special relativity is a logical and mathematically rigorous argument based on the principles of the theory. Each step of the derivation follows logically from the previous steps and is consistent with the underlying assumptions and postulates of special relativity. While the concepts of time dilation and length contraction may seem counterintuitive at first, they have been extensively tested and confirmed by numerous experiments and observations.
To provide a demonstration of one step of the derivation, let's consider the time dilation equation derived by Einstein. In his paper, Einstein starts with two inertial reference frames, one at rest (frame S) and the other moving relative to it (frame S'). He assumes that the two frames are in relative motion along the x-axis with a constant velocity v.
Einstein introduces two events: the emission of a light signal at point A and its reception at point B. He then considers the time intervals Δt and Δt' as measured by an observer at rest in frame S and an observer moving with frame S', respectively.
The key assumption made by Einstein in this derivation is that the speed of light in a vacuum is constant and the same for all observers, regardless of their relative motion. This is known as the principle of the constancy of the speed of light.
Using this assumption, Einstein derives the relationship between the time intervals Δt and Δt' by considering the propagation of the light signal. By analyzing the geometry of the events and applying the principles of special relativity, he arrives at the equation:
Δt' = Δt / γ
where γ (gamma) is the Lorentz factor given by:
γ = 1 / sqrt(1 - v^2/c^2)
Here, c represents the speed of light, and v is the relative velocity between the two frames.
Each step of this derivation can be justified mathematically and logically, and the result aligns with experimental evidence. The time dilation effect has been observed in various experiments, such as particle accelerators and high-precision atomic clocks.
While the concepts of time dilation and length contraction may seem counterintuitive, they are fundamental consequences of the theory of special relativity and have been confirmed by a wide range of experimental observations.