iSoul In the beginning is reality.

# 2D light clock

The famous Michelson-Morley experiment used what could be described as a 2D light clock since their interferometer combined two light clocks at right angles. Their hypothesis was that this would show the Earth moving through the aether, but they failed to detect any motion. Einstein explained this failure as a feature of relativity. In other words, the expected difference between the two light clocks was “corrected” by relativity.

The reasoning of the Michelson-Morley experiment went like this:

Light is sent from the source and propagates with the speed of light c in the aether. It passes through the half-silvered mirror at the origin at T = 0. The reflecting mirror is at that moment at distance L (the length of the interferometer arm) and is moving with velocity v. The beam hits the mirror at time T1 and thus travels the distance cT1. At this time, the mirror has traveled the distance vT1. Thus cT1 = L + vT1 and consequently the travel time T1 = L / (cv). The same consideration applies to the backward journey, with the sign of v reversed, resulting in cT2 = LvT2 and so T2 = L / (c + v). The longitudinal travel time T|| = T1 + T2 is:

Michelson obtained the longitudinal expression correctly in 1881 but his equation for the transverse direction was corrected by Alfred Potier (1882) and Hendrik Lorentz (1886).

The derivation of the travel time in the transverse direction can be given as follows (analogous to the derivation of time dilation using a light clock): The beam is propagating at the speed of light c and hits the mirror at time T3, traveling the distance cT3. At the same time, the mirror has traveled the distance vT3 in the x direction. In order to hit the mirror, the travel path of the beam is L in the y direction (assuming equal-length arms) and vT3 in the x direction. This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame. Therefore, the Pythagorean theorem gives the actual beam travel distance of √(L² + (vT3)²) and consequently the travel time T3 = L / √(c² − v²), which is the same for the backward journey. The transverse travel time T = 2T3 is:

Compare this to the (proper) time in the rest frame: T = 2L/c. We then have these ratios:

T = γT and

T|| = γT.

These equations express time dilation, the first relative to the rest frame and the second relative to the transverse frame.