A previous post on this subject is here. One reference for this post is V. A. Ugarov’s Special Theory of Relativity (Mir, 1979).
A light clock is a device with an emission-reflection-reception cycle of light that registers the current timeline point and placeline point in units of cycle length and duration. Consider two identical light clocks, at first in their reference frames at rest, K, K´. Then, as the light clock in K´ moves transversely relative to K with uniform motion at velocity v (right), from K one observes the following:
The illustration above shows one cycle length of the light path (i.e., wavelength), X, on the left and one cycle duration (i.e., period), T, on the right at rest in reference frames K, K´. For the reference frame K´, in motion relative to reference frame K, call the arc length of one cycle of the light path x. Call the distance between the beginning and ending place points of one cycle x⊥. For the reference frame K´ relative to reference frame K, call the arc time of one cycle of the light path t. Call the distime between the beginning and ending timepoints of one cycle t⊥.
Following Ugarov: Observing clock time rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock time from one frame with readings of several clock times from another frame, because two clock times from different reference frames occur at the same place point only once. In one of the frames there must be at least two clock times which are supposed to be synchronized. For the sake of definiteness we shall be comparing one clock time, t, from the frame K´ with two clock times from the frame K, at the place point in the beginning and end of a cycle.
Let a clock and a light source be located at the origin O´ of the frame K´. A mirror is set perpendicular to the L axis at the distance X/2 from the light source (and the clock). A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin place point O´ with the period T = X/c. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.
Now let us consider the propagation of the same signal in the frame K relative to which the source and the mirror move to the right together with the frame K´ at velocity v. Although the signal was sent from the two coincident origins, O and O´, the reflection from the mirror will occur at another place point x⊥/2 of the frame K and the reception of the reflected signal at the place point x⊥ of the axis. In this way the path of the signal in the frame K traces out two sides of an isosceles triangle.
As the path traveled by light in the frame K is greater than that in the frame K´, one can expect that the period T between the sending and reception of the signal, when measured in the frame K, will be greater than t. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different place points of space. The period T between these two events in the frame K will be measured in this case by the two clocks removed from each other by the distance vt⊥ along the direction of motion. The velocity of light is equal to c in all reference frames. With Euclidean space we obtain:
(x/2)² = (ct/2)² = (vt⊥/2)² + (X/2)².
Given t = t⊥ and collecting t from this equation, we get
t² (1 − v²/c²) = (X/c)²,
t = (X/c)/√(1 − v²/c²) = γ (X/c)
where γ = 1/√(1 − v²/c²).
Considering that X/c = T, then
t = γ T.
Since γ > 1, for spacetime a clock (or any body) moving transversely relative to an observer in an inertial frame will run slower (dilate) than at rest in its frame of reference.
Following Ugarov but with Euclidean time: Observing distance change rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one distance change rate from one frame with readings of several distance change rates from another frame, because two distance change rates from different reference frames occur at the same timepoint only once. In one of the frames there must be at least two distance change rates which are supposed to be synstancized. For the sake of definiteness we shall be comparing one distance change rate x⊥ from the frame K´ with two distance change rates from the frame K, at the timepoint of the first and last points of a cycle.
Let a clock and a light source be chronated at the origin timepoint O´ of the frame K´. A mirroring event occurs parallel to the t⊥ axis at the distime T/2 from the light source (and the clock) perpendicular to the relative motion. A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin timepoint O´ with the cycle length X = cT. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.
Now let us consider the propagation of the same signal in the frame K relative to which the source and mirror move to the right together with the frame K´ at velocity v. Although the signal was sent from the two coincident origin timepoints, O and O´, the reflection from the mirror will occur at another timepoint t⊥/2 of the frame K and the reception of the reflected signal at the timepoint t⊥. In this way the path of the signal in the frame K traces out two sides of an isosceles triangle.
As the time path travelled by light in the frame K is greater than that in the frame K´, one can expect that the cycle length L between the sending and reception of the signal, when measured in the frame K, will be greater than x. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different timepoints of time. The cycle length X between these two events in the frame K will be measured in this case by the two distance change rates removed from each other by the distime x⊥/v along the direction of motion. The velocity of light is equal to c in all reference frames. With Euclidean time we obtain:
(t/2)² = (x/2v)² = (x⊥/2c)² + (T/2)².
Given x = x⊥ and collecting x from this equation, we get
x² (1 − v²/c²) = (cT)²,
x = (cT)²/√(1 − v²/c²) = γ (cT),
where γ = 1/√(1 − v²/c²).
Considering that cT = X, then
x = γ X.
Since γ > 1, in chron a body or wave moving transversely relative to an observer in an inertial frame will be longer (expand) than at rest in its frame of reference.