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 time and stance in units of cycle length and duration. Consider two identical light clocks, at first in their reference frames at rest, K, K´ (left). Then, as the light clock in K´ moves relative to K with uniform motion at velocity v (right), from K observes the following:

The left illustration shows one cycle length of the light path (i.e., wavelength), L, and one cycle duration (i.e., period), T, at rest in reference frames K, K´ (left). 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 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 instants 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 point in space 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 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 L/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 point O´ with the period T = L/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 the velocity v. Although the signal was sent from the two coincident origins, O and O´, the reflection from the mirror will occur at another point x_⊥/2 of the frame K and the reception of the reflected signal at the point x_⊥ of the axis. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the path travelled 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 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 motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(x_</2)² = (ct_</2)² = (vt_⊥/2)² + (L/2)².

Given t_< = t_⊥ and collecting t_< from this equation, we get

t_<²(1 − v²/c²) = (L/c)²,

  t_<= (L/c)/√(1 − v²/c²) = γ (L/c)

where γ = 1/√(1 − v²/c²).

Considering that L/c = T, then

t_< = γ T.

Since both events occurred at the same point in the frame K´, they were registered by means of the same clock. A time interval between events registered by means of the same clock (which implies that the events occurred at the same point of space) is referred to as a proper-time interval between these events. Of course, a time interval of which the initial and the final moments are registered at different points of the reference frame and, consequently, by means of different clocks will not be a proper-time interval between events.

Following Ugarov but with Euclidean time: Observing clock stance rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock stance from one frame with readings of several clock stances from another frame, because two clock stances from different reference frames occur at the same instant in time only once. In one of the frames there must be at least two clock stances which are supposed to be synstancized. For the sake of definiteness we shall be comparing one clock stance x_⊥ from the frame K´ with two clock stances from the frame K, at the instant of the first and last points of a cycle.

Let a clock and a light source be chronated at the origin instant 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 instant O´ with the wavelength L = 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 the velocity v. Although the signal was sent from the two coincident origin instants, O and O´, the reflection from the mirror will occur at another instant t_⊥/2 of the frame K and the reception of the reflected signal at the instant t_⊥. In this way the path of the signal in the frame K traces out two sides of an equilateral 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 wavelength 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 instants of time. The wavelength L between these two events in the frame K will be measured in this case by the two clock stances removed from each other by the distime x_⊥/v along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, 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

x_<²(1 − v²/c²) = (cT)²,

x_<= (cT)²/√(1 − v²/c²) = γ (cT),

where γ = 1/√(1 − v²/c²).

Considering that cT = L, then

x_< = L/γ.

Since both events occurred at the same instant in the frame K´, they were registered by means of the same clock stance. A length interval between events registered by means of the same clock stance (which implies that the events occurred at the same instant of time) is referred to as a proper-length interval between these events. Of course, a length interval of which the initial and the final moments are registered at different instants of the reference frame and, consequently, by means of different clock stances will not be a proper-length interval between events.

~

The moving light clock has x_< = vt_⊥. and x_< = ct_<. Note: if c = ∞, then t_|| = 0; x_< = x_⊥; and t_< = t_⊥. If v = 0, then t_⊥ = 0; x_< = x_||; and t_< = t_||.

From the Euclidean metric for space we have: (x_</2)² = (x_⊥/2)² + (x_||/2)² . Combine this with the above to get:

(ct_</2)² = (vt_⊥/2)² + (ct_||/2)².

Divide by c² to get:

(t_<)² = (βt_⊥)² + (t_||)².

If x_< = x_⊥ above, then

(x_<)² = (x_</β)² + x_||², or

(x_<)² (1 − 1/β²) = x_||².

From the Euclidean metric for time we have: (t2)² = (t_⊥)² + (t_||)². Combine this with the above to get:

(x_</c)² = (x_⊥/v)² + (x_||/c)².

Multiply by c² to get:

(x_<)² = (x_⊥/β)² + x_||²,

which is a weighted metric.

Can we infer x_|| = ct_||?