Visualizing Proper Time in Special Relativity [with a Light Clock]

The Principle of Relativity

...have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.

Galileo's Principle of Relativity

As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. At this juncture the question of the validity of the principle of relativity became ripe for discussion, and it did not appear impossible that the answer to this question might be in the negative.

Nevertheless, there are two general facts which at the outset speak very much in favour of the validity of the principle of relativity. Even though classical mechanics does not supply us with a sufficiently broad basis for the theoretical presentation of all physical phenomena, still we must grant it a considerable measure of “truth,” since it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful. The principle of relativity must therefore apply with great accuracy in the domain of mechanics. But that a principle of such broad generality should hold with such exactness in one domain of phenomena, and yet should be invalid for another, is a priori not very probable.

Einstein's Principle of Relativity

Light travels at 3.00 x 10⁸ m/s

On a spacetime-diagram with the axes calibrated as above, a light-ray [shown in yellow] always makes at angle of 45^o with the time-axis [which is, by convention, oriented vertically upward].

When a flash of light is emitted by a source light-rays spread out isotropically in space, tracing out a cone on a spacetime-diagram. This cone is called the "[future] light-cone of the emission event".

The light-cone depends only on the emission event. It does not depend on the motion of the source of the light.

The source [modeled as a point-particle in space] traces out a curve (called its "worldline") on a spacetime-diagram. The green source is at rest, and the red source is in motion (with v=0.5c).

The Light Clock [at Rest]

When we place a mirror at a distance L away from the source, a light ray travels at the speed of light for a time t=(L/c) to get to that distant mirror. Then, upon reflection, a light ray travels at the speed of light for an additional time t=(L/c) to get back to the source.

The duration of the round-trip is T=(2L/c). This defines one "tick" of this clock.

In these diagrams, we have chosen
L=(0.5 m).

The Light Clock [in Uniform Motion]

What is the duration of one tick of this moving clock? We measure the duration of that red tick to be longer than the duration of our our green tick.

Why? Because [to us] the moving mirror is moving away from the source, the light ray (traveling at the speed of light) that reaches that moving mirror must travel a longer distance for a longer duration.

This is time-dilation.

Let us refer to this type of light clock as a "transverse light-clock". Transverse means that the light clock is oriented perpendicular to the direction of relative-motion.

(In the next section, we will also use a light clock that is oriented parallel to the direction of relative-motion. We refer to this type of clock as a "longitudinal light-clock".)

A Michelson-Morley apparatus [at Rest]

Observe:

Light rays, emitted by the source at a common event then reflected back by the equidistant mirrors, are received by the source at the same reception event.

[To us,] the reflections on our distant green mirrors occurred at the same value of t. Let us refer to this as our "half-tick".

A Michelson-Morley apparatus [in Uniform Motion]
(attempt #1; in conflict with the Principle of Relativity)

We already know that the light ray (traveling at the speed of light) must travel a longer distance for a longer duration to reach the "transverse mirror".

In fact, the light ray (traveling at the speed of light) that reaches the "longitudinal mirror" moving away from the source must travel an even longer distance for a even longer duration. [It had a headstart along the direction of motion!]

Thus, the outgoing light ray reaches the moving longitudinal mirror after the light ray reaches the moving transverse mirror.

Upon reflection, the return trip to the moving source doesn't take as long since the moving source is moving toward the incoming light ray.

Although these are interesting observations (which we will deal with later), a more serious observation is this:

Note carefully that the reflections are not received at the same event. The longitudinal reflection is received first, followed by the transverse reflection.

This situation, however, violates the principle of relativity!

The red clock in motion does not receive its reflections simultaneously. In fact, the time-difference between the two receptions can be used to calculate the velocity of the moving red clock.

The experiment of Michelson and Morley effectively tried to measure this time difference... but they always measured zero. (experimental data)

Apparently, there is something wrong with this proposed model of the moving clock.

A Michelson-Morley apparatus [in Uniform Motion]
(attempt #2; in accordance with the Principle of Relativity)

This is length-contraction.

Watch: VPT - Michelson-Morley [with contraction] (AVI) (0.7 MB) VPT - Michelson-Morley [with contraction] (mp4) (0.1 MB)

Absolute Simultaneity Lost

However, observe that we do not regard the "reflection events at his distant red mirrors" to be simultaneous (i.e., they do not occur at the same value of t for us).

In accordance with the Principle of Relativity, he regards the "reflection events at his distant red mirrors" to be simultaneous (i.e., they occur at the same value of [call it] t' for him).

The next visualization will elaborate on this.

Circular Light Clocks
(Visualizing Proper Time)

This provides us with a visualization of one tick of proper time.

Since each mirror traces out its worldline, the collection of mirrors traces out a "worldtube".

Consider the light-cone of the first light-flash
and the worldtube of the green light-clock, which is at rest. The intersection of that light-cone and the green worldtube represents a set of "reflection events for the green light-clock". These events are "simultaneous for the green light-clock".

Observe that these reflection events actually trace out a circle [for the green light-clock]. Consider the unique plane that contains this circle. Realize that all events on this plane are simultaneous [for the green light-clock]. One could call this the "t=0.5-plane [for the green light-clock]".

Following the reflected rays leads to the "first tick". (By the way, these reflected rays trace out the "[past] light-cone of the reception event".)

Then, the procedure can be repeated for the other ticks. Upon considering all of the ticks for this clock, one obtains a "family of parallel planes of simultaneity, marking the "half-ticks" [for the green light-clock]".

The same procedure applies to the red light-clock, which is in motion.

The intersection of that first light-cone and the red worldtube represents a set of "reflection events for the red light-clock". These events are "simultaneous for the red light-clock but NOT for the green light-clock".

After tracing out the paths of all of these light-rays, the measurement of the intervals of proper-time is now reduced to the "counting of the ticks".

By counting ticks, it is evident that, for the green light-clock, "when 3 red-ticks have elapsed, 5 green ticks have elapsed".

The ratio 5/3 is the time-dilation factor

There are a few other neat things [like the Doppler effect, the measurement of lengths, and the symmetry of the observers] that one can "read off" such diagrams. When I have some more time, I will update this page. For now, you can have a look at my paper below.

The Clock Effect / Twin Paradox

Here's how the story runs (adapted from Spacetime Physics, ex. 27):

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On their twenty-first birthday, Peter leaves his twin Paul behind on earth and goes off in the x-direction for (3 years) of his time at v=(0.8c), then reverses direction and in another (3 years) of his time returns at the same speed.

• What is Peter's age at the moment of reunion?
• What is Paul's age at the moment of reunion?

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You and I meet. We synchronize our watches, and agree to meet up later (here or at some other designated location). When we meet that second time, we will find that our watches are no longer synchronized.

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(The issue of the [false] "paradox" arises when one attempts to assert the brothers are "equivalent", so that, by the supposed symmetry, the age difference must be zero [which is at odds with the notion of time-dilation].

The paradox is resolved when it is realized that the brothers are, in fact, not-equivalent. Paul has been an inertial observer (traveling with a constant velocity-vector ["zero", in this case]) between the separation and reunion events. Peter, however, has not been an inertial observer---he has been for most of his trip. However, he changed his velocity-vector when he turned around. That event disqualified him from being "inertial".)

Visualizing Proper Time (the paper)