A long time ago (when I was in the sixth or seventh grade) in a galaxy far, far away (namely Southern California), I was introduced to Einstein’s Special Theory of Relativity by way of a story about two astronauts on different spacecraft watching a bouncing-light-beam clock, and I was really impressed. But as I was chewing my cud (See definitions of “ruminate”) as I was wont to do, and walked home from school, it didn’t seem to make as much sense upon further review. I developed some questions but didn’t know where to get answers, and as life pressed on my attention wandered elsewhere, and everyone lived happily ever after. . .

Until recently. In the last year, the subject has come up several times, the questions seem to be the same, and I still don’t know where to turn.

### The Story

Although not exactly as I remember it, www.dummies.com1 describes a similar thought experiment in the second section, “UNIFYING SPACE AND TIME”, with a spacecraft traveling at ½ the speed of light, but doesn’t give much explanation. A more detailed explanation can be found in The Star Garden2.

### The Problem

“Time Dilation”, Section 7.2.2 of Reference 2 concludes

“The time between heartbeats is also slower, and so from the perspective of a stationary person, a moving person appears to be living their life at a slower rate. Conversely, from the perspective of the moving person, the stationary person seems to live their life as if it is being fast-forwarded. If they travel fast enough, then they could see the stationary person age before their eyes.”

My problem with that conclusion is that based on the second paragraph of Section 7.1 at the beginning the article, which states

“there’s no such thing as absolute speed or velocity, and something can only be said to be moving at a constant velocity relative to something else. In the same way, something can only be said to be stationary relative to something else”,

how do we really know which astronaut is supposed to be aging before our eyes? What if we put a bouncing-light clock on each spacecraft? Would it explode?

The authors of Reference 2 seem to address this issue at the bottom of the next section, 7.2.3, where they say

The twin paradox asks why the astronaut can consider themselves to be moving and the Earth to be stationary, when Galileo’s relativity shows that there’s no such thing as absolute velocity. Why can’t the astronaut consider themselves to be stationary while the Earth moves away at tremendous speeds?

The answer is acceleration. Galileo’s relativity applies to inertial – that is non-accelerating – reference frames. The fact that the astronaut must have accelerated before getting to such high speed means that they know they are the one that is moving.”

To me, this sounds bogus; any acceleration before or after the experiment should be immaterial. Let’s have three observers; one person remains on Earth while two astronauts board different spacecraft, each leaving the Earth in opposite directions and reaching similar stable speeds well in excess of ½ the speed of light (meaning their relative speed would exceed the speed of light in a non-relativistic world). Each of the observers has their own bouncing-light clock. If you start counting after their speed stabilizes, exactly how do each of the three observers see the ages change for the other two?

### One Last Question

A question that one might ask in each of these scenarios is “how does the state of the bouncing light in one spacecraft become known to the other observers?” Reference 1 states that Amber, on a different spacecraft, would see the bouncing light travel further between bounces, as if Amber had super X-ray vision and/or was otherwise experiencing the light beam in real time. How does that work? If she had to wait for reflected light rays from the event to reach her eyes, would that affect the apparent outcome in any way?

One phenomenon that may or may not have anything to do with the solution to this problem involves ocean waves. In deep water, a wave’s speed is nearly proportional to the square root of its wavelengthA,

$S \cong 1.251 \sqrt{W}$

where S is the wave’s speed (measured in meters per second) and W is its wavelength (in meters).
For shallow water waves, the speed is proportional to the square root of the depth.

$S \cong 3.1 \sqrt{d}$

where d is the water’s depth (in meters).

but in all cases, it is much less than the speed of light. If an observer were to watch the crest of a wave as it moved along a seawall, or along any imaginary line that wasn’t along the wave’s direction of travel (directly away from a point source, or in the direction of the wind, or perpendicular to the wavefront, etc.), then the apparent speed would be greater than the calculated or expected speed, and as the angle of that reference line approached 90° to the direction of travel, the apparent speed would approach infinity.
$A = \frac{S}{\cos \theta}$