# The Relativity of Simultaneity

## Simultaneity Ain't what It Used to Be

One of the most fundamental deductions Albert Einstein made from the finite speed of light in his theory of special relativity is the relativity of simultaneity—because light takes a finite time to traverse a distance in space, it is not possible to define simultaneity with respect to a universal clock shared by all observers. In fact, purely due to their locations in space, two observers may disagree about the order in which two spatially separated events occurred. It is only because the speed of light is so great compared to distances we are familiar with in everyday life that this effect seems unfamiliar to us. Note that the relativity of simultaneity can be purely due to the finite speed of light; while it is usually discussed in conjunction with special relativity and moving observers, it can be observed in situations where none of the other relativistic effects are present. The following animation demonstrates the effect.

(The animation is a rather large file and may take a while to download; please be patient if it doesn't start immediately.) In this model we will consider what three widely separated observers (represented by the yellow, blue, and grey spheres), none moving with respect to one another, observe when two lights, one red and the other green, illuminate at equal distances on either side of the yellow observer. When the lights come on, wavefronts of green and red light begin to spread out spherically from them at the speed of light. (Of course, the wavefronts appear as circles in this two-dimensional projection. Note also that we are observing this event from a large distance, equidistant from the red and green lights.)

At the moment the lights illuminate and the wavefront begins to propagate outward from each, we start a “progress meter” for each observer, with a time indicator bearing the observer's colour code showing local elapsed time. When the wavefront from the red light reaches an observer, we begin to plot a red line above the time line to indicate that the observer now sees the red light, and when the wavefront from the green light arrives, a green line begins to be drawn for the observer. The animation repeats continuously and pauses at the start and end.

When the animation reaches the end of a cycle, compare the progress meters for the three observers. Solely due to their respective distances from the red and green lights, each sees the lights come on in a different order: the yellow observer sees both illuminate simultaneously; the blue observer sees the red light first, while the grey observer sees the green light first. The lesson of the finite speed of light and special relativity is that not only isn't there a universal time valid everywhere, observers cannot even agree on the order in which they observe events to occur when the distances between them are significant compared to the speed of light.

Technical note: Some physicists prefer to reserve the term “relativity of simultaneity” for cases where observers are in motion relative to one another and the effects of special relativity obtain. In a case like the example above, where all the observers and sources are at rest with respect to one another, it is possible, using Einstein's definition of simultaneity, for spatially separated observers to synchronise their clocks and define simultaneity in terms of the spacetime interval between events. When observers are in relative motion, however, it is impossible, even in principle, to synchronise their clocks, so no definition of simultaneity is possible. But to me, the phrase “relativity of simultaneity” means precisely what it says, notwithstanding relativistic effects or their absence. In this case, three observers of the same two events see three different orders in which they appeared to occur from their particular vantage points; hence their perception of simultaneity is relative even though it is entirely due to light travel time instead of motion.

## “The Eagle has landed”

You're probably familiar with the following audio clip of the touchdown of Apollo 11 on the Moon. This is from the Public Affairs Officer (PAO) feed provided to news media covering the landing and broadcast worldwide in real time over television and radio. The feed included occasional commentary by the PAO, but during the landing only the “air to ground” (so-called, despite the conspicuous absence of air in the vicinity of the Moon) downlink and CAPCOM (the astronaut in mission control designated to speak to the crew, Charles Duke in this instance) are heard.

Buzz Aldrin does most of the talking in this sequence. As lunar module pilot, his responsibility was to provide Neil Armstrong, who was flying the spacecraft and looking out the window searching for a landing site, with a running commentary of instrument readings and status indications, which was also transmitted over the air to ground link to mission control. Armstrong does not key his microphone until he makes the post-landing transmission, “Houston, Tranquility Base here. The Eagle has landed.” (Actually, if you listen very carefully, you can hear Armstrong confirm “Out of detent” after Aldrin calls “ACA out of detent” immediately after engine stop; his voice was picked up by Aldrin's microphone.) An annotated transcript of this sequence appears from vehicle elapsed time 102:45:40 through 102:46:06 in the indispensable Apollo Lunar Surface Journal edited by Eric M. Jones.

## “Houston, uhhh…”

Every time I've listened to this sequence, I've been a bit puzzled why Armstrong paused so long between “Houston” and the rest of the “Tranquility Base…” transmission. He was certainly excited at the time: telemetry records his heart rate at touchdown at 150 beats per minute, and most people (although perhaps not steely-nerved engineering test pilots) tend to speak rapidly in such circumstances.

Thinking about this, it occurred to me to ask what Neil Armstrong heard through his headphones immediately after landing on the Moon. This isn't what listeners on Earth heard, due to relativity of simultaneity. At the time of the Moon landing (which I take as 20:17:43 UTC on 20th July 1969, Julian Day 2440423.34564, for the purposes of this document) the Moon was 385693 km from the Earth, which distance light took about 1.2865 seconds to traverse. Consequently, when you listen to the Earth-based recordings of the Apollo communications, you're hearing the CAPCOM in Houston as soon as he speaks, but transmissions from the Moon more than 1¼ seconds after they were spoken.

## Microphone on the Moon

Digital audio editing tools make it easy (although still a bit tedious) to transform a recording of transmissions from widely separated sources into how they would be received at any given location. In particular, by extracting transmissions from the LM from those originating in mission control onto separate tracks with the Audacity audio editor, I was then able to time-shift transmissions originating from the Earth by the light delay of 1.2865 seconds to reproduce what Buzz Aldrin and Neil Armstrong heard through their headphones in the cabin of the Eagle lunar module on the surface in Mare Tranquillitatis. During the landing phase, an on-board tape recorder in the lunar module captured the voices of Armstrong and Aldrin even when they were not transmitting on the air to ground link. From this noisy source, I have restored the few remarks by Armstrong which were only heard within the cabin. This is, then, the lunar touchdown as heard by the astronauts who performed it.

Now it's obvious what happened to Armstrong's post-landing transmission! Right before he began the call, Duke's message, sent a second and a quarter earlier, arrived at the Moon. While, from an earthly perspective, this was spoken well before Armstrong said “Houston”, on the Moon this message “stepped on” the start of Armstrong's transmission (especially considering human reaction time), and caused him to pause before continuing with his message. Note also that on the Earth-based recording, Duke's response occurs almost immediately after the end of Armstrong's transmission, but on the Moon, the astronauts had to wait for the pokey photons to make it from the home planet to their high gain antenna on its distant satellite.

## Apollo 11 Moon Landing: You are there

I have taken the liberty of preparing an audio track of the entire terminal descent and landing of the lunar module Eagle on the Moon as heard by the astronauts on board.

In this stereo presentation, Aldrin's transmissions appear in the centre, with Armstrong's remarks captured only on the onboard recorder in the left channel and CAPCOM Charles Duke's transmissions in the right channel. Consider, as you listen to this audio, that apart from other folks who clicked on this link before you, only Neil Armstrong, Buzz Aldrin, and this humble scrivener have ever listened to this event from the lunar surface's unique perspective in spacetime. The transcript for this sequence occurs between vehicle elapsed time 102:42:25 and 102:47:15 in the Apollo Lunar Surface Journal.