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16 June 2007

Space and time are truely intertwined that it makes to sense to speak of time as being separate from space. The reasons they must be taken together will be explored in more detail later. But, this essay concentrates on understanding the interesting effects that must be explained. Some people, rather than simply not understanding Relativity no matter how many times it is explained, actually refuse to accept it as real. They will insist that time is absolute, that time is unique and distinct from space, that time does not make sense as a “dimension”, or that the interesting observations are illusions or artifacts and not the fundimental nature of reality.

The idea was certainly mindblowing when it was first advanced. Albert Einstein developed the
theory of special relativity, publishing it in 1905. But the great mathematician Hermann Minkowski
realized in 1907 that the math came out so much simpler and extrodanarily elegant if expressed
as a 4-dimensional system unifying space and time. Not only the math, but the *understanding*
is vastly simplified using this concept.

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. —Hermann Minkowski, 1908

As we shall see, the concept of time being relative is directly analogous to the concept of “up” being relative that was explored in part 1.

No matter how fast you move, in any direction, you will observe the same speed for light. If you accelerate in a space ship to half the speed of light, and measure the speed of light in a lab onboard the ship, you will get the identical results that was measured on Earth. Ah, but so will any measurement for the speed of sound. The medium of air is moving with the lab, so that doesn’t prove anything as some hypothetical medium for light could be doing the same.

More telling, if you measure the speed of light coming from the Sun receeding
behind you and half the speed of light, what do you get? If you expected half the
Earthly speed of light (hensforth abbreviated as .5 *c*) you are stuck in
the 19^{th} century or are one of those reality denyers. Today everyone
knows that you will measure the light from the Sun at precicely the one true speed of
light, or 1 *c*.

Likewise, if the speeding ship shines a light ahead of itself, two telling observations
can be made. First, if someone in the ship measures the speed of that light when it
reflects off an upcoming asteroid (the “light” was RADAR in this case), he will find it
to be 1 *c*, even though the emitter is moving at .5 *c* relative
to his ship. Meanwhile, an observer at rest outside the ship will measure the same speed,
relative to himself, and perceive that the leading edge of the RADAR pulse is moving
.5 *c* faster than the ship chasing behind it.

Back in the shipboard lab, under more controlled conditions, the same thing happens.
The measured speed of light is *c*, as mentioned earlier. But if the experiment
were also visible to an observer outside the ship at rest, he would see the same speed
for the light! If the lab contained a moving medium like sound, the outside observer should
see that particular light pulse move at 1.5 *c* in one direction and 0.5 *c*
in the other. This paradox is illustrated below.

The ship-board observer sees the light beam cover a distance of *2d*.
A naïve speculation about the outside observer, seeing that the apparatus is moving at half the speed of
the light beam, will conclude that he will measure a total distance of *2⅔d*, which is definitly
a longer path.

But, the different observers *do* measure the same speed for light, always.
Note that I’m not talking about light slowing down when moving through water or air or diamond,
but the unfettered speed of light in a vacuum. If two observers measure the same speed,
but have different distances, something’s got to give.

When you actually do the experiment and see what happens, something interesting
indeed shows up. Two somethings interesting, in fact. First, you will notice that the
length of the ship, and of the apparatus within it, is squished! What the shipboard
observer measured as *d*, the outside observer measures as *0.87 d*
(^{√3}⁄_{2} to be precise).

Since the distance between the emitter/detector and the mirror is closer together,
the ship will move less in the time it takes a beam of light to go from the emitter to the mirror.
Travelling at a velocity of .5 *c*, half the speed of the beam of light, the entire
apparatus will have moved its own length when the light reaches the mirror, regardless of
what that length is. If the apparatus is shorter, so is the distance it must move to
move its own length. Likewise, on the return trip the reflected light travels twice as far
as the detector when the meet in the middle.

So, the outside observer actually measures a total trip distance of 2.31 *d*,
which is 87% of what you originally supposed before doing the experiment.

That is an improvement (2.31 instead of 2.67, compared with the other observer’s 2.0), but does not fully resolve the paradox. How can both observers measure the same speed of light, when they measured different distances for its path?

That brings us to the second effect, which is more subtle and does not show up in this diagram. It is not until the two observers compare notes and their scratch work, not just their results, that the second effect becomes apparant.

- The shipboard observer measured distance 2
*d*and elapsed time 2*t*, giving the speed of one*d*per*t*. - The outside observer measured distance 2.31
*d*and elapsed time 2.31*t*, also giving the speed of one*d*per*t*.

Not only do the two observers measure a different length of the apparatus, but they experience a different time duration! During the flight of the beam, one observer lives for 15.5% longer than the other. Their watches or other clocks, their beating hearts, their thoughts, and any physical processes at all will show that the shipboard observer was experiencing less time than the outside observer*.

The fact that for a moving object **length contracts** and **time slows down** explains
the fact that the speed of light is measured to be the same velocity. Velocity is
distance over time. Both factors change by the same amount, giving the same
quotient.

The combination of length contraction and time dialation explain completely how different observers can measure the same speed of light. But, you can’t really call that an explanation, since you are just replacing one bizzare effect with another. What it does do is resolve the apparant paradox. A constant speed of light, by itself, does’t make sense and leads to contradictions. But length/duration change implies the constant speed of light and everything works out consistantly.

The constant speed of light clearly is not a single isolated effect, but merely one interesting effect of a richer whole with many interrelated effects.

With the flat-Earth analogy, imagine being confronted *only* with the results that
when surveying large areas of land, you get triangles whose interior angles don’t add up to 180°
and circles whose circumference is shorter than π times the diameter. Trying to comprehend how
that can be so, with no concept of anything other than plain geometry, would be mind-boggling.
But if you try it on a toy globe, you see that these impossible shapes work out naturally. Impossible
triangles or circles can’t exist on their own, but are part of a larger range of effects stemming from
a single deeper mechanism.

Length contraction and time dialation were presented above as two separate effects, that just happen to work together and always hold opposite values. Probing the effects even deeper, it turns out that length contraction can be explained as just an effect of relative simultaniousness. It is hard to understand the two ends of the ship (points separated by some distance in space) as being seen in different times, because the ship is one rigid object. So let’s come back to that later, and start with a different example.

In the illustration below, you see that observer 1 (let’s use names instead and call him Albert) has prepared a series of space buoys. They all lie in a straight line, and are not moving relative to each other or with Albert, who waits a short distance off. In other words, the buoys are not moving in Albert’s reference frame.

The buoys are equipt with flashing lights, and Albert synchronizes them to flash at the same time. To verify the setup, Albert uses RADAR to check the position of each buoy. From the length of time it takes the signal to return, and the direction of the beam, he can plot their positions. It agrees with how he positioned them in the first place. When they flash, he sees the nearer ones first because the speed-of-light delay is shorter. Noting the time at which he sees each one flash, and compensating for each one’s distance from him, he determines that they all flash at the same time and at the expected time based on his clock.

Observer 2, Benjamin, is in a ship approaching the line of buoys from the left, meaning is is moving right in the diagram, at relativistic speed.

In Ben’s reference frame, the buoys are approching him at high speed and he considers himself stationary. His RADAR results are harder to figure out, since the target is moving. One measure of the speed is from the doppler shift of the returning pulse. Another is the change in position from two successive pulses. He knows the distance at the point of RADAR beam reflection, and that it continues to move after that. So, he can build up a picture of the buoys all moving towards him in a line, not changing their place in formation, and know where each will be at any given time on his clock.

Meanwhile, observer 3, Carl, is doing the same thing but approaching from the right. He is moving left in the diagram, at relativistic speed.

Now at the appointed time on Albert’s clock, all the buoys flash in unison. What does Ben experience? First, his clock doesn’t show the same time as Albert’s because it is running slow. He can correct for that, but it doesn’t really matter to this observation. We can suppose that Ben is not told when the flash will occur, so he just has to watch out for it.

As you would expect, he *sees* the flash from the nearer buoys first, because
there is less of a delay for the light to reach him. But Ben compensates for this, noting the
time on his clock when each flash arrives, and his knowledge of the trajectory of the line of
buoys, and calculates backwards to determine where each buoy was when the light pulse
left it, and what time it was when that happened.

He double-checks all his work, and reports in. Besides noting that the light from the buoys is
bluer in frequency than Albert, and that the spacing of the buoys in the line is closer together than
what Albert measured, he also reports that **the buoys did not flash all in sync**. He reports that
the closest buoy flashed earliest, and then each flashed in turn with the rightmost (on our diagram)
flashing last.

At the same time, Carl is also making measurments. His report is identical with Ben’s, except
that when he reports that the closest buoy flashed first and the farthest last, he means that
the right one on our diagram was first, and the left last. He determines that **the flashes
took place in the opposite order** as Ben.

To summarize the situation, we have three observers reporting on the same events. In one reference frame, all the events took place at the same time. In the second, the events took place at different times, with some short interval between each, with the leftmost first. In the third reference frame, the time interval was the same but the rightmost was first.

From the careful considerations described, you should be convinced that this time difference is a real physical effect on the flowing of time itself, and is not a measurement artifact or optical illusion. When the leftmost buoy flashes, the event of the middle one flashing is in Ben’s future, Carl’s past, and Albert’s present! All of those are correct statements!

This may seem utterly bizzare and worse than clocks running slow, but it is analogous
to the earlier example of the round Earth. Each observer disagreed on the direction in which
Saturn’s ring plane was tilted; in other words, they had a different ordering for higher/lower.
Here, we have a different ordering for past/future. How can the two observers disagree
as to which item is above the other? A flat-Earth beleiver would be just as befuddled.
But the diagram of “up” being different for each observer, and the digram, along with the
*acceptance that this is so*, makes it all clear.

This section might have been titled *Time Is an Axis in 4 Dimensions*, but I wanted
to preserve the analogy with part 1, and didn’t want to scare anyone off continuing.

the shipboard observer was experiencing less time than the outside observer

This is actually the twin paradox. Don’t worry about it in the example, and suppose that the ship came back and the observers are comparing notes while both are in the observer’s reference frame.