# Special Relativity: Physics at the Speed of Light

Setting the Stage

You awaken, and your mind clears. Yes, you are traveling on the inter-stellar freighter Hyperion, outbound to mine anti-matter from a galactic vortex. The automated systems have just revived you from suspended animation. Your assignment – perform periodic ship maintenance.

Climbing out of your hibernation chamber, you punch up system status. All systems read nominal, no issues. That is good. Your ship extends 30 kilometers. Just performing routine maintenance exhausts the mind and body; you don’t need any extra work.

You contemplate the task of the freighter. The Hyperion, and its three sister ships, fly in staggered missions to harvest energy, in the form of anti-matter. Each trip collects a million terawatt-hours, enough to support the 35 billion human and sentient robots in the solar system for a full year.

Looking up at the scanner screen, you see the mid-flight space buoy station about a light-hour ahead. The station contains four buoys, configured in a square, 30 kilometers on a side. A series of eleven stations keeps your ship on course during its two year travel out from Earth.

You check the freighter’s speed relative to the buoys – about 50 percent of the speed of light, but constant, i.e. no acceleration or deceleration. That makes sense – at mid-flight, the freighter has entered a transition phase between acceleration and deceleration.

The Theory of Relativity

Either through deliberate study, or general media coverage, you likely have heard of the Theory of Relativity, the master piece of Albert Einstein. Einstein built his theory in two phases. The first, Special Relativity, covered non-accelerating frames of reference, and the second, General Relativity, dealt with accelerating and gravity-bound frames of reference.

Special Relativity gave us the famous E=MC squared equation, and covers the physics of objects approaching the speed of light. General Relativity helped uncover the possibility of black holes, and provides the physics of objects in gravity fields or undergoing acceleration.

Here we will explore Special Relativity, using our hypothetical ship Hyperion. The freighter’s speed, a significant fraction of that of light, dictates we employ Special Relativity. Calculations based on the laws of motion at everyday speeds, for example those of planes and cars, would produce incorrect results.

Importantly, though, our freighter is neither accelerating nor slowing and further has traveled sufficiently into deep space that gravity has dwindled to insignificant. The considerations of General Relativity thus do not enter here.

Waves, and Light in a Vacuum

Special Relativity starts with the fundamental, foundational statement that all observers, regardless of their motion, will measure the speed of light as the same. Whether moving at a hundred kilometers an hour, or a million kilometers an hour, or a billion kilometers an hour, all observers will measure the speed of light as 1.08 billion kilometers an hour.

A caveat is that the observer not be accelerating, and not be under a strong gravitational field.

Even with that caveat, why is this case? Why doesn’t the speed of the observer impact the measured speed of light? If two people throw a baseball, one in a moving bullet train, while the other stands on the ground, the motion of the bullet train adds to the speed of the throw ball.

So shouldn’t the speed of the space ship add to the speed of light? You would think so. But unlike baseballs, light speed remains constant regardless of the speed of the observer.

Why?

Let’s think about waves. Most waves, be they sound waves, water waves, the waves in the plucked string of a violin, or shock waves travelling through solid earth, consist of motion through a medium. Sound waves consist of moving air molecules, water waves consist of moving packets of water, waves in a string consist of motion of the string, and shock waves consist of vibrations in rocks and soil.

In contrast, stark contrast, light waves do not consist of the motion of any underlying substrate. Light travel does not need any supporting medium for transmission.

In that lies the key difference.

Let’s work thought that in the context of the inter-stellar freighter. You rise from suspended animation. Acceleration has stopped. In this case, no buoys exist near-by.

How do you know you are moving? How do you even define moving? Since you reside in deep space, and you are away from the buoys, no objects exist near-by against which to measure your speed. And the vacuum provides no reference point.

Einstein, and others, thought about this. They possessed Maxwell’s laws of electromagnetism, laws which gave, from first principle, the speed of light in a vacuum. Now if no reference point exists in a vacuum against which to measure the speed of a physical object, could any (non-accelerated) motion be a privileged motion? Would there be a special motion (aka speed) at which the observer gets the “true” speed of light, while other observer’s moving at a different speed would get a speed of light impacted by that observer’s motion.

Physicists, Einstein especially, concluded no. If a privileged reference frame exists, then observers at the non-privileged speed would find light violates Maxwell’s laws. And Maxwell’s laws stood as so sound that rather than amend those laws, physicists set a new assumption – relative speed can’t change the speed of light.

Ahh, you say. You see a way to determine whether the Hyperion is moving. Just compare its speed to the buoys; they are stationary, right? Really? Would they not be moving relative to the center of our galaxy? Doesn’t our galaxy move relative to other galaxies?

So who or what is not moving here? In fact, if we consider the whole universe, we can not tell what “true” speeds objects possess, only their speed relative to other objects.

If no reference point provides a fixed frame, and if we can only determine relative speed, Maxwell’s laws, and really the nature of the universe, dictate all observers measure light as having the same speed.

Contraction of Time

If the speed of light remains constant, what varies to allow that? And something must vary. If I am moving relative to you at near the speed of light (remember, we CAN tell speed relative to each other; we can NOT tell absolute speed against some universally fixed reference) and we measure the same light pulse, one of use would seem to be catching up to the light pulse.

So some twist in measurement must exist.

Let’s go back our freighter. Imagine the Hyperion travels right to left, with respect to the buoys. As noted, the buoys form a square 30 kilometers on each side (as measured at rest with respect to the buoys).

As the Hyperion enters the buoy configuration, its front end cuts an imaginary line between the right two buoys. It enters at a right angle to this imaginary line, but significantly off center, only a few hundred meters from one right buoy, almost 30 kilometers from the other right buoy.

Just as the front of the freighter cuts the line, the near right buoy fires a light pulse right across the front of the freighter, to the second right buoy, 30 kilometers away.

The light travels out, hits the second right buoy, and bounces back to the first right buoy, a round trip of 60 kilometers. Given light travels 300 thousand kilometers a second, rounded, or 0.3 kilometers in a micro-second (one millionth of a second), the round trip of the light pulse consumes 200 micro-seconds. That results from dividing the 60 kilometer round trip by 0.3 kilometers per micro-second.

That calculation works, for an observer stationary on the buoy. It doesn’t work for you on the Hyperion. Why? As the light travels to the second right buoy and back, the Hyperion moves. In fact, the Hyperion’s speed relative to the buoys is such that the back of the freighter arrives at the first right buoy when the light pulse returns.

From our vantage point, on the freighter, how far did the light travel? First, we realize the light traveled as if along a triangle, from the front of the ship, out to the second right buoy and back to the back of the ship. How big a triangle? The far right buoys sits 30 kilometers from the first right buoy, so the triangle extends 30 kilometers high, i.e. out to the second right buoy. The base of the triangle also extends 30 kilometers – the length of the ship. Again, let’s picture the light travel. In the Hyperion’s reference frame, the light passes the front of the ship, hits the second right buoy, and arrives back at the back of the freighter.

Some geometry (Pythagorean theory) shows that a triangle 30 high and 30 at the base will measure 33.5 along each of the slanted sides. We get this by splitting the triangle down the middle, giving two right triangles 15 by 30. Squaring then summing the 15 and 30 gives 1125 and the square root of that gives 33.5.

In our reference frame then, the light travels 67 kilometers, i.e. along both the slated sides of the triangle. At 0.3 kilometers per micro-second, we measure the travel time of the light pulse at just over 223 micro-seconds.

Remember, our observer stationary on the buoy measured the time travel at 200 micro-seconds.

This reveals a first twist in measurements. To keep the speed of light constant for all observers, clocks moving relative to each other will measure, must measure, the same event as taking different amounts of time. In particular, to us on the Hyperion, the clock on the buoys is moving, and that clock measured a shorter time. Thus, clocks moving relative to a stationary clock tick slower.

Again, that is the twist. Clocks moving relative to an observer tick slower than clocks stationary with respect to that observer.

But wait. What about an observer on the buoy. Would they not say they are stationary? They would conclude stationary clocks tick slower.

We have a subtle distinction. We can synchronize clocks at rest relative to us. Thus we can use two clocks, one at the back of the Hyperion and the other at the front, to measure the 223 micro-second travel time of the light beam. We can not synchronize, or assume to be synchronized, moving clocks. Thus, to compare the travel time of the light in moving verses stationary reference frames, we must measure the event in the moving reference frame with the same clock.

And to observers on the buoy, the Hyperion was moving, and on the Hyperion the event was measured on two different clocks. Given that, an observer on the buoys can not use our two measurements to conclude which clocks tick slower.

Uncoupling of Clocks

This uncoupling of clock speeds, this phenomenon that clocks moving relative to us run slower, creates a second twist: clocks moving relative to us become uncoupled from our time.

Let’s step through this.

The Hyperion completes its freight run, and once back home in the solar system, the ship undergoes engine upgrades. It now can now reach two-thirds the speed of light at mid-flight. This higher speed further widens the differences in measured times. In our example above, at about half the speed of light, the moving reference frame measured an event at 89% of our measurement (200 over 223). At two-third the speed of light, this slowing, this time dilation, expands to 75%. An event lasting 200 micro-seconds measured on a moving clock will measure 267 micro-seconds on a clock next to us on the freighter.

We reach mid-flight. As we pass the right buoy, we read its clock. For ease of comparison, we won’t deal with hours and minutes and seconds, but rather just the position of a hand on a micro-second clock.

As the front of the Hyperion passes the buoy, the buoy clock reads 56 micro-seconds before zero. Ours reads 75 micro-seconds before zero. The buoy clock thus now reads slightly ahead of ours.

Now remember, we think we are moving. However, from our perspective, the buoy clock moves relative to us, while clocks on our freighter stand stationary relative to us. So the buoy clocks are the moving clocks, and thus the clocks that run slower.

With the Hyperion at two thirds of the speed of light relative to the buoy, the buoy travels past us at 0.2 kilometers per micro-second (speed of light is 0.3 kilometers per micro-second). Thus by our clocks, the buoy travels from the front of the freighter to the midpoint in 75 micro-seconds (15 kilometers divided by 0.2 kilometers per micro-second). The freighter clocks are synchronized (a complex procedure, but feasible), and thus we see the micro-second hand at zero micro-seconds on our clock.

What do we see on the buoy? We know its clocks run slower. How much slower? By a “beta” factor of the square root of (one minus the speed squared). This beta factor falls right out of the Pythagorean math above, but the details, for this article, are not critical. Simple remember the key attributes, i.e. a moving clock runs slower and that an equation – one tied to the (relatively) simple Pythagorean Theorem – exists to calculate how much slower.

The beta factor for two thirds the speed of light equates to just about 75%. Thus, if our clocks advanced 75 micro-seconds as the buoy traveled from front to mid-section, the buoy clocks advanced 75% of 75 or 56 micro-seconds. The buoy clock read 56 micro-seconds before zero when that clock passed the front of the Hyperion, so it now reads zero.

The buoy now travels farther and passes the back of the Hyperion. That is another 15 kilometers. Our clocks advance to 75 micro-seconds, while the buoy clock moves up to only 56 micro-seconds.

This progression reveals a key phenomenon – not only do moving clocks tick slow, those clocks read different times. At some points, those moving clocks read an earlier time than clocks stationary to us, and at times, they read a time later than clocks stationary to us.

We thus see moving objects in what we would consider our past or future. Very spooky.

Do we have some type of vision into the future then? Could we somehow gather information about the moving reference frame, and enlighten them on what will come? Or have them enlighten us?

No. We might see the buoy at a time in our future (as the buoy passes the front of the Hyperion, its clock reads 56 micro-seconds before zero, or19 micro-seconds earlier than our clock). We however do not also simultaneously see the buoy at our present, i.e. 75 micro-seconds before zero. To cheat time, to tell the buoy about its future, we need to take information from one point in time and communicate that information to another point in time.

And that never happens. We see the buoy in our future, then in our present, and then our past, but as that happens we do not see the buoy at another point in time. We thus cannot communicate any future knowledge to the buoy.

Length Contraction

Let’s summarize quickly. The laws of nature dictate all observers, regardless of motion, will measure light at the same velocity. That dictate implies and requires that clocks moving relative to an observer will tick slower, and further implies and requires that time registering on moving clocks will be uncoupled from time registering on clocks stationary to us.

Do we have more implications? Yes.

The constancy of light speed requires and dictates that moving objects contract in length.

As the buoys speed by, at a particular instant, the Hyperion should align with the buoys. Our 30 kilometer length equals the 30 kilometer buoy separation. Thus, when our ship aligns itself side-by-side with the buoys, observers at the front and back of the Hyperion should see the buoys.

But this doesn’t happen. Our observers on the Hyperion don’t see the buoys when the mid-ship point of the Hyperion aligns with the midpoint between the buoys. In fact, at this alignment, the Hyperion observers must look towards mid-ship to see the buoys. At alignment of mid-ship of the Hyperion to midpoint between the buoys, each of the buoys lies over 3 kilometers short of the ends of the Hyperion.

What happened? Why do we not measure the buoys 30 kilometers apart? What caused the 30 kilometer separation to shrink almost 7 kilometers?

What happened, what we have encountered, represents another ramification of the constancy of the speed of light, specifically that we measure a moving object as shorter than when we measure the object at rest.

How does that occur? Let’s uncover that by assuming that we had measured the moving buoys as still 30 kilometers apart, then by doing some math with that assumption. We will find that we will run right into a contradiction. That will indicate our assumption can not be right.

Let’s run the calculations. As noted above, we will assume we measure the buoys 30 kilometers apart. The buoys, under this assumption, will align with the ends of the Hyperion. For our experiment, at that instant of alignment, we fire light beams from the ends of the Hyperion towards the middle.

To keep things straight, we need distance markers on the Hyperion, and on the buoys. We will label the two ends of the Hyperion plus 15 kilometers (the right end) and minus 15 kilometers (the left end), and by extension, the middle of the ship will be zero. The Hyperion clocks will read zero micro-seconds when light beams start.

We will also mark the buoys as being at minus 15 and plus 15 kilometers, and by extension, a point equidistant between the buoys as distance zero. A clock will be placed at the buoy zero point. That clock will read zero micro-seconds when the mid-ship on the Hyperion aligns with the midpoint of the buoys.

Now let’s follow the light beams. They of course race towards each other until they converge. On the Hyperion, this convergence occurs right in the middle, at distance marker zero. Each light beam travels 15 kilometers. Given light travels at 0.3 kilometers per micro-second, the light beams converge in 50 micro-seconds.

The buoys move past the Hyperion at two thirds the speed of light, or 0.2 kilometers per micro-second. In the 50 micro-seconds for the light to converge, the buoys move. How much? We multiply their speed of 0.2 kilometer per micro-second times the 50 micro-seconds, to get 10 kilometers. With this 10 kilometer shift, when the light beams converge, our zero point aligns with their minus 10 kilometer point. Remember, if the Hyperion travels right-to-left, then on the Hyperion, we view the buoys at traveling left-to-right.

On the Hyperion, we see the light beams each travel the same distance. What about observers in the moving frame, i.e. moving with the buoys?

They see the light beams travel different distances.

The light beam starting at the right, at plus 15, travels all the way to minus 10 kilometers, in the buoy reference frame. That represents a travel distance of 25 kilometers. The light starting at the left, at minus 15, travels only 5 kilometers, i.e. from minus 15 kilometers to minus 10 kilometers. These unequal travel distances occur, of course, because the buoys move during the light beam travel.

In the buoy frame of reference, one light beam travels 20 kilometers farther than the other. For them to meet at the same time, the beam traveling the shorter distance must wait while the other light beam covers that extra 20 kilometers. How much of a wait? At the 0.3 kilometers per micro-second that is 66.7 micro-seconds.

Let’s contemplate this. In our stationary reference frame, the light beams each start at time equal zero on clocks on both ends of the Hyperion. For the buoys though, light leaves one buoy, the buoy at distance plus 15, 66.7 micro-seconds earlier, than the one that leaves the buoy at distance minus 15.

At the start of this experiment, we set the clock at the mid-point between the buoys at time equal zero. By symmetry, with this 66.7 micro-second difference, the clock at the minus 15 point must have read plus 33.3 micro-seconds, and the clock at the plus 15 point must have read minus 33.3, when the light beams left.

What about the meet point, at minus 10 in the buoy reference frame? What was the time at the meet point in the reference frame of the buoys, when the light beams left? Remember, the meet point in the buoy frame of reference is minus 10 kilometers. If the minus 15 point is 33.3 micro-seconds, the minus 10 point is 22.2 micro-seconds.

We now pull in that clocks run slower in the moving frame. At two thirds the speed of light, clocks run at 75% (or more precisely 74.5%) the rate of clocks in our stationary frame. Given our clocks measured 50 micro-seconds for the light travel time, the clocks on the buoys measure a light travel time of 37.3 micro-seconds.

A bit of addition gives us the meet time in the buoy reference frame. The clocks at the meet point read plus 22.2 micro-seconds when the light started, and advance 37.3 micro-seconds during the light travel. We thus have a meet time of 59.5 micro-seconds in the moving reference frame, i.e. the buoy reference frame.

The light started from the minus 15 point at 33.3 micro-seconds, and arrives at the minus 10 point at 59.5 micro-seconds. Let’s call that a 26 micro-second travel time. The travel distance was 5 kilometers. The implied speed, i.e. 5 kilometers divided by the 26 micro-second travel time, comes out to 0.19 kilometers per micro-second.

From the other end, the light traveled 25 kilometers, in 92.8 micro-seconds (from minus 33.3 to plus 59.5). The implied speed, i.e. 25 kilometers divided by the 93 micro-second travel time, comes out to 0.27 kilometers per micro-second.

No good. Light travels at 0.3 kilometers per micro-second. When we assumed that we would measure the buoys 30 kilometers apart, and adjusted the clocks to try to fit that assumption, we did NOT get the speed of light.

Remember critically that all observers must measure the speed of light as the same. Clock speeds, and relative time readings, and even measured distances, must adjust to make that happen.

How far apart DO the buoys need to be, for the buoys to align with the ends of the Hyperion? They need to be 40.2 kilometers apart. With the buoys 40.2 kilometers apart, the front and back of the Hyperion will align with the buoys, when the mid-ship (of the Hyperion) and the midpoint (of the buoys) align.

Amazing, almost incomprehensible. The need for all observers to measure the same speed of light dictates that we measure moving objects shorter, significantly shorter, than we would measure them at rest.

What will the buoy clocks read, if we adopt this 40.2 kilometers spacing? When the ship and the buoys align, the left buoy clock will read plus 44.7 micro-seconds and the right buoy clock will read minus 44.7 micro-seconds. Since the light beams fire when the ships and buoys align, the light beam on the right leaves 89.4 micro-seconds before the light beam on the left, in the buoy frame of reference.

That time difference equates to the right beam traveling 26.8 kilometers before the left beam starts, as seen in the buoy frame of reference. Both beams then travel 6.7 kilometers until they met. The 26.8 plus 6.7 twice totals to the 40.2 kilometer between the buoys.

The left beam starts at location minus 20.1, at time plus 44.7 micro-seconds, and travels 6.7 kilometers. Light needs 22.4 micro-seconds (6.7 divided by 0.3) to travel the 6.7 kilometers. Thus, the clock at the minus 13.4 point (minus 20.2 kilometers plus the 6.7 kilometers the left light beam traveled) should read 67.1 micro-seconds when the left light beam gets there.

Does it?

By proportions, when the buoys and the Hyperion align, a clock at the minus 13.4 point would read plus 44.7 minus one-sixth of 89.4. One-sixth of 89.4 is 14.9, and 44.7 minus 14.9 would be 29.8 micro-seconds.

Remember now that the buoy clocks must advance 37.3 micro-seconds during the travel of the light beams. That occurs because on the Hyperion, the light beam travel requires 50 micro-seconds, and the buoy clocks must run slow by a factor of 75 percent (or more precisely 74.5 percent).

Add the 29.8 and the 37.3, and we get 67.1 micro-seconds. We stated earlier that the clock at minus 13.4 kilometers should read 67.1 micro-seconds when the left light beam arrives. And it does. A separation of the buoys by 40.2 kilometers thus aligns the clocks and distances on the buoys so that they measure the correct speed of light.

What Really Happens

But do moving objects really shrink? Do the atoms of the objects distort to cause the object to shorten?

Absolutely not. Think about what we were reading on the clocks. While the clocks on the Hyperion all read the same time, the clocks in the moving reference frame all ready different times. Moving distances shrink because we see the different parts of the moving object at different times. With the buoys 40.2 kilometers apart (measured at rest), we saw the left buoy at plus 44.7 micro-seconds (in its reference frame) and the right buoy at minus 44.7 micro-seconds.

Let’s look at another way to conceive of length contraction, in a more down-to-Earth example.

Picture a long freight train, four kilometers long, moving at 40 kilometers an hour. You and a fellow experimenter stand along the tracks three kilometers from each other. When the front on the train passes you, you signal your partner. Your partner waits 89 seconds and takes note of what part of the train now passes in front of him. What does he see? The end of the train.

The four kilometer train fit within the three kilometer separation between you and your fellow experimenter. That occurred because your partner looked at the train later than you.

This is NOT precisely how fast moving objects impact measurements. In our train example, we created two different times of observation by waiting. In the Hyperion situation, we didn’t need to wait – the near light passing speed of the buoys created a difference in the clock observation times.

Though not an exact analogy, the simplified train example DOES motivate how measuring the length of something at two different times can distort the measurement. The train example also demonstrates that we can shorten the measured length of an object without the object physically shrinking.

While the shrinkage does not really happen, the time stamps differences are real. In our Hyperion example, with the light beams, if we went back and picked up the clocks on the buoys, those clocks would record that the light beams we fired really did start 89.4 micro-seconds apart. We would look at our Hyperion clocks, and our Hyperion clocks would really show that in our reference frame the light beams started at the same time.

Are the Clocks Smart?

How do the clocks “know” how to adjust themselves? Do they sense the relative speeds and exercise some type of intelligence to realign themselves?

Despite any appearances otherwise, the clocks do not sense any motion or perform any adjustments. If you stand beside a clock, and objects zip by you at near the speed of light, nothing happens to the clock next to you. It makes no adjustments, changes, or compensations for the sake of passing objects.

Rather, the geometry of space and time cause an observer to see moving clocks ticking slower, and moving objects measuring shorter.

If you move away from me, and I measure you against a ruler held in my hand, your measured height shrinks proportional to your distance from me. Your looking smaller results from the smaller angle between the light from you head and the light from your feet as you move away. The light didn’t need to know what to do, and the ruler didn’t adjust. Rather, the geometry of our world dictates that as you move away you will measure shorter.

Similarly, if I place lens between you and a screen, I can expand or shrink your height through adjustments of the lenses. The light doesn’t need to know how adjust; the light simply follows the laws of physics.

So using distance and lens, I can make the measurement of you height change. I could readily write formulas for these measurement changes.

Similarly, moving clocks read slower from the nature of time. We think clocks need to “know” how to adjust, since our universal experience at low velocities indicates clocks run at the same rate. But if we were born on the Hyperion and lived our lives traveling at near light speeds, the slowing of clocks due to relative motion would be as familiar to us as the bending of light beams as they travel through lens.

All observers must measure the speed of light as the same. That attribute of nature, that fact of the geometry of space and time, creates counter-intuitive but nonetheless real adjustments in observations of time and space. Moving clocks run slower, they become uncoupled from our time, and any objects moving with those clocks measure shorter in length.