Fun Fact - Special Relativity

In this article, I will attempt to explain einstein's theory of special relativity using concepts that are accessible to an advanced high school or first year college student. Yes, there is some calculus, but it is basic differentiation and integration. I hope I am successful in this endeavor, because the theory is as beautiful as a symphony, a symphony that everyone should hear.

This is of course a simplified version of a very complicated topic. The math is correct, but the reasoning that leads to the math is not quite as simple and/or Newtonian as I make it seem. For example, Einstein was somewhat uncomfortable with the notion of variable mass, calling it a mathematical and conceptual convenience. It is perhaps more accurate to describe the phenomenon as changes in relativistic energy and momentum. However, I will retain the notion of a change in mass, at least for this article. It provides some much needed intuition, and with all the math flying about, we really need some intuition. It's a bit like electrons revolving around the nucleus of an atom in their orbital shells - not 100% accurate, but helpful and explanatory nonetheless.

Why does time slow down when objects move, and why does this imply a change in length and in mass? Why is all this implied by a constant speed of light? I will approach these questions with just a bit of math, and if you follow along, then imagine how much more there is to Einstein's work. This is merely the first bite of the appetizer in a four course meal.

Let me begin with my own thought experiment, at the age of 12, while staring at the fan. Yes, my fascination with fans is never-ending. As the wind blew my hair back, I wondered what would happen if the blades were suddenly transformed into a smooth circular mirror. There would be no wind of course, and if I sang into the fan there would be no flutter in my voice. The mirror would simply spin in silence, and reflect the image of my face back to me. Suppose the mirror spins extremely fast, and by some magic it does not fly apart. Could it shift my image a few degrees clockwise, so that my face was tilted looking back at me? Perhaps I could rev it up even faster, so that my face was upside down. Is that possible, even in theory?

I thought, and imagined, and then I did something rather insightful for a 12 year old boy - I changed my frame of reference. Suddenly I was sitting on the spinning mirror, like a fly going for a ride. A beam of light that heads directly into the fan will strike this spot on the mirror at an apparent angle - the angle implied by the speed of light and the motion of the mirror. Whatever the angle coming in, the light bounces off at the same angle going out. This is illustrated by a ping pong ball bouncing off the table. It approximates the letter V, with the bottom of the V touching the table. In an ideal world, the table only reverses the up-down motion of the ball; the forward motion is unchanged. The ball is moving down and toward your opponent, then it strikes the table and is moving up and toward your opponent. That's what happens to a beam of light when it strikes a mirror. It bounces off like a perfect ping pong ball. In optics this principle is stated as, "The angle of incidence equals the angle of reflection.", and it holds to an almost arbitrary precision, else we would not be able to build the primary mirror for the Hubble space telescope. Since the angle is the same in and out, relative to the moving fan blade, the light bounces straight back to me, without any rotational shift. I see my face straight on, as though the mirror was standing still. The image is unchanged, even if the mirror is whizzing around at nearly the speed of light.

Now let's move over to some thought experiments that changed the world. Einstein began with the premise that the speed of light was constant, which is entirely counterintuitive. If you are riding a bike at 10 miles per hour, and you throw a baseball forward at 10 miles per hour, the ball is moving forward, relative to the sidewalk, at 20 miles per hour. Velocity is additive. We learned that in junior high, and it is born out by all our real world experiences. Even at high speeds attained by our spacecraft, velocity remains additive. But somehow light is different. It travels at an extremely high speed, which I will call c. This is fast, but still finite. Turn on your flashlight while riding your bike, and the beam of light still travels at c relative to the ground. It does not move forward at a speed of c+10. The speed of the bike is not added to the beam of light. Light travels at the same fixed speed c, whether you are standing still, or walking forward, or biking, or traveling at 17,500 mph on the International Space Station.

You never notice this effect in your day to day activities because c is so large, approximately 300,000 kilometers per second, or 186,000 miles per second. If confined to a tube, a beam of light could circle the earth 7 times a second. Or a beam of light could race from here to the moon in just over a second. That's pretty fast! You certainly can't tell the difference between c and c+10, thus we had no idea that velocity is not additive when it comes to light.

In 1887, Michelson and Morley proved that the speed of light does not vary with direction, even as the earth rotates on its axis and revolves about the sun. At any time of the day, or year, you could point your flashlight in any direction and the speed of light is the same. This did not by itself prove that light would not speed up as the light source moved toward you, but it suggested a uniformity of light that was not typical of matter. Other experiments added weight to the hypothesis that c was invariant, even if the light source was moving. By 1905 Einstein was willing to consider the possibility that c was fixed in all inertial frames. Then, using nothing but his mind, he rewrote the laws of physics.

Here is Einstein's first thought experiment, connecting light, speed, and time. Amy is riding a train with a large open window, so Bob can see her as she passes by. Amy has a flashlight, which she points straight up to the ceiling. The roof of her train car contains a mirror, which reflects the light back down again. Amy also has a high tech watch that can measure time in nanoseconds. As she sits quietly in her train, she snaps on the light. The last digit on her high precision watch is 7. The beam travels up to the mirror, which is about a foot, or 30 cm, above her flashlight. When the beam strikes the overhead mirror, her watch clicks over, so that the last digit is 8. The beam bounces back down and returns to her flashlight as her watch clicks over to 9. Light is traveling at 3E10 centimeters per second, or about 30 cm per nanosecond, just as it should. All is well.

What does Bob see as he watches Amy ride by from left to right? The beam of light leaves her flashlight, travels up and to the right (accounting for the forward motion of the train), bounces off the mirror, travels down and to the right, and returns once again to Amy's flashlight, Amy having moved forward as well. The beam traces the legs of an isosceles triangle, while the flashlight travels along the base.

Look at the first leg of this triangle, as the beam leaves the flashlight and rises to the ceiling. Again, Amy's watch shows 7 at the start, and 8 when the light reaches the ceiling. From Bob's point of view the light travels farther; it travels at an angle instead of a direct path up to the roof. Yet the speed of the light beam is the same. If the speed is the same, and the distance is longer, then more time is required. That's the bottom line. Bob has the same kind of watch as Amy, and his watch also reads 7 when Amy turns on her flashlight. But the beam requires more than a nanosecond to reach the roof, from Bob's point of view. When Bob's watch clicks over to 8, the beam has not quite reached the ceiling, and Amy's watch still shows 7. A fraction of a nanosecond later, the light reaches the ceiling and Amy's watch clicks over to 8. Amy's watch is running slower than Bob's, even though they are identical time pieces. The key insite is that this phenomenon is not restricted to Amy's watch. Time itself runs slower. Amy's heart beats slower, her hair grows slower, her neurons run slower, her thoughts run slower. Time is "dilated" within the moving train, as seen in Bob's reference frame.

Can we calculate the change in time? Let the train move across Bob's path at a speed of v. As Bob looks in, the light travels up the hypotenuse at a speed of c. By the pythagorean theorem, the upward velocity is sqrt(c2 - v2). Let l be the vertical distance from flashlight to roof. This was 30 cm in our example, but it could be anything. The elapsed time for Amy is l/c, but the elapsed time for Bob is l/sqrt(c2 - v2). The time dilation factor, the ratio of Bob's time to Amy's time for the same event, often denoted γ in the literature, is c / sqrt(c2 - v2), or (multiplying top and bottom by 1/c), 1 / sqrt(1 - v2/c2).

Our fastest spacecraft barely reach 0.01% the speed of light. when v is this small, i.e. when v/c is a tiny fraction, the time dilation is 1 + ½(v/c)2, a good approximation to γ. If Bob watches Amy fly by in one of these space ships, he only sees time stretch by a factor of 1.000000005. That is too small to notice, though it is not too small to measure using state of the art clocks. In 1971, Hafele and Keating flew cesium-beam clocks around the world in commercial airplanes. Even at the plodding speed of 500 mph, they were able to measure and confirm the changes in time, as dictated by relativity. Deviations were in the range of 100 nanoseconds, yet such can be measured by our atomic clocks. Of course this is orders of magnitude below human perception.

As a thought experiment, rev up Amy's train to half the speed of light. Now γ is 1.154, whence time is stretched by 15%. Events that are 1 second apart for Amy are 1.154 seconds apart for Bob. This is still a small change as humans perceive time. If you really want to slow Amy down, you have to run her train at nearly the speed of light. At 99% of c, γ = 7. At 99.9% of c, γ = 22. Bob's clock ticks off 22 seconds while Amy's clock advances by one second. As Amy approaches the speed of light, time slows to a crawl. If she could ever reach the speed of light, she would appear frozen in time as Bob sees her. Of course everything seems normal in Amy's world.

There is a beautiful symmetry here. Shift your frame of reference, so that Amy is sitting still in her train and the world is whizzing by. This is equally valid. As she looks out at Bob through the window, his watch is running slower than hers. He doesn't have a flashlight shining up at a mirror, but he could, and if he did, the geometry and the math would be the same. So each person sees the other running slower in time. Both watches show 7 as they pass each other, but Bob sees his watch click over to 8 before Amy's, and Amy sees her watch click over to 8 before Bob's. Time is no longer an absolute throughout the universe.

This seems like a contradiction, but Amy and Bob are flying apart, so the differences in time do not matter. "Well," you may ask, "what happens if the train brings Amy back home? Are the two watches both slower than each other?" This is called the twin paradox, and it is normally described like this.

Amy and Bob are twins, and when they are 20, Amy leaves earth on a very fast rocket that flies at nearly the speed of light. She zooms away for quite some time, turns around, and zooms back home. For most of the journey, Amy and Bob are in motion relative to each other. Bob ages 10 years, 5 years as Amy flies away and 5 years as Amy flies back home. But he can watch her through a powerful telescope. Time dilates by a factor of 10, so she only ages a year. At the same time, Amy watches Bob back on earth through a telescope, and due to time dilation, he ages only a year. When she returns, she is 21 years old, and he is 30, but at the same time, Bob is 21 years old and Amy is 30. They can't both be right.

Some of this confusion can be resolved within the purview of special relativity, once you allow for length contraction. I'll deal with that next. But another aspect of the twin paradox will have to wait for general relativity, which is not addressed in this article.

For notational convenience, let α be the reciprocal of γ. Thus α = sqrt(1 - v2/c2). This is the ratio of Amy's time to Bob's time. While Bob ages 1 second, Amy, in her moving train, ages α seconds. As v approaches c, α approaches 0, and γ approaches infinity.

Now let's look at length contraction. Amy's train, and everything in the train, including Amy, is shorter in the direction of travel, as Bob sees it. Assume Amy's train travels at 80% the speed of light. Verify that α = 0.6. Bob's watch advances one minute while Amy's watch advances 36 seconds from Bob's point of view. Bob holds a meter stick level with the ground, along the direction of the train's travel. The train has a red dot on it, on the side of Amy's car. Bob notes the time when the red dot reaches the left end of the meter stick, and again he notes the time when the red dot reaches the right end of the meter stick. The train is traveling at 80%c, hence the red dot is traveling at 80%c. With c = 3E8 m/s, the red dot travels a meter in 4.16 nanoseconds. When the red dot is at the right end of the meter stick, Bob places a blue dot on the train at the left end of the meter stick. You might think the blue dot and the red dot are a meter apart on the train, and I suppose they are, in Bob's reference frame, but they are not as seen by Amy. She looks out her window and sees bob, holding his meter stick aloft. The red dot on her train passes by point x, the left end of his meter stick, and then, according to bob's watch, the blue dot on her train passes by point x 4.166 nanoseconds later. But according to Amy, bob's watch is running slow. Her watch records 4.16 × γ = 6.94 nanoseconds. She knows that the world, along with point x, is racing past her at 80%c. She does the math, and the blue dot and the red dot are 1.66 meters apart. A distance of 1.66 meters along Amy's train is 1 meter according to Bob's measurements. The train, and everything and everyone on board, is compressed in the forward direction by a factor of α. If the train had a meter stick pasted to its side, it would only be 0.6 meters long. Amy looks a little thinner as she flies by. At even higher speeds, the entire train might have the thickness of a piece of paper, front to back. Of course it is just as wide and tall as it was before.

By symmetry, Amy sees the same length contraction in Bob, and his meter stick, and the world around her.

Now let's return to the twin paradox. Morph Amy's train into a space ship, because she is going to the nearest star, Alpha Centauri, approximately 4 light years away. At 80%c it takes her 5 years to get there and 5 years to return. Bob is 10 years older when Amy comes back home, but her ship time is multiplied by α, or 0.6, thus she is 6 years older when she comes home. Next, move to Amy's reference frame. The universe is passing by her at 80%c. By length contraction, Alpha Centauri is 4 × 0.6 = 2.4 light years away. She gets there in 3 years, and comes home in 3 years, thus she is 6 years older. Amy and Bob both agree on her age. They don't agree, however, on his age. He is 10 years older, but for 6 years Amy observes his time passing at 0.6 normal, thus he is 3.6 years older. Her observation is considered null and void, because she switched reference frames halfway through the journey. She had to accelerate, at Alpha Centauri, to turn around and come home. This change in velocity, in fact a reversal of velocity, modifies her observations of the universe, and of Bob back on earth. Again, the precise mechanism will have to wait for another article.

Special relativity comprises a triad: ship time is multiplied by α, length is multiplied by α, and mass is multiplied by γ. A person moving relative to you appears slower, thinner, and more massive. Let's tackle the change in mass now.

Defining mass rigorously is not easy, and it was even harder in Newton's day, when no one could imagine anything other than the constant and unchanging gravity field of earth. But mass isn't the same as weight. We know that now, but what a leap of genius that was for Newton. Mass is a resistance to being pushed, as Newton summarized in his first law of motion: "An object that is in motion tends to remain in motion, and an object at rest tends to remain at rest." If you're floating on the Space Station, nothing has any weight, but try pushing a refrigerator versus a toaster. The former barely moves, while the latter flies across the cabin. Your arm imparts the same force to both objects, but the masses are different. Thus we find Newton's second law of motion: "Force = mass times acceleration." This is also a pretty good definition of mass, mass = force divided by acceleration.

Apply this to Amy's space ship, as it heads out towards the stars. In her reference frame, the engine applies thrust sufficient to provide 1 G of acceleration, as though she was back on earth. The instantaneous change in velocity, as she sees it, is 1 G. By equivalence, the speed of the universe going past her equals the speed of her ship traveling through the universe. Both speeds have to be the same. As her engines fire, Bob sees the same increase in Amy's velocity, however, the change in velocity occurs over a much longer time in his frame of reference. Acceleration is delta v divided by time, and time is multiplied by γ, thus acceleration is multiplied by α. In our description of mass, acceleration is in the denominator, thus mass is multiplied by γ. (This isn't the best analysis, but it is intuitive, and it gives the right answer.) Set all this math to the side and think of it this way, the same force takes longer to change the speed of the ship from Bob's point of view, thus the ship has more mass. That completes the triad of special relativity: Amy runs slower, looks thinner, and has more mass.

What is the energy of a pitched baseball? Newton said it was ½mv2. This is the result of the integral of the force applied to the baseball, over the distance that the force is applied, as the baseball goes from at-rest to velocity v. It doesn't matter if you apply a small force over a long time, or a lot of force over a short time, the integral comes out the same, and the kinetic energy is the same. To illustrate, apply a constant force f, imparting a constant acceleration a. At time t, the baseball is at location ½at2. (This is the result of another integral). Set t = v/a, to attain the desired velocity. We are integrating force, or m×a, from 0 to ½a(v/a)2. The result is ½mv2.

Now assume an arbitrary force profile, a little force here, a lot of force there. The force function is f(t), hence the acceleration function a(t) equals f(t)/m. The distance function is d(t) and the velocity function is v(t). At time t0 the object is at rest at location 0. In other words, d(0) = 0 and v(0) = 0. At time t1, and distance d1, the object reaches the desired velocity v1. We want to integrate force over distance, that is, f(d), as d runs from 0 to d1. By variable substitution, this is f(d(t)) times the derivative of d(t), or f(d(t))×v, as t runs from 0 to t1. Consolidate f(d(t)) into f(t), force as a function of time. Thus the integrand is f(t)v. With f = ma, the integrand is mav, or mva, or mvv′. Change variables again, to v, and kinetic energy is the integral of linear momentum, mv, with respect to velocity. As mentioned above, this comes out ½mv2, regardless of the force profile. All good.

To review, mv is momentum, mass times velocity, force is the time derivative of momentum (since acceleration is the time derivative of velocity), and kinetic energy is the integral of force with respect to distance, or force times velocity with respect to time. Let's try to retain these concepts as v approaches c.

How does Einstein compute the energy of a moving object, knowing that his objects can approach the speed of light? Momentum is again mass times velocity, but mass is adjusted by γ.

p = mγv

We want to integrate fv with respect to t, knowing that f is the time derivative of momentum. The integrand is p′ times v, and that suggests integration by parts. The result is pv - the integral of pv′. Expand p, i.e. relativistic momentum, to get this.

mγv2 - ∫{ mγ vv′ }

mγv2 - ½m ∫{ γ (v2)′ }

mγv2 - ½mc2 ∫{ γ (v2/c2)′ }

The derivative of 1 is 0, so take the next step.

mγv2 + ½mc2 ∫{ γ (1-v2/c2)′ }

Let h = 1 - v2/c2. Thus γ = h.

mγv2 + ½mc2 ∫{ hh′ }

As time and velocity increase from 0, h decreases from 1, and γ increases from 1. Run the integral over h, rather than t. The integral becomes 2h½. If h is the endpoint and 1 is the start, the result is 2h½ - 2.

mγv2 + ½mc2 × (2h½ - 2)

mγv2 + mc2h½ - mc2

mh½ × (v2/h + c2) - mc2

mh½ × (c2/h) - mc2

mc2γ - mc2

c2 × (mγ - m)

The kinetic energy is c2 times the increase in mass due to the relative velocity of the object. In other words, speed corresponds to an increase in mass, corresponds to the energy of motion. Now how cool is that!

Since γ approaches infinity as v approaches c, It would take an infinite amount of energy to reach the speed of light. Chuck Yeager broke the sound barrier, but nobody is going to break the light barrier. Even a subatomic particle, revved up to speed in a particle accelerator, can never reach c, although the Large Hadron Collider, LHC, accelerates protons to a speed of 0.999999991 c, just 3 meters per second shy of c. The relative mass of the traveling proton is about 55 million times its rest mass.

If you've been taught all your life that the earth is round, then it somehow seems expected, even mundane, that someone would discover this truth of nature. But this is our privileged perspective. It's hard to imagine a world where everyone believes the earth is flat. In that context, it is truly a leap of genius to deduce that the earth is round, and offer extraordinary evidence to support this extraordinary claim. In the same way, imagine a world where energy and matter are completely unrelated concepts. In fact it is self-evident. Heat, light, motion, electricity; these are obviously not matter. Such was the case in 1905 and all years prior. Enter Einstein, with another leap of genius. If the energy of motion is proportional to the increase in mass caused by that motion, then perhaps the energy of a body at rest is proportional to its rest mass. Perhaps energy and matter are really the same thing, with a currency exchange rate of c2. This gives the formula that everyone has seen.

e = mc2

If you thought c was a large number, then think about c2. A tiny bit of matter is equivalent to a huge amount of energy. No wonder nobody noticed. On a good day, a double A battery can deliver 18 kilojoules of energy. This is 18,000 kilograms meters2 / seconds2. Notice that Einstein's formula also gives the proper units, mass times velocity squared. Divide by c2 to see how much mass is converted into energy as the AA battery drains.

1.8E3 / 3E82 = 2E-14

The battery is 2E-11 grams lighter after it has discharged its energy. A tiny bitt of mass, 20 pikograms, is converted into energy. Atoms have not disappeared within the battery, but the resulting compounds, after discharge, are just a bit lighter than the reactants - in the same way that water is just a bit lighter than equivalent amounts of hydrogen and oxygen. Of course we can't measure the mass of a battery down to the pikogram, so this cannot be directly confirmed. However, the mass energy equivalence has been confirmed by hundreds of experiments since its proposal in the special theory of relativity.

Stepping up to a higher level, the atomic bomb that fell on Nagasaki converted one gram of matter into energy. Hold a piece of paper in your hand; if the mass of that paper was converted entirely into energy it would obliterate a city.

This is paltry compared to a star. Each second the sun converts about 4 million metric tons of matter into energy. In other words, the sun is 4 million tons lighter each second, as it radiates its energy out into space. (The sun is 227 tons, so the decrease in mass, even over several billion years, is negligible, about 0.1%.) The helium nucleus is 0.7% lighter than the 4 protons that combine to form this nucleus, as nuclear fusion powers the sun. Prior to this understanding, intelligent and educated scientists questioned the age of the earth, geological evidence notwithstanding. It was entirely reasonable to remain skeptical. The earth can't be older than the sun, and the sun, if it burns via a chemical process, like a fire, would exhaust its fuel in just a few million years. This was a paradox that stood, like the elephant in the room, until Einstein and others revealed the enormous energy potential of nuclear fusion, as it converts matter into energy through the multiplier of c2. The sun really can burn for 10 billion years, and the earth really is 4.5 billion years old. Conundrum resolved.

Einstein derived special relativity and more using nothing but pencil and paper and his thought experiments. He asked himself, "I wonder what would happen if the speed of light was constant in all reference frames.", and then he derived time dilation, length contraction, and finally energy / mass equivalence. And this is just the tip of the iceberg. I haven't even touched general relativity, which unites space, time, motion, gravity, and acceleration. If you could stand next to a black hole, time is so distorted that you could literally watch the universe run down. Perhaps I will address this in a future article.

Example with Constant Power

Suppose an infinitely long linear particle accelerator is capable of pushing a proton to arbitrarily high speeds, closer and closer to c. To keep things simple, the proton has a rest mass of 1 kilogram, (a rather large proton), and the accelerator delivers energy into the moving proton at a rate of c2 watts, or c2 joules per second. After t seconds, our ideal machine has dumped tc2 joules into the proton. The kinetic energy of the proton is tc2. Divide by c2, and the change in mass, due to the relative motion of the proton, is t. The moving mass of the proton is t+1, and γ = t+1. Turn this around, and the ratio r, shorthand for v/c, = sqrt(1 - 1/(t+1)2). When t = 0, r = 0, and the proton is at rest. As t approaches infinity, r approaches 1, and v approaches c. Of course v never reaches c, even though we may push the proton along its guideway forever.

If Amy rides along with the proton, Bob sees her time as α = 1/(t+1). Integrate this as t runs from 0 to infinity, and the result is infinite. From Amy's point of view, it still takes forever to reach the speed of light.

Write mass, velocity, momentum, and acceleration as functions of time.

m = t + 1

v = sqrt(1 - 1/(t+1)2) c

p = mv = (t+1) × sqrt(1 - 1/(t+1)2) c

a = (t+1)-3 (1-1/(t+1)2) c

If you calculate the derivative of momentum with respect to time, the result is force, but it is not the effective mass times acceleration. (I'll spare you the calculations.) Suppose the two results were the same. Differentiate momentum, (mγ)v, with respect to v, then multiply by dv/dt by the chain rule to find dp/dt. Of course dv/dt = a, acceleration. So if m never changes, if γ = 1, then dp/dt is mass times acceleration no problem. But if mass changes as well, via the factor γ, then force serves to increase the velocity and the moving mass. Force plays two roles here, and is best viewed as the time derivative of momentum, rather than the simpler mass times acceleration. This is how we derived kinetic energy above, leading to the mass energy equivalence principle.

Further Reading

Relativistic time and mass
Lorentz Factor Kinetic energy Momentum
Fastest spacecraft
Hafele Keating experiment
Mass energy equivalence
Large Hadron Collider
AA battery
Mass energy conversion of the sun is modest relative to the solar mass