Copyright © Karl Dahlke, 2023

Einstein developed special relativity, and even the beginnings of general relativity, by pure thought. No laboratory, and no experiments; though he did draw upon the experiments of other scientists in the late 1800's.

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 mirrors for our space telescopes. 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 fast, 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 concede 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. Alice is riding a train with a large open window, so Bob can see her as she passes by. Alice 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. Alice also has a high tech watch that can measure time in nanoseconds, that is, billionths of a second. 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 Alice 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 Alice's flashlight, Alice 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, Alice'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 Alice, and his watch also reads 7 when Alice 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 Alice's watch still shows 7. A fraction of a nanosecond later, the light reaches the ceiling and Alice's watch clicks over to 8. Alice'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 Alice's watch. Time itself runs slower. Alice'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(c^{2} - v^{2}).
Let l be the vertical distance from flashlight to roof.
This was 30 cm in our example, but it could be anything.
Remember that time is distance divided by velocity, distance divided by how fast you are going.
The elapsed time for Alice is l/c, but the elapsed time for Bob is l/sqrt(c^{2} - v^{2}).
The time dilation factor,
the ratio of Bob's time to Alice's time for the same event,
often denoted γ in the literature,
is c / sqrt(c^{2} - v^{2}).
Multiply top and bottom by 1/c and get
1 / sqrt(1 - v^{2}/c^{2}).

Our fastest spacecraft barely reach 0.01% the speed of light. If Bob watches Alice 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. This is orders of magnitude below human perception, but it verifies the theory. Einstein was right! Time is not constant throughout the universe; it changes with your frame of reference.

As a thought experiment, rev up Alice'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 Alice are 1.154 seconds apart for Bob. This is still a small change as humans perceive time. If you really want to slow Alice 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 Alice's clock advances by just one second. As Alice 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 Alice's world.

There is a beautiful symmetry here. Shift your frame of reference, so that Alice 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 Alice's, and Alice 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 Alice and Bob are flying apart, so the differences in time do not matter. "Well," you may ask, "what happens if the train brings Alice back home? Are the two watches both slower than each other?" No, they get back in sync, because of the acceleration needed to slow the train down and bring Alice back home. This is part of general relativity, and beyond the scope of this book.

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 Alice'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 Alice's velocity, however, the change in velocity occurs over a longer time in his frame of reference. Acceleration is delta v divided by time, and time is multiplied by γ, thus acceleration is divided by γ. The force is the same, but the acceleration is less, so the masss must be greater. The mass is multiplied by γ. (This isn't the best analysis, but it is intuitive, and it gives the right answer.) The same force takes longer to change the speed of the ship from Bob's point of view, thus the ship has more mass. In summary, time runs slower on Alice's ship, and everything, including Alice, has more mass.

Since γ approaches infinity as v approaches c, It would take an infinite amount of energy to reach the speed of light. The object gets heavier and heavier, so to speak, the faster it goes. It takes ever more energy to push it towards the speed of light. Chuck Yeager broke the sound barrier, but nobody is going to break the light barrier. It is physically impossible. 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.

Einstein's general theory of relativity also affects time, in an unexpected way. Time itself slows down under gravity. Put Bob on top of a mountain, and put Alice down in the valley. Alice's watch runs slower than Bob's, because she is experiencing slightly more gravity. This too has been verified by our atomic clocks. Einstein was right again!

If Alice could stand next to a supermassive black hole, she would seem frozen in time, as Bob looks on. Her heart might beat once every million years. In contrast, Alice watches the universe run down before her eyes. Civilizations rise and fall, and stars run out of fuel and wink out of sight.

Special relativity was symmetric - Bob saw Alice, on her moving train, the way Alice saw Bob - but general relativity is not symmetric. Alice experiences hyper gravity, while Bob does not. Bob sees Alice frozen, like a statue, while Alice sees Bob live out his entire life in the wink of an eye.

If only we could realize significant time dilation in the real world. Imagine a fridge that does not keep food cold; it simply slows down time. Your lunchmeat does not spoil, because a week of your time is only a minute in the fridge. We'd be drinking warm milk, (have to get use to that), but it would never spoil. I doubt we will ever have the technology to do this, and that's a shame, because there are many applications.