Ask a Mathematician / Ask a Physicist

Physicist: The best way to think about it is; there is a speed (C) that is the fastest speed and, by the way, light goes that fast. There’s nothing special about light, it’s just a useful way of describing C (“the speed of light”). Photons are just another podunk massless particle, whipping around the universe as fast as fast can be.
Historically, the derivation of the strange properties of C (relativity) relies on a pretty straight forward piece of Einsteinian logic, based in part on an understanding of light.

1) All the laws of physics work the same, whether you’re moving or not. There is no experiment that can tell you whether or not you’re moving.
2) Light is an electromagnetic wave, and the velocity of these waves can be derived from Maxwell’s laws.

3) Maxwell’s laws, like all physical laws, are independent of how fast you’re moving. So the speed of light must also be independent of how fast you’re moving.

4) So, there exists a speed (the speed that light travels at) that is the same to everyone, no matter how fast they themselves are moving. Holy crap! There’s your special relativity!

So when you see equations like (“energy equals mass times the speed of light squared”), you may ask yourself “what in the hell does light have to do with how much energy is stored in the mass of an object?” Well, the answer is it doesn’t. C is just a speed, and and all the other equations with C would stay the same even if light didn’t exist at all.

So why is C the fastest speed? A good way to think of it is to first ask; how do you know when you’re moving faster than something else? If you’re driving down the highway and you’re moving faster than the car in front of you, then eventually you’ll pass that car. However, C is the same to everyone, no matter what. So, say a photon goes past you, and you try to catch up. But no matter how much you speed up, the photon will always be moving away at the speed of light. You can never catch up (or even come close to starting to catch up). So, regardless of perspective, the photon is always moving faster than you.
Some of this may seem seem contradictory, but surprisingly, it’s all self consistent. Very surprisingly.

The original question was: … also brings up the famous Einstein analogy of a bowling ball in a mattress as bending spacetime. What confuses me is that this seems circular- using the analogy, say we put a bowling ball on a mattress and then roll a marble past it. The marble will fall in towards the bowling ball. But what’s causing it to fall in? Gravity!

Physicist: Here’s what this is about. Way back in the day a popular demonstration used to explain how the presence of matter creates gravity was to drop a heavy ball onto a sheet of some kind, and then roll a smaller ball around the inside of the indentation that is made. If you were to try this demonstration while floating around on the space station you’d be wasting time that could be better spent putting on pants two legs at a time (no gravity to pull the bowling ball and make the indentation in the first place).

In Newtonian mechanics gravity is a spooky, unexplained force. In Einstein’s General Relativity gravity is caused by the curvature and stretching of space and time. Objects move in straight lines like always, but the messed up spacetime they move through makes it appear as though they’re changing direction (that is to say: falling). What’s weird as hell is that they really are moving in straight lines locally, but not globally. If you carefully try to draw a straight line on a bowl you’ll find that it may be straight if you look at a tiny piece of it, but if you stand back it’s curved.

The bowling-ball-mattress thing is another example of how messed up geometry can create “force”. It’s just a bad metaphor. In the one case the pull toward the center is a result of the object in question following a straight line through a messed up spacetime, and in the other it’s trying to roll downhill.
Different.

Physicist: In terms of things like space travel, the difference between 100mph and light speed is academic.  Everything out there is really far apart.  The speed of light, “C”, is woven into the laws of the universe from top to bottom, mostly in the context of electro-magnetism.  Changing the speed of light would have profound effects on chemistry and the fundamental forces.

But those changes are boring.  What’s more interesting is the effects that special relativity would have on every day life.

For what follows, the speed of light is now C = 100 mph (161 km/h for our Canadian or otherwise foreign readers).

"Gamma" is a measure of how much velocity dilates time and shrinks distances. Most of the action happens beyond 90% of C.

Movement? Nopers: If you’ve taken intro physics you may have learned that the kinetic energy of an object is .  But this is just a low-velocity approximation of the true equation (found by Einstein), which is .

The first term is the famous rest mass energy (E=mc²), the second term is the regular kinetic energy, and the third, fourth, fifth (and so on) terms are only important when the velocity is a substantial fraction of light speed (so Newton can be forgiven for getting this one wrong).  But if C=100mph, then suddenly those later terms become important even at low speeds, and you’ll find that moving as fast as 0.01mph would require something like a rocket or a nuclear-powered car.

But that’s boring, so let’s pretend that it isn’t the case.

No long range communication: 100mph is about 45m/s, so having a conversation with someone who isn’t close at hand will result in really annoying delays.  It would be like those satellite interviews, only in person.  To send a message to someone on the other side of the world would take at least 5 days and 4 hours at the speed of light.

I’m ignoring the effects, by the way, of the Earth rotating at about 1,000 mph (at the equator).

Leave your watch at home: The act of walking around would cause you to lose about half a second for every mile you walk, which isn’t to bad.  But if you started moving around in a car at highway speeds (65 mph), then you could expect to lose about 17 seconds for every mile you travel.

“Super Speed”: One of the slick things about traveling at relativistic speeds is that, although you can only pass things at up to 100mph, you can actually cover more distance than the 100mph speed limit might imply.  There are two ways to look at this.

From you’re point of view the world around you undergoes length-contraction.  So, for example, at about 87mph you would see the world contracted by a factor of 2.  So while you’d see things pass by at 87mph, you’d be eating up distance as though you were traveling at 174mph (2 x 87mph).

From everyone else’s point of view, you’re traveling through time slower.  At 87mph they’d see your watch ticking at half the usual rate, so the trip will only take half the time it should.

Pretty colors: Even at running speed there would be enough relativistic doppler shift to change the colors around you.  If you were driving past a yellow field of grain, it would appear blue in front of you and fade to deep red as it passed behind you.

There are just a hell of a lot of other effects, so if you’re wondering about any of them, just ask in the comments.

Physicist: From a classical (Newtonian) view point this is a completely solid question.  However, in the context of special relativity the question itself is (unfortunately) non-sense.  For many practical purposes, the speed of light (hereafter I’ll call it “C”) is “infinitely fast”.  If you define infinitely fast as the speed you’ll be going if you accelerate forever, then C is exactly that.

Normally when you want to figure out “the behavior at infinity” you can “take a limit”.  For example; the limit as x goes to infinity of 1/x is 0.  This statement just means that as x gets bigger and bigger 1/x gets closer and closer to zero.  So by looking at the behavior at larger and larger finite values you can talk about what happens at infinity.  C, on the other hand, is fundamentally different from all other speeds.

At a basic level, speed is just distance traveled over time taken (as in “miles per hour”).  Due to the laws of special relativity, movement affects both the relative distances and relative time between two reference frames.

As a quick aside, a “reference frame” is just the set of all things that are moving at the same speed or, equivalently, are stationary with respect to each other.  So if you’re traveling down the highway you’re in the same frame as all the other cars around you (if everyone’s going the same speed), while the repair teams and clean-up crews on the shoulder are in a different reference frame.

It may seem silly to say it, but no matter how fast you move you still see things passing by, and it still takes at least a little time to get where you’re going.  At C however, the distance to your destination is always zero due to length contraction, while the time it takes to get there is also zero due to time dilation.  If you were to calculate your own speed you would say , which makes no damn sense.  I mean, what is that?

The universe: As seen by something traveling slower than C, and something traveling at C.

Also, consider this: at any other speed you can speed up or slow down, but at C you genuinely don’t have time to step on the brakes or the gas.  Literally, “time” and “distance” are phenomena that only make sense if you’re talking about them at speeds slower than C.  Stuff in the universe is divided into two categories: “massive” and “massless”.  Massive objects (anything with mass) always travel slower than C, while massless things must travel at C.

All that being said, you can wave your hands and talk about what life is like for a photon, that can’t exist at sub-light speeds (after all, what speed would you expect light to move at?).  When a photon is generated it immediately takes off at C, and never slows down until it runs into something.  Photons never experience time or distance.  As far as they’re concerned they are emitted and absorbed at the same place and time.  Many of the radio photons hitting you right now (about a third of them), have been traveling for around 15 billion years, but they think that the beginning of the universe just happened (or would, if they could think).

Physicist: Back in the day, Galileo came up with the “Galilean Equivalence Principle” (GEP) which states that all the laws of physics work exactly the same, regardless of how fast you’re moving, or indeed whether or not you’re moving.  (Acceleration is a different story.  Acceleration screws everything up.)  What Einstein did was to tenaciously hold onto the GEP, regardless of what common sense and everyone around told him.  It turns out that the speed of light can be derived from a study of physical laws.  But if physics is the same for everybody, then the speed of light (hereafter “C”) must be the same for everybody.  The new principle, that the laws of physics are independent of velocity and that C is the same for everybody, is called the Einstein Equivalence Principle (EEP).

Moving faster makes time slow down: I’ve found that the best way to understand this is to actually do the calculation, then sit back and think about it.  Now, if a relativistic argument doesn’t hinge on the invariance of C, then it isn’t relativistic.  So ask yourself “What do the speed of light and time have to do with each other?”.  A good way to explore the connection is a “light clock”.  A light clock is a pair of mirrors, a fixed distance apart, that bounce a photon back and forth and *clicks* at every bounce.  What follows is essentially the exact thought experiment that Einstein proposed to derive how time is affected by movement.

The proper time "τ" is how long it takes for the clock to tick if you're moving with it. The world time "t" is the time it takes for the clock to tick if you're moving with a relative velocity of V.

Let’s say Alice is holding a light clock, and Bob is watching her run by, while holding it, with speed V.  Alice is standing still (according to Alice), and the time, , between ticks is easy to figure out: it’s just .  From Bob’s perspective the photon in the clock doesn’t just travel up and down, it must also travel sideways, to keep up with Alice.  The additional sideways motion means that the photon has to cover a greater distance, and since it travels at a fixed speed (EEP y’all!) it must take more time.  The exact amount of time can be figured out by thinking about the distances involved.  Mix in a pinch of Pythagoras and Boom!: the time between ticks for Bob.  So Bob sees Alice’s clock ticking slower than Alice does.  You can easily reverse this experiment (just give Bob a clock), and you’ll see that Alice sees Bob’s clock running slow in exactly the same way.

It turns out that the really useful quantity here is the ratio: .  This equation is called “gamma”.  It’s so important in relativity I’ll say it again: .

It may seem at first glance that the different measurements are an illusion of some kind, like things in the distance looking smaller and slower, but unfortunately that’s not the case.  For Alice the light definitely travels a shorter distance, and the clock ticks faster.  For Bob the light really does travel a greater distance, and the clock ticks slower.  If you’re wondering why there’s no paradox, or want more details, then find yourself a book on relativity.  There are plenty.  Or look up Lorentz boosts.  (The very short answer is that position is also important.)

The lower the slower: Less commonly known, is that the lower you are in a gravity well, the slower time passes.  So someone on a mountain will age (very, very slightly) faster than someone in a valley.  This falls into the realm of general relativity, and the derivation is substantially more difficult.  Einstein crapped out special relativity in a few months, but it took him another 10 years to get general relativity figured out.  Here’s a good way to picture why (but not quite derive how) acceleration causes nearby points to experience time differently:

Redder light at the top, bluer light at the bottom.

Alice and Bob (again) are sitting at opposite ends of an accelerating rocket (that is to say; the rocket is on, so they’re speeding up).  Alice is sitting at the Apex (top) of the rocket and she’s shining a red light toward Bob at the Bottom of the rocket.  It takes some time (not much) for the light to get from the Apex of the rocket to the Bottom.  In that time Bob has had a chance to speed up a little, so by the time the light gets to him it will be a little bit blue-shifted.  Again, Alice sees red light at the Apex and Bob sees blue light at the Bottom.

Counting the blue crests is faster than counting the red crests. However, since it's all the same light beam the number of crests has to be the same to everybody.

The time between wave crests for Bob are short, the time between wave crests for Alice are long.  Say for example that the blueshift increases the frequency by a factor of two, and Alice counts 10 crests per second.  Then Bob will count 20 crests per second (No new crests are being added in between the top and the bottom of the rocket).  Therefore, 2 seconds of Alice’s time happens in 1 second of Bob’s time.  Alice is moving through time faster.

Einstein’s insight (a way bigger jump than the EEP) was that gravitational acceleration and inertial acceleration are one and the same.  So the acceleration that pushes you down in a rocket does all the same things that the acceleration due to gravity does.  There’s no way to tell if the rocket is on and you’re flying through space, or if the rocket is off and you’re still on the launch pad.

It’s worth mentioning that the first time you read this it should be very difficult to understand.  Relativity = mind bending.