Relativity: A Short Guide

Kevin Hainline

Dartmouth College

kevin.n.hainline (at sign) dartmouth (dot) edu

So, relativity is a tricky subject, that requires hours of careful thought, an open mind, and a dark room to slowly cry in case it gets too confusing. This short guide is intended to clear up some of the weirder ideas in an attempt to clarify what is admittedly a completely crazy (some might even say "made-up") subject. I assure you, however, that this is all real science, and while it does sound insane, it is backed by scientific evidence. Ok! On to the science.

Relativity is split up into two subjects:

1. Special Relativity

2. General Relativity

Special relativity is the subject dealing with the constant velocity motion and the idea that there is no preferred reference frame. It combines this idea with the fact that light goes at the same speed for every reference frame, no matter how fast you are traveling.

General relativity is the subject dealing with the idea of acceleration and it's relationship to gravity. It proposes that there is no preferred reference frame with respect to gravity or acceleration, and also, that gravity is just a warping of a higher-dimensional "space-time."

So, let's go over this in a little more detail.

1. Special relativity.

When you move, you generally are thinking: "oh man, I am totally moving. I like moving, and I will continue to do this." But how do you know you're moving?

"Well...my feet are pushing at the ground under me. This makes me move past things...like trees, or cars."

Ok, but what if you were on a treadmill and while you're walking, you're not going anywhere. Are you "moving" then?

The idea of "moving" is a really weird concept. You probably realize that you have to have something to measure motion by in order to understand if you're moving (a tree, the sidewalk, some grass, a building). If that is coming at you, and you're moving your feet in a walking motion, you're probably moving. But...what if you're in space, where you have nothing to measure your motion by? How would you know you're moving then?

You'd need something else to be moving, and then you'd see how you were moving in comparison. If you were in empty outer space, then if something were coming at you, you could try to figure out if you're moving. But! If you're in space, how do you know if something is coming at you...or you're moving towards that something?

Well, we define a point of view for these things as a "frame of reference." In your "frame of reference," the object is moving towards you. IMPORTANT POINT - you are always stationary in your own frame of reference. (have you ever seen those shows where they put a camera on someone's head and it's pointed towards their face? You know, like one of those survivor shows where they have someone running through a forest scared, but you can see their frightened face? Isn't that creepy, to see their head stay still while the world jostles around? It's weird to be in the head's "frame of reference").

Anyway, Einstein decided (actually, Galileo came up with this first) that there is no frame of reference that is "right." For instance, you can bicker all day with a friend (if you're floating around in space) over who is the one who is moving and who is staying still. Motion is measured relative to a frame of reference. On the Earth, we measure almost everything based on the frame of reference of the ground.

This was the first part of the special theory of relativity - there is no preferred ("right") frame of reference. The second part is that light moves the same speed no matter what frame of reference you're in. If you're moving at a super high speed and you're trying to catch light, you won't, since in your super fast frame of reference (where you aren't moving) light is still moving away from you at c, the speed of light. (This is weird! On Earth, if you are chasing someone in a car, and you're driving 80 miles an hour, but the car is going 90 miles an hour, you will see the car ahead of you slowly moving away from you at 10 miles an hour based on your frame of reference. This does not work with light! Light will still be going away from you at velocity c no matter how fast you go to try to catch up.)

This creates a lot of weirdness. Since this is text, I don't have pictures, so I will list the weirdness that comes from this idea:

1. If you're going at a velocity V, your clock will tick slower than someone who is going at a slower velocity. For every second that the slower person experiences, you will experience more than a second. So, if you're going fast enough, what seems like an hour for you might be days for someone who is not moving as fast as you. This is called "time dilation."

2. If you're going at a velocity V, you will become squished along the direction of motion. This is called "length contraction." So, if you are holding a meter stick, and someone else has a meter stick and they are flying at you at a high speed V, you will hold up your meter stick, and as they pass by, their meter stick will be shorter. Similarly, they will think your meter stick is shorter as well! Weird idea.

3. If you're going at a velocity V, an outside observer will detect that you will gain mass. This one is kind of a weird one, and is the reason you can't travel at the speed of light - as you go faster, you gain mass, and therefore it becomes harder and harder to travel faster, until you get to the point where it requires an infinite amount of energy to travel at the speed of light. Light, made up of photons, is massless, and can travel at c.

4. Simultaneous events are dependent on your reference frame. Whoa, this one is a doozy. What it means is that if you think that you are doing something at the same time as someone else, you only are in your own reference frame. In another reference frame, you are not. This comes from the fact that time is weird in other reference frames. This is a super important idea, and most weird paradoxes in relativity can be solved by understanding this fact.

For instance, you might have thought: "Hey! I don't buy what you said up there in number 2, about me and a friend measuring each other as being squished! Who's actually correct?" Well, the trick here is to remember that when you measure something, you look at one end of the thing and then the other. Even if you could measure something instantaneously, this simultaneous measurement changes in another frame, which leads to the fact that both of you measure a smaller distance.

I know it's confusing. But all you need to remember is that sometimes, you might think two things are happening at the same time - and somebody standing right next to you would go: "yes. Those two things happened at the same time." However, someone moving in a different frame might beg to differ, saying one happens before the other. This person would call you and your friend idiots. That's why you have a friend though, to beat this guy up. If you can catch him.

So, let's illustrate some of the weirdness in relativity with a problem:

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1. The barn-bus paradox. (also called barn-pole, or barn-ladder paradox). So let's say a bus that's 30 feet long is driving towards a barn that 20 feet long, with two doors on the barn. Inside the barn are two people who are manning the controls for each door. The bus is going really fast, and therefore, is experiencing length contraction, and get this, in the reference frame of the people in the barn - the bus looks only 15 feet long! This means that the bus could actually fit INSIDE the barn! (Think about this for a second. This is just a consequence of weirdness number 2 above)

So, they decide that once the now-squished bus is fully inside the barn, they are going to close the doors at the same time, and then open them quickly, so that for a split second, the bus can fit inside the barn. Easy enough.

But, let's now switch viewpoints, and say we are on the bus. (Again, this is an important point. Most of special relativity involves going into different frames, and remember, in the buses frame of reference it is not moving, but the barn is coming towards it) So, we're on a 30 foot long bus, and from our perspective, there's a barn moving towards us, and it is all squished, so it looks like it's only 10 feet long! The bus we're on is 30 feet long!

So, to recap:

1. Frame of the barn - Bus is squished, and can fit inside the barn.
2. Frame of the bus - Barn is squished, no freakin' way can the bus fit inside the barn.

Who is right? They both are! The bus moves through the barn, and the doors work, and everyone is happy.

In the frame of the barn, they close the doors and open them, and the 30 foot bus does indeed contract to 15 feet and fit inside the 20 foot long barn.

The trick comes when we consider the frame of the bus. Now, the bus is moving towards this small squished barn, where two guys are going to try (and actually succeed) to close the doors. But remember what I said above - simultaneous events depend on the reference frame. So, for the bus, first one door closes and then opens and then only after the bus passes by the front door, does the other door open and close. Instead of them closing at the same time, one closes before the other, and then instead of them opening at the same time, one opens before the other, and the bus makes it through.

You can find some pictures here.

But...what if the bus, after passing this first barn, comes to a second barn 20 foot long barn where there is only one door, and entrance, and there is no exit. In this second barn's frame of reference, he is going to close the door when the squished bus enters the barn, and then leave it closed.

In the bus frame of reference, there is a barn with a wall at the end moving towards it, and this barn is squished to ten feet. What happens?

Well, in the reference frame of the barn, the bus comes in, hits the wall, and the door closes behind it. Once the bus stops, it is not length contracted, so it busts out the back door!

But...in the reference frame of the bus, how did it ever get into the barn, if the barn were so squished so small?

Well, the trick here comes from the fact that the bus goes into the barn and hits the back wall. Remember though, that nothing can travel faster than the speed of light - not even the bus crash - so the signal from the front of the bus (all of the molecules squishing together as it crashes) actually goes back through the bus at some velocity. Meanwhile, since the back of the bus has no idea it has crashed (since the signal hasn't gotten there yet), it keeps right on driving in. Here, the bus will keep driving in and eventually make it all the way into the barn before the door is closed and the signal finally reaches the back of the bus, which stops, and then breaks through the back wall.

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Now, there are a lot of paradoxes that all work the same way, but the important point to remember for most of them is that length is contracted, and simultaneous events depend on frame of reference. (What you think might happen at the same time doesn't happen that way for someone moving quickly.)

2. General Relativity

So, now we will look at Albert Einstein's theory of gravity. Before we get into the nitty gritty - think about what you know about gravity? What have you been taught? You know that gravity is...

- what keeps us on the ground.
- what keeps planets around the sun
- a force between two bodies

if you're sharp, you'll remember that Newton came up with a formula to describe gravity, and the force between any two objects is the product of their masses divided by the distance between them squared (all multiplied by a constant, G). This is a good way of looking at gravity, and certainly, it is the way that most physicists mathematically see gravity.

Einstein decided to look at it in a different way. See, Einstein did a lot of thinking about gravity. If you remember, in special relativity, there is no "preferred" reference frame, and so you can't tell, if you're all the way out in the middle of space, if you're moving or stationary. Well, Einstein got a little worried about this whole deal when he really started to think about it.

What if you're in a spaceship and the spaceship is accelerating? Well, if you remember Newton's law F=MA, you'll realize that if you're accelerating you'll feel it, since there's a force! Uh-oh, this means that you can tell if you're undergoing an acceleration! Einstein freaked out about this, but finally turned to his friend gravity for the solution.

Let's say you woke up tomorrow and you were in an empty locked room. As you walked around, I guarantee that since you're totally in love with physics, you'll probably think: "hrm, this room has gravity in it!" Well, this might be the case...or...you could be in a spaceship accelerating upward (think about how you feel a pull in an elevator). You could even do some physics experiments in your room and never tell that you were in a spaceship or on Earth. Einstein showed that the effects of gravity are exactly equivalent to the effects of acceleration. This is what was known as the "equivalence principle." It's really important, so remember it. Acceleration effects = gravity effects.

Now, before we look at how Einstein viewed gravity, I want to define something pretty weird: "spacetime." Ok, so what is "spacetime?" Well, we know that a 1-dimensional object is a line. It only has one dimension, and you can only move forward or back. A two dimensional object would be a flat plane. We can move right, left, up, or down. A three-dimensional world has an extra dimension, and you could move in three directions, forward, back, right, left, up, or down.

(If you remember your old video games, in Pong, you could only move in 1-D, up or down the screen. In Pac-Man, you could move right and left, up and down in 2-D. In modern day games like Halo, you can move in 3-D.)

Now, I would try to describe higher dimensions, but our brain can't imagine higher dimensions. When Einstein talked about "spacetime," what he meant is imagining our world as being 4-dimensional, with the fourth dimension (which we can't imagine) being time!

Whoa!

I know, you're probably shaking your head, but seriously, you guys have probably always thought about it this way! Trust me. If you ask a friend (providing you have friends) to meet you at a baseball game, then you would say something like:

"Meet me on the upper deck, row D, seat 113 at 8:00."

Now, each piece of information you have to give here is one dimension. In this case (and in all cases) you are specifying four things, here. Upper deck tells you the first dimension, the up/down dimension. Row D tells you a forward/backward dimension, seat 113 tells you a right/left dimension, and 8:00 tells you the 4-th dimension, time!

So, Einstein used the idea of "spacetime" to show that gravity can be looked at as a (and get this, it's kind of weird), warping and curving of spacetime.

Whoa! (again)

I know you wish you could understand what that looked like, but again we can't imagine four dimensions. So, let's look at it a different way, by going down a dimension. Let's say that we live in a 2-dimensional plane. When I say that we live on a plane, I mean that we are living actually in the plane, so we can only look forward, backward, left and right. We can't look up and down (out of the plane) and so we have no idea that those dimensions exist.

Well, what Einstein proposed is that gravity is a bending of the 2-dimensional plane when a mass is introduced, much like how you can place a heavy object in a sheet, it stretches the sheet in the middle and sags. Now, if we lived in the plane, we'd have no idea that the sheet was all saggy. (Remember, you can't look up or down, so you don't know that things are weird, to you, there's just a mass in the middle)

So, to recap, we're living in a plane, and there's a mass in the middle. To someone looking at us living in the plane, they can see that the mass is actually stretching it and causing it to sag. Got it?

Well, for us living in the plane, if we were to throw a rock towards the mass (we fear this mysterious mass!), and we missed (we are bad at throwing!), we would be INCREDIBLY surprised because the rock would actually BEND around the mysterious mass. What? We would be confused. To that same person looking in on us from above, it would be obvious! The rock is actually curving because of the saggy part of the plane! (Think about rolling a ball across your bedsheet with something heavy in the middle of the sheet, it would curve because of the sag of the bedsheet)

Now, to cut to the chase - this stretching of the 2-dimensional sheet is gravity! Gravity, according to Einstein, is a curving of 4-dimensional spacetime because of a mass. The more the mass, the more the curvature!

Well, now you're probably thinking: "Kevin...if spacetime is 3 dimensions and a fourth dimension plus time...and spacetime is all curvy around masses...then shouldn't that screw around with time a bit when you get close to a mass? Yes!

There's a concept called "gravitational time dilation," and this is exactly what it is - time runs slower the more curvy spacetime is. So, closer to a mass, where spacetime is more warped, time moves slower. The more massive an object, the more stretched spacetime gets, and the slower time ticks for a person living on the surface. For an object like a black hole, where spacetime is curved an insane amount (infinitely curved at one point), time actually slows to a stop! Weird!

You can actually prove that time runs slower closer to a gravitating body with the equivalence principle. If you were to try to have a strobe light rave in an accelerating spaceship, with you at the top of the ship (moving upwards) and your friend at the bottom, because you are accelerating upwards away from his strobe light, his strobe light will look to you like it's strobing slower than yours. The light from your friend's strobe is having to travel slightly longer than it has to travel from your own strobe light, and you interpret this as his clock being slower. Now, if you take this entire accelerating spaceship and just put it on the earth (which we can do! Remember the equivalence principle! gravity effects = acceleration effects), then you would have the same weird slow clock effect for your friend at the bottom. The moral of the story is to not have any raves in tall buildings.

So, now you're probably completely baffled. This all seems new and scary and mysterious...but do you buy it? Well, you should, since there are a variety of things that prove that general relativity is true:

1. For years, scientists could not figure out why they couldn't accurately predict the orbit of Mercury. They thought they were accounting for everything...but when they finally took into account general relativity (and the fact that Mercury moves deeper within the curvy spacetime as it gets closer to the sun), things worked out.

2. Light bends around masses in curvy spacetime, since light always wants to move on the path that is the shortest distance. Think back to us living in the 2-D world - we would normally think that the shortest distance between us and someone on the other side of that mysterious mass would be straight ahead, right through the mass. But this isn't the case, since we would be moving farther due to the sagging of the plane that we live in. But if we were to go around it, where the sagginess is a little less, it would take less time. So, light bends AROUND massive objects.

Well, we can actually detect this light bending around things like the Sun! In other cases, we detect the light from distance objects getting "gravitationally lensed" (the fancy term for 'light bending around an object') around galaxies that are closer to us.

4. Also, since I mentioned that time is slower for people living closer to a massive object, where space is more curvy, this has effects for someone trying to flash a light from the surface. For someone who's not so close, when they get the light, they'll find that it's actually redshifted, since the clock that emitted it was slow (if it was emitted with 10 waves/second on the surface, where a second is slower, then the frequency will go down when it gets to somewhere where the seconds are faster.) We detect this gravitational redshift from things like the sun!

5. Finally, as masses move through space, they change the curvy-ness of spacetime. This creates waves of gravity which actually carry energy away from things moving around. Astronomers have looked at two stars orbiting around each other and thought: "I bet that they're kicking up a lot of gravitational waves." If this is true, then they're giving up a lot of energy, and their period must be decreasing as they get closer together. Well, we've detected this! Score one more for Einstein.

So, that's special and general relativity. Believe it? Maybe. It's good to be skeptical, especially since this is really kind of a weird thing. But the important thing is that it works, weird or not. So, study up, and slowly you'll realize that it's actually quite inspired...this Einstein guy was a pretty smart guy.