Einstein’s theory of special relativity is one that interests a lot of people.
Judging from the views and comments that I’ve been getting, you guys would like to see a
lot more of them.
So I’ve decided to do something different.
What I’m going to do is make a series of videos that strings together a longer and
more coherent narrative.
This one will be the first of the series, even though I’ve already made some others.
That might seem a little weird, but, well, I’m sure it will all work out.
So in this video, I thought I’d start, like all good stories start- at the beginning.
I’m going to tell you the central meaning of relativity, the crucial assumptions, and
I will end with the two core equations.
Now I don’t intend to derive those equations in detail.
I’ll point out the important features and direct you to a different video where I showed
the basic method of how the derivation would go.
So let’s start by demystifying relativity.
At its very center, it isn’t about clocks running slow or objects shrinking.
It’s not about some people aging faster than others.
Those are all consequences of the theory, but not what relativity is all about.
Relativity is really about something much more basic.
It’s about how different people see the same situation.
Relativity is not just an Einstein thing.
So let’s start with something way more familiar that will let you get comfortable with the
idea.
Suppose you have two guys, one standing on the side of the road and the other in a car
driving at sixty miles per hour.
Now assume that both the standing guy and the driving guy are raging egomaniacs who
both claim that they are the center of the universe.
This is the key point- both of them adamantly claim that they’re not moving.
On the other hand, while they are both egomaniacs, they are at least willing to try to understand
how the other person views the world.
And that is the heart of relativity.
So let’s flesh this out a bit.
The car guy believes that he is stationary and the world is zooming by him.
Not only that, he thinks that other positions are also stationary.
For instance, the location a mile ahead of him is always precisely a mile ahead of him.
It doesn’t move.
Likewise, the position a mile behind him is also stationary and unmoving.
So he can specify any location using the symbol x and we’ll add the subscript ‘car’
to distinguish it from the standing guy.
We can even hang some numbers on that.
We’ll say that the car guy is at location x sub car equals zero.
That means that he is at the center of his universe and, remember, we did say that he
was a raging egomaniac.
Now the location a mile ahead of him is simply x sub car equals plus one mile, while the
location behind him is x sub car equals minus one mile
So this establishes the world from the viewpoint of the car guy.
How does the standing guy see the same things?
Well the standing guy is also an egomaniac and insists that he is stationary.
But he wants to know the location of the car guy and he says that the location of the car
guy is changing.
So the standing guy says that his location is x sub standing equals zero.
And that means when the car passes his position, that’s the one time they agree on position,
since they both think their location is x equals zero and they are in the same place
at the same time.
We’ll get back to time later.
Okay, so the standing guy thinks that the car guy is moving and his location after half
an hour is plus 30 miles away and after an hour is plus 60 miles.
In fact, we can write a simple equation for what the standing guy calls the car guy’s
location and it is simply his velocity times time.
Now that’s for the car guy’s location, but we can generalize that.
If the car guy picks a location that is stationary as far as he is concerned- say a mile ahead
of him, then the standing guy also sees that location moving.
At the moment the car passes the standing guy, that location is at x sub standing equals
plus one mile.
After half an hour, that location is at plus 31 miles.
After an hour, that location is at plus 61 miles, and so on.
We can thus make a general equation that tells us how to convert between these two egomaniac’s
viewpoints.
If the car guy picks any location that is stationary with respect to him and call that
location x sub car, the standing guy will say that this location changes with time and
can be written as x sub standing equals x sub car plus the car’s velocity times time.
Now this equation is called a Galilean transform, named after Galileo Galilei.
His heyday was like in 1600 or so, so this isn’t exactly new.
In general, we don’t want equations that specify that this is a situation involving
a car and a standing person.
So, textbooks simply call the two points of view- what we call frames- as the primed frame
and the unprimed frame.
So I am adding here the equation like you’ll see in textbooks.
For us, the primed frame is the point of view of the standing guy, while the unprimed frame
is the point of view of the car guy.
But the idea works for any two frames that are moving with respect to one another.
I’ll give you a second to take that in.
Okay, so that’s a core point.
Now we have to talk about time.
In our intuitive and familiar world, we say that the two people- standing guy and car
guy, or primed frame and unprimed frame- experience time the same way.
So we can write that here.
t sub standing equals t sub car or, equivalently, t prime equals t.
By the way, you can do the same exercise starting from the point of view of the standing guy.
He can pick positions and then we can ask what the car guy thinks.
I leave that to you to think through.
But if you do that, remember from the point of view of the car guy, the standing guy is
moving backwards at sixty miles per hour.
This means that his velocity is negative, at least as far as car guy is concerned.
It’s probably worth playing with these ideas on your own.
Alright- so that’s Galilean relativity and it’s not that hard.
This relativity just tells two pig-headed egomaniacs what the other guy thinks about
locations and time.
How is Einstein’s special theory of relativity different?
I’m not going to show you a derivation of Einstein’s equations because I did something
similar in a different video.
But I am going to tell you the two core assumptions that went into Einstein’s equation and then
show you them.
The first core assumption is that the laws of physics are the same for both people or
what we call observers in relativity lingo.
That also means that both of them can say that they are the unmoving person.
In fact, both of them has to insist that they are the unmoving person.
Note that this is true for both Galilean and Einsteinian equations.
The thing that is unique to Einstein is that he said the speed of light was the same for
all observers.
This flies in the face of common sense.
After all, if you have two cars moving in opposite directions at 60 miles per hour on
the highway, their closing speed is 120 miles per hour.
But that’s not how it is for light.
Everybody sees the same speed.
If you take these two assumptions, you can derive Einstein’s transformation equations.
Actually, they were originally derived by physicists long before Einstein.
They are named Lorentz Transforms after Hendrik Lorentz.
But it was Einstein who embraced them in a modern way.
Like I said before, I am not going to derive the equations here.
There are plenty of places you can look them up.
And if you want to see the most important points, I made a video about the gamma, which
is what we call the Lorentz factor that gives the highlights.
It’s also on the Fermilab YouTube channel.
So let’s talk about Einstein’s relativistic equations.
Suppose there are two women in empty space moving either towards or away from one another
at some velocity v. Call them just the primed person and unprimed person.
In my example, there are no external reference points, so you can’t know if one person
is moving or the other is.
So they both say that they aren’t moving.
This is the raging egomaniac thing again.
Now assuming that they use the time where they pass one another as the zero point of
both their time and location, you can transform between the two viewpoints using these two
equations.
If we start with the unprimed woman and she picks a time and a location that is stationary
with respect to her and call them t and x, then we can determine the position and location
for the primed person and those are embodied in these equations here.
For maximum clarity, let’s explicitly go through the equations.
For position, we have x prime equals gamma times the quantity x plus their velocity times
the elapsed time seen by the unprimed woman and denoted t.
For time, we have t prime equals gamma times the quantity t plus the velocity times x divided
by c squared- and, of course, c is the speed of light.
Gamma is called the Lorentz factor and is one over the square root of one minus the
quantity v squared over c squared.
And I show where that factor comes from in the Lorentz factor video.
You’ll notice how the position equation is pretty similar to the Galilean transforms,
but the time one is very different.
Now I’ll use these equations to show you some cool things in another video, but I want
to emphasize some things before we wrap it all up.
These are the core two equations of relativity.
Everything else derives from them.
And it’s super important to remember exactly what they are, which are equations that tell
you how to convert from the point of view of one person to the point of view for another
person.
That’s it.
It’s just like the car we talked about, where the standing guy says the location of
the car changed as the guy drove, while the car guy said that the car was stationary and
the standing guy was moving away from the car.
That’s it- that’s all that’s going on here.
So this video has gotten a little long, but here’s what I’m going to do.
In the next video, I’ll guide you on how to use them.
Some of you are already asking about the twin paradox and other oddities of relativity and
we’ll get there a couple of videos from now.
In the meantime, let’s just take a look at these two magnificent equations, shall
we?
Okay, so this relativity series is a departure from how I’ve done things in the past.
If you like the idea of a series of connected videos, let me know in the comments.
Of course, you guys are all pros, so I don’t need to tell you to like, subscribe and share.
We want the Fermilab YouTube channel to be the best on the internet.
And, as always, remember, that physics is everything.