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.
