There are lots of really common misconceptions when one starts playing with Einstein’s
theory of special relativity.
I’ve talked about some of the biggies, like how the way that you experience space and
time depend on your perspective.
But there is a much simpler phenomenon that so many people get wrong.
And that’s how to add velocities.
You know how to add velocities in the familiar world.
If two cars head towards one another, each moving 60 miles per hour, their relatively
closing velocity is 120 miles per hour.
By the way, that’s 100 and 200 kilometers per hour for you guys who aren’t bilingual.
So that’s super easy.
If two people are heading towards one another with two different velocities- call them big
V and little v, then, if you ask one of them what their closing speed is, it’s just v
closing equals big V plus little v.
With that easy idea, we could be done with this video.
We’d set a record for being short.
Of course, it’s not that easy in the relativity world.
To see why, let’s consider two people heading towards one another not at the paltry speed
of 60 miles per hour, but each moving at 60 percent the speed of light.
Taking the same approach as we did with cars, each person would say that their closing speed
relative to each other was 1.2 times the speed of light.
So that’s a problem.
Measurement tells us that this isn’t true.
You can’t go faster than the speed of light.
And I’ve seen people post ideas like this in the comments section of some of my other
relativity videos.
It’s a pretty common misconception.
So, how do we do this the right way?
This actually is a pretty easy derivation to do.
Let’s give it a go.
We’ll start out by setting up the problem.
You have two people heading towards one another, Ron and Don.
Ron is traveling with velocity little v and Don with velocity big V. Let’s try to figure
out how fast Ron thinks Don is going.
To do this, you need to remember the definition of velocity, which is just the change in position
over change in time.
That means that you can simply pick two arbitrary times- call them 1 and 2- and write Don’s
velocity as big V equals x2 minus x1, divided by t2 minus t1.
We can also write this as Delta x divided by Delta t.
I just did that to make the equations a little more compact.
If you like, you can keep it in the format using the 1’s and 2’s.
Also, I’d like change things around and write this as Delta x equals big V times Delta
t.
That'll be handy in a minute.
So you’re the unprimed observer and you want to know what Ron sees.
Ron is the primed observer.
We can write how he sees Don’s velocity in the same way, but this time with the primed
x’s and t’s.
I’ve said many times in this video series that if you want to do relativity correctly,
you can’t go wrong if you go back to the Lorentz Transform equations, which we can
see here.
I’ve explained them many times, so I won’t go through each term.
I made a video called “What is relativity all about?”
if you don’t remember.
So we can substitute in the Lorentz transforms for the primes and we get this ugly looking
thing here in terms of unprimed quantities.
But now it’s just some simple algebra to get us the rest of the way.
We can cancel the gammas, which helps.
We can also remember that equation we had for Delta x, which was just big V times Delta
t and substitute that in.
We can then pull out the Delta t’s and cancel them too.
And finally, we are just left with the answer, which we see here.
All of those primes and unprimes might be confusing, so let me remind you that big V
prime is Don’s velocity according to Ron.
The unprimed velocities are little v, which is Ron according to you, and big V, which
is Don according to you.
I wrote it this way to avoid confusion with the ones and twos in the derivation, but you
don’t usually see it written in the textbooks with big V and little v. What you usually
find is a simpler and cleaner notation.
So, let me tell you the cleaner notation and then we’ll work out the implications.
If you see two people, call them 1 and 2, moving towards or away from one another at
velocity v1 and v2, the two of them will see their relative velocity as simply v relative
equals v1 plus v2 all divided by 1 plus v1 times v2 divided by c squared.
c is, of course, the speed of light, and it is a ginormous number, specifically 186,000
miles per second or 300,000 kilometers per second.
It’s fast enough to circle the Earth about eight times in a single second.
Because c is so large, c squared is larger still.
Unless the two velocities v1 and v2 are very large, that whole v1 v2 over c squared is
basically zero.
And we see that this then reduces to the intuitive idea that the two velocities just add.
Okay, so that's pretty easy.
Let's look at how it works.
Suppose you have two people that are heading towards one another at half the speed of light
according to you.
According to the intuitive equation, they should have a closing velocity of the speed
of light.
But when we put the numbers into the correct equation, we see that the two of them see
a closing velocity of 0.8 times the speed of light.
And if they are heading towards one another, each with the speed of light according to
you, we see that they each see the other person moving towards them at not twice the speed
of light, but rather just the speed of light.
This chart shows you the whole story.
This is for two people heading towards one another according to you.
On the bottom axis is their speed as a fraction of the speed of light according to you.
On the vertical axis is the speed that they see the other guy going.
The blue line is what they would see if relativity didn’t exist.
You see that if they were moving towards one another at near the speed of light, they’d
think that the other person was moving at twice the speed of light.
And, finally, the red line is what they would see if relativity is true.
Even though both of them were moving at the speed of light as far as you’re concerned,
they’d see each other moving no faster than light- just as relativity says.
So, you might be saying “So what?
That’s just a prediction.
What’s real?"
Well, it turns out that particle physicists like myself can and do make measurements that
show very clearly which of these two predictions are true.
There are many such experimental examples, but one such example is when a highly energetic
Z boson is made.
Z bosons are very heavy and decay into, for example, a matter/antimatter pair of electrons.
As far as the Z boson is concerned, which means in the frame where the Z boson isn’t
moving, the electron moves at essentially the speed of light.
However, as far as we're concerned, the Z boson is also moving an appreciable fraction
of the speed of light.
Classically, we’d expect that we’d see that the electron would be moving faster than
light.
But it doesn’t.
It moves slower than the speed of light as far as we're concerned.
And that, as they say, is that.
Relativity wins.
Really, was there ever any doubt?
Relativity is full of mind benders, that’s for sure.
But it’s nice to see that it all hangs together, both in making self-consistent predictions
and in agreeing with data.
If you liked this video, be sure to like, subscribe and share.
Tell all your friends!
And remind them, in case they forgot of course, that physics is everything.