I’ve done a couple of videos about relativity and I’ll do some more. I guess that you
could say that I’m on a relativity kick.
In one of my videos, I showed that relativistic mass wasn’t real. It’s just a pedagogical
device that has some value in terms of building intuition, but ultimately is more trouble
than it’s worth.
In the comments of that video, people asked about the physical meaning of the gamma term
in the momentum equation. By the way, another name for gamma is the Lorentz factor.
And that’s a really good question. After all, while equations are the language of precise
science, equations should lead to an understanding and not just be accepted as is.
It turns out that there is more than one answer to the physical meaning of gamma and in what
is no doubt going to annoy some of you, I’m going to save my favorite interpretation for
the next video.
But wait! Don’t click away! There’s a reason for that!
The reason is that unless you’ve had a class in special relativity, my favorite interpretation
just kind of comes at you from out of the blue. So, I thought I’d tell you in this
video a perfectly fine way to understand the Lorentz gamma. And you even get the anticipation
of knowing that in an upcoming video you’ll see something even more interesting.
So, what I thought I’d do in this episode is show you how it is derived in an introductory
class in special relativity. Now you guys said you wanted to see some equations, so
we’re going to go for it.
Einstein started with two postulates when he did his derivation of special relativity.
The first is that the laws of physics are the same for any two people who are moving
with respect to one another at constant speed. These are called inertial observers. And the
second postulate is that the speed of light is the same for all observers.
Now that last one is the thing that causes relativity skeptics so much trouble. After
all, it flies in the face of common sense. I mean- if I sit with a ball in my hand, I
will say that the ball isn’t moving. However, if I am sitting in a car, driving 60 miles
per hour, a person on the side of the road will say that the ball is moving 60 miles
per hour. By the way, that's 100 kilometers per hour for the pro-metric crowd.
Common sense says that Einstein’s second postulate is wrong. But we can simply assume
that he is right and see the consequence. So, let’s do that.
We’ll start with a railroad car, a mirror, a flashlight, a stopwatch and two observers.
One observer is in the railroad car. She simultaneously starts the stopwatch and shines the flashlight
across the car into the mirror, which then reflects the beam of light back to her. When
it returns, she stops the stop watch. She’s got really fast reflexes.
If the width of the train car is W, then the distance the beam of light travelled is 2W,
because it goes there and back. And if you wanted to figure out how long it took for
the light beam to travel back and forth, the time would be the distance divided by the
velocity. We use the symbol c for the speed of light and we see that the transit time
T is 2 times W, divided by c.
You’ll note that the T has a subscript “stationary” to imply that the woman doesn’t see the
mirror move. And, in case you are wondering why the symbol “c” is used for the speed
of light, it’s from the Latin word celeritas, which means “quickness.”
Okay , so now let’s have the train move and ask how long an observer outside the train
thinks it takes for the light to leave the woman, bounce off the far side of the train,
and return to her. Suppose that we say that the train is moving at a velocity that we
call v.
Now the absolutely, key, crucial, central and other similar adjectives, point is that
the person outside the train also sees light moving at the speed of light, which is c.
And that changes everything.
Classically, the way you learn about all other velocities, you’d say that the velocity
of light would be faster than c to the person seeing the train moving. That’s because
the stationary observer said that light moved at c across the train. Because the train is
also moving at velocity v, we would then say by the Pythagorean theorem that light was
moving along the hypotenuse of the triangle at a speed S, which is equal the square root
of c-squared plus v-squared. But this is simply not right. It fails Einstein’s second postulate.
Remember that the outside observer also says that light is traveling at the speed of light
along the path they see, which is the hypotenuse of the triangle. So if light is moving along
the hypotenuse at c and along the motion of the train at v, then it's moving towards the
mirror at the speed of the square root of c-squared minus v-squared.
And if the velocity of light moving across the train and back is that, then the time
it takes for light to cross the train and back according to the person seeing the train
move is two times W divided by that speed, which is the square root of c-squared minus
v-squared.
Now if we remember our equation for T-stationary, we can do some simple algebra and come up
with an equation for 2W, which we can then put into the T-moving equation.
And finally we can factor out the c, and we can finally put in the gamma.
So that’s where the gamma comes from. Or at least that’s one way to do it. There
are lots of similar derivations.
What does it mean? Well if you put in any non-zero velocity less than the speed of light,
you find that gamma is a number greater than 1. That means that the person watching the
train move will think that the round trip time for the light to bounce off the mirror
is longer than the person sitting in the train.
Now that might sound just utterly ridiculous- in fact, let me share a secret with you- when
I first heard this when I was a teenager, I thought it was all so much malarkey or other,
less family friendly, words. It’s partly why I became a physicist, because I thought
that the world’s scientists had gone mad- mad I tell you! Mwahahahaha!
But it turns out that we can prove that this bizarre-oid prediction is actually true. In
fact, I made another video showing how physicists prove that this difference in how time is
perceived every single day. If you haven’t watched it, it’s worth your time.
So this brings us to the question I asked at the beginning of the video. What is gamma?
Well, we can solve for gamma and we see that it is simply the ratio of time a person experiences
when they see a clock moving to the time experienced by someone who is moving along with the clock.
Now, like I said before, this isn’t exactly my favorite explanation of what gamma is physically.
There’s another explanation that I like better. But the derivation is way harder,
so I thought I’d show you this approach as a starter. The next video will tell you
some eye-opening truths about E = m c squared and why it’s actually wrong or at least
not the whole story.
Okay, so you guys said that you were willing to see some equations, so I made a video that
had more of them. I’m curious to know what you thought. Hopefully you loved it. If you
did, please like, subscribe and share. And let me know what you think in the comments.
Well, I’ll see you next time, and remember- physics is everything.