text: 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.
  