Lorentz transformation derivation part 1

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.419
- [Voiceover] So in all of our
videos on special relativity

00:00:02.419 --> 00:00:04.754
so far, we've had this
little thought experiment,

00:00:04.754 --> 00:00:09.679
where I'm floating in space,
and right at time equals zero,

00:00:09.679 --> 00:00:11.985
a friend passes by in her spaceship.

00:00:11.985 --> 00:00:14.400
She's traveling in the
positive x direction,

00:00:14.400 --> 00:00:19.148
velocity is equal to v,
and we draw space-time

00:00:19.148 --> 00:00:20.541
diagrams for both of us.

00:00:20.541 --> 00:00:23.246
First, I draw my space-time
diagram in white,

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and then I overlay her space-time diagram.

00:00:27.065 --> 00:00:30.107
And the angle that is formed between

00:00:30.107 --> 00:00:34.368
the time axes and the position
axes, right over there,

00:00:34.368 --> 00:00:36.178
that's going to be dictated by

00:00:36.178 --> 00:00:40.357
how fast v is, how fast
she is actually traveling.

00:00:40.357 --> 00:00:43.503
And we give the space-time diagram from

00:00:43.503 --> 00:00:46.127
her frame of reference, we
see with the little primes

00:00:46.127 --> 00:00:47.625
right over there.

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Now, one thing that you might
have been thinking about

00:00:49.203 --> 00:00:50.940
throughout this entire series, is well,

00:00:50.940 --> 00:00:55.781
if I perceive her traveling
with a velocity of v

00:00:55.781 --> 00:00:58.428
in the positive x direction,
if we took her point of view,

00:00:58.428 --> 00:01:00.123
and that's what I have right over here.

00:01:00.123 --> 00:01:02.688
If she views herself as
just floating in space,

00:01:02.688 --> 00:01:06.160
what she will see me as
right at time equals zero,

00:01:06.160 --> 00:01:08.388
actually I'd say right at
t prime is equal to zero.

00:01:08.388 --> 00:01:13.310
We're saying that t prime and
t equals zero are coinciding.

00:01:14.230 --> 00:01:17.548
Right at that moment,
she will see me fly by

00:01:17.548 --> 00:01:21.820
at negative v, going in
the negative x direction.

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Once again, there is no
absolute frame of reference.

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These frames of reference
are all absolute...

00:01:28.565 --> 00:01:31.705
I'm sorry, these frames of
reference are all relative.

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And so you could imagine
what it would look like,

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'cause if we drew her
space-time diagram where her

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ct prime axis and x
prime axes that they are

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perpendicular to each other
and then based on that,

00:01:43.028 --> 00:01:47.704
my space-time diagram
would be at an angle.

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It's at an angle like this.

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You can kind of see the positive ct axis

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is in the second quadrant here because

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I'm traveling with the
velocity of negative v,

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but these angles are going to be the same.

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This is going to be alpha
and that is going to be,

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let me write this,

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is going to be alpha
and this is going to be,

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and this right over here
is going to be alpha.

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Now what I want to do in this video is

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use this symmetry, use these two ideas to

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give us a derivation of
the Lorentz Transformation

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or the Lorentz Transformations.

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And the way we might
start, and this is actually

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a reasonable way that the
Lorentz Transformations

00:02:30.667 --> 00:02:32.502
were stumbled upon, is to say,

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all right, we could start with
the Galilean Transformation,

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where we could say, all right,

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the Galilean Transformation
would be x prime is equal to,

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is going to be equal to

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x minus v times t.

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V times t.

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Now we already know that if you just use

00:02:54.386 --> 00:02:55.922
the Galilean Transformation,

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then the speed of light
would not be absolute,

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it would not be the same in
every frame of reference.

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And so we had to let go of the constraints

00:03:03.534 --> 00:03:05.612
that time and space are absolute, and

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so there's going to be some
type of scaling factor involved.

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And so we can call that
scaling factor, gamma.

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So, we could say all
right, let's just postulate

00:03:15.160 --> 00:03:19.769
that x prime, if we assume the
speed of light is absolute,

00:03:19.769 --> 00:03:24.680
is going to be some scaling
factor, gamma times x minus vt.

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Well, you could make the same
argument the other way around.

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If you view it from
her frame of reference,

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and you're trying to translate
it to my coordinates,

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you could say, well, x,
instead of just using the

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Galilean Transformation that
x is going to be equal to,

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x is going to be equal to x prime,

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and now instead of a v, we
have a negative v, right?

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So if you subtract a negative v, in fact,

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let me just write it that way.

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X minus negative v times t prime,

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that would be the Galilean Transformation,

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but whatever scaling factor we used here,

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there's a symmetry here.

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I shouldn't have to use a
different scaling factor

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if I assume a different, kind of,

00:04:07.321 --> 00:04:09.887
if I'm in a different frame of reference.

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So, if we assume the absoluteness
of the speed of light,

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we're going to have some
other scaling factor

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just like that or we
could rewrite this as x,

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let me do that same color,

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we could rewrite it as x is equal to

00:04:27.589 --> 00:04:29.455
this scaling factor.

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I'm really having trouble
changing colors today.

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It's gonna be equal to that scaling factor

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times x prime,

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subtract the negative plus vt prime.

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And if you ignore the scaling
factor right over here,

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this is the Galilean Transformation

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from the primed frame of reference

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to the non-primed frame of reference.

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So an interesting thing is
what is this scaling factor?

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How do we figure out

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what that scaling factor is going to be?

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And so we can do a little bit
of interesting algebra here.

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What we could do is,

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Actually, let me just
write what I just wrote,

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let me write it right below here.

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So we could say that x...

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Once again changing colors is difficult.

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We could write that x is
equal to our scaling factor,

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gamma times x prime,

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times x prime

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plus vt prime.

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And now what I'm going to
do in order to have myself

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an equation that involves all
of the interesting variables,

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I'm gonna multiply both
sides of this equation by,

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one way to think about
it, I'm going to multiply

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both sides of this top equation by x.

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So if I multiply the left-hand side by x,

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I'm going to have x times x prime,

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x times x prime.

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And then the right-hand
side of the equation,

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I could multiply by x,
but x is the same thing.

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I'm saying it's the same thing as

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gamma times all of this business.

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So, I'm just going to
multiply the left-hand sides

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of the equation and I'm
going to multiply the

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right-hand sides of the equation.

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So if I multiply the right-hand
sides of the equation,

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I am going to get gamma squared times,

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and I'm gonna have a big expression here,

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and so just really applying the
distributive property twice,

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x times x prime,

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x times x prime,

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and then x times positive vt, so,

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that prime doesn't look like a prime,

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x times x prime

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plus x times positive vt

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plus x times, actually
positive vt prime I should say,

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gotta be careful here.

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X times positive vt prime and then

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I have negative vt times x prime.

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So it's gonna be negative vt

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times x prime, times x prime.

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And then finally, I'll
have negative vt times

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positive vt prime.

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So that's going to be, I could
write that as a negative.

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Let's write that as vt squared,

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I'm sorry, negative v squared.

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And actually, let me
delete this parentheses.

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I don't wanna force myself
to squeeze for no reason.

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So I'm gonna have negative v times v,

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so that's negative v squared times t,

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times t prime.

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Times t prime.

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And now let me place my parentheses.

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So how can I use all
of this craziness here

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to actually solve for gamma?

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And here we're going to go back

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to one of the fundamental postulates,

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one of the assumptions
of special relativity

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and that's the speed of light is absolute.

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You're going to measure it to be

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the same in any frame of reference.

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And to think about that,
let's imagine an event

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that is connected with the
origin with a light beam.

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So let's say right at time
and t prime is equal to zero,

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I were to shoot my flashlight and

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let's say it hits something at some point.

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We look at some event right over there and

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they're connected by a
light beam, by photons.

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So let me connect them.

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So let me connect them.

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And so if you say, and once again

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this could be me turning on my flashlight

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and the photon at some future,

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at some forward distance
and some forward time,

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it could just be at some position or

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maybe it hits something, it
triggers some type of reaction.

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Who know what it does, but
we're going to talk about

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this event right over there.

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That event in my frame of reference,

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its coordinates are going to be x and ct.

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And since we know the
speed of light is absolute

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and the way that we've
set up these diagrams,

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any path of light is always
going to be at a 45 degree

00:09:09.351 --> 00:09:11.266
or a negative 45 degree angle,

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we know that x is going to be equal to ct.

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X is going to be equal to
ct for this particular case.

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I could draw it on this
diagram as well, if I'd like.

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Just to show that I can,
so let me draw that.

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So it would look like this.

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It would look like this and

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we would once again have x equaling ct.

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How would you read that?

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Well, to get the x coordinate,

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you go parallel to the ct axis.

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So that would be the x
coordinate on this diagram.

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And then the ct coordinate,
you go parallel to the x axis.

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So, just like that.

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But, once again, x is
going to be equal to ct,

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and, similarly, because the speed of light

00:09:56.235 --> 00:09:59.488
is going to be absolute
in any frame of reference,

00:09:59.488 --> 00:10:04.479
if we look at x prime, x
prime is going to be the same,

00:10:04.479 --> 00:10:09.359
is going to need to be
equal to ct prime, ct prime.

00:10:09.359 --> 00:10:13.899
If we look at over here, x prime is going

00:10:13.899 --> 00:10:16.732
to be equal to ct prime.

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Once again, because
light, this is going to be

00:10:19.552 --> 00:10:21.096
at a 45 degree angle.

00:10:21.096 --> 00:10:24.370
So x prime is equal to ct prime.

00:10:24.370 --> 00:10:26.066
They're connected by light events.

00:10:26.066 --> 00:10:28.237
So if you take your change in x divided by

00:10:28.237 --> 00:10:31.277
your change of time is going
to be the speed of light.

00:10:31.277 --> 00:10:33.657
So what we can do is use this information,

00:10:33.657 --> 00:10:35.171
for this particular event,

00:10:35.171 --> 00:10:38.329
if gamma's going to be true
for all transformations,

00:10:38.329 --> 00:10:40.946
it definitely should be true
for this particular event.

00:10:40.946 --> 00:10:45.832
I can use this information
to substitute back in

00:10:45.832 --> 00:10:47.515
and then solve for gamma.

00:10:47.515 --> 00:10:49.477
And that's exactly what I'm
gonna do in the next video,

00:10:49.477 --> 00:10:51.172
although, I encourage
you to try it on your own

00:10:51.172 --> 00:10:53.825
before you watch the next video.


Deriving Lorentz transformation part 2

WEBVTT
Kind: captions
Language: en

00:00:00.319 --> 00:00:02.014
- [Voiceover] We left
off in the last video

00:00:02.014 --> 00:00:04.092
trying to solve for gamma.

00:00:04.092 --> 00:00:06.286
We set up this equation
and then we had the insight

00:00:06.286 --> 00:00:08.922
that look, we could
particular, we could pick

00:00:08.922 --> 00:00:13.125
a particular event that is
connected by light signal,

00:00:13.125 --> 00:00:16.317
and in that case x would be equal to c t,

00:00:16.317 --> 00:00:19.807
but also x prime would
be equal to c t prime.

00:00:20.091 --> 00:00:22.749
And if gamma's gonna hold
for any transformations

00:00:22.749 --> 00:00:26.116
between events, between x and
x prime and t and t prime,

00:00:26.116 --> 00:00:28.659
it should definitely hold
for this particular event,

00:00:28.659 --> 00:00:32.188
and so maybe we can use
this to substitute back in

00:00:32.188 --> 00:00:34.127
and solve for gamma, so
that's exactly what we're

00:00:34.127 --> 00:00:35.126
gonna do right now.

00:00:35.126 --> 00:00:38.510
So for all the xes I'm gonna
substitute it, with the c ts,

00:00:39.167 --> 00:00:43.794
so I'm gonna substitute with
a c t, so x, substitute ct.

00:00:43.794 --> 00:00:46.405
x, I'm gonna substitute c t.

00:00:46.919 --> 00:00:48.924
x, substitute c t,

00:00:49.392 --> 00:00:50.785
and that's it.

00:00:50.785 --> 00:00:52.793
And then all the x primes
I'm gonna substitute

00:00:52.793 --> 00:00:56.343
with a c t prime, with a c t prime.

00:00:56.845 --> 00:00:59.542
So x prime, c t prime.

00:01:00.119 --> 00:01:03.268
x prime, c t prime.

00:01:03.579 --> 00:01:07.402
And then I have an x prime here,
so it's gonna be c t prime.

00:01:07.852 --> 00:01:10.487
So let's simplify, and
now I'm gonna switch to

00:01:10.487 --> 00:01:11.871
a neutral color.

00:01:12.217 --> 00:01:15.096
So I'm gonna now have, I'm now gonna have

00:01:15.096 --> 00:01:18.327
c times t times t times t prime.

00:01:18.893 --> 00:01:22.669
So that's gonna be c
squared, actually let me just

00:01:23.060 --> 00:01:27.693
keep using the t, t primes, t, t primes.

00:01:27.693 --> 00:01:29.527
OK, I'll still do a little
bit of color coding.

00:01:29.527 --> 00:01:34.148
T prime is equal to gamma squared,

00:01:34.148 --> 00:01:39.144
gamma squared times, so
it's gonna be c squared

00:01:39.686 --> 00:01:43.836
c times c is c, let me do
that in the yellow color,

00:01:45.027 --> 00:01:48.434
so c squared, t times t prime,

00:01:50.251 --> 00:01:52.602
so t times

00:01:54.291 --> 00:01:55.710
t prime.

00:01:56.126 --> 00:01:57.904
and then we have plus

00:01:59.423 --> 00:02:04.101
c times v times t times t prime.

00:02:05.054 --> 00:02:09.105
plus c times v,

00:02:09.593 --> 00:02:11.021
I'll just write it this way.

00:02:11.021 --> 00:02:14.004
C times t,

00:02:15.828 --> 00:02:17.104
times v,

00:02:18.823 --> 00:02:20.072
times t prime.

00:02:21.261 --> 00:02:24.774
And then we have minus, minus, let's see,

00:02:24.774 --> 00:02:26.075
we're gonna have a c here.

00:02:26.451 --> 00:02:28.838
So minus c,

00:02:29.923 --> 00:02:33.134
minus c times t

00:02:34.717 --> 00:02:35.363
t

00:02:36.378 --> 00:02:39.473
times v, times t prime

00:02:40.441 --> 00:02:43.211
times v times t prime.

00:02:43.553 --> 00:02:45.860
I wrote this v in blue just
so it matches up with this.

00:02:45.860 --> 00:02:48.150
And we see something
interesting is about to happen.

00:02:48.150 --> 00:02:50.801
And then finally we have minus v squared.

00:02:51.587 --> 00:02:53.134
Minus v squared,

00:02:53.723 --> 00:02:55.340
times t times t prime.

00:02:55.859 --> 00:02:58.243
Times t times t prime.

00:02:59.052 --> 00:03:01.328
It doesn't look that much
simpler, but we're about

00:03:01.328 --> 00:03:03.305
to simplify it a good bit.

00:03:04.660 --> 00:03:07.086
And so we're gonna get
these two middle terms

00:03:07.086 --> 00:03:08.274
to cancel out.

00:03:08.653 --> 00:03:13.301
So plus c t v t prime,
minus c t v t prime.

00:03:13.924 --> 00:03:16.525
So those are going to cancel out.

00:03:16.525 --> 00:03:19.497
And then every other term has a

00:03:19.497 --> 00:03:20.658
t, t prime in it.

00:03:20.658 --> 00:03:23.808
So let's divide both sides of
this equation by t t prime.

00:03:24.513 --> 00:03:26.353
And so we're gonna get, if
we divide the left-hand side

00:03:26.353 --> 00:03:29.076
by t t prime, we're just
gonna be left with c squared,

00:03:29.453 --> 00:03:32.524
and then we're just gonna
divide everything by t t prime,

00:03:32.524 --> 00:03:35.728
and there our whole thing
has simplified quite nicely.

00:03:35.728 --> 00:03:38.770
Our equation is now, I'll
continue it over here.

00:03:39.617 --> 00:03:43.437
Our equation now is, c squared is

00:03:43.925 --> 00:03:48.925
equal to gamma squared,
is equal to gamma squared,

00:03:49.590 --> 00:03:51.808
times c squared,

00:03:52.690 --> 00:03:57.242
times c squared minus v squared.

00:03:58.588 --> 00:03:59.773
Minus v squared.

00:04:00.759 --> 00:04:02.037
Minus v squared.

00:04:03.522 --> 00:04:06.123
Close the parentheses.
And now we can divide

00:04:06.123 --> 00:04:08.514
both sides by c squared minus v squared,

00:04:08.514 --> 00:04:11.034
and we would get gamma squared,

00:04:11.487 --> 00:04:13.484
and I'm gonna swap the sides, too.

00:04:13.484 --> 00:04:15.574
So gamma squared is equal to

00:04:16.084 --> 00:04:20.102
c squared, c squared over,

00:04:20.624 --> 00:04:22.922
c squared, I'll write
it all in one color now.

00:04:22.922 --> 00:04:25.140
c squared minus v squared.

00:04:25.657 --> 00:04:27.775
Now if we like we can divide the numerator

00:04:27.775 --> 00:04:29.552
and the denominator by
c squared, in which case

00:04:29.552 --> 00:04:32.037
this will be equal to one over

00:04:32.419 --> 00:04:34.428
c squared divided by c squared is one,

00:04:34.428 --> 00:04:37.807
and then, v squared divided by c squared,

00:04:38.317 --> 00:04:39.757
and we are in the home stretch now,

00:04:39.757 --> 00:04:42.416
we can just take the
square root of both sides,

00:04:42.416 --> 00:04:45.016
and we get, we deserve a
little bit of a drum roll.

00:04:45.016 --> 00:04:46.491
Actually let me continue it up here where

00:04:46.491 --> 00:04:48.174
I have some real estate.

00:04:48.174 --> 00:04:51.576
We get gamma is equal to
the square root of this,

00:04:51.576 --> 00:04:53.921
well the square root of one is just one,

00:04:53.921 --> 00:04:56.127
over the square root of the denominator.

00:04:56.127 --> 00:05:00.737
square root of one minus v squared

00:05:00.737 --> 00:05:02.385
over c squared.

00:05:02.385 --> 00:05:03.987
So hopefully you found that as satisfying

00:05:03.987 --> 00:05:07.853
as I did, because all we
did, we just thought about

00:05:07.853 --> 00:05:11.417
the symmetry of x prime is
gonna be some scaling factor

00:05:11.417 --> 00:05:13.716
times the traditional
Galilean transformation,

00:05:13.716 --> 00:05:16.224
and x is going to be
some scaling factor times

00:05:16.224 --> 00:05:18.650
the traditional Galilean
transformation from

00:05:18.650 --> 00:05:20.090
the prime coordinates.

00:05:20.090 --> 00:05:22.760
We use that and it's important that we use

00:05:22.760 --> 00:05:24.328
one of the fundamental assumptions of

00:05:24.328 --> 00:05:26.325
special relativity,
that the speed of light

00:05:26.325 --> 00:05:28.821
is absolute in either frame of reference,

00:05:28.821 --> 00:05:32.849
that x divided by t is c, that
x prime divided by t prime

00:05:32.849 --> 00:05:35.554
is going to be equal to c,

00:05:35.554 --> 00:05:38.016
for some event that's associated with a,

00:05:38.016 --> 00:05:39.235
a light beam.

00:05:39.595 --> 00:05:41.417
We use that to substitute back in,

00:05:41.417 --> 00:05:43.972
and we were able to solve for gamma.

00:05:44.355 --> 00:05:45.934
So this looks pretty neat.

00:05:46.259 --> 00:05:47.609
And so some of y'all might be saying,

00:05:47.609 --> 00:05:49.404
well what about, what about,

00:05:49.404 --> 00:05:52.052
so we've been able to do the derivation

00:05:52.052 --> 00:05:54.119
for the x coordinates,

00:05:54.119 --> 00:05:56.174
but what about the, the
Lorentz transformation

00:05:56.174 --> 00:05:59.552
for the t and t prime coordinates?

00:05:59.552 --> 00:06:01.886
And I'll let you think
about how we do that.

00:06:01.886 --> 00:06:03.813
And I'll give you a clue,
it's just going to be

00:06:03.813 --> 00:06:05.241
a little bit more algebra,
and we're going to do that

00:06:05.241 --> 00:06:06.446
in the next video.


Algebraically manipulating Lorentz transformation

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:02.099
- What I hope to do in
this video is get even more

00:00:02.099 --> 00:00:05.681
algebraically familiar with
the Lorentz transformation,

00:00:05.681 --> 00:00:08.028
so that we can recognize
it in its different forms

00:00:08.028 --> 00:00:11.660
and start to build our
intuition for how it behaves.

00:00:11.660 --> 00:00:13.516
So let's just write down
the Lorentz transformation,

00:00:13.516 --> 00:00:14.995
or at least the way
that I like to write it.

00:00:14.995 --> 00:00:16.635
So let's just remind ourselves,

00:00:16.635 --> 00:00:18.356
if I'm in my frame of reference,

00:00:18.356 --> 00:00:21.054
I am floating through
space, so I could say s

00:00:21.054 --> 00:00:23.122
for Sal's frame of reference.

00:00:23.122 --> 00:00:25.910
Some event in space time,
from my point of view,

00:00:25.910 --> 00:00:30.269
is going to have some x-coordinate,

00:00:30.269 --> 00:00:34.042
I will do that in green,
and it's going to have

00:00:34.042 --> 00:00:38.452
some ct coordinate, I
can do that in orange.

00:00:42.632 --> 00:00:44.764
So the Lorentz transformations
are going to go from

00:00:44.764 --> 00:00:46.489
coordinates in my frame of reference,

00:00:46.489 --> 00:00:48.663
space-time coordinates for an event,

00:00:48.663 --> 00:00:51.608
to my friend's frame of reference,

00:00:51.608 --> 00:00:55.868
so we can say that's the
s-prime frame of reference,

00:00:55.868 --> 00:00:58.048
and her frame of reference, the event,

00:00:58.048 --> 00:01:01.615
will have space-time coordinates x-prime,

00:01:01.615 --> 00:01:06.355
let me write it this way: x-prime,

00:01:07.985 --> 00:01:10.615
comma, ct-prime,

00:01:10.615 --> 00:01:12.704
I'm really having trouble
switching colors today,

00:01:12.704 --> 00:01:16.175
ct-prime, so let's just
write it down the way

00:01:16.175 --> 00:01:17.814
I've written it down
in the previous videos,

00:01:17.814 --> 00:01:20.099
and then I'll do a little bit
of algebraic manipulations,

00:01:20.099 --> 00:01:22.564
so we can recognize its different forms.

00:01:22.564 --> 00:01:24.000
So if we want to get x-prime,

00:01:27.230 --> 00:01:29.851
we see that it's going to be based on

00:01:29.851 --> 00:01:34.441
the Lorentz factor times x

00:01:36.024 --> 00:01:40.234
minus a scaled version of ct,

00:01:40.234 --> 00:01:42.291
and the scaling factor is beta,

00:01:42.291 --> 00:01:44.966
and we will redefine beta in a second,

00:01:44.966 --> 00:01:48.157
so beta times ct,

00:01:49.257 --> 00:01:51.897
where our Lorentz factor,

00:01:51.897 --> 00:01:53.126
let me write it over here,

00:01:53.126 --> 00:01:56.913
the Lorentz factor is
one over the square root

00:01:56.913 --> 00:02:00.617
of one minus v-squared over c-squared,

00:02:00.617 --> 00:02:03.688
or we could write it as
one over the square root

00:02:03.688 --> 00:02:05.906
of one minus beta-squared,

00:02:08.805 --> 00:02:11.335
where beta is equal to

00:02:13.635 --> 00:02:16.157
v over c. So there you go.

00:02:16.157 --> 00:02:18.820
That's how we get x-prime
and it's gonna be based on

00:02:18.820 --> 00:02:21.862
the Lorentz factors dependent on v

00:02:21.862 --> 00:02:23.534
and of course the rest
of this is going to be

00:02:23.534 --> 00:02:26.120
dependent on x and ct.

00:02:26.120 --> 00:02:28.383
And so how do we get ct-prime?

00:02:28.383 --> 00:02:31.929
Well ct-prime is going to be equal to,

00:02:31.929 --> 00:02:33.773
I'll just write it right over here,

00:02:33.773 --> 00:02:37.526
ct-prime is going to be equal to

00:02:37.526 --> 00:02:41.018
the Lorentz factor times,

00:02:41.018 --> 00:02:42.242
and once again this is going to be

00:02:42.242 --> 00:02:44.441
the nice symmetry we talked about,

00:02:44.441 --> 00:02:48.603
times ct, we'll do that in orange,

00:02:49.306 --> 00:02:53.363
so ct minus

00:02:53.363 --> 00:02:57.475
beta times x.

00:02:57.475 --> 00:02:59.986
And like I said before, I
like to write it this way.

00:02:59.986 --> 00:03:01.156
I find it easier to remember.

00:03:01.156 --> 00:03:02.731
I find it easier to
remember because it has this

00:03:02.731 --> 00:03:05.057
nice, beautiful symmetry to it.

00:03:05.057 --> 00:03:06.927
When I'm trying to solve for x-prime,

00:03:06.927 --> 00:03:09.383
it's x minus beta times ct.

00:03:09.383 --> 00:03:13.783
When I'm solving for ct-prime,
it's ct minus beta times x.

00:03:13.783 --> 00:03:16.637
And in both cases, I'm
scaling by the Lorentz factor.

00:03:16.637 --> 00:03:18.472
Well let's manipulate this a little bit,

00:03:18.472 --> 00:03:19.771
just to understand a little better,

00:03:19.771 --> 00:03:21.793
and reconcile with what you might see

00:03:21.793 --> 00:03:24.586
with other sources,
including, say, your textbook.

00:03:24.586 --> 00:03:27.482
Well we know that beta
is equal to v over c,

00:03:30.382 --> 00:03:32.920
and this is v over c,

00:03:32.920 --> 00:03:36.618
and so we can write this
as v over c times c,

00:03:36.618 --> 00:03:38.691
this c's going to cancel with that c.

00:03:38.691 --> 00:03:40.635
And so we could rewrite this as

00:03:40.635 --> 00:03:45.245
x-prime is equal to the Lorentz factor,

00:03:45.245 --> 00:03:49.553
gamma times x minus vt.

00:03:54.340 --> 00:03:58.396
V times t.

00:03:58.396 --> 00:04:00.558
Now this is really
interesting right over here,

00:04:00.558 --> 00:04:04.853
'cause if you'd ignored gamma,
or if gamma was one here,

00:04:04.853 --> 00:04:08.222
this is essentially the
Galilean transformation.

00:04:08.222 --> 00:04:10.806
If it was just x minus vt, that was just

00:04:10.806 --> 00:04:12.824
our intuition about our everyday life,

00:04:12.824 --> 00:04:15.443
but Newtonian physics
would actually tell us,

00:04:15.443 --> 00:04:17.280
and so when you view it in this form,

00:04:17.280 --> 00:04:19.214
you really think, okay, you're
just going to scale that

00:04:19.214 --> 00:04:20.883
by this Lorentz transformation,

00:04:20.883 --> 00:04:22.594
which has this interesting behavior

00:04:22.594 --> 00:04:26.275
that if v is much, much smaller
than the speed of light,

00:04:26.275 --> 00:04:28.325
well then, this whole factor

00:04:28.325 --> 00:04:29.889
is going to be pretty close to one,

00:04:29.889 --> 00:04:32.238
and that's why the Galilean
transformation's worked for us,

00:04:32.238 --> 00:04:34.992
for kind of everyday velocities.

00:04:34.992 --> 00:04:39.558
But then if v starts to
approach the speed of light,

00:04:39.558 --> 00:04:42.698
this thing blows up and we
get a very different result

00:04:42.698 --> 00:04:46.283
than with our traditional
Galilean transformation.

00:04:46.283 --> 00:04:48.572
Well let's think about
what happens over here,

00:04:48.572 --> 00:04:51.874
and over here, instead
of staying in ct-prime,

00:04:51.874 --> 00:04:54.244
I'm gonna also divide both sides by c,

00:04:54.244 --> 00:04:56.113
so we're just solving for time

00:04:56.113 --> 00:04:57.909
as we normally associate it.

00:04:57.909 --> 00:05:01.970
Just the t-prime variables
as opposed to ct-prime.

00:05:01.970 --> 00:05:04.527
So let's also divide both sides by c,

00:05:04.527 --> 00:05:07.706
so you divide by c there, and
you can divide by c there,

00:05:07.706 --> 00:05:10.207
and you can divide by c there,

00:05:10.207 --> 00:05:12.776
those c's cancel, those c's cancel,

00:05:12.776 --> 00:05:16.119
and so we're left with t-prime is equal to

00:05:16.119 --> 00:05:18.081
the Lorentz factor,

00:05:20.176 --> 00:05:25.176
times t times t minus,

00:05:25.185 --> 00:05:27.090
now you're going to get v times x,

00:05:27.090 --> 00:05:30.160
and you're going to divide by
c twice, so over c-squared.

00:05:30.160 --> 00:05:32.588
So vx,

00:05:32.588 --> 00:05:37.588
v times x over c-squared.

00:05:41.213 --> 00:05:43.481
And this is actually a more
typical way, both of these,

00:05:43.481 --> 00:05:45.956
of seeing the Lorentz transformation.

00:05:45.956 --> 00:05:48.337
The reason why I don't
like this form as much,

00:05:48.337 --> 00:05:50.573
even though this does
have the neat, kind of,

00:05:50.573 --> 00:05:52.603
when you look at it, it looks
like you're just scaling up

00:05:52.603 --> 00:05:54.093
the Galilean transformation,

00:05:54.093 --> 00:05:56.272
is it you no longer
see the symmetry there,

00:05:56.272 --> 00:05:57.584
and you should see the symmetry there,

00:05:57.584 --> 00:05:58.826
because we're talking about space-time.

00:05:58.826 --> 00:06:02.257
We're not talking about this
independence of space and time.

00:06:02.257 --> 00:06:05.112
We saw how the angles in
the Minkowski diagram,

00:06:05.112 --> 00:06:09.078
how those were symmetrical,
the angles between the regular,

00:06:09.078 --> 00:06:12.331
or the unprimed axes, and the primed axes.

00:06:12.331 --> 00:06:13.603
And so what I don't like about this

00:06:13.603 --> 00:06:15.376
is you no longer see the symmetry,

00:06:15.376 --> 00:06:17.713
while you did see it the
first way that I wrote it.

00:06:17.713 --> 00:06:20.183
And frankly I find this
harder to remember.

00:06:20.183 --> 00:06:21.940
But let's just think
about what happens here

00:06:21.940 --> 00:06:25.390
when v is a very small
fraction of the speed of light.

00:06:25.390 --> 00:06:28.290
Well as we've already
said, our Lorentz factor

00:06:28.290 --> 00:06:30.254
is going to be pretty close to one,

00:06:30.254 --> 00:06:33.352
and if v is a very small fraction of c,

00:06:33.352 --> 00:06:36.945
well then this second
term right over here,

00:06:36.945 --> 00:06:39.098
is going to be pretty close to zero.

00:06:39.098 --> 00:06:43.473
And so if v is a small fraction of c,

00:06:43.473 --> 00:06:45.982
then this thing is going to be get pretty,

00:06:45.982 --> 00:06:46.899
just let me just write this down,

00:06:46.899 --> 00:06:51.899
so if v is much less than c,

00:06:52.669 --> 00:06:54.702
then this is going to reduce to

00:06:54.702 --> 00:06:57.972
t-prime being approximately equal to,

00:06:57.972 --> 00:06:59.241
because our Lorentz factor

00:06:59.241 --> 00:07:00.575
is going to be pretty close to one,

00:07:00.575 --> 00:07:02.586
this is going to be pretty close to zero.

00:07:02.586 --> 00:07:04.678
So this is going to be pretty close to t.

00:07:04.678 --> 00:07:08.877
Likewise, for v

00:07:08.877 --> 00:07:11.547
much lower than c over here,

00:07:11.547 --> 00:07:14.401
our Lorentz factor is going
to be pretty close to one,

00:07:14.401 --> 00:07:16.309
and so x-prime is going to be

00:07:16.309 --> 00:07:21.309
approximately equal to x minus vt.

00:07:21.342 --> 00:07:23.022
So for small v's,

00:07:23.022 --> 00:07:25.223
and small can even be
the speed of a bullet,

00:07:25.223 --> 00:07:26.809
or even the speed of the space shuttle,

00:07:26.809 --> 00:07:29.175
or things that are much, much smaller

00:07:29.175 --> 00:07:30.271
than the speed of light.

00:07:30.271 --> 00:07:34.243
Three times 10 to the
eighth meters per second,

00:07:34.243 --> 00:07:36.225
that's why the Galilean transformations

00:07:36.225 --> 00:07:38.813
are pretty good approximations.

00:07:38.813 --> 00:07:39.864
So hopefully this starts to give you

00:07:39.864 --> 00:07:41.089
a little bit of intuition.

00:07:41.089 --> 00:07:43.884
Start evaluating this,
evaluate this for v's

00:07:43.884 --> 00:07:46.062
in our everyday life,
and then see what happens

00:07:46.062 --> 00:07:49.883
when v starts to approach
.8 times the speed of light.

00:07:49.883 --> 00:07:54.883
.9 c, .99 c, think about what
happens to the Lorentz factor.

00:07:55.095 --> 00:07:57.445
And hopefully you'll
get an appreciation for

00:07:57.445 --> 00:07:59.520
how this whole thing behaves.


Evaluating a Lorentz transformation

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.184
- [Voiceover] Let's now
dig a little bit deeper

00:00:04.184 --> 00:00:06.180
into the Lorentz Transformation.

00:00:06.180 --> 00:00:07.759
In particular, let's
put some numbers here,

00:00:07.759 --> 00:00:09.570
so that we're, we get a little bit more

00:00:09.570 --> 00:00:10.503
familiar manipulating and

00:00:10.503 --> 00:00:12.883
then we'll start to get a
little bit more intuition

00:00:12.883 --> 00:00:15.263
on how this transformation
or sometimes it's

00:00:15.263 --> 00:00:19.355
spoken of in the plural,
the transformations behave.

00:00:19.720 --> 00:00:22.965
So let's pick the scenario in
which our friend passes us by,

00:00:22.965 --> 00:00:24.729
and this is the same scenario
that we've been doing

00:00:24.729 --> 00:00:26.993
in previous videos, with
a relative velocity,

00:00:26.993 --> 00:00:30.698
from my frame of reference,
at half the speed of light.

00:00:30.698 --> 00:00:33.623
So the magnitude of her velocity
is half the speed of light.

00:00:33.623 --> 00:00:37.698
She is moving in the
positive x direction and

00:00:37.698 --> 00:00:42.040
our space-time diagrams,
they coincide at the origin.

00:00:42.040 --> 00:00:44.942
And so let's pick an event in space-time.

00:00:44.942 --> 00:00:47.252
And so let's say in my coordinate system,

00:00:47.252 --> 00:00:50.422
in my frame of reference,
this event that we focused on

00:00:50.422 --> 00:00:53.252
in the last video, let's say that is at

00:00:53.252 --> 00:00:56.398
x is equal to one meter...

00:00:56.398 --> 00:00:58.616
Let's use the same color...

00:00:58.616 --> 00:01:01.920
X is equal to one meter and let's say

00:01:01.920 --> 00:01:05.855
that time or ct is also
equal to one meter.

00:01:05.855 --> 00:01:08.573
And like we said in, I think
it was several videos ago,

00:01:08.573 --> 00:01:10.211
we could do this as a light meter,

00:01:10.211 --> 00:01:12.637
the time it takes for
light to go one meter.

00:01:12.637 --> 00:01:15.213
So we will also say this is one meter.

00:01:15.213 --> 00:01:17.606
So in my coordinate system,
in my frame of reference,

00:01:17.606 --> 00:01:21.053
this would be the point one comma one.

00:01:21.053 --> 00:01:26.053
One meter in the x direction,
one meter in the ct direction.

00:01:26.091 --> 00:01:29.051
Now, based on that, think about what

00:01:29.051 --> 00:01:31.467
would be the prime coordinates.

00:01:31.467 --> 00:01:33.869
What would be the coordinates
in her frame of reference?

00:01:33.869 --> 00:01:36.012
And I encourage you to pause the video,

00:01:36.012 --> 00:01:39.731
evaluate the Lorentz Factor using v and c,

00:01:39.731 --> 00:01:44.027
and then evaluate what x
prime and ct prime would be.

00:01:45.207 --> 00:01:46.756
All right, I'm assuming
you've had a go at it.

00:01:46.756 --> 00:01:48.230
Now let's work through this together.

00:01:48.230 --> 00:01:49.912
So first let's figure out
what the Lorentz Factor...

00:01:49.912 --> 00:01:52.153
Actually, let's first figure
out what beta would be.

00:01:52.153 --> 00:01:53.871
That will simplify everything.

00:01:53.871 --> 00:01:57.772
So, beta, we'll do it in the blue color,

00:01:57.772 --> 00:02:02.569
beta in this case is going to
be equal to zero point five c.

00:02:04.299 --> 00:02:07.724
That's her relative velocity
in my frame of reference.

00:02:07.724 --> 00:02:09.615
The ratio between that
and the speed of light,

00:02:09.615 --> 00:02:13.864
so that's just going to be
equal to zero point five.

00:02:13.864 --> 00:02:15.763
You could just view beta as

00:02:15.763 --> 00:02:16.953
what fraction of the speed of light

00:02:16.953 --> 00:02:21.770
is that person traveling
in my frame of reference

00:02:23.420 --> 00:02:25.112
since we're using that as kind of the

00:02:25.112 --> 00:02:27.271
non-primed frame of reference.

00:02:27.568 --> 00:02:30.634
And so let's now think about
what gamma is going to be,

00:02:30.634 --> 00:02:31.795
the Lorentz Factor.

00:02:31.795 --> 00:02:34.849
The Lorentz Factor is going to be...

00:02:34.849 --> 00:02:37.441
We'll do it in that reddish
color not the magenta...

00:02:37.441 --> 00:02:40.704
it is going to be, so gamma is going to be

00:02:40.704 --> 00:02:45.638
one over the square root
of one minus beta squared.

00:02:46.508 --> 00:02:50.626
Beta squared is, zero point five squared

00:02:50.626 --> 00:02:54.828
is going to be zero point two five.

00:02:54.828 --> 00:02:56.876
Actually let me just write
zero point five squared

00:02:56.876 --> 00:02:58.640
just so you can see what I'm doing.

00:02:58.640 --> 00:03:02.333
So this is zero point five squared

00:03:02.333 --> 00:03:03.819
and if we were to evaluate that...

00:03:03.819 --> 00:03:08.079
Let's see this is going to be
one minus zero point two five.

00:03:08.079 --> 00:03:09.972
So that's going to be point seven five.

00:03:09.972 --> 00:03:14.917
This is going to be equal
to one over the square root

00:03:16.307 --> 00:03:19.242
of point seven five,
one over the square root

00:03:19.242 --> 00:03:22.028
of zero point seven five.

00:03:22.028 --> 00:03:23.247
Let me get my calculator out.

00:03:23.247 --> 00:03:24.628
We can at least approximate it.

00:03:24.628 --> 00:03:28.819
So point seven five, let's
take the square root.

00:03:28.819 --> 00:03:32.278
And I'll just take the reciprocal
of that, so approximately

00:03:32.278 --> 00:03:35.529
one point one five.

00:03:36.659 --> 00:03:41.510
So our Lorentz Factor is
approximately one point one five.

00:03:43.788 --> 00:03:45.415
And now using that we can figure what

00:03:45.415 --> 00:03:47.927
x prime and ct prime are going to be.

00:03:50.418 --> 00:03:54.602
X prime is going to be
equal to my Lorentz Factor

00:03:54.602 --> 00:03:57.307
which is approximately one point one five.

00:03:57.307 --> 00:03:59.942
So maybe I'll write it as approximately

00:03:59.942 --> 00:04:02.763
going to be equal to one,
we'll do that same color,

00:04:02.763 --> 00:04:06.908
it's going to be one point,
I'm having trouble switching

00:04:06.908 --> 00:04:10.849
colors today, one point one five.

00:04:11.319 --> 00:04:15.463
One point one five times, now we're saying

00:04:15.463 --> 00:04:20.463
x is one meter, so x is
one meter minus beta,

00:04:21.477 --> 00:04:26.477
beta is zero point five,
so zero point five.

00:04:26.641 --> 00:04:30.193
And ct we're also saying is one meter,

00:04:30.193 --> 00:04:32.567
so times one.

00:04:34.001 --> 00:04:37.913
And then t prime or I should say ct prime,

00:04:37.913 --> 00:04:42.603
ct prime is going to be
approximately the Lorentz Factor

00:04:42.603 --> 00:04:45.981
one point one, I always have
trouble switching colors

00:04:45.981 --> 00:04:48.535
for the Lorentz Factor,
it's going to be approximate

00:04:48.535 --> 00:04:50.802
to one point one five times...

00:04:51.902 --> 00:04:53.772
Well, ct is one.

00:04:53.772 --> 00:04:55.327
I think you see a little
bit of symmetry here.

00:04:55.327 --> 00:04:56.874
This one in particular because

00:04:56.874 --> 00:05:00.004
it had the same x and ct coordinates.

00:05:00.004 --> 00:05:03.725
So one minus beta, one minus beta,

00:05:05.832 --> 00:05:10.749
so zero point five times
x, which is once again one.

00:05:11.508 --> 00:05:13.172
So times one.

00:05:13.172 --> 00:05:16.434
So in this particular
case, it simplifies to half

00:05:16.434 --> 00:05:20.393
of the Lorentz Factor because
this one minus zero point five

00:05:20.393 --> 00:05:23.469
times one that's just going
to be zero point five.

00:05:23.469 --> 00:05:25.849
And same thing over here, zero point five.

00:05:25.849 --> 00:05:30.626
So these things are going
to be approximately equal to

00:05:30.626 --> 00:05:33.410
zero point five times the Lorentz Factor.

00:05:33.410 --> 00:05:34.827
We already had the Lorentz
Factor in my calculator,

00:05:34.827 --> 00:05:39.215
so let me just multiply
by zero point five.

00:05:39.215 --> 00:05:43.347
And I get, it's approximately
point five, I'll just say

00:05:43.347 --> 00:05:45.083
point five eight.

00:05:46.623 --> 00:05:51.623
So, zero point five eight,
zero point five eight.

00:05:52.129 --> 00:05:54.787
And once again the units are in meters.

00:05:54.787 --> 00:05:57.318
So even though this is one meter and

00:05:57.318 --> 00:06:02.156
one meter, this over here x
prime is zero point five eight,

00:06:03.916 --> 00:06:05.882
zero point five eight meters and

00:06:05.882 --> 00:06:10.807
ct prime is also zero
point five eight meters.

00:06:11.686 --> 00:06:15.860
So this is also equal to
zero point five eight meters.

00:06:16.353 --> 00:06:19.650
So one way to think about
it, if right when our two,

00:06:19.650 --> 00:06:22.506
right as she was passing
me at x equals zero,

00:06:22.506 --> 00:06:25.037
time equals zero in my frame of reference,

00:06:25.037 --> 00:06:28.025
If I were to shoot my lazer gun or

00:06:28.025 --> 00:06:30.497
I were to turn my flashlight on and

00:06:30.497 --> 00:06:33.938
that very first photon starts traveling

00:06:33.938 --> 00:06:36.852
and so I could think about
it's path through space-time,

00:06:36.852 --> 00:06:40.013
it would look, that very first photon

00:06:40.013 --> 00:06:43.328
would look something like that.

00:06:45.759 --> 00:06:49.544
When I think that a, when
I think that that photon

00:06:49.544 --> 00:06:53.793
has traveled one meter in
the positive x direction

00:06:53.793 --> 00:06:57.079
and one light meter of time has passed,

00:06:57.079 --> 00:06:59.585
from my friend's frame of reference,

00:06:59.585 --> 00:07:01.305
she would say, "No, no, no, no."

00:07:01.305 --> 00:07:04.067
At exactly that moment,
let's say it hits an asteroid

00:07:04.067 --> 00:07:06.005
at that moment, it lights up an asteroid.

00:07:06.005 --> 00:07:07.867
She would say, "No, no, no, no, no."

00:07:07.867 --> 00:07:10.673
That happened zero point
five eight light meters

00:07:10.673 --> 00:07:12.635
after she passed me up.

00:07:12.635 --> 00:07:15.770
And it happened zero
point five eight meters

00:07:15.770 --> 00:07:17.953
in the positive x direction.

00:07:17.953 --> 00:07:21.075
So something very, very,
very interesting is going on.

00:07:21.075 --> 00:07:23.954
And I encourage you to think
about what's actually going on

00:07:23.954 --> 00:07:27.576
with these different parts of
the Lorentz Transformations.

00:07:27.576 --> 00:07:29.654
The most interesting is
what's going on, well,

00:07:29.654 --> 00:07:30.687
actually it's all interesting.

00:07:30.687 --> 00:07:32.099
In fact, the symmetry's interesting.

00:07:32.099 --> 00:07:34.816
But the Lorentz Factor, think
about what's happening here.

00:07:34.816 --> 00:07:37.497
Think about what's happening
here for low velocities when

00:07:37.497 --> 00:07:40.195
v is a very, very, very small fraction

00:07:40.195 --> 00:07:41.530
of the speed of light.

00:07:41.530 --> 00:07:44.513
Well then beta is going to
be pretty close to zero,

00:07:44.513 --> 00:07:45.667
and then the Lorentz Factor

00:07:45.667 --> 00:07:48.767
is going to be pretty close to one.

00:07:48.767 --> 00:07:49.747
And think about what happens when

00:07:49.747 --> 00:07:52.347
v approaches the speed of light.

00:07:52.347 --> 00:07:54.216
Well then this thing just booms.

00:07:54.216 --> 00:07:57.374
This thing gets larger and
larger and larger as we see

00:07:57.374 --> 00:08:01.007
this denominator getting
smaller and smaller and smaller.

00:08:01.007 --> 00:08:03.817
If v were actually equal
to the speed of light,

00:08:03.817 --> 00:08:06.882
well then you're going
to be dividing by zeros.

00:08:06.882 --> 00:08:09.068
Well, you know, that's when

00:08:09.068 --> 00:08:10.961
all sorts of silliness starts to happen.

00:08:10.961 --> 00:08:13.562
So I really encourage you to
try out different numbers.

00:08:13.562 --> 00:08:16.635
We tried very high relative velocity,

00:08:16.635 --> 00:08:17.657
half the speed of light,

00:08:17.657 --> 00:08:19.972
incredibly, incredibly high velocity.

00:08:19.972 --> 00:08:21.876
Try it out for something more mundane

00:08:21.876 --> 00:08:24.592
like the speed of a bullet
or something like that.

00:08:24.592 --> 00:08:26.415
But definitely get very
familiar with this.

00:08:26.415 --> 00:08:28.098
And also manipulate it algebraically.

00:08:28.098 --> 00:08:29.654
In fact, maybe in the next
video I'll manipulate this

00:08:29.654 --> 00:08:31.917
a little bit algebraically
so that you can reconcile

00:08:31.917 --> 00:08:33.811
the way I've written the
Lorentz Transformation or

00:08:33.811 --> 00:08:36.131
the Lorentz Transformations
with the way that

00:08:36.131 --> 00:08:39.365
you might see it in your
textbook or other resources.


Introduction to the Lorentz transformation

WEBVTT
Kind: captions
Language: en

00:00:00.324 --> 00:00:01.740
- [Voiceover] So we've
already been able to explore

00:00:01.740 --> 00:00:03.934
a lot with our little thought experiment

00:00:03.934 --> 00:00:06.559
where I am floating in space.

00:00:06.559 --> 00:00:09.240
I'm at the center of
my frame of reference,

00:00:09.240 --> 00:00:10.923
and right at time equals zero

00:00:10.923 --> 00:00:12.455
in my frame of reference,

00:00:12.455 --> 00:00:16.994
a friend comes in a
spaceship passing me by

00:00:16.994 --> 00:00:18.817
with a velocity v,

00:00:18.817 --> 00:00:20.454
I'll say the magnitude is v,

00:00:20.454 --> 00:00:22.694
and it's going in the
positive x direction.

00:00:22.694 --> 00:00:23.346
We're just going to focus,

00:00:23.346 --> 00:00:24.646
and we have been focusing,

00:00:24.646 --> 00:00:27.444
just on the x dimension for simplicity.

00:00:27.444 --> 00:00:31.228
And we've thought about reconciling space

00:00:31.228 --> 00:00:33.027
and time in my frame of reference

00:00:33.027 --> 00:00:35.128
relative to her frame of reference.

00:00:35.128 --> 00:00:36.220
The conundrum that we've faced

00:00:36.220 --> 00:00:37.926
in previous videos is how do we reconcile

00:00:37.926 --> 00:00:40.109
that the speed of light is always

00:00:40.109 --> 00:00:43.580
going to be the same in
every frame of reference?

00:00:43.580 --> 00:00:46.586
And to reconcile them, we
had to essentially come

00:00:46.586 --> 00:00:49.329
up with the idea of spacetime.

00:00:49.329 --> 00:00:51.546
I should say it even faster, spacetime.

00:00:51.546 --> 00:00:54.866
Spacetime, let me write it out, spacetime.

00:00:54.866 --> 00:00:57.652
And the first time I
heard about spacetime,

00:00:57.652 --> 00:00:58.976
I assumed that people were just talking

00:00:58.976 --> 00:01:01.936
about space and time as independent things

00:01:01.936 --> 00:01:05.429
and just plotting your
point in space and time.

00:01:05.429 --> 00:01:07.613
But when people talk about spacetime,

00:01:07.613 --> 00:01:09.772
they're really talking about
this continuum of one thing,

00:01:09.772 --> 00:01:11.571
and we're just talking
about different directions

00:01:11.571 --> 00:01:12.860
in spacetime.

00:01:12.860 --> 00:01:13.766
They could have called
this something else.

00:01:13.766 --> 00:01:15.647
They could have called this spime,

00:01:15.647 --> 00:01:19.733
or tace, or stace,

00:01:19.733 --> 00:01:22.009
or a lot of different things.

00:01:22.009 --> 00:01:23.146
But this is spacetime,

00:01:23.146 --> 00:01:25.411
and it's this idea that
it's this continuum.

00:01:25.411 --> 00:01:28.998
And when we started to
make spacetime diagrams,

00:01:28.998 --> 00:01:31.842
we realized in order
for the speed of light

00:01:31.842 --> 00:01:34.698
to be absolute, that time and space

00:01:34.698 --> 00:01:36.090
weren't as independent of each other

00:01:36.090 --> 00:01:38.110
as we thought, and they
weren't as absolute

00:01:38.110 --> 00:01:39.213
as we thought.

00:01:39.213 --> 00:01:41.291
And we constructed these Minkowsky

00:01:41.291 --> 00:01:43.291
spacetime diagrams for each

00:01:43.291 --> 00:01:44.788
of our frames of reference.

00:01:44.788 --> 00:01:45.868
So my frame of reference,

00:01:45.868 --> 00:01:47.075
the spacetime diagram,

00:01:47.075 --> 00:01:48.155
is here in white,

00:01:48.155 --> 00:01:50.674
and for my friend's frame of reference,

00:01:50.674 --> 00:01:52.729
her spacetime diagram is here

00:01:52.729 --> 00:01:53.948
in this blue color.

00:01:53.948 --> 00:01:57.732
The angle formed between these axes,

00:01:57.732 --> 00:01:59.474
between the x and the x prime axis

00:01:59.474 --> 00:02:02.956
and the ct and the ct prime axis,

00:02:02.956 --> 00:02:04.570
this angle alpha here,

00:02:04.570 --> 00:02:07.032
this is going to be dependent on her

00:02:07.032 --> 00:02:10.446
relative velocity in
my frame of reference.

00:02:10.446 --> 00:02:13.825
So if her velocity is v,

00:02:13.825 --> 00:02:15.183
or the magnitude of her velocity is v

00:02:15.183 --> 00:02:16.483
in my frame of reference,

00:02:16.483 --> 00:02:17.760
this angle we've already seen

00:02:17.760 --> 00:02:19.200
is going to be the inverse tangent

00:02:19.200 --> 00:02:21.706
or the arctangent of the ratio

00:02:21.706 --> 00:02:24.109
between her relative velocity

00:02:24.109 --> 00:02:25.549
and the speed of light.

00:02:25.549 --> 00:02:29.130
So this is going to be
equal to the inverse tangent

00:02:29.356 --> 00:02:31.677
of v over c.

00:02:31.677 --> 00:02:33.430
So the faster she goes,

00:02:33.430 --> 00:02:38.430
these two things are going
to start squeezing together,

00:02:38.515 --> 00:02:40.248
and if somehow she were
to approach the speed

00:02:40.248 --> 00:02:43.906
of light, they would both
approach a 45 degree angle

00:02:43.906 --> 00:02:46.263
and actually start to coincide

00:02:46.263 --> 00:02:48.190
if they were actually able to approach

00:02:48.190 --> 00:02:49.676
the speed of light.

00:02:49.676 --> 00:02:51.474
And that's all interesting already,

00:02:51.474 --> 00:02:54.005
this idea that space and time are not

00:02:54.005 --> 00:02:56.125
as independent, that it's all a continuum

00:02:56.125 --> 00:02:58.360
called spacetime, but some
of you have probably said,

00:02:58.360 --> 00:03:00.275
well, I want to deal with some more

00:03:00.275 --> 00:03:01.993
tangible numbers here.

00:03:01.993 --> 00:03:06.824
For example, if this event right over here

00:03:06.824 --> 00:03:09.273
in spacetime, we can think about it

00:03:09.273 --> 00:03:10.562
from my frame of reference,

00:03:10.562 --> 00:03:11.363
and we can think about it

00:03:11.363 --> 00:03:12.883
from her frame of reference.

00:03:12.883 --> 00:03:14.403
In my frame of reference,

00:03:14.403 --> 00:03:16.892
I would view the coordinates here.

00:03:18.211 --> 00:03:20.009
This coordinate would be x,

00:03:20.009 --> 00:03:22.226
and this coordinate right over here

00:03:22.226 --> 00:03:23.642
would be ct.

00:03:23.642 --> 00:03:25.222
We had a whole video on why we think

00:03:25.222 --> 00:03:26.906
of time in terms of ct.

00:03:26.906 --> 00:03:28.971
The units here literally would be meters.

00:03:28.971 --> 00:03:31.850
We could think of it as
light-meters if we like.

00:03:31.850 --> 00:03:32.825
So that would be the coordinates

00:03:32.825 --> 00:03:34.532
in my frame of reference.

00:03:34.532 --> 00:03:35.484
Well, what would be the coordinates

00:03:35.484 --> 00:03:37.167
in her frame of reference?

00:03:37.167 --> 00:03:39.593
Well, we've already
thought about how to read

00:03:39.593 --> 00:03:42.112
these Minkowski spacetime diagrams.

00:03:42.112 --> 00:03:43.843
To find her x prime coordinate,

00:03:43.843 --> 00:03:48.243
we would just go parallel
to the ct prime axis.

00:03:48.243 --> 00:03:50.169
So that would be the x prime coordinate

00:03:50.169 --> 00:03:51.492
in her frame of reference.

00:03:51.492 --> 00:03:53.245
And to figure out the ct prime coordinate,

00:03:53.245 --> 00:03:56.504
we would just go parallel
to the x prime axis.

00:03:56.504 --> 00:04:01.504
So this would be the ct prime coordinate.

00:04:02.732 --> 00:04:05.473
Now how do you actually go in between,

00:04:05.473 --> 00:04:08.096
transform, from x to x prime

00:04:08.096 --> 00:04:11.195
and from ct to ct prime?

00:04:11.195 --> 00:04:13.331
And to do that, we're going to introduce

00:04:13.331 --> 00:04:17.505
in this video the Lorentz transformations.

00:04:23.954 --> 00:04:25.812
And what they do is they allow us to do

00:04:25.812 --> 00:04:27.936
exactly what we just needed to do.

00:04:27.936 --> 00:04:32.614
They'll allow us to go from x, ct

00:04:32.614 --> 00:04:36.581
to x prime

00:04:37.704 --> 00:04:40.484
and ct prime.

00:04:40.484 --> 00:04:41.756
And to help us think about it,

00:04:41.756 --> 00:04:43.739
I'm going to introduce some variables,

00:04:43.739 --> 00:04:45.509
and hopefully it will show the symmetry

00:04:45.509 --> 00:04:46.894
of the Lorentz transformations.

00:04:46.894 --> 00:04:48.635
You might see them written in other ways

00:04:48.635 --> 00:04:51.027
in other sources, and we'll reconcile all

00:04:51.027 --> 00:04:52.768
of those in the future.

00:04:52.768 --> 00:04:54.126
But the Lorentz transformations,

00:04:54.126 --> 00:04:56.332
we'll start with what we
call the Lorentz factor

00:04:56.332 --> 00:04:57.354
because this shows up a lot

00:04:57.354 --> 00:04:58.247
in the transformation.

00:04:58.247 --> 00:05:00.267
So I'll just define this ahead of time.

00:05:00.267 --> 00:05:04.726
So the Lorentz factor,
denoted by the Greek letter

00:05:04.726 --> 00:05:06.468
gamma, lowercase gamma,

00:05:06.468 --> 00:05:10.669
it is equal to one over the square root

00:05:10.669 --> 00:05:15.669
of one minus v squared over c squared.

00:05:15.951 --> 00:05:19.387
Now sometimes you might
even see it written like,

00:05:19.387 --> 00:05:22.046
well, I'll write it another way.

00:05:22.046 --> 00:05:25.656
Sometimes you might see
it written as gamma,

00:05:25.656 --> 00:05:27.303
let me do it in that same color,

00:05:27.303 --> 00:05:29.022
same reddish color,

00:05:29.022 --> 00:05:32.528
gamma is equal to one over the square root

00:05:32.528 --> 00:05:34.850
of one minus beta squared.

00:05:34.850 --> 00:05:36.462
You might say, well, what is beta?

00:05:36.462 --> 00:05:38.970
Well, beta is another
variable that shows up

00:05:38.970 --> 00:05:41.721
a lot when we're thinking
about special relativity.

00:05:41.721 --> 00:05:46.160
And beta is just the ratio

00:05:47.261 --> 00:05:50.085
between the relative velocity,

00:05:50.085 --> 00:05:52.267
her relative velocity in
my frame of reference,

00:05:52.267 --> 00:05:53.741
and the speed of light.

00:05:53.741 --> 00:05:54.542
It shows up a lot,

00:05:54.542 --> 00:05:55.900
even this angle alpha here,

00:05:55.900 --> 00:05:58.997
we could have said this
is the inverse tangent

00:05:59.685 --> 00:06:01.864
of beta.

00:06:03.111 --> 00:06:04.539
This becomes a little bit
simpler when you write

00:06:04.539 --> 00:06:06.397
the Lorentz factor.

00:06:06.397 --> 00:06:08.383
And when we look at the
actual transformation

00:06:08.383 --> 00:06:10.181
between the coordinates,
we'll see that beta

00:06:10.181 --> 00:06:11.517
becomes useful again, at least the way

00:06:11.517 --> 00:06:12.898
I like to write it.

00:06:12.898 --> 00:06:15.939
So if we want to think about what x prime

00:06:15.939 --> 00:06:19.260
is going to be, so we can write

00:06:19.260 --> 00:06:22.453
x prime is going to be equal

00:06:22.453 --> 00:06:25.953
to the Lorentz factor,

00:06:27.380 --> 00:06:29.187
let me do it in that red color,

00:06:29.187 --> 00:06:32.989
is going to be equal to the Lorentz factor

00:06:32.989 --> 00:06:36.145
times x

00:06:37.937 --> 00:06:40.852
minus, and now we're going to say

00:06:40.852 --> 00:06:45.618
beta times ct,

00:06:47.991 --> 00:06:50.760
and ct prime

00:06:52.855 --> 00:06:57.313
is going to be equal to
the Lorentz factor gamma,

00:06:57.313 --> 00:06:59.606
let me do that same color again,

00:06:59.787 --> 00:07:01.772
switching colors is sometimes difficult,

00:07:01.772 --> 00:07:04.105
gamma times, and we'll see it's just

00:07:04.105 --> 00:07:05.730
the other way around.

00:07:05.730 --> 00:07:10.730
It is going to be c,

00:07:10.791 --> 00:07:12.323
let me do it in that green color,

00:07:12.323 --> 00:07:14.234
ct

00:07:16.190 --> 00:07:18.187
minus, you might even guess what I'm

00:07:18.187 --> 00:07:21.391
about to write based on the symmetry

00:07:21.391 --> 00:07:22.552
that we see here,

00:07:22.552 --> 00:07:27.552
ct minus beta times x.

00:07:28.147 --> 00:07:29.866
And I really want you to appreciate this

00:07:29.866 --> 00:07:34.788
because it really shows
that space and time

00:07:34.788 --> 00:07:37.145
are really just different directions

00:07:37.145 --> 00:07:39.361
in the spacetime and
there's a nice symmetry

00:07:39.361 --> 00:07:40.244
to them right over here.

00:07:40.244 --> 00:07:45.244
Notice we have an x and we have an x.

00:07:45.364 --> 00:07:49.890
We have a ct and we have a ct.

00:07:49.890 --> 00:07:52.132
So when we're thinking about x prime,

00:07:52.132 --> 00:07:56.009
it's x minus beta times ct,

00:07:56.009 --> 00:07:58.714
and we're scaling it
by the Lorentz factor,

00:07:58.714 --> 00:08:00.745
and then when we're thinking about time,

00:08:00.745 --> 00:08:01.711
well we do it the other way around.

00:08:01.711 --> 00:08:03.174
We're still scaled by the Lorentz factor,

00:08:03.174 --> 00:08:07.225
but now it is ct minus beta times x.

00:08:07.225 --> 00:08:09.292
Now this might all seem like Greek to you,

00:08:09.292 --> 00:08:11.091
and we actually are using Greek letters,

00:08:11.091 --> 00:08:11.731
but in the next video,

00:08:11.731 --> 00:08:14.285
I'll actually use some
sample numbers here,

00:08:14.285 --> 00:08:16.604
and you'll see that evaluating these

00:08:16.604 --> 00:08:19.711
is just a little bit of
straightforward algebra.


Angle of x' axis in Minkowski spacetime

WEBVTT
Kind: captions
Language: en

00:00:00.474 --> 00:00:02.505
- [Voiceover] We have been
doing some interesting things

00:00:02.535 --> 00:00:04.691
in the last few videos.

00:00:04.964 --> 00:00:07.446
We let go of our Newtonian assumptions

00:00:07.476 --> 00:00:09.426
that the passage of time is the same

00:00:09.743 --> 00:00:11.876
in all initial frame of reference

00:00:11.906 --> 00:00:13.772
that time is absolute that one second

00:00:13.802 --> 00:00:15.223
in my frame of reference is the same

00:00:15.253 --> 00:00:17.285
as one second passing in
your frame of reference.

00:00:17.572 --> 00:00:21.051
We even let go of the idea
that space is absolute

00:00:21.081 --> 00:00:22.638
that one meter in a certain direction

00:00:23.006 --> 00:00:25.447
in my frame of reference
is going to be the same

00:00:25.477 --> 00:00:28.293
as one meter in your frame of reference

00:00:28.323 --> 00:00:30.419
if we're are moving at relative velocities

00:00:30.449 --> 00:00:31.987
with respect to each other.

00:00:32.256 --> 00:00:34.659
And what that allows us to
do, it allowed us to reconcile

00:00:34.689 --> 00:00:36.876
what is actually observed in the universe.

00:00:37.007 --> 00:00:41.428
And that's the idea that
speed of light is an absolute.

00:00:41.458 --> 00:00:44.932
That regardless of what inertial
frame of reference we're in

00:00:45.075 --> 00:00:47.599
regardless of the speed
of the source of the light

00:00:47.831 --> 00:00:51.443
that we will always measure
light travelling at three times

00:00:51.473 --> 00:00:54.683
or roughly three times 10 to
the eighth meters per second.

00:00:54.944 --> 00:00:56.426
And when we let go of our assumptions

00:00:56.456 --> 00:00:58.680
about absolute time and absolute space

00:00:59.030 --> 00:01:02.764
and we did assume absolute
velocity for light.

00:01:02.902 --> 00:01:05.256
It gave us a very interesting diagram

00:01:05.677 --> 00:01:09.920
because it took our x
prime axis from being

00:01:10.059 --> 00:01:13.606
on top of the x axis and
put it an angle relative

00:01:13.636 --> 00:01:15.564
to the x-axis.

00:01:15.725 --> 00:01:17.263
Now, one question that you might have

00:01:17.293 --> 00:01:19.167
is well what is this angle,
the way I've drawn it,

00:01:19.197 --> 00:01:21.333
it looks like the angle
between x prime and x

00:01:21.564 --> 00:01:24.857
is the same as the angle
between t prime and t.

00:01:24.887 --> 00:01:27.874
Or as we did in the last videos,
we turned our units in time

00:01:27.904 --> 00:01:31.574
where the units along
this axis, the ct-axis,

00:01:31.604 --> 00:01:33.044
instead of saying seconds,

00:01:33.261 --> 00:01:34.806
we're now going to measure it in meters,

00:01:34.867 --> 00:01:36.499
and once again watch the last video

00:01:36.529 --> 00:01:39.388
if you're having an issue with that.

00:01:40.055 --> 00:01:43.607
But it looks like this angle
and this angle is the same.

00:01:43.870 --> 00:01:46.311
What I want to do in this video
is feel good about the fact

00:01:46.341 --> 00:01:48.783
that they really are the same.

00:01:49.626 --> 00:01:52.197
So, one of the convenient
things when a couple videos ago,

00:01:52.227 --> 00:01:53.574
we went through this thought experiment

00:01:53.604 --> 00:01:55.215
of in my friend's frame of reference,

00:01:55.369 --> 00:01:57.464
her admitting a photon of light

00:01:57.856 --> 00:01:59.179
bouncing off of the spaceship

00:01:59.209 --> 00:02:00.619
that's three times 10 to the eighth meters

00:02:00.649 --> 00:02:02.554
in front of her, and
then coming back to her.

00:02:02.762 --> 00:02:06.590
As we assumed that as
we mark off the meters,

00:02:06.620 --> 00:02:08.511
and as we mark off the seconds

00:02:08.655 --> 00:02:10.872
that the distance from the origin

00:02:10.901 --> 00:02:13.091
for three times 10 to the
eighth meters is the same

00:02:13.349 --> 00:02:16.195
as the distance from the
origin is one second.

00:02:16.225 --> 00:02:18.296
And now it makes even more
sense 'cause now we call this

00:02:18.326 --> 00:02:19.473
three times 10 to the eighth meters

00:02:19.503 --> 00:02:21.846
and this times three times 10
to the eighth meters as well

00:02:22.134 --> 00:02:22.981
and so the speed...

00:02:23.011 --> 00:02:26.359
When we depict the path
of light of a photon

00:02:26.496 --> 00:02:28.726
in either of these space time diagrams,

00:02:28.894 --> 00:02:31.096
it's always going to be at a either

00:02:31.126 --> 00:02:35.289
a positive 45 degree angle
or a negative 45 degree angle

00:02:35.319 --> 00:02:37.682
depending on the direction
that light is traveling in.

00:02:37.712 --> 00:02:40.180
In fact, that was what
we used to establish

00:02:40.308 --> 00:02:43.166
that this x prime axis is going to be,

00:02:43.319 --> 00:02:45.197
it's not going to sit
on top of the x-axis,

00:02:45.619 --> 00:02:46.950
it's going to be at an angle.

00:02:47.096 --> 00:02:49.091
But I don't actually
think about that angle.

00:02:49.524 --> 00:02:54.524
So, if we say that this angle
right over here is alpha

00:02:55.767 --> 00:02:59.041
and if we were to continue
this 45-degree line

00:02:59.071 --> 00:03:01.138
that we had to remember,
any path of a photon

00:03:01.168 --> 00:03:04.337
is gonna be at 45 degree
angle, so let me continue that.

00:03:04.998 --> 00:03:07.812
So, this is going to look like that

00:03:07.961 --> 00:03:09.396
and then we could go back

00:03:09.426 --> 00:03:11.733
and continue it right like that.

00:03:11.892 --> 00:03:15.517
Well, we know that this
going to be 45 degrees.

00:03:15.743 --> 00:03:18.834
45 degrees right over there
this is going to be...

00:03:19.159 --> 00:03:23.078
Whoops, this is going to be
45 degrees right over there

00:03:23.334 --> 00:03:27.768
and we know, we that this
green triangle that I'm...

00:03:28.071 --> 00:03:29.749
It's not green, it's not different enough,

00:03:29.779 --> 00:03:31.157
let me get a better color.

00:03:31.425 --> 00:03:33.896
This??, no that's still green.

00:03:34.336 --> 00:03:38.213
We know that this, this purplish triangle

00:03:38.495 --> 00:03:39.989
that I'm setting up right here.

00:03:40.123 --> 00:03:42.633
This is an isosceles triangle.

00:03:42.779 --> 00:03:45.083
How do I know that this
is an isosceles triangle?

00:03:45.408 --> 00:03:49.260
Well, this hash mark right
over here, on the ct prime axis

00:03:49.609 --> 00:03:51.789
that is three times 10
to the eighth meters

00:03:52.063 --> 00:03:54.685
and this hash mark on
the x prime axis is also

00:03:54.715 --> 00:03:56.447
three times 10 to the eighth meters.

00:03:56.725 --> 00:03:59.674
And, so, this side is equal to that side

00:03:59.892 --> 00:04:02.797
and so we know that the base
angles of an isosceles triangle

00:04:02.827 --> 00:04:03.693
are going to congruent.

00:04:03.723 --> 00:04:07.257
So, this angle is going to
congruent to that angle.

00:04:07.584 --> 00:04:09.120
Well, if those two angles are congruent,

00:04:09.548 --> 00:04:13.438
then this angle is going
to congruent to that angle.

00:04:13.468 --> 00:04:17.103
Because they are supplementary
to those two base angles.

00:04:17.394 --> 00:04:19.425
And so notice, if we look at...

00:04:19.716 --> 00:04:21.139
If we look at this triangle...

00:04:21.601 --> 00:04:23.052
Let me do this in a new color...

00:04:23.660 --> 00:04:27.570
If we look at this
triangle right over here

00:04:27.725 --> 00:04:30.809
and compare it to this
triangle right over here

00:04:31.106 --> 00:04:33.471
notice they both have a 45 degree angle,

00:04:33.885 --> 00:04:36.628
they both have this blue
angle which is congruent

00:04:36.897 --> 00:04:38.684
so the third angle has to be congruent.

00:04:38.714 --> 00:04:40.374
They always have to add up to 180 degrees

00:04:40.404 --> 00:04:42.490
if you have two angles of
two different triangles

00:04:42.520 --> 00:04:44.589
of the same, then the third
angle has to be the same

00:04:44.746 --> 00:04:47.783
and so, if this is alpha, I shall do...

00:04:48.229 --> 00:04:49.411
Let's see, I've already used...

00:04:49.544 --> 00:04:52.929
I'll do four arcs there,
if that angle is alpha

00:04:53.199 --> 00:04:57.440
then this angle is going
to be alpha as well.

00:04:57.835 --> 00:04:58.821
And, so, this is really interesting,

00:04:58.851 --> 00:05:00.581
it's a beautiful kind of symmetry,

00:05:01.033 --> 00:05:02.540
and it really comes out of the fact

00:05:02.570 --> 00:05:05.244
that the speed of light
should always be measured

00:05:05.274 --> 00:05:07.832
at three times 10 to the
eighth meters per second

00:05:07.986 --> 00:05:10.008
in any inertial frame of reference.

00:05:10.546 --> 00:05:12.791
Notice, if we go back
to my frame of reference

00:05:13.200 --> 00:05:16.321
that photon that I emit
from my, my flashlight

00:05:16.927 --> 00:05:18.568
it will look like this.

00:05:18.844 --> 00:05:20.218
It will look like this.

00:05:20.248 --> 00:05:23.485
It's path in this Minkowski space time

00:05:23.617 --> 00:05:25.194
from my point of view will look like that

00:05:25.728 --> 00:05:28.286
and notice, from my friend's point of view

00:05:28.316 --> 00:05:32.396
the one that's travelling
on a ship for every amount

00:05:32.426 --> 00:05:34.989
let's say, let's pick a
certain point in space time

00:05:35.019 --> 00:05:36.063
right over there.

00:05:36.481 --> 00:05:40.776
Well, her x time coordinate is
going to be right over there

00:05:41.039 --> 00:05:44.882
and her ct prime coordinate is
going to be right over there

00:05:45.257 --> 00:05:46.470
so that makes sense.

00:05:46.500 --> 00:05:49.847
In one, in one I guess
we could say light meter

00:05:49.877 --> 00:05:51.505
that thing travels a meter.

00:05:51.535 --> 00:05:53.897
So, it is still travelling
at the speed of light

00:05:53.927 --> 00:05:55.595
from her point of view.

00:05:55.745 --> 00:05:58.942
Even though, she is moving
at half the speed of light

00:05:59.253 --> 00:06:00.487
relative to me.

00:06:00.628 --> 00:06:02.180
And you can think about what would happen

00:06:02.210 --> 00:06:03.651
if she was moving even faster.

00:06:04.047 --> 00:06:05.317
Someone moving even faster.

00:06:05.753 --> 00:06:08.351
Their ct prime axis
would be yet at an even,

00:06:08.810 --> 00:06:10.434
at an even more severe angle

00:06:10.671 --> 00:06:14.609
so it might look something like that.

00:06:14.751 --> 00:06:16.373
And then, what would their
x prime angle look like?

00:06:16.403 --> 00:06:18.471
Well then, their x prime
angle is going to symmetric

00:06:18.862 --> 00:06:22.757
around that line that
shows the path of light,

00:06:22.787 --> 00:06:23.889
so it would look like this.

00:06:23.919 --> 00:06:26.471
So, this is someone,
let's say who's moving

00:06:26.501 --> 00:06:29.435
even faster relative to me,
let's call that ct prime prime

00:06:29.687 --> 00:06:32.052
and this would be their x prime prime.

00:06:32.323 --> 00:06:33.500
As you can imagine, you get close

00:06:33.530 --> 00:06:36.072
as that second frame of reference

00:06:36.102 --> 00:06:38.011
gets closer and closer to speed of light

00:06:38.407 --> 00:06:41.419
from my frame of reference,
their coordinate axis

00:06:41.449 --> 00:06:43.435
are just going to get
more and more squanched up

00:06:43.716 --> 00:06:48.236
around, around this 45 degree
angle, so that's really neat.

00:06:48.692 --> 00:06:50.002
Now, another thing I wanted to think about

00:06:50.032 --> 00:06:51.374
and now, I won't draw it in this video

00:06:51.404 --> 00:06:54.975
just 'cause I think I've
given you enough to digest is

00:06:55.652 --> 00:06:56.798
there's a symmetry here.

00:06:57.215 --> 00:06:59.477
If my friend is moving at
half the speed of light

00:06:59.753 --> 00:07:01.715
moving at half the speed of light

00:07:01.745 --> 00:07:03.688
in the positive extraction
according to me,

00:07:03.988 --> 00:07:05.768
then from her point of view,

00:07:05.798 --> 00:07:07.542
I'm moving at half the speed of light

00:07:07.783 --> 00:07:11.099
in the negative x prime direction,
so you should think about

00:07:11.129 --> 00:07:13.009
well what would if I were to,

00:07:13.039 --> 00:07:14.547
if we were to look at this right over here

00:07:14.577 --> 00:07:17.401
what would my frame of reference look like

00:07:17.431 --> 00:07:19.772
if I were to project it on top of that.


Measuring time in meters in Minkowski spacetime

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.106
- [Voiceover] I am about to do something

00:00:02.106 --> 00:00:04.927
that most of you will
probably find disconcerting,

00:00:04.927 --> 00:00:07.307
in fact, the first time
that I saw someone do this,

00:00:07.307 --> 00:00:10.081
I too found it disconcerting.

00:00:10.081 --> 00:00:12.101
So, just as a little bit of background,

00:00:12.101 --> 00:00:13.599
so far, when we've been thinking about

00:00:13.599 --> 00:00:15.688
our space axis, and we've really just been

00:00:15.688 --> 00:00:17.488
focusing on one dimension of space,

00:00:17.488 --> 00:00:19.484
we've been focusing on the x dimension,

00:00:19.484 --> 00:00:21.400
or the x prime dimension, depending on

00:00:21.400 --> 00:00:24.929
whose frame of reference
we are talking about.

00:00:24.929 --> 00:00:27.483
We measured that in terms of meters.

00:00:27.483 --> 00:00:29.166
We measured that in terms of meters,

00:00:29.166 --> 00:00:30.699
and then when we thought about time,

00:00:30.699 --> 00:00:32.939
well, we said, initially,
we said, "Well, time is

00:00:32.939 --> 00:00:34.959
"fundamentally something
different than space,"

00:00:34.959 --> 00:00:37.304
so we had a different set
of units called seconds,

00:00:37.304 --> 00:00:39.440
in fact, you know, this
goes well before we,

00:00:39.440 --> 00:00:41.681
well before the 20th century when we got

00:00:41.681 --> 00:00:43.979
special relativity, this
goes all the way back to well

00:00:43.979 --> 00:00:46.905
before even that, where we said
time is something different.

00:00:46.905 --> 00:00:49.508
We measure it in units
of time, in seconds,

00:00:49.508 --> 00:00:51.669
or minutes, or hours, or days.

00:00:51.669 --> 00:00:53.219
While distance, which
is different than time,

00:00:53.219 --> 00:00:55.466
we measured in meters, or feet, or miles,

00:00:55.466 --> 00:00:58.686
or kilometers, or whatever else.

00:00:58.686 --> 00:01:00.975
But as we started to
get into this world of

00:01:00.975 --> 00:01:04.572
special relativity and
we started to see that

00:01:04.572 --> 00:01:07.662
space and time are not each absolute.

00:01:07.662 --> 00:01:10.576
And, in fact, we can
actually think of all events

00:01:10.576 --> 00:01:14.314
as happening in this
continuum called spacetime.

00:01:14.314 --> 00:01:16.043
And I say it fast, "spacetime".

00:01:16.043 --> 00:01:18.683
Because it's not saying "space-time".

00:01:18.683 --> 00:01:20.118
Two different things.

00:01:20.118 --> 00:01:22.277
It's saying that they're really
just different directions

00:01:22.277 --> 00:01:24.483
in this continuum spacetime.

00:01:24.483 --> 00:01:26.759
And so, if they are all
the same thing why do we

00:01:26.759 --> 00:01:30.473
use different units for space and time?

00:01:30.473 --> 00:01:33.863
Why do we use seconds for
time and meters for space?

00:01:33.863 --> 00:01:37.323
Or at least in the examples
that I've been doing so far.

00:01:37.323 --> 00:01:39.459
And so to fix that,
instead of calling this

00:01:39.459 --> 00:01:43.139
the "t-prime axis" or the "t
axis," instead of labeling

00:01:43.139 --> 00:01:45.519
it in terms of seconds, what we can do

00:01:45.519 --> 00:01:47.074
and this is the part that many of you

00:01:47.074 --> 00:01:49.698
will find disconcerting, let's call this

00:01:49.698 --> 00:01:52.171
the "c times t-prime axis."

00:01:52.171 --> 00:01:55.096
And let's call this the "c times t axis."

00:01:55.096 --> 00:01:58.114
And the "c times t-prime axis."

00:01:58.114 --> 00:01:59.600
Well, what's that going to do?

00:01:59.600 --> 00:02:01.841
Well, we know that the speed
of light is an absolute.

00:02:01.841 --> 00:02:04.012
It is, if we are measuring it in,

00:02:04.012 --> 00:02:05.746
and I'll do approximately
because it's actually

00:02:05.746 --> 00:02:07.355
two point nine something.

00:02:07.355 --> 00:02:10.320
But, approximately three
times 10 to the 8th

00:02:10.320 --> 00:02:11.260
meters per second.

00:02:11.260 --> 00:02:13.158
For the sake of all of these
videos I'm just assuming

00:02:13.158 --> 00:02:14.820
it's three times ten to
the 8th meters per second.

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For simplicity it's roughly that.

00:02:17.374 --> 00:02:19.672
So if we were to take c times time

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instead of this being one
second, in terms of seconds,

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well, we multiply it times

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three times ten to the
8th meters per second.

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Well, the seconds cancel
out and we're left...

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If we want to measure
time in terms of meters,

00:02:31.897 --> 00:02:35.077
would be three times
ten to the 8th meters.

00:02:35.077 --> 00:02:38.600
So this is three times
ten to the 8th meters,

00:02:38.600 --> 00:02:40.069
instead of calling it one second.

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This over here is negative.

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So that over there is negative

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three times ten to the 8th meters.

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Along the "c t-prime axis."

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Likewise, this what we called one second,

00:02:53.350 --> 00:02:55.916
right over here, instead
we can call this as

00:02:55.916 --> 00:02:59.375
three times ten to the 8th meters.

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This we could call negative

00:03:01.764 --> 00:03:04.774
three times ten to the 8th meters.

00:03:04.774 --> 00:03:07.165
Now this will be counterintuitive to you

00:03:07.165 --> 00:03:10.311
because you've always
viewed time as fundamentally

00:03:10.311 --> 00:03:11.669
something different than meters.

00:03:11.669 --> 00:03:14.165
You haven't been thinking
in terms of spacetime.

00:03:14.165 --> 00:03:16.974
In fact, in our normal human
experience we don't experience

00:03:16.974 --> 00:03:19.203
the world in terms of space time.

00:03:19.203 --> 00:03:21.223
Time is something that we
are just falling forward into

00:03:21.223 --> 00:03:24.810
and space is something that
we feel like we have more

00:03:24.810 --> 00:03:27.109
agency and we can move in
the different dimensions

00:03:27.109 --> 00:03:29.744
of space more easily, while
time just feels like we're

00:03:29.744 --> 00:03:32.786
plummeting forward in that dimension.

00:03:32.786 --> 00:03:34.725
But now we're thinking
in terms of spacetime

00:03:34.725 --> 00:03:37.046
and this makes our units the same.

00:03:37.046 --> 00:03:39.264
Now if it helps you,
you could view this as

00:03:39.264 --> 00:03:42.224
three times ten to the 8th light meters.

00:03:42.224 --> 00:03:44.848
So you could think of this
as the time it would take

00:03:44.848 --> 00:03:48.342
for light to go three times
ten to the 8th meters.

00:03:48.342 --> 00:03:50.687
Likewise, if we wanted
everything in terms of what

00:03:50.687 --> 00:03:53.508
we traditionally conceive
of as our time dimensions,

00:03:53.508 --> 00:03:55.923
we could have kept this as one second,

00:03:55.923 --> 00:03:58.212
instead of calling this three
times ten to the 8th meters

00:03:58.212 --> 00:04:00.449
we could have called
this one light second,

00:04:00.449 --> 00:04:02.678
which would be the
distance that light travels

00:04:02.678 --> 00:04:07.058
in what we measure, or what
we consider to be one second.

00:04:07.058 --> 00:04:09.842
But the benefit of this
is now we're consistent.

00:04:09.842 --> 00:04:12.640
We're measuring different
directions in spacetime,

00:04:12.640 --> 00:04:14.521
which is kind of a continuum,

00:04:14.521 --> 00:04:16.888
there is no separate time and space.

00:04:16.888 --> 00:04:19.547
We're measuring them
all in the same units,

00:04:19.547 --> 00:04:22.600
which we will find is
very, very convenient.

00:04:22.600 --> 00:04:24.481
So, I know this is going to
take a little bit of time

00:04:24.481 --> 00:04:26.698
to get your head around,
and I'll maybe do a few

00:04:26.698 --> 00:04:28.822
more videos to make you
feel comfortable with

00:04:28.822 --> 00:04:31.888
this type of idea, but this will be a

00:04:31.888 --> 00:04:32.908
convenient thing for us.

00:04:35.158 --> 00:04:39.921
Especially as we start
to have a metric in our,

00:04:40.551 --> 00:04:44.320
what we would call, our
Minkowski spacetime.

00:04:44.320 --> 00:04:46.215
Because we are going to be
operating in the different

00:04:46.215 --> 00:04:49.417
dimensions as if they have the same units.

00:04:49.417 --> 00:04:52.226
So, hopefully, I encouraged
you to kind of sit and ponder

00:04:52.226 --> 00:04:53.944
and think about this a little bit,

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and hopefully it doesn't
bother you too much.