[SOUND] So we're going to adopt Einstein's solution as verified by the
Michelsonu-Morley experiment, and many others since, that claims the speed of
light as measured by all observers as constant.
Since we derived the velocity addition formula by simple logic, we're going to
have to change simple logic and intuition.
There are many approaches to doing this. I find that the most useful one is
actually a little bit algebra heavy but on the other hand leads to clarity.
We're going to derive the Lortentz transformations that are the replacement
of Galilean transformations. And so the next clip is going to use a
bit of algebra. If you're uncomfortable or if you feel
that this drowns things out. You can, feel free to skip it, the next
clip we'll present the result, and then, discuss it's consequences.
But here, we're going to use only the assumption of the principle of
relativity, as applied to electromagnetism therefore to the speed
of light, to derive the corrections together on relativity, in the way that
Einstein did. So what are the Lorentz transformations?
They are derived by the following idea. We have the statement that the speed of
light in constant as measured by observers no matter what their relative
velocity to each other. So, we do the following think, thought
experiment. We take our two observers, remember we
had our two observers, the green axis and the blue axis.
So, we have one observer x and t whose measurements will be represented in
black, and we have another observer which was the blue observer who was moving
relative to the first observer at some speed.
So, this is the t prime axis. And, what we are going to do.
Is that, at the same time as their, at the same instant as their meeting at t
equals t prime equals x equals x prime equals zero.
They both are at the same place, at the same time.
At that same instant one of them, and it doesn't matter who, releases a pulse of
light, moving say to the right at speed c and so, if you want you can also emit a
pulse, to the left. the red line is therefore given by
X, and as far as the, green observer measures it, the red line is described by
the straight line X=CT because, the light is moving relative to the observer with
speed C. But Einstein tells us it will also be
described by X primeCT equals CT prime, and this is going to be inconsistent of
course with Galilean relativity. What does it tell us?
Well. Let's do some algebra.
So we release this light pulse one goes to the left one goes to the right.
x is plus or minus ct focusing on one of them will be quite enough we want to
understand how. These two observers described this
position and time of the same event, some birthday party that happened somewhere,
and so I am going to claim that the position and the time as measured by the
moving blue observer are going to be in some way dependent on the position and
time as measured by the, stationary if you wish, the black observer, and these
transformations have a few properties that I am going to insist upon.
One is, of course that I want to insist that when x equals t equals zero.
That was the instant that they met. At that time, at that event.
Is also. The same.
As x prime equals t prime equals zero. This is just a way of synchronizing their
clocks and, indeed, these transformations have these properties.
But this is not the most general functional dependence that has this
property this is called a linear dependence.
I am assuming that x prime is some number times x plus another number times t and,
likewise for t prime. The reason I want to do this is because
this has the property that straight lines in the xt plain will be transformed in
the straight lines in the x prime t prime frame.
We want this because we want physics to be preserved by these
relativistic transformation. In other words, we want to know that an object
that would be interpreted by the black observer as inertial, no forces act upon
it, would also be interpreted by the blue observer as inertial the property of
moving at a constant speed of having a straight world line needs to be preserved
by our relativistic transformations so this is the most general transformation
that has these two properties. And now,
what are these numbers, A, B, C and D? Well, of course, in general, A, B, C and
D will depend upon something. What will they depend on?
Well, they relate the black observer's observations to the blue observer's
observations, what relates these two observers is the
relative velocity V, so A, B, C and D are arbitrary, initially, functions of V, and
our job is to find these four functions of the relative velocity V.
So let's do it. We expect that we think we know about the
relation between the black guy's observations and the blue guy's
observations. Well, the first is that if you said X
prime equal to zero, that's the here of the blue observer.
Well that here should be moving to the right, say, with respect with the black
observer, x beat v, that's what defines the relative velocity v. And so setting x
prime to zero means we should discover that x is equal to vt.
Okay, put that into our equation, set x prime to zero, and what you see is that x
prime equals zero means that ax plus bt is equal to zero.
That's the equation x prime equal to zero.
I can solve this and this says that x is equal to minus bt, divided by a. This
indeed describes something that is moving at a constant velocity.
That constant velocity is negative b over a.
So, we discover that v must be negative b over a.
The dependence of these coefficients on v is such that negative b over a is equal
to v or more elegantly, b is equal to negative a times v.
Okay, we learned something about these coefficients.
If I want I can go back to my original equation and cross this guy out, and
write minus A times V times T. Okay.
What else do we know? Well, there's this relativity story.
If the observer in black sees the observer in blue moving with speed V,
then, of course, the observer in blue must see the observer in black moving
with speed negative V. In other words, if you look at the
position X equal to zero, then, that moves relative to the blue system with a
velocity negative V. So lets plug that into our equations and
see what it tells us. So, when I set x0, equal to zero, I can
see from my equation, that when x0, equal to zero, I find that X prime is equal to
B times T. When X is equal to zero, I find the T
prime is equal to D times T. Remember,
the idea is, any event, X equals zero at any time should be such
that x prime is related to t prime. By this relation the world line of the
black observer is of something moving with velocity minus v.
So I set these two to satisfy that relation and I find that this means that
x prime indeed is d over b imes t prime but that means that d over b should be
negative v or, written more elegantly, b is negative d times v.
And so that tells me, in fact, comparing these two, that d is the same as a.
I can rewrite the equations implementing both of those results already and I see a
simplified form. A appears twice three times in fact.
we still haven't determined the coefficient c, but we haven't used
relativity yet. We haven't used the idea that, that light
pulse is moving with speed C. Notice so far,
I haven't found any contradiction with Galilean relativity yet.
But now I will, and
what I'm going to find is the following story so I want to insist that the light
pulse which is described remember by x is equal to ct that's that line moving world
line of something moving with speed c is also described by x prime equal to ct
prime with the same coefficient c. Hm, so now I need to plug all that into
the current form of my equations. Let's try to do that.
So if x is equal to ct, I find that x prime is equal to a times x of ct.
Ct minus vt. And on the other hand, T prime is equal
to c times x, which is ct, sorry for the double use of c at least it's clear in
the type set version plus A times T. Now I'm not so interested in T prime, I
want to study c times t prime to make the comparison, so let's put another c here.
Multiplying this by C, I get a C here. And now I see that when I said this is
equal, supposed to be equal to that the ACT terms agree so I can cross them out.
The leftover pieces should agree with each other, so minus A times V times T
should be C times the square of C minus the speed of light times T.
Again I can cancel off the Ts and I find that C is equal to negative A times V
over C squared. Aha!
That nails everything in terms of this one coefficient A.
So let's write all that out. And we can plug that back into these
equations, and this is what we've discovered so far.
And this is what we've discovered, I have put back the velocity dependence of a,
now that it's not too cumbersome there's only one coefficient that contains all
the information. Alright, how do I determine this,
velocity dependence of A? Well,
what I do here is I have here two equations determining x prime and t prime
in terms of x and t. I can invert them just as I get for the
Galilean transformation and solve for x and t as functions of x prime and t
prime. Little bit of algebra, you're more than
encouraged to do it yourself, and you find that these are the inverse
transformations that follow. I mean, if x prime and t prime satisfy
the, these two equations, you can solve for x and t and get these two equations.
Now what do you expect? Remember x and t are the,
position in time of a particular event, as measured by the black observer, for
who the blue observer observes position x prime and
time, T prime. And so, remember the black observer is
moving relative to the blue observer with velocity minus V so we know, if Lorentz
transformations, if our relativistic transformations are going to be correct.
We know how to relate X and T to X prime and T prime.
We basically repeat the calculation we get here but with relative velocity
negative V. In other words, this should be the same
as that with, the sign of V changed, so that's what we expect, that the
transformation from X prime and T prime to X and T will be the same as this, with
everything everywhere, V replaced by, relative velocity negative V.
Okay. That's great,
because now, lots of things look consistent.
Look the, X prime plus vt prime cancels.
That's, that's reasonable. The dependence here is reasonable.
It's only a question of this, a of v and a of -v.
What do we know about a of v and a of minus v?
A of v is the coefficient corresponding to moving with velocity v,
say to the right, or of speed v say to the right.
If minus v is the coefficient corresponding to moving with that same
speed to the left. The universe doesn't care if you're
moving to the left and right. The universe as far as we know is
isotropic, left and right are the same thing.
So we in fact expect A of V to be the same as A of minus V.
That's because which direction you're moving in should not change the nature of
relativity. What that allows us is to put a plus sign
here, because we don't care. Multiplying through by A of V, we find
that A of V squared, therefore, is equal to 1 over 1 minus V squared over C
squared, taking the square root we have our answer A of V is 1 over the square
root of 1-V squared over C squared. Plugging that back in we have derived
explicitly the Lorentz transformations, that form the core of the Theory of
Special Relativity and we will, be studying them carefully.
We will be using them everywhere, and note that they are not the equations of
Galilean relativity. Of course they're not, they have the
property that in when this change the speed of a pulse of light as measured by
the two observers will be c and that violates the velocity addition.
So, of course we made a change, but Newtonian and Galilean physics worked
very well. what is, where is the limit in which we
reproduce Newtonian physics? The limit is clearly that if V, the
relative motion, is very small compared to C, then I can pretty much neglect
these denominators. This is much smaller than one, These
denominators are going to be approximately one and I find that X prime
is indeed approximately X minus VT. And T prime would be approximately T
because V over C is a small number. So,
if I don't look too far away, and we'll talk about what that means in the next
clip, then I am reproducing the galilean
transformations that makes sense. the difference between relativity and
galilean relativity shows up at. Velocities who which are not negligible
to the speed of light the reason that common sense does not agree with what
Einstein predicts is because common sense is the intuition that we have developed
over the course of our life and over the course of our life we have not been
walking around at speeds comparable to the speed of light we have no intuitive
sense for what this means. So we'll have to use math, and Lorentz
transformations are the math that will make everything clear.