[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.