So like it or not, intuitive or not, 
those are the rules of the games. 
The universe is invariant under Lorentz 
transformations and not under Galileo's 
transformations. 
If we're going to do physics, we're going 
to have to understand how to do physics 
in a way that is Lorentz invariant. 
So, one of the important things that we 
had discussed is our conservation laws. 
And, we run into trouble with our 
conservation laws, in the sense that the 
Newtonian conservation laws are not 
invariant under Lowen's transformation 
laws. 
What does that mean? 
That means that if you write momentum 
equal to mv, 
we can put little vectors there if you 
want to be fancy. 
And energy, kinetic energy for the 
moment, equal to half MV squared. 
those are not, not cons, not in Lorentz 
invariant quantities in the following 
sense. 
Of course momentum of the momentum of a 
given particle is different as observed 
by different observers because it's 
velocity is different. 
That was true in Galilean relativity as 
well, but in Galilean relativity you can 
show, very directly, that if momentum is 
conserved in some process in one frame, 
then momentum is conserved for any other 
observer moving at a constant velocity 
relative to the first observer who 
declared momentum to be conserved. 
And likewise energy. 
It's not trivial, but it's true. 
On the other hand we can use Roman 
transformations. 
You can find that one observer would 
claim that momentum is conserved and the 
other that it isn't. 
That's impossible. 
Momentum conservation thus cannot be a 
law of nature. 
Laws of nature have to be true as 
measured by an observer. 
On the other hand. 
The untonian conservation laws worked 
well for objects that were moving slowly 
relative to the speed of light. 
Then you can of course pick frames to 
observe them in, such that none of the 
objects move relative to you at close to 
the speed of light. Newtonian physics has 
to work. 
So we need and expect to find 
conservation laws that are extensions or 
modifications of the conservation of 
energy and momentum, but that are valid 
in a Lorentz invariant way, and in fact 
we do. 
The two quantities we find that extend it 
are a momentum and energy, or three 
components of momentum if you want. 
And the relativistic co-variant momentum 
is NV divided by the square root of one 
minus V squared over C squared, and the 
energy is MC squared divided by the 
square root of one minus V squared over C 
squared. 
Now, 
these are the objects, of course again it 
is not true that momentum is measured by 
each Lorentz, each, observer will be the 
same, but what is true is if these 
quantities are conserved, momentum and 
energy for one Lorentz observer, then 
they are conserved for any Lorentz 
observer. 
And therefore, the, the conservation of 
these is consistent with Lorentz 
invariance, 
and it is, in fact, the conservation law 
that obtains to nature. 
Now for the, we expect to recover 
Newtonian Conservation Laws when objects 
are not moving much faster. 
But at speeds comparable, to the speed of 
light, when they're moving much slower. 
And indeed, it's not too difficult to see 
that when v squared is much less than c 
squared in the case of momentum, we drop 
this factor and momentum at 
Velocities much smaller than the speed of 
light is mv and then we can use our 
Newton's approximation, to write remember 
the denominator. 
Here is one minus v squared over c 
squared to the power minus one half which 
is approximately one plus v squared times 
a half divided by2, c squared, the minus 
sign is cancelled. 
And so there is the first relativistic 
correction to, the Newtonian momentum 
conservation at velocities much smaller 
than the speed of light. 
Newtonian momentum will be conserved. 
At higher velocities, you have to 
consider this, factor, 
and remember, this approximation is valid 
when v2 squared over c2 squared is small. 
At velocities higher yet, you have to use 
this complete, relativistic expression. 
Very nice. 
So we have the, understanding of 
relativistic momentum. 
Notice that what this means is that as 
you accelerate a particle to speeds 
closer and closer to the speed of light, 
its momentum actually becomes closer and 
closer to infinity. 
It takes an infinite amount of momentum 
as you try to accelerate something to the 
speed of light. 
This is reasonable. 
It means if you go around giving 
something a kick and each kick you give 
it adds a certain quantity of momentum. 
You could ask well why don't I keep 
kicking it until its moving faster than 
the speed of light? 
The answer is that if each kick adds a 
certain amount of momentum that will only 
make its velocity asymptotically, as the 
number of kicks grows without bound 
become closer and closer to the speed of 
light, but never accelerated past the 
speed of light. 
This is sometimes stated as saying that 
you rewrite this as something times v, 
and interpret the something as some kind 
of relativistic mass. 
I don't like that point of view, and most 
physicists do not. 
We will not treat this as a change in the 
mass. 
We just treat it as the correct 
relativistic definition of momentum, 
so that the mass for us is going to be 
the object that's over there, and we'll 
see in a minute how we interpret it, and 
it's a property of the particle. 
So extracting Newtonian, the Newtonian 
limit of momentum conservation is a 
simple matter of setting v squared over c 
squared less than one. 
In the case of energy, this is a lot less 
obvious. 
What, what is this? 
when I drop the v squared over c squared 
I get nc squared. 
That doesn't look at all like the kinetic 
energy of anything. 
We were supposed to recover the kinetic 
energy conservation law, but if we follow 
through the Newton's approximation to the 
term, thing get a little better thus we 
have the same half V square over C 
square. 
And that ends up being, well expanding 
this the first term is mc squared. 
And the second term, holy moly, a half 
and v squared. 
And of course this is valid for v squared 
much less than c squared. 
For higher velocities you have to use all 
of this. 
But when you think about it this way, you 
see the kinetic energy here, 
and then this famous e equals mc sqared. 
The conservation law if mass is conserved 
and this is just a constant as long as 
the particle exists this is just a 
constant and remember shifting energy by 
a constant makes no difference but. 
The fact that 
in relativistic dynamics, it is passable 
because of the modified momentum 
conservation in fact. 
For mass to change the total mass of 
particles coming in a process need not be 
the sa, the total mass of the particles 
coming out of a collision. 
This in fact was not possible in 
Newtonian physics. 
you'll show it in the homework, but it is 
possible in relativistic physics, in a 
way that conserves both energy and 
momentum. 
If the mass changes, then we see here the 
rate of conversion of mass to energy, if 
the particle of mass end disappears, and 
it's mass becomes converted to energy. 
And this is the energy, and here is the 
kinetic energy contribution at slow 
velocities, 
and at higher velocities this is the more 
accurate description. 
So we have our relativistic conservation 
laws. 
few remarks about this. 
One is that if you compute just from the 
expressions [INAUDIBLE] wrote, 
E squared minus P squared C squared, you 
get M squared C to the fourth. 
It is this is the reason why. 
What is the mass of particle? 
No matter what momentum you give it the 
mass of a particle characterizes this 
property. 
To give it some momentum you'll have to 
give it some energy, 
and E squared can always be written as P 
squared C squared plus M squared C to the 
fourth. 
And this combination of ENP is a property 
of the particle. 
We call it the invariant mass, and that 
is what I will use to define the mass. 
this is a property of the particle and 
not of it's motion. 
Good. 
So a particle has a mass as long as it 
exists. 
The other thing to note, is I did, I will 
not prove it, 
but I said, of course, different 
observers will measure different momenta. 
Different observers will measure 
different, energies for the same system. 
But, in fact, these four quantities. 
E and then c2 squared times p, transform 
under Lorentz transmissions, in exactly 
the same way that t and x do. 
In other words, 
momentum along the direction of motion 
and energy get mixed in the same way 
precisely that t and x do. 
so you can compute east prime and p prime 
in terms of e and p, using the Lorentz 
transformations. 
And in particular, if the change in 
energy and momentum is zero in one frame, 
it's zero in any other frame. 
Because the Lorentz transformation takes 
xt=0 equals t equal zero, to x prime 
equals t prime equals zero. 
This is why energy momentum conservation 
is consistent. 
if you use this definition of energy. 
Okay. 
Now, an important canard, 
this E equals M C squared is sometimes 
invoked as the reason why nuclear, 
processes release so much energy. 
Oh, because they convert mass to energy, 
and C squared is such a large number. 
That's a canard. 
In fact, these conservation laws. 
Relativity was always true, even before 
we discovered it. 
if Lavoisier in 1777, when he made his 
ground breaking measurement, that said if 
you weigh the gas products of combustion, 
and the ashes, you recover the weight of 
the wood that you started burning. 
Had he measured very precisely, he would 
have discovered this is not true. 
Why? 
Because the wood, in burning, lost some 
heat. 
The heat radiated away. 
That means that the total energy of the 
system has been reduced. 
Converting by dividing by c squared, he 
would have found a very small reduction 
in mass. 
The point is that when wood burns, it 
doesn't produce that much energy, and 
therefore, it loses very little of its 
mass, because you are harnessing weak 
electric, magnetic forces at atomic 
radii. 
If you, on the other hand managed to 
harness the strong force, or forces that 
nuclear distances, the loss of the energy 
released is much larger. The change in 
the mass is actually measurable, and this 
is why the mass of four protons is lager 
than the mass of an alpha particle by a 
measurable amount. 
And, but, but this has, the statement 
that the sun is fuelled by the conversion 
of mass to energy is a canard, in the 
same way that the statement that my 
fireplace warms me up by converting mass 
to energy as a canard. 
The statement that is true is that the 
sun produces energy by harnessing a very 
strong nuclear force. 
the other thing is that the decay of a 
particle not conserving mass is 
consistent and it happens. 
We know of 
neutrons decaying to protons, electrons 
and neutrini. 
The sum of the masses of the constituents 
is not, is less than the sum, the mass of 
the neutron. 
Which is why they come off with some 
kinetic energy. 
Because of the conservation of, total 
energy mass. 
And, all our other conserved quantities. 
Electric charge, electron number and so 
on, are actually invariant. 
They are numbers, they are identical. 
The electric charge in a given region as 
measured by different observers is the 
same. 
So conservation of electric charge, 
electron number and so on, works very 
well. 
We don't have a problem. 
We have our conservation laws. 
Now let's write down the laws of physics. 
As I said electromagnetism is tooken care 
of, 
so to speak. 
It's taken care of. 
Electromagnetism, Maxwell's equations 
were Lorentz invariances. 
That's where we all started from. 
What about the nuclear interactions? 
Well, we know a nuclear, a, a way of 
writing the nuclear interactions, and 
though its in variant form, though that's 
not the way they were initially written. 
In fact. 
Are study of the nuclear interactions and 
of, electromagnetism, is often inherently 
in quantum, and by the 1940s, and re, for 
nuclear fie, interaction by the 1970s. 
A relativistic, version of, lumison 
variant version of, quantum mechanics 
called quantum field theory had been 
developed, and we have a quantum 
relavistic version, of electromagnetism, 
the strong nuclear force, the weak 
nuclear force. 
Physics, is very nicely consistent with, 
relativity, 
which leaves the question of gravity. 
Well turn, an, an, and this is the 
question with which Einstein was stuck. 
Newtonian gravity, our poster child for a 
theory of physics, the first one to 
really be understood the first 
fundamental force to be properly 
understood. 
the one that was our poster child for 
gallilaian invariance is not invariant as 
well will see and through lobeman's 
transformations it is not consistent with 
relatively Einstein understood this in 
1905 when he figured out the special 
theory of relativity. 
And it took a whole decade of Einstein to 
figure out the generalization. 
So we're going to spend some time trying 
to understand what's wrong, why gravity 
is not consistent with relativity, and 
how Einstein figured out how to resolve 
it. 
As for the quantum version of 
relativistically invariant gravity, 
that'll be left for your homework.