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.