Okay. 
I hope you realize that last week's 
discussion of stellar evolution barely 
scratched the surface. 
There are a lot of things that could and 
I hope you will, follow up on. 
But, 
we were left with an interesting question 
at the end. 
If you have a stellar core whose mass is 
below the [UNKNOWN] limit of about 1.4 
solar masses, it forms a white dwarf. 
If it's more massive, electron degeneracy 
will not hold the core. 
And the core collapses further, forming a 
neutron star with a density much, much 
larger than a white dwarf. 
But, 
neutron degeneracy will maintain the, the 
the mass of a neutron star up to maybe 
two or possibly three, depending on what 
you think of the nuclear equation of 
state solar masses. 
What happens if the solar core is more, 
stellar core is more massive than the 
neutron star limit? 
What happens then? 
As many times in this class, the answer 
to this is going to require us to step 
back and start over learning some new 
fundamental physics. 
The physics we're going to learn in order 
to understand this is going to be of 
course the theory of relativity. 
And we're going to step all the way back 
to enunciate the principle of relativity 
and see where it takes us. 
So, 
what is this principle of relativity? 
It's a thing that was known in ancient 
times. 
It was articulated very clearly by 
Galileo, to my knowledge. 
But not very concisely. 
What Galileo was saying in this lengthy 
quote from his dialogue from 1632 is that 
if you are in an enclosed room on a ship 
in a cabin and you do whatever physics 
experiment you want, 
you will not be able to distinguish the 
situation where the ship is at rest at 
anchor or the ship is slowly moving 
through calm seas. 
The laws of physics do not allow you to 
make an experiment that tells you whether 
you are at rest or moving at a constant 
velocity. 
Now, Galileo didn't know the math that 
underlies this, as we know. 
The math was articulated by Newton. 
But, we understand it. 
Newton's law FMA, = ma, this is 
everything remember? 
Determines, says that the laws of physics 
determine the accelerations of objects. 
What it turns out is that when you 
measure the acceleration of something 
from a laboratory that is at rest, or you 
might measure the acceleration of the 
same object making the same motion from a 
lab that is moving with a constant 
velocity, you get exactly the same 
answer. 
There's a bit of calculus hiding under 
there, let's do a graphic demonstration 
to make this clearer. 
So, what we have here is the essentially 
the simulation we used to describe a 
uniform circular motion. 
We have here a red dot that is moving 
uniformly around a cir, in a circle 
around that point in the center. 
we have a red arrow pointing to the dot 
which tells us its location relative to 
the center. 
The blue arrow remember, is its velocity 
vector. 
Its speed is constant, but its velocity 
is changing. 
And the green arrow, therefore, signifies 
the acceleration that you measure when 
you see something moving in a circle. 
And the acceleration points at the 
circle. 
If this were a planet orbiting the star, 
we would ascribe this acceleration to 
Newtonium gravity 
an attractive force between the star and 
the planet. 
Now, 
this is all drawn in a situation where 
the star is addressed. 
Question now is, what happens if we 
observe this same physical phenomenon but 
we are sliding to our left or to the left 
in this image, at a constant speed or 
constant velocity, since the direction is 
constant? 
Well, in that case, what we would observe 
is essentially the same motion, 
but shifting slowly to the right. 
And here is what it would look like. 
This is the same object but superimposed 
on that is a slow drift to the right 
which is what you would observe if you 
were drifting slowly to the left. 
And you see that the motion is a little 
bit more complicated. 
In this image, we've added on, as we did 
then, the velocity vector. 
And you see that unlike the previous 
case, the magnitude of velocity, the 
speed is not constant, 
the object is moving faster at this point 
and slower at this point. 
The motion is more complicated but it 
becomes clearer if you do what we did 
last time and pull off the velocity 
vector. 
And, draw in the bottom picture the 
velocity vector is the blue vector. And 
you see that the blue vector indeed is 
moving around the circle, but the circle 
is simply shifted. 
The 
velocity with which this planet is now 
moving is 
given by some constant magnitude vector 
which rotates, that's the black arrow 
that as the auxiliary arrow in the image 
at the bottom. 
Added on to a constant shift to the right 
which takes account of the fact that we 
are drifting to the left. 
And indeed, if I then compute the rate of 
change of the velocity, the acceleration 
then in the bottom main panel here, you 
can see the green arrow which indicates 
the rate of change of that blue arrow. 
And you notice that the magnitude of that 
green arrow does not change. And, in 
fact, its direction is always tangential 
to the circle. And if you move it, what 
you see in the upper panel is that if we 
were observing the acceleration of this 
planet, we would observe an acceleration 
that is always directed towards the 
center of the circle as the circle drifts 
to the right. 
And we would say, aha, there is a 
gravitational force between something at 
the center of that circle and the planet. 
We would derive the same physical laws 
from this drifting system that we would 
have had the star been at rest. 
So, that helped you visualize the fact 
that because Newton's law determines an 
acceleration, you would observe the same 
laws of physics when you added a constant 
drift velocity to whatever it is that you 
were observing. 
So, we're going to elevate this 
observation, if you wish, into a 
principle. 
And the principle says that, since there 
are no measurements you can make that 
will detect whether you are at rest or 
moving with a constant velocity. 
In fact, since you can't measure it, 
the question of are you at rest or moving 
with a constant velocity? 
Or what is your constant velocity? 
Is, in fact, not a proper physical 
question. 
There is no such thing as being at rest. 
It makes no sense. 
There is no cosmic rest frame. 
Though maybe there is, 
we'll come back to that in the last week 
of the class. 
But at the level of the laws of physics 
moving at a constant velocity is all you 
can say. You can claim that you are not 
accelerating. If you were accelerating, 
you would note the effects of 
acceleration. We saw that the effects of 
accelleration are measurable. when the, 
someone hits the brakes in your car and 
you go flying on the wind, at the 
windshield, you know that the car is 
accelerating. But, 
a constant velocity of motion is 
unmeasurable and therefore the only thing 
you can ask is, what is your velocity 
relative to someone, some other system 
and absolute risk does not make sense. 
This is the principle of relativity and 
it's going to be the central 
philosophical point we follow this week. 
And of necessity, this week, veers a 
little into philosophy. 
What we're going to do is we're going to 
take this principle adopted as a property 
of the universe, and ride it hard. 
We're going to see what it tells us. 
And as we'll see, Newtonian physics is 
very nicely consistent with this. 
We'll do some explicit studies to see 
that. 
But, Maxwellian electromagnetism is not 
consistent with the principle of 
relativity. 
unless you make modifications to what you 
mean by space and time. 
And after some hesitation, we will adopt 
Einstein's solution, which is Maxwellian 
electrodynamics is correct. 
Our ideas of space and time are, need 
modification, that modification is 
Einstein's special theory of relativity. 
And I hope that we'll be able to really 
understand what that theory means and 
what its implications are, and how this 
is derived both philosophically and at 
the level of being able to compute things 
and solve some problems. 
Once we have this, we can follow in 
Einstein's footsteps and say, okay, now 
we understand a modified principal of 
relativity. 
All physics must now obey this modified 
principal of relativity. 
Electrodynamics is fine. 
It'll turn out the nuclear forces are 
fine. 
Good old Newtonian gravity is not. 
Newtonian gravity will be discovered to 
be inconsistent with our new principle of 
relativity. 
And modifying this is what led Einstein, 
after a decade of hard work, to the 
general theory of relativity, which is a 
modern view of gravity. 
Now, 
there are two things I want to emphasize 
here. 
One is that the special theory of gravity 
is something that we will be able to 
actually understand at a technical level. 
And your average working physicist has a 
good understanding of the special theory 
of relativity. 
The general theory is mathematically and 
conceptually much more involved. 
We will hopefully be able to understand 
some of the conceptual issues. 
The technical aspects of GR are, 
unfortunately, something you will not be 
able to master. 
So we will be a little bit more 
impressionistic in our description of 
general relativity. 
But, we will describe what the theory 
says and how it's constructed and some of 
its more astronomically interesting 
consequences. 
And finally, we'll apply it to the 
important question of what happens to a 
star whose core is too massive. 
I want to make one other philosophical 
point before we go on back to our work 
which is, 
this is going to be a great example this 
week of a way that science proceeds by a 
sequence of successively better 
approximations to the truth. 
what Einstein's theory of relativity is 
going to do is in a philosophical sense 
overthrow Newtonian ideas completely. 
We're going to have to say that Newton 
was philosophically fundamentally wrong. 
But on the other hand, we're going to 
have to also figure out that all of the 
calculations and the predictions and the 
measurements of stellar masses and so on 
that we very successfully performed and 
verified observationally using Newtonian 
physics are still valid. 
The way that this works is that, if you 
take a suitable limit in the parameter 
space under suitable circumstances, 
Einstein's theory of relatively 
reproduces the results of Newtonian 
physics. And we're going to have to keep 
that constantly in mind because Newton's 
physics is not wrong. 
It may be philosophically misguided. 
But at a technical level, it produces 
correct predictions when applied under 
the right circumstances. 
And indeed, Einstein's theory will 
reproduce the results of the old theory, 
in the suitable limits. 
And this is definitely the way that new 
scientific theories do not completely 
overthrow the old, but actually 
understand them as being valid in a 
limiting case.