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Okay. 
I hope you realize that last week's 

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discussion of stellar evolution barely 
scratched the surface. 

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There are a lot of things that could and 
I hope you will, follow up on. 

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But, 
we were left with an interesting question 

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at the end. 
If you have a stellar core whose mass is 

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below the [UNKNOWN] limit of about 1.4 
solar masses, it forms a white dwarf. 

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If it's more massive, electron degeneracy 
will not hold the core. 

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And the core collapses further, forming a 
neutron star with a density much, much 

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larger than a white dwarf. 
But, 

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neutron degeneracy will maintain the, the 
the mass of a neutron star up to maybe 

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two or possibly three, depending on what 
you think of the nuclear equation of 

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state solar masses. 
What happens if the solar core is more, 

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stellar core is more massive than the 
neutron star limit? 

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What happens then? 
As many times in this class, the answer 

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to this is going to require us to step 
back and start over learning some new 

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fundamental physics. 
The physics we're going to learn in order 

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to understand this is going to be of 
course the theory of relativity. 

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And we're going to step all the way back 
to enunciate the principle of relativity 

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and see where it takes us. 
So, 

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what is this principle of relativity? 
It's a thing that was known in ancient 

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times. 
It was articulated very clearly by 

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Galileo, to my knowledge. 
But not very concisely. 

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What Galileo was saying in this lengthy 
quote from his dialogue from 1632 is that 

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if you are in an enclosed room on a ship 
in a cabin and you do whatever physics 

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experiment you want, 
you will not be able to distinguish the 

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situation where the ship is at rest at 
anchor or the ship is slowly moving 

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through calm seas. 
The laws of physics do not allow you to 

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make an experiment that tells you whether 
you are at rest or moving at a constant 

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velocity. 
Now, Galileo didn't know the math that 

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underlies this, as we know. 
The math was articulated by Newton. 

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But, we understand it. 
Newton's law FMA, = ma, this is 

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everything remember? 
Determines, says that the laws of physics 

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determine the accelerations of objects. 
What it turns out is that when you 

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measure the acceleration of something 
from a laboratory that is at rest, or you 

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might measure the acceleration of the 
same object making the same motion from a 

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lab that is moving with a constant 
velocity, you get exactly the same 

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answer. 
There's a bit of calculus hiding under 

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there, let's do a graphic demonstration 
to make this clearer. 

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So, what we have here is the essentially 
the simulation we used to describe a 

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uniform circular motion. 
We have here a red dot that is moving 

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uniformly around a cir, in a circle 
around that point in the center. 

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we have a red arrow pointing to the dot 
which tells us its location relative to 

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the center. 
The blue arrow remember, is its velocity 

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vector. 
Its speed is constant, but its velocity 

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is changing. 
And the green arrow, therefore, signifies 

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the acceleration that you measure when 
you see something moving in a circle. 

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And the acceleration points at the 
circle. 

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If this were a planet orbiting the star, 
we would ascribe this acceleration to 

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Newtonium gravity 
an attractive force between the star and 

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the planet. 
Now, 

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this is all drawn in a situation where 
the star is addressed. 

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Question now is, what happens if we 
observe this same physical phenomenon but 

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we are sliding to our left or to the left 
in this image, at a constant speed or 

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constant velocity, since the direction is 
constant? 

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Well, in that case, what we would observe 
is essentially the same motion, 

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but shifting slowly to the right. 
And here is what it would look like. 

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This is the same object but superimposed 
on that is a slow drift to the right 

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which is what you would observe if you 
were drifting slowly to the left. 

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And you see that the motion is a little 
bit more complicated. 

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In this image, we've added on, as we did 
then, the velocity vector. 

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And you see that unlike the previous 
case, the magnitude of velocity, the 

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speed is not constant, 
the object is moving faster at this point 

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and slower at this point. 
The motion is more complicated but it 

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becomes clearer if you do what we did 
last time and pull off the velocity 

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vector. 
And, draw in the bottom picture the 

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velocity vector is the blue vector. And 
you see that the blue vector indeed is 

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moving around the circle, but the circle 
is simply shifted. 

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The 
velocity with which this planet is now 

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moving is 
given by some constant magnitude vector 

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which rotates, that's the black arrow 
that as the auxiliary arrow in the image 

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at the bottom. 
Added on to a constant shift to the right 

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which takes account of the fact that we 
are drifting to the left. 

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And indeed, if I then compute the rate of 
change of the velocity, the acceleration 

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then in the bottom main panel here, you 
can see the green arrow which indicates 

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the rate of change of that blue arrow. 
And you notice that the magnitude of that 

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green arrow does not change. And, in 
fact, its direction is always tangential 

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to the circle. And if you move it, what 
you see in the upper panel is that if we 

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were observing the acceleration of this 
planet, we would observe an acceleration 

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that is always directed towards the 
center of the circle as the circle drifts 

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to the right. 
And we would say, aha, there is a 

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gravitational force between something at 
the center of that circle and the planet. 

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We would derive the same physical laws 
from this drifting system that we would 

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have had the star been at rest. 
So, that helped you visualize the fact 

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that because Newton's law determines an 
acceleration, you would observe the same 

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laws of physics when you added a constant 
drift velocity to whatever it is that you 

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were observing. 
So, we're going to elevate this 

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observation, if you wish, into a 
principle. 

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And the principle says that, since there 
are no measurements you can make that 

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will detect whether you are at rest or 
moving with a constant velocity. 

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In fact, since you can't measure it, 
the question of are you at rest or moving 

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with a constant velocity? 
Or what is your constant velocity? 

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Is, in fact, not a proper physical 
question. 

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There is no such thing as being at rest. 
It makes no sense. 

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There is no cosmic rest frame. 
Though maybe there is, 

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we'll come back to that in the last week 
of the class. 

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But at the level of the laws of physics 
moving at a constant velocity is all you 

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can say. You can claim that you are not 
accelerating. If you were accelerating, 

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you would note the effects of 
acceleration. We saw that the effects of 

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accelleration are measurable. when the, 
someone hits the brakes in your car and 

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you go flying on the wind, at the 
windshield, you know that the car is 

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accelerating. But, 
a constant velocity of motion is 

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unmeasurable and therefore the only thing 
you can ask is, what is your velocity 

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relative to someone, some other system 
and absolute risk does not make sense. 

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This is the principle of relativity and 
it's going to be the central 

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philosophical point we follow this week. 
And of necessity, this week, veers a 

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little into philosophy. 
What we're going to do is we're going to 

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take this principle adopted as a property 
of the universe, and ride it hard. 

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We're going to see what it tells us. 
And as we'll see, Newtonian physics is 

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very nicely consistent with this. 
We'll do some explicit studies to see 

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that. 
But, Maxwellian electromagnetism is not 

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consistent with the principle of 
relativity. 

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unless you make modifications to what you 
mean by space and time. 

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And after some hesitation, we will adopt 
Einstein's solution, which is Maxwellian 

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electrodynamics is correct. 
Our ideas of space and time are, need 

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modification, that modification is 
Einstein's special theory of relativity. 

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And I hope that we'll be able to really 
understand what that theory means and 

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what its implications are, and how this 
is derived both philosophically and at 

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the level of being able to compute things 
and solve some problems. 

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Once we have this, we can follow in 
Einstein's footsteps and say, okay, now 

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we understand a modified principal of 
relativity. 

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All physics must now obey this modified 
principal of relativity. 

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Electrodynamics is fine. 
It'll turn out the nuclear forces are 

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fine. 
Good old Newtonian gravity is not. 

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Newtonian gravity will be discovered to 
be inconsistent with our new principle of 

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relativity. 
And modifying this is what led Einstein, 

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after a decade of hard work, to the 
general theory of relativity, which is a 

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modern view of gravity. 
Now, 

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there are two things I want to emphasize 
here. 

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One is that the special theory of gravity 
is something that we will be able to 

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actually understand at a technical level. 
And your average working physicist has a 

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good understanding of the special theory 
of relativity. 

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The general theory is mathematically and 
conceptually much more involved. 

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We will hopefully be able to understand 
some of the conceptual issues. 

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The technical aspects of GR are, 
unfortunately, something you will not be 

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able to master. 
So we will be a little bit more 

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impressionistic in our description of 
general relativity. 

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But, we will describe what the theory 
says and how it's constructed and some of 

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its more astronomically interesting 
consequences. 

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And finally, we'll apply it to the 
important question of what happens to a 

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star whose core is too massive. 
I want to make one other philosophical 

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point before we go on back to our work 
which is, 

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this is going to be a great example this 
week of a way that science proceeds by a 

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sequence of successively better 
approximations to the truth. 

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what Einstein's theory of relativity is 
going to do is in a philosophical sense 

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overthrow Newtonian ideas completely. 
We're going to have to say that Newton 

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was philosophically fundamentally wrong. 
But on the other hand, we're going to 

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have to also figure out that all of the 
calculations and the predictions and the 

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measurements of stellar masses and so on 
that we very successfully performed and 

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verified observationally using Newtonian 
physics are still valid. 

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The way that this works is that, if you 
take a suitable limit in the parameter 

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space under suitable circumstances, 
Einstein's theory of relatively 

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reproduces the results of Newtonian 
physics. And we're going to have to keep 

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that constantly in mind because Newton's 
physics is not wrong. 

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It may be philosophically misguided. 
But at a technical level, it produces 

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correct predictions when applied under 
the right circumstances. 

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And indeed, Einstein's theory will 
reproduce the results of the old theory, 

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in the suitable limits. 
And this is definitely the way that new 

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scientific theories do not completely 
overthrow the old, but actually 

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understand them as being valid in a 
limiting case.