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.