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PROFESSOR: We do have a little bit
of finishing to do because we didn't quite finish
the dynamics of homogeneous expansion last time.
So we'll begin by finishing that after a brief review of where
we were last time.
And then we'll move on to discuss non-euclidean spaces,
which I hope will be still the bulk of today's lecture.
OK.
In that case, let's get going.
Again, as I said, I want to begin
by just reviewing some of the things we
talked about last time.
And you should consider this a good opportunity
to ask questions if you discover there are things that you're
not really sure you understood as well as you'd like.
We were talking about the evolution of a closed universe.
And to summarize that calculation we first
re-shuffled the first order Friedmann
equation, the equation for a dot over a quantity squared,
by bringing all the d t's to one side and all the d a's
on the other side after doing a little bit of rescaling.
And we got this equation.
Which we then said we can integrate.
And the integral from time will go from time 0
to some arbitrary final time that we called t tilde sub f,
where the "tilde" indicates that we multiplied by c.
And the sub f means it's the final time of our calculation.
And on the other side we have to integrate
with corresponding limits of integration.
Corresponding to t equals 0 is a equals zero.
So the lower limit of integration is 0.
And the final limit of integration
is just the value of a tilde at the final time, whatever
that is.
Then we discovered that we could actually
do the integral on the right if we made a substitution.
And in the lecture last time we did it in two stages.
But the combined substitution is just
to replace a tilde by cosine theta,
according to this formula, if we combine the two substitutions
that we made last time.
And if we do that we could integrate the right hand side.
And the integral then gives us t tilde
f is equal to the integral of that,
which is just this expression.
And the expression below that is just the substitution itself.
How to relate a tilde to theta, according to substitution
which gave us that formula.
So these two formulas together allow
us to determine t tilde sub f, and a tilde
sub f in terms of theta and alpha.
And once we had that we realized we no longer needed
to keep those sub f's because that was really
just a way of keeping track of our notation
during the calculation.
It holds for any theta sub f.
So it holds for any theta.
So then we just rewrote those same formulas without the sub
f's.
And here we wrote it removing the tildes,
replacing them by their definitions.
And this was the final answer.
This describes the evolution of a closed universe expressed
in this parametric form.
That is, we were not able to explicitly write
a as a function of t, which is what would have liked
to determine how the expansion varied as a function of time.
But instead we introduced the auxiliary variable theta,
often called the development angle.
And in terms of theta we can express both t and a,
and thereby indirectly have an unambiguous relationship
between a and t.
Any questions about that?
OK.
Then we noticed that those were, in fact,
the equations of a cycloid.
And I won't go through the argument again,
but the key point is that the evolution
of our closed universe scale factor, as a function of time,
is described by what would happen
if you had a disk rolling on the horizontal axis
with a point marked on the disk which was initially
straight down, and then as the disk rolls that point
traces out a cycloid.
And that is exactly the evolution of a closed universe.
It starts at size zero, goes to a maximum size,
and then contracts again in a completely symmetric way.
The contraction phase is the mirror image
of the expansion phase.
And then we went on to calculate the age of a closed universe.
And that was really more of an exercise in algebra
than anything else.
The age is really given by this formula to start with.
This expresses the age, t, in terms of alpha and theta.
And the only problem with that is that nobody really
knows what alpha and theta mean.
It's much more useful to have an expression for the age in terms
of things that astronomers directly measure.
And one needs two things to replace alpha and theta.
And the kinds of things astronomers directly measure
are things like the Hubble parameter
and the mass density, where they often express the mass density
in terms of omega where omega is the mass density divided
by the critical density.
And that's what I chose to do in the formulas
that we went on to derive.
So our goal is simply to take that formula for t
and figure out how to express the alpha and the theta that
appear in it so that we can get an expression
in terms of h and omega.
And to do that, we'll go one step at a time.
We started by just writing down the Friedmann equation.
And the Friedmann equation has an a in it.
And everything else in it is rho,
which can be expressed in terms of omega without any trouble,
and h itself.
So everything else in it consists of variables
that we're accepting to be part of our final answer.
So we could solve the Friedmann equations for a or a tilde
and that will allow us to eliminate
any a tilde's from our expressions.
Next, alpha was originally defined
in terms of this formula which only involves rho and a tilde.
We now know what to do with a tilde.
We could substitute that formula.
And rho, we always knew what to do with.
We could express that in terms of omega.
So we can instantly solve that equation
and get an equation for alpha in terms of the quantities
that we want to appear in our final answer.
And there's still one more thing we need.
We need an expression for theta.
And theta we can get by looking at the other of those
two parametric equations, the one that's
not the "t equals" equation, but rather the "a equals" equation.
And in this equation we know everything except theta itself.
So we could substitute for a over root k from up here- that.
And on the right hand side we can
place alpha by that expression and then the 1
minus cosine theta stays.
And now we could solve this equation for theta.
I might just mention that, in lecture last time I
actually mis-copied this equation.
I forgot the factor of omega in the numerator.
So if any of you were taking notes
you can go back and correct your notes.
But it's correct in the printed notes.
And now it's correct on a screen.
So this you could either solve for cosine theta--
initially what I did was to solve for cosine theta.
But then it turned out to be more
useful to know what sine theta is,
because sine theta appears in that parametric expression
for the age.
So if you know cosine theta you can, of course,
get sine theta because sine theta is just
the square root of 1 minus cosine squared theta.
And that's what we did.
And that's how we got a square root here.
And since square roots can have either sign,
there's a plus or minus there which
depends on where you are in the evolution of our universe.
The right hand side here is always positive
because we define that square root symbol
to mean the positive square root,
where the plus or minus, in the end, tells you the sine.
And sine theta itself-- we know what theta does.
It goes from 0 to 2 pi.
So sine theta starts out positive
and after theta crosses pi, sine theta is
negative for the second half of the evolution.
Meaning, sine theta is negative for the contracting phase
and positive for the expanding phase.
We then put all that together into the formula for CT,
which is alpha theta minus sine theta.
The alpha becomes this factor.
And the theta minus sine theta becomes the arc sine
of this expression, minus or plus that expression,
where this is just sine theta from that formula.
And it's minus sine theta, which is why the plus minus becomes
minus or plus.
The signs get a little tricky, but it's, in principle,
straightforward.
It all just follows from this formula.
And if you know the sine of theta
you have all your problems solved.
So we put all that together into a table.
This is just a copy of the same formula
that we had on the previous slide.
Theta is what appears in this right hand column.
It's indicated as the sine inverse of some expression.
It refers to that expression-- an abbreviation.
So we know what theta does.
Theta just goes from 0 to 2 pi in quadrants.
At least, it pays to divide it up into quadrants.
Sine theta is always positive for the first half
and negative for the second half.
And that means that we have the plus sign, or the upper sign,
for the first half of the motion and the minus
sign for the second half of the motion.
Omega we could just calculate in terms of theta.
No problem about filling in the omega column.
And we know that we're expanding for the first half
and contracting for the second half,
so that really completes the table.
And the important thing when you're actually
using this formula is to decide what theta is.
And once you have that, the sine of that--
the inverse sine of that-- well, no.
Excuse me.
Theta itself appears there, and the sine of theta
appears there.
And theta itself you have to figure out
which is the relevant value in terms
of where you are in the evolution.
The point is that sine theta itself goes--
does not uniquely determine what theta is.
OK.
That, I think, completes our discussion
of the evolution of closed universes.
I think it completes everything that we did last time.
So are there any questions?
OK.
Good.
To finish our discussion of the evolution
of matter-dominated universes, we
go on to discuss open universes.
And open universes are really the same algebra
as closed universes.
They just differ by the sine of k.
Because one doesn't like to deal with negative numbers,
I defined kappa to be equal to minus k,
so that for our open universes kappa
was positive while k was negative.
Then I used a different substitution for a tilde.
Instead of a tilde being a divided
by the square root of k, which in this case
would be the square root of a negative number-- which, one
doesn't like to deal with imaginary numbers
if one doesn't have any need to.
So instead, for the open universe
I'm defining a tilde to be a divided by the square
root of kappa, so that a tilde is again real.
Then I'm going to skip all the algebra here.
There's a little bit more of it shown in the printed lecture
notes.
But there are no new concepts here.
Everything is really the same, conceptually,
as it was for the closed universe.
One difference is that this time,
instead of getting trigonometric functions,
we find that we get hypergeometric functions.
Hyper-- yeah.
Hyper trigonometric functions, I think, is the right word.
That is, sinhs and coshs instead of sines and cosines.
The formulas are very similar.
These are the formulas we get for the open universe case,
compared to those formulas for the closed universe case.
We get a change in the order of-- instead
of theta minus sine theta, we get sinh theta minus theta.
But that's what you have to get if it's
going to turn out to be positive.
Sine theta is always less than theta.
So this is a positive quantity.
Sinh theta is always greater than theta,
so this is a positive quantity.
And same for the second lines.
Cosh theta minus 1 is always positive,
and one minus cosine theta is always positive.
So you really know which order to write them in just
by knowing that you want to write down
something that's positive.
So these formulas describe the evolution of the open universe
exactly the same way as those formulas describe the evolution
of a closed, matter-dominated universe.
So any questions?
OK.
Next step, just repeating what we did for the closed universe,
we can derive a formula for the age of an open universe.
And again, it's really just a matter
of substituting into the formula we already
have to be able to re-express it in terms of useful quantities.
Which, again, I choose to be the Hubble expansion
rate and omega.
And here I've put together all three formulas for the age.
The flat universe, the first one we
did, where t is just h inverse if we bring it
to the other side, times 2/3.
And the open universe on the top, and the closed universe
on the bottom.
Now one of the, perhaps surprising, things
that one finds here is that all three of these expressions
look fairly different from each other.
And you might think that that would
give some kind of a jagged, discontinuous curve.
But you can go ahead and plot it, which I did.
And there's the plot.
It's one nice, smooth curve.
And we won't go into this in detail, but many of you
have had courses in complex functions, functions
of a complex variable.
If you know about functions of a complex variable
you can tell that these are, in fact, all the same function.
That is, if omega is, say, bigger than 1,
this formula can be evaluated straightforwardly.
It involves only things like square roots
of positive numbers.
But you could also try evaluating this formula
for omega bigger than 1.
And then you have square roots of negative numbers appearing.
But square roots of negative numbers
are OK if you know about complex numbers.
They're just purely imaginary.
And then you get trigonometric functions, and even
inverse trigonometric functions, or inverse hyperbolic trig
functions of imaginary arguments.
But all those are well-defined.
And if you work through what the definitions are,
the top line really is identical to the bottom line.
Those really are the same function.
And that's why one expects that, when you plot them
they will join together smoothly, as they clearly do.
The point in the middle here is the first age
that we derived for the flat universe
where omega is equal to 1 and h t is just equal to 2/3.
That is, t is equal to 2/3 h inverse
for a flat universe, which is the middle dot.
And the closed universes are on the right
and open universes are on the left.
OK.
Any questions about these age calculations?
Yes?
AUDIENCE: I noticed that, for the open solution,
there's a case where you get some imaginary numbers?
PROFESSOR: If you use these formulas
you don't get any imaginary numbers.
But if you tried to use, say, the bottom formula
for a value of omega less than 1 it
would give you imaginary numbers.
And it would, in fact, if you trace
through what those imaginary numbers do,
it would give you the formula on the top.
So it's all consistent.
It's most straightforward to use the formula
on the top for omega less than 1 and the formula
on the bottom for omega greater than 1.
And then one never confronts imaginary numbers.
AUDIENCE: OK.
PROFESSOR: Any other questions?
OK.
Where are we going next?
Finally, just actually one last graph
on the evolution of matter-dominated universes,
which is the final form of a of t
for a matter-dominated universe.
If you re-scale things by dividing a over root
k by alpha, and c t over alpha-- if you look back
at our equations-- let me go back
to where the equations were.
We're really just graphing these equations.
If I divide this equation by alpha,
I just get a pure function of theta
with dimensionless variables.
And similarly here, if I divide a over root k by alpha
I just get 1 minus cosine theta, which is a pure number.
So that's what I've chosen to do.
And the dimensionality works the same for the open case as well.
So it allows you to draw a plot which
is just independent of parameters.
All the parameters are absorbed into the way
the axes are defined, which are both
defined as dimensionless numbers.
And in that case the closed universe
survives for a duration of 2 pi.
The axis here, at least, has the same duration as theta.
It's not actually theta, because t is not linear in theta.
But this point does change by 2 pi
as theta goes from 0 to 2 pi.
And one can see here, the three possible curves.
A closed universe which starts and then falls back,
a flat universe which goes off to the right
and has actually a constant slope as you go out here,
and an open universe which behaves slightly differently.
Actually, I think I was wrong when
I said the flat universe has a constant slope.
But the open and the flat case are similar to each other.
They both go off to infinity, but in different ways.
And all three of them merge as you go backwards in time.
That's not something that might have been obvious
before we wrote down the equations.
But in very early times all universes
look like they're flat universes,
if you go to early enough times.
And that actually is an important point
which we'll talk about later in terms
of what's called the flatness problem of cosmology.
That basically is the consequences of that fact.
Yes?
AUDIENCE: Why didn't you just say
all of them looks like open universes [INAUDIBLE]?
PROFESSOR: [LAUGHS]
AUDIENCE: I mean, what's special about flat universes?
PROFESSOR: Well, actually, there is
something special about flat.
Which is that, if we plotted omega as the function of time
all of them approach omega equals 1 as time goes to 0.
So there is a very definite meaning
of saying that they're all approaching flat,
rather than they're all approaching open or closed.
Good question, though.
Because from this graph you really
can't tell anything special about the flat.
Any other questions?
Yes?
AUDIENCE: So does this mean that the open and flat solutions
will extend indefinitely?
PROFESSOR: Yeah.
The open and flat solutions extend indefinitely in time.
That's right, they do.
And one can see that from the formulas or the graph.
Yes?
AUDIENCE: So for plotting omega as a function of time,
I'm a little bit confused as to how
it changes as-- for example, an open universe expands,
or a closed universe-- because it seemed
like, from our calculations, that when the universe was
expanding, omega was increasing?
Is that-- at least for a closed universe?
PROFESSOR: That is true for a closed
universe during the expanding phase.
For a closed universe during the expanding phase,
omega does increase.
It starts out as 1 and then it rises
to infinity when the universe reaches its maximum size
and is about to turn around and go back.
Because it doesn't mean the mass density increases.
That's maybe what's confusing you.
The actual mass density always decreases as it expands,
but the critical density decreases even faster.
So the ratio, omega, actually rises for a closed universe
as it expands.
For an open universe it's the reverse.
For an open universe omega starts
out being 1 at early times, as it
does for any matter-dominated universe.
And as the universe expands, omega
gets smaller and smaller for an open universe.
And it follows-- I don't have them on slides here,
but we do have in the notes formulas
that we derived, that we did on the blackboard, that give us
omega as a function of theta.
And those formulas, you can just look at them
and see how omega behaves as the universe evolves.
Because as the universe evolves, theta just increases.
And those formulas do trivially show
what I said about how omega evolves.
Any other questions?
Yes?
AUDIENCE: Why a over alpha root k, but for a flat universe k
is 0?
PROFESSOR: That's right.
I didn't really say that, but for the flat universe
there's an arbitrary normalization
that one chooses in drawing this graph.
And it was really an arbitrary choice for me
to draw the flat case to join on smoothly
with the open and closed cases.
I could have put any constant in front of t to the 2/3.
And you're right, then they would not necessarily
mesh unless I chose that constant in the right way.
Yes?
AUDIENCE: Based on that, is there a particular reason
you decided to chose [INAUDIBLE] a flat universe looks
like this?
Is there a particular thing you're
trying to show in choosing [INAUDIBLE]?
PROFESSOR: OK.
The question is, is there a particular reason
why I chose the normalization that I chose?
If I did not choose it, it would only
differ by an overall factor.
So it would still look-- the flat line
by itself would look indistinguishable, really.
It would just be higher or lower.
So the only question is how it meshes with the others.
And since the flat case really is
the borderline between the other two cases,
and since this constant that appears in front of t
to the 2/3 has no physical meaning whatever,
it seems the sensible thing to do
is to plot it so that it looks like the limit
of an open or closed universe.
Because physically it is the same as the limit of an open
or a closed universe coming from either side.
Any other questions?
Those are good questions.
OK.
In that case, we are finished with the evolution
of matter-dominated universes, and ready to start
talking about non-euclidean spaces.
So what we'll be doing next is kind of a mini introduction
to general relativity, which how non-euclidean spaces
enter physics.
Now, needless to say, general relativity
is an entire course separate from this course.
And of course that even has more prerequisites
that this course has, so we're not
going to duplicate what would be taught
in the general relativity class.
But it turns out that the discussion
of general relativity does, in fact, divide pretty
cleanly into two major issues.
And we will be dealing with one of those issues but not
the other.
In particular, what we will be doing in this class
is learning how the formulas of general relativity
is used to describe curved spaces.
And we will learn how particles move in curved spaces.
So we'll be able to analyze trajectories
in any curbed space if somebody tells you
what the curved space itself looks like.
What we will not be doing is we won't even
attempt to describe how general relativity predicts that matter
should cause space to curve.
That would be left entirely for the GR course
that you may or may not take.
But it will not be discussed here.
There's only one point where we will
need a result of that sort.
We'll want to know how the matter in our Friedman Roberson
Walker universe affects the curvature.
And there I would just give you the result.
I'll try to make it sound plausible.
But I won't make any pretense being
able to drive that result.
That is, we will not be able to drive how much space curves
as a consequence of the matter that's in it.
But we will write down the answer
so you at least know what the answer is
for a homogeneous isotropic universe.
So here's a picturesque slide about curve space.
Four years ago, I think it was, a postdoc
here named Mustafa Amin gave this lecture for me
because I was out of town.
And he had much more colorful transparencies than I ever do.
So I'll be using some of his transparencies here.
And this is one of his opening slides.
So this is what we want to talk about-- curved space
as illustrated in that nifty picture.
So I want to begin with a kind of an historical introduction.
To be honest, I'm pretty much following the logic
of the first chapter of Steve Weinberg's General Relativity
book.
Non-Euclidean geometry of course starts
by thinking about Euclidean geometry and then
how one might be move away from it.
And historically, there's kind of a clear cut
path, which was followed.
Euclid based his geometry, as described in Euclid's elements,
in terms of five postulates.
The first of which is that a straight line segment
can be drawn joining any two points.
Sounds clear enough.
Second, any straight line segment
can be extended indefinitely in a straight line.
Also sounds obvious, which is what Euclid was banking on.
Third, given a straight line segment,
a circle can be drawn having the segment as a radius and one
endpoint as the center.
That also-- you can imagine yourself doing that--
seems straightforward enough.
But then we come to the fifth postulate,
which still sounds pretty obvious.
But it's certainly much more complicated than the others.
The fifth postulate says that if a straight line falling
on two straight lines makes the interior angle on the same side
are less than two right angles, the two straight lines--
if produced indefinitely-- meet on
that-- I think this is mis-typed.
Mustafa typed it.
I guess he's typed this one, too.
This one shows the picture.
Yeah, this should be on that side in which
the angles are less than two right angles.
So let me just explain it, independent of the text.
The question is, what happens if you
have one line-- the line that's shown more or less vertical
here-- and two lines that cross it such
that the interior angles-- here shown as a and b--
are on one side less than pi.
Less than two right angles is the way you could describe it.
Then, as you can see from the picture,
these lines will meet on this side
and will not meet on that side.
And that's what the postulate says
that under those circumstances where two lines cross a given
line such that the sum of the two angles
adds up to less than pi on one side
that the lines will be on that side
and will not be on the other side.
Yes?
AUDIENCE: What was his motivation
for making this the fifth postulate?
It seems kind of arbitrary.
PROFESSOR: OK, the question is what was Euclid's motivation
for making this the fifth postulate?
Well, I have to admit I haven't had many conversations
with Euclid so I'm not sure I know
the answer to that question.
But it was what was needed to basically complete geometry
as we know it.
So much of what you've learned in geometry
would not be there if there was not something equivalent
to this postulate.
But actually what I'm going to be talking about next
is there has been discovered there
are a number of substitutes for this fifth postulate.
Mathematicians studied for a long time
whether or not this postulate could
be derived from the others since it seems so much more
complicated than the others.
And there was a strong desire during thousands
of years really-- at least over 1,000 years--
among mathematicians to try to prove the fifth postulate
from the first four postulates.
And nobody ever succeeded in doing that.
And we now are pretty clear that it's not possible to do that.
That the postulate is independent
of the other postulates.
It was discovered along the way that there
are a number of equivalent statements to fifth postulate.
And you could equally well have chosen any one of these four
statements that are illustrated in these pictures.
And I'll give you words one by one
for what these alternative versions of the fifth postulate
would be.
A, up here, illustrated there, says that if a straight line
intersects one of two parallels, meaning two parallel lines--
so this is the one line intersecting
two parallel lines.
If it intersects one of them as the heavy part of the line
shows, then the theorem says that if you continue that line
it will always intersect the other.
And certainly obvious from the picture, that's the way
it works.
But that's equivalent to the fifth postulate
and not provable from the other four postulates.
A second statement-- b is the one
that's illustrated there-- is the one that I remember
learning when I was in high school, I think,
which says that if you have one line and another line parallel
to it-- or rather, I'm sorry another point off the line
that there's one and only one line through that point
parallel to the original line.
So that is yet another statement of this famous fifth postulate.
Number c is less obvious.
But it turns out that once you go away
from Euclidean geometry, your space always
has a built in scale.
So things are not scalable.
One example I might mention at this point of occurred space
is, say, the surface of a sphere.
And the service of a sphere has some fixed size.
So if you have a figure of one size,
and you wanted, on the surface of this sphere,
to make a figure 5 times bigger, it might not fit on the sphere
anymore.
So you can't always make a figure bigger
on the surface of the sphere.
In fact, you could never make a figure
bigger without bending in some way
on the surface of the sphere.
So that gives rise to this third statement
of the fifth postulate, which is that, given any figure,
there exists a figure similar to it of any size.
And by similar it means that for polygons they're
similar if the corresponding angles are equal to each other
as they're supposed to be on those two images.
And the corresponding sides are proportional to each other.
So a similar figure is just a blow up--
a rescaling-- of the original figure.
And you can only do it if the fifth postulate holds.
You can do it on a flat space but not on a curved space.
And I think that is a less obvious version
of the fifth postulate.
And finally, the fifth postulate is
linked to the behavior of triangles.
We all learned in Euclidean geometry
that the sum of the three angles of a triangle is 180 degrees.
That is a crucial theorem of Euclidean geometry that
depends directly on the fifth postulate and is, in fact,
equivalent to the fifth postulate.
So you could assume it and forget the fifth postulate
and still prove everything.
So the fact that-- actually, I'm sorry.
It is equivalent to the fifth postulate.
But you don't need to assume that it's
true for every triangle to prove the fifth postulate.
It turns out that sufficient and mathematicians always
look for the minimum axiom to be able to prove
what they want to prove.
The minimum version of the axiom for triangles
is to simply assume that there's just one triangle who's
angles add up 180 degrees.
And if there exist just one triangle whose angles add up
to 180 degrees, then the fifth postulate
has to hold-- turns out-- which is not so obvious, again.
So these are all different, equivalent ways
of staying in the fifth postulate.
But it's pretty clear now that it's not
possible to prove the fifth postulate from the first four.
Any questions about that?
OK, so a little bit of history now.
The first person, apparently, to seriously explore
the possibility that the fifth postulate might be wrong
was a Jesuit priest named Giovanni Geralamo Saccheri.
I'm sure other people can pronounce
it much better than I can.
And in 1733, which is the same year he died,
he published a study of what geometry would be like
if the fifth postulate did not hold.
And he titled it- this is a lot title, which
I don't know how to pronounce really,
but in English it's apparently translated
as Euclid Freed of Every Flaw, treating the fifth postulate
as kind of a flaw in Euclid's axiomization of geometry.
Saccheri was, in fact, convinced that the fifth postulate
was true.
He didn't really want to consider the possibility
that it was false.
But he was nonetheless exploring the possibility
that it was false because he understood
the concept of a proof by contradiction.
He was looking for a contradiction
to be able to prove that mathematics would not
be consistent if you assume the fifth postulate was false.
And therefore, the fifth postulate
would be proven to be true.
So he was exploring what would happen
if the fifth postulate was false,
looking all the time to find some inconsistency,
and was not able to find any.
So he considered all of this a failure.
But he published it anyway.
And that's the publication front page.
The next person to enter the stage-- or actually
three people together, but I only
have a nice transparency on Gauss.
Gauss, Bolyai, and Lobachevsky went on
to seriously explore the possibility of geometry
without the fifth postulate, actually assuming
that the fifth postulate is false,
developing what we call GBL, Gauss-Bolyai-Lobachevsky
geometry.
Gauss was a German mathematician.
These were the years he lived.
He, in fact, was born the son of a poor, working class
parents, which I found a little surprising.
We kind of think of scholars in those early years
as being gentleman who were part of the nobility.
But Gauss was not but, nonetheless,
went on to be one of the most important mathematicians
of his age.
One of the other things that surprised me
and to be honest I just learned all this from Wikipedia.
I'm no real expert on the history.
But they gave a list of Gauss' students.
And they included that Richard Dedekind, Bernhard Riemann,
Peter Gustav Lejeune Dirichlet, which is the name I remember,
Kirchhoff, and Mobius.
So quite a list of famous mathematicians.
So I have to admit, when I read that, I was just
about to send off an email to all of my former students
saying, look, what's happening here?
You're not competing at all. [LAUGHING]
But I decided not to do that.
It would be impolite.
And who knows?
Maybe 100 years from now my students
will be as famous as these guys.
You never know.
We can plan, I hope.
OK, so the other guys involved in this and they were all
working independently were Bolyai who was, I think,
a Prussian military man, actually, and Lobachevsky
who was also a professional mathematician working
in the university.
The three of them independently developed a geometry
in which the fifth postulate was assumed to be false.
There are two ways it could be false.
In version with the triangles, for example, a triangle
could have more than or less than 180 degrees.
Since there were assuming that the fifth postulate was false,
it meant they had to be assuming that every version that we just
talked about of the fifth postulate
is false because they are all equivalent to each other
and these people realize that.
So in particular in the Gauss-Bolyai-Lobachevsky
geometry, there are infinitely many lines parallel
to a given line.
And no figures of different size are similar.
And the sum of the angles of a triangle
are always less than 180 degrees or pi,
depending on whether you're a radian person or a degree
person.
Now I should mention here that the surface of the sphere
is, in fact, a perfectly good example
of the non-Euclidean geometry.
But for some reason it was not taken seriously
by mathematicians until long after these guys
were doing their work.
And part of the reason, I guess, is that the surfaces three
evaluates not just one of Euclid's axioms
but two if we go back to Euclid's axioms.
The second of Euclid's axioms said that any straight line
segment can be extended indefinitely
in a straight line.
And the surface of the sphere and the analog
of the straight line is a great circle.
And if you extend the great circle,
it comes back on itself.
So the surface of the sphere violates the fifth postulate.
But it also violates the second postulate.
But still perfectly consistent geometry,
and it is a non-Euclidean geometry.
And on the surface of this sphere,
these statements all kind of reverse.
Instead of having infinitely many lines parallel to a given
line, you have no lines that are parallel to a given line.
Remember, lines are great circles and lines are parallel
if they never meet.
But any two great circles meet.
So there are no lines parallel to a given line
in the geometry of the surface of the sphere.
It's, again, true that no figures of different size
are similar.
That has to be true for any any geometry
where the fifth postulate was false.
The last one, again, has a choice.
And it's the opposite choice for a sphere
as for the Gauss-Bolyai-Lobachevsky
geometry.
In the Gauss-Bolyai-Lobachevsky Bolyai- geometry,
the sum of the angles is always less than 180 degrees.
If you picture a triangle and a sphere,
you can imagine that the edges look like they bulge.
And the sum of the angles on the surface
of the sphere of a triangle are always
greater than 180 degrees.
The easiest way to convince yourself
that that's true in at least one important case
is to imagine a triangle.
Here's a sphere.
Everybody see the sphere?
There's the North Pole.
There's an equator.
Imagine a triangle where one vertex
is at the North Pole and the liver disease or on the equator
and the triangle looks like this.
And these are both 90 degree angles down here.
So you already have 180 degrees.
And any angle you have on top just
adds to the 180 degrees putting you
above the Euclidean value of 180 degrees.
So for a sphere it's always the opposite
of this greater than 180 degrees.
OK, continuing with Gauss, Bolyai and Lobachevsky,
their work was based on exploring the axioms of Euclid,
assuming the reverse of the fifth postulate
in any one of its forms.
And they proved theorems and wrote,
sort of, their own versions of Euclid's elements.
But that still left open the question
whether or not all of this was really consistent.
That is, from the thinking of Saccheri
that we already talked about, there's always the chance
that you might find the contradiction even if you
haven't found a contradiction yet.
And Gauss, Bolyai and Lobachevsky
didn't really have any way of answering that.
They didn't really have any way of knowing
whether if they continued further
they might find some contradiction.
So what they did certainly extracted the properties
of this non-Euclidean geometry.
But it did not really prove that the non-Euclidean geometry
was consistent.
That happened slightly later in an argument
by Felix Klein, who was the same Klein as the Klein bottle,
by the way.
And that was his famous paper on this
was published in 1870 somewhat later than the early work
that we're talking about.
And what he did is he gave an actual construction of the GBL
geometry.
And by construction I mean in terms
of coordinates using coordinate geometry ideas, which
were originally developed by Descartes.
That's why we call them Cartesian coordinate systems.
So what he gave as a description of the Gauss-Bolyai-Lobachevsky
geometry was a space that consisted of a disk
with coordinates x and y just as Descartes would have done.
He restricts himself to the inside of the disk.
So he restricts himself to x squared
plus y squared less than 1.
And what he gave was a function of two points
in this disk where the function represents
the distance between those two points.
He decided that distances don't have
to be Euclidean distances if we're
trying to explore non-Euclidean geometry.
So he invented his own distance function.
And it's pretty complicated looking.
The function he came up with was that the cosh of the distance
between points 1 and 2 divided by sum number a-- and a
could be any number-- is equal to 1 minus x1 times x2--
these are the two x-coordinates for points 1
and 2-- minus y1 y2 over the square root of 1 minus
x1 squared minus y1 squared.
Sorry, it's the product of two square roots in the denominator
here.
And the second square root is the same thing
for the two coordinates 1 minus x2 squared minus y2 squared.
So this formula is certainly not obvious to anybody.
But Klein figured out that this actually
does-- the geometry described by these formulas--
reproduce completely the postulates
of the Gauss-Bolyai-Lobachevsky geometry,
including the failure of Euclid's fifth postulate.
And since this boils down to just algebra,
if algebra is consistent, it proves
that the Gauss-Bolyai-Lobachevsky
geometry is consistent.
Now, as I understand it, nobody can prove that any of this
actually is consistent.
People prove relative consistency.
So in the assumption that algebra is consistent,
theorems about the real numbers are consistent.
Felix Klein was able to prove that
the Gauss-Bolyai-Lobachevsky geometry is consistent.
And this was really the beginning
of coordinate geometry.
I'm sure all of you are rather familiar
with coordinate geometry.
It's a standard topic now in math courses
and even in high school.
And this is really where it began.
Euclid had no idea that it was any value
in trying to represent geometric quantities as equations.
Euclid did everything in terms of theorems.
But this opened up a whole new door
for how to discuss geometry.
So the geometry is a slide that just shows the same formula.
And I guess this is supposed to be an image of the disk.
One thing I should point out, which
I forgot to point out, which is that although this disk looks
finite, it really is an infinite space that's being described.
And one can see that by looking carefully
at the distance functions.
As either one of these two points--
point 1 or point 2-- approaches the boundary
x square plus y square equals 1, this square
root denominator blows up.
So the distance between a point and another point that's
approaching the boundary goes to infinity
as that point approaches the boundary.
So boundary's actually infinitely far away
even though in coordinates they are still
x squared plus y squared equals 1.
So this introduces another important concept,
which we'll be using in general relativity, which
is the coordinates don't represent distances.
Distances could be very different
from the way the coordinates look.
So that boundary, even though it looks
like it's right there on the blackboard,
is actually very far away from the other points.
OK, so after Klein, the important new idea
that Klein introduced was, first of all,
the explicit construction but also the general idea
that you can describe geometry not by giving postulates
but instead by actually doing a construction where
you've given names to the points in terms of coordinates
and you describe what happens geometrically
in terms of some distance function which describes
the distance between two arbitrary points.
And Gauss went on to make two other very important
observations about geometry, which
has become essential to our current understanding
of non-Euclidean geometry.
So let me mention two other ideas that Gauss introduced.
The first one was the distinction
between what he called inner and outer properties
of a curved surface.
His curved spaces were all two dimensional,
so they were surfaces.
So this is most easily described for say
a surface of a sphere where we can all
visualize what we're talking about.
The surface of a sphere we visualize in three Euclidean
dimensions and we think of its properties
as being determined by that three dimensional space.
And the geometry of that three dimensional space.
And of course the three dimensional space
is Euclidean that we're embedding our sphere into.
But the non-Euclidean aspects are all
seen in the geometry of the two dimensional surface.
Figures drawn on the surface as if the rest of the three
dimensional space did not exist.
And that is this key idea of inner versus outer properties.
The outer properties of the sphere
are properties that relate to the three dimensional
space in which the sphere is embedded.
But what Gauss realized is that there's
a perfectly well defined mathematics contained entirely
in the two dimensional space of the surface of the sphere.
You could discuss it without making any reference
to the three dimensional space around it if you wanted to,
it's just a little more complicated to be
able to do that.
But we will in fact be doing it explicitly very shortly.
And all it amounts to from our point of view
is assigning coordinates to the points on the surface
of the sphere and the distance function for those coordinates.
And then one has a full description
of this two dimensional space consisting
of the surface of the sphere which no longer needs make
any reference to the third dimension
that you imagined the sphere embedded in.
So the study of the properties of that two dimensional space
is the study of the inner properties of the space.
And if you care about how it's embedded,
that's called the outer properties.
And Gauss made it clear in the way he described things
that from his point of view, the real thing that mathematicians
should be doing is studying the inner properties.
The outer properties are not that interesting.
So that's one key idea that Gauss introduced.
And the second is the idea of what we now call a metric.
And there are really two pieces to this.
The first of which is that instead
of giving macroscopic distances, which is what Klein did,
he told you how to write down a formula for the distance
between any two points, Gauss realized
that things could become more interesting
if instead of trying to immediately write down
a function for the distance between two arbitrary points,
you can find your attention to very nearby points.
And consider the distance between two
far away points as just the length
of a line that joins them where every little segment
in the line is a short distance which is governed by the rules
that you've made up for short distances.
So if you understand short distances,
long distances are obtained just by adding.
That's the basic idea.
So only short distances are needed.
And then there's another important idea
which is that the short distances themselves
for an interesting class of spaces
should not be some arbitrary function of the coordinates
of the two points but should in fact be a quadratic function.
And in two dimensions, a quadratic function
mean something like this, where gxx is just
any function of x and y, x and y are the two coordinates
of the space.
gxy is also just any function of x and u.
gyy is any function and gxx multiplies the x squared,
xy multiplies the x times dy and g sub yy multiplies dy squared.
And there's no terms portion for dx by itself or dy by itself
or dz by itself.
And there's no terms proportional
to the cubes of those quantities, it's all quadratic.
That's the assumption.
Now what underlies that assumption is not
that all spaces have to have this property.
This does narrow one down to a particular class of spaces.
But one way of characterizing that class
is that what Gauss understood is that that class, the class
of spaces described by a metric like this,
are precisely the class of spaces which are locally
Euclidean, meaning that even though the surface is curved,
any tiny little patch of it looks flat and can be covered
by Euclidean coordinates, redefining coordinates
in that one little patch only where the metric in the one
little patch would just be the Euclidean metric given
by the Pythagorean theorem of Euclidean geometry,
dx prime squared plus dy squared,
which would be the distance function for a flat space
and ordinary Cartesian coordinates.
And it turns out that saying that this
is true in every microscopic region
is equivalent to saying that the metric over the entire space
can be written as a quadratic form,
meaning one that looks exactly like that.
And spaces that have this property
are now usually called Riemannian spaces,
even though it was Gauss who first identified them.
And all the spaces that we deal with in physics,
in particular the spaces we deal with general relativity,
will be Riemannian spaces.
Or sometimes they're called pseudo Riemannian spaces
because time is treated a little bit differently in physics.
And the word "Riemannian" was really
built on spatial geometry.
But the word "pseudo" only changes
the fact that this becomes the Lorenz metric, if you know what
that means, instead of the Euclidean metric.
But the same idea holds, which is
that the spaces that we're interested in
are spaces which locally look exactly like flat space.
And that implies that globally, you can always write down
a metric function which is quadratic, as Gauss said
So the metric should be a quadratic function that
specifies the distance only between infinitesimally
separated points, not finitely separated points
and should have the form of ds squared
is equal to some function of x and y times
dx squared plus a different function of x and y times dx
times dy plus a different function of x and y times dy
squared.
So I'm just writing the same form that was on the side.
This is a very important formula.
Incidentally, the two that appears here
is only because when we write this in a different notation,
this term will occur twice, once as dx times dy,
and once as dy times dx.
And the two times it occurs are equal to each other,
so here they're just collected together with a factor of two
in front.
Yes.
STUDENT: Just to make sure, those three different functions
you wrote, those subscripts aren't
supposed to mean partial derivatives, right?
PROFESSOR: No, that's right.
The subscripts and this expression
only mean that this multiplies dx squared,
so it has subscripts xx.
And this multiplies dx times dy, so it has subscripts xy.
That's what the subscripts mean and nothing more.
So the subscripts only mean that they're
the things that appear in that equation.
Yes.
STUDENT: It seems like the metric is giving us
distance in terms of an infintesimal displacement,
but then a locally Euclidean space is already tangent
infintesimally, so how are we relating the local metric
with the global metric?
PROFESSOR: OK, the question is how
do we relate the local metric which I say
is Euclidean to the global metric?
And the answer I think for now I will
stick to just giving kind of a pictorial answer based
on the picture here.
That is once you know the distances between any two
points in a tiny little patch here,
it's then always possible to construct coordinates,
here called x prime and y prime, such
that the distance between any two points as calculated
from the original metric, which is the one here,
is exactly the same as the distance you
get using this metric.
And the claim is that you can always
define coordinates x prime and y prime which make that true.
That claim is not absolutely obvious.
But it's something you can probably visualize if you just
imagine that every little tiny piece of this curved surface
looks like it was just a flat surface and then
a flat surface you know that you can write down
a Cartesian coordinate system, which will have the Euclidean
metric.
But it's only an intuitive statement,
proving it is actually harder.
Yes.
STUDENT: [INAUDIBLE] bottom formula [INAUDIBLE]
with positive curvature if we analogize to second derivative.
PROFESSOR: I'm sorry, say that again.
STUDENT: If we analogize gxx, gyy,
in the bottom formula there, [INAUDIBLE] positive curvature
[INAUDIBLE] second derivative--
PROFESSOR: Yeah OK.
So you're asking does this mean that we
have two states of positive curvature?
STUDENT: Bottom right.
PROFESSOR: Bottom right, oh this.
These are Mustafa's slides.
I forgot to say that.
You can tell from the style, these are not my slides,
these are Mustafa's slides.
And I don't know what he meant by this.
You're right, this does-- well you
do want the metric to be positive
definite, which is not the same as saying the curvature
is positive.
And I think this might just be the condition
that the metric is positive definite,
that this expression will always be positive.
Yeah.
I'll bet that's what it is, I don't know for sure.
I'll bet that's what that condition is about.
And you do want that, the metric had better
be positive definite for spatial geometries.
In fact, in general relativity, where it's a space time metric,
it will not be positive definite.
For reasons that we'll talk about later.
But for geometry, the metric should be positive definite.
All distances should be positive.
OK.
So that ends my slides.
So now I'll continue on the blackboard.
OK next I wanted to say a few general comments
about general relativity.
General relativity was of course invented by the Einstein
in 1916.
It's a theory that he was working on for about 10 years
after he invented the theory of special relativity.
To understand what's going on there,
you want to recognize that special relativity is
a theory of mechanics and electrodynamics
which was designed to be consistent with the principle
that the speed of light is always
the same speed of light independent of the speed
of the source of light or the speed of the observer
of the light.
And it's of course not easy to do that,
because you think that if you move relative to something
else that's moving, that you would see its velocity change.
So in order to make a theory that where the speed of light
was an absolute invariant, Einstein
had introduced a number of things.
And we talked about those at the beginning
of the course, the three primary effects built
in to special relativity.
Time dilation, length contraction,
and new rules about simultaneity,
and how things that look simultaneous to one observer
will not look simultaneous to other observers in a very
definite, well defined way.
So by inventing these rules, Einstein
was able to devise a system which
was consistent with the idea that the speed of light
always looked the same to all observers.
And at similar times, the Michelson Morley experiment
seemed to show that that was in fact the case
and ultimately there's a tremendous amount of evidence
verifying that what the hypothesis that Einstein was
pursuing is the right one for the way nature behaves.
The speed of light is invariant.
But what was lacking in Einstein's formulation
of special relativity was any version
of a consistent theory of gravity.
Gravity had a well defined description known since Newton.
But Newton's description was a description
of a force at a distance.
And that is intrinsically inconsistent with a theory
like special relativity, which holds that simultaneity
is itself relative.
In Newtonian physics, the force of gravity
is equal to Newton's constant times
the product of the two masses that
are interacting with each other divided
by the square of the distance and then times some union
vector pointing along the line that joins the two particles.
But for that formula to make any sense at all,
you have to know where the two particles are
at the same instant of time.
And this r is the distance between the locations
of the two particles at some instant of time.
And this r hat is a unit vector that
points along the line joining those two particles where
you've pinned down where those particles are
at one instant of time.
But we know from the very beginning
that in special relativity the notion of two things happening
at distant points at one instant of time is ambiguous.
Different observers will see different notions
of what it means for two events to happen
at the same time across this distance.
And that means there's no way to make sense out
of Newton's Law of gravity in the context
of special relativity.
You can't modify it by just changing
the way the force depends on distance.
You really have to change the variables
that it depends on from the beginning.
Yes.
STUDENT: Is that equivalent to saying the other particle can't
know how much mass of one is [INAUDIBLE]
limit of the information of the speed of light?
PROFESSOR: Is that equivalent to saying that one particle can't
know what the mass of the other particle
is because of limitation.
Yeah it's the distance, it's not the mass.
Because the mass is preserved in time.
So one particle could've measured
the mass of the other particle at an earlier time
and it would be reasonable to infer,
given the laws of physics, that we
know that would stay the same.
But one particle has no way of where
the other particle is at the same time.
And not only does the particle not have any way of knowing,
but even an external observer can't know.
Because different external observers
will have different definitions of what
it means to be at the same time.
So there really is no way that this could work.
Now I should mention that it is still
possible to have action at a distance theories
which are consistent with special relativity.
And in fact, Maxwell's equations can be reformulated that way.
The easiest way to make things consistent
with special relativity is to describe interactions
in terms of fields.
And that is really what Einstein did originally
in special relativity.
He was thinking about light, he was
thinking about Maxwell's equations
and he was thinking very explicitly about Maxwell's
equations.
And in the Maxwell description of electromagnetism,
particles at a distance don't directly
interact with each other.
But rather, each particle interacts
with the fields around it and that is a local interaction.
A particle interacts directly only
with the fields at the same point.
But then those fields obey wave equations
that can propagate information.
And the fact that you have an electron
here can create a field which then
exerts a force on an electron there.
And the force on the electron here
depends only on the electric fields here
or the magnetic fields as well.
The particle is moving, but everything is completely local
and the description of electromagnetism as
given in the form of Maxwell's equations.
And Maxwell's equations are completely relativistically
invariant.
And that was part of Einstein's-- it was really
the key part of Einstein's motivation in constructing
the theory of special relativity in the first place to make
Maxwell's equations invariant.
That they held in every frame.
It is still possible though and worth
recognizing that it's possible to reformulate electromagnetism
as an action at a distance theory.
And it is in fact described that way
in volume one of the Feynman lectures
for those of you who've looked at the Feynman lectures.
In order to make that work, you have
to complicate things in a pretty significant way.
So I'm going to draw here just a space time diagram.
X and CT, CT going up that way, x going that way.
So in this diagram the speed of light
would be a 45 degree line.
And let's suppose we have two particles traveling
in this space.
A particle that I will call a and a particle
that I will call b.
Being very original with these names.
If we wanted to know the force on particle a
at a certain time t indicated by this dotted line,
I guess I'll label t to make it as clear as possible.
Feynman gives us a formula where we can determine it
solely in terms of the motion of particle B
without talking about fields at all.
But the formula does not depend on where b is at the same time.
And it could not if this is going
to be a relativistic description.
But instead, the way this action and the distance formulation
works is when imagine drawing a 45 degree line backwards,
meaning a line that light could travel on,
and one sees where that intersects
the trajectory of particle B. And that time is t prime.
And the word that's used for that symbol is retarded time.
It's an earlier time.
It's exactly that time which has the property that if particle
b emitted a light beam at that time,
it would be arriving at particle a at just the time
t that we're interested in the time when we're
trying to calculate the force on particle a.
And what Feynman gives you in volume one if you look at it
is a very complicated formula that
determines the force on particle a
in terms of not only the position of this particle
at time t prime but also its velocity and even
its acceleration.
But if you do know the position, the velocity,
and the acceleration of this particle,
and of course the velocity of particle a,
you can determine the force on a.
Not obvious, but it's true.
But that's certainly not the easiest way
to formulate electromagnetism.
And that's not the way most of us have learned,
unless you've learned by starting
by reading volume one of Feynman.
But most of us learn Maxwell's equations
as differential equations.
Where information is propagated by the field from one
point to another.
In the case of general relativity,
one has the same problem.
How can you describe something which--
and the only approximate you initially
know is an action and the distance,
how can you describe it in a way that's
consistent with relativity?
And the idea that simultaneity is not a well defined concept.
So that was the problem was Einstein
was wrestling with for 10 years, how
to build a theory of gravity that
would be consistent with the basic principles
of special relativity.
And the result of those 10 years of cogitating
is the theory that we call general relativity.
And it's essentially a theory which
describes gravity as a field theory
similar to Maxwell's field theory
where all interactions are local.
Nothing interacts at a distance, but particles interact
with fields at the same point, the fields
can propagate information by obeying wave equations,
and the fields then at a distant point
can exert forces on other particles.
But in the case of general relativity, what Einstein
concluded was that the fields that were relevant,
the fields that described gravity,
were in fact the metric of space and time.
So general relativity is the field theory
of the metric of space and time.
And gravity is described solely as a distortion
of space and time.
And that's what general relativity is about
and that's what we will be learning more about.
Now as I said earlier, now I can say it perhaps more
explicitly, what we will be learning about
is how to describe the curvature of space time
as general relativity describes it.
We will learn how that curvature affects things
like the motions of particles.
But we will not in this course learn
how the presence of particles and masses
affects the curvature of space time.
That you'll have to take it in a relativity course to learn.
OK, I think that's where we'll be stopping today.
Just doesn't pay to start a new topic with a minute and a half
left.
But let me just ask if there are any questions.
OK, well class over, I will see you
folks in a week, because there's no class next Tuesday.