So having established that if we're going to study the entire universe, we're going
to assume that we are studying a homogeneous isotropic place.
We are going to try to understand what that tells us about the nature of general
relativity in a space that is isotropic and homogeneous.
And so the first thing we need to ask is, what does this tell us about space-time?
So what kind of a space time does describes a universe in which space is
homogeneous and isotropic at all times. And it turns out that in the context of
general relativity. Isotropic and homogeneous means that
there is a particular set of coordinates that we can find in which the curvature
is constant at every point, and this need not mean by the way, that just because a
space is the same everywhere, that it is flat.
In fact, I have drawn here, or someone has drawn here on the right, figures of
three two-dimensional spaces which have the property that they look the same as
observed from every point. These are three surfaces of constant
curvature. The one on the bottom is of course the
Euclidean Plane. It's curvature is exactly zero, and, this
is a familiar space, and it's one of the properties of the Euclidean plane, that
it's invariant under translations. It looks the same from every point.
It's invariant under rotations, has no preferred direction.
Over here at the top, we have a 2 sphere, a 2-dimensional surface given by the
outside of a ball. This is again, a universe if this were
our 2-dimensional space, in which every point is the same as every
other point. In this case of the sphere, this is
obvious because of its invariance under rotations in 3 dimensions.
You can just take this ball and twist it around, making any point the North Pole.
The North Pole is not distinguished by anything.
Unless you're on earth, in which case it's distinguished by the rotation.
But, the ball in itself has no distinguished North Pole.
And, this is a surface of constant positive curvature and then over here is
this, saddle-like shape which has the property that its curvature is constant
and negative. It is far less obvious, why it is that all points on this shape are
the same as every, as all other points, or why there is not a preferred
direction, but there is indeed a, notion of rotation in this, hyperbolic plane
that, preserves, all of the geometry. And so these in fact in 2 dimensions are
the only 3 examples of spaces of constant curvature.
Any space of constant curvature is one of these 3 and in 4 dimensions, in 3
dimensions sorry. We live in a 3 dimensional space, so our space at any
given time, it turns out is analogous, is one of three types of spaces, analogous
to these three. It can be a space of constant positive
curvature, which is a three dimensional extension of the surface of a ball.
In fact, it's called the three sphere, if you could imagine.
A 3-dimsion- a 4-dimensional ball. This would be the 3-dimensional outside
of a 4-dimensional ball and like the 2-dimensional outside of a 3-D ball it
has positive curvature that's constant and all points in all directions are the
same as all others. There's a negative sort of hyperbolic
version of the same thing and then there's plain old flat Newtonian,
Euclidian 3-dimensional space. And space at any given time is one of
these, how do we distinguish these? Well here remember objects are drawn as
surfaces embedded. You're meant to imagine that ball sitting
inside flat three-dimensional space because that's where we're used to
thinking. Remember that I am talking about the ball
in and of itself. You cannot, there is no inside to this
ball or outside. There's no, anymore than there is below
or above the plane. Our, universe is 3 dimensional.
There, it's not sitting, as far as I know, inside any higher dimensional
universe. Though, maybe,
on an optional talk, we'll have that discussion about maybe it is.
at the moment, as far as we know it is not,
so all there is is the 3 dimensions. So what intrinsic property of these
spaces can tell us about the curvature? remember, that it's the properties of the
geodesics, the weird straight lines on curved spaces, and the easiest sort of
most beautiful distinction in the, that the geodesics tell us between 0,
negative, and positive curvature is that while geodesics in the plane are our
favorite Euclidean straight lines and they satisfy the nice axioms of euclidean
geometry. For example, the sum of the angles of a triangle is 180 degrees.
on this saddle point, and it's not completely clear in this picture, but on
the saddle points, geodesic. I'll, I'm going to exaggerate the effect.
Geodesics, in fact, diverge, which tells you that the sum of the angles of a
triangle drawn on a space of negative curvature is going to be less than 180
degrees. On the other hand, on the sphere,
currently geodesics converge. Think about two lines of longitude
leaving the pole. They meet again at the other pole.
And so, the sum of the angles, of a triangle drawn on a sphere will in fact
be larger than 180 degrees. But note that a very small triangle drawn
on the sphere, or a very small triangle drawn on the hyperbola, will appear
approximately to have, will have a sum of the angles, which is very close to 180
because a small segment of anything looks like flat space. the order of magnitude
that tells you how big a triangle needs to be before you measure distinguished
before you measure a distinction between flat space and one of these curved
spaces, is the magnitude of the curvature denoted here by R0.
There is no curvature here. If R0 is big, then the curvature is
large. If R0 is big here, this is a very curved
sphere, which means roughly that the ball it is wrapping is rather small, so curve,
it's more curved. And so all of the positive curvature,
positive constant curvature spaces that can possibly exist are not exactly all
the same. They are all a 3-dimensional version of
the sphere, but of course they can have different sizes.
And so space that if, you decide that space is positively curved, then space at
any given time is given by from the 3-dimensional analog of the sphere, and
then the size of the sphere can change. And as the size of the sphere changes its
curvature can change. A bigger sphere, as I said before, has a
smaller curvature. You need a bigger triangle before you
measure deviation from 180 degrees. Likewise, you can make this hyperbola,
bigger or smaller. And a bigger hyperbola had, had a smaller
curvature. And if you look at the dimensions, it's
the area of the triangle that determines the deviation of the angles.
So if you make this feel twice as big. The curvature becomes 1/4.
the curvature it was previously, so we have these 3 types of universes.
Positive curvature. Positive curvature universes are closed
just like this 2 sphere in the sense that there is no way you can get 2 observers
cannot get as far as they wish from each other.
If you try to go too far. You'd come back around the world, and
come back to where you were. this is not the case for the hyperbola.
In the hyperbola, it is clearly open and infinite.
there is no way to close that. and the geodesics are diverging.
And then we could possibly live in a flat, three dimensional space.
If we live in a flat three dimensional space.
Then, it looks at least infinite and open and so, the meaning of this question mark
will become clear. So, space at any given time in a universe
that is homogeneous and isotropic Is either positively curved with constant
curvature negatively curved with constant curvature or flat.
And our job will be to determine which of these three situations obtains in our
universe and if it's not flat then to determine the actual curvature r 0.
Now the space time geometry of a space time that at any given time is given by
one of these kinds of spaces was described by a whole collection of
people, Friedman, Lemaitre, Robertson, Walker, and goes under the name
Robertson-Walker geometry. Robertson-Walker geometry is the geometry
of space time. So now we are talking about 4-dimensional space-time, in which
space is homogeneous. So at any given time space is say, a
3-sphere, but the size of the sphere, 3-sphere, can vary with time, and so
there is a scale factor that relates. Pick the size of one sphere to be one,
and then bigger spheres will have a bigger than one, smaller times one, space
is smaller, will have a smaller than one. And, because all we're doing in scaling
all of space, the distances between observers that occupy in some sense a
fixed position on the sphere. Pick a point on the sphere and then stay
at that same point when you enlarge the sphere.
the distances between observers sitting at fixed points on this fixed space, so
label points of space by observers and then distances between them will all
scale together when I make space bigger. And by convention we choose the scale
factor A, normalized so that A at T 0 which we call the present, is 1.
In other words, D 0 are the distances at T=0 and then if A of at some future time
is 2, that means that by that time, all distances have doubled and so on.
Now in this Robertson-Walker geometry, you can solve the equations of
electromagnetism for example and you find that light moves and this is not an
obvious fact but it's true, despite the fact that space is growing, light moves
along geodesics. In other words, the motion of a light beam is along these
straightest straight lines that you can draw in space.
In, if space is flat, like moves simply along straight lines,
if space is a sphere, like moves along great circles, and so on.
And it moves of course, this is a physical obvious fact,
with velocity c as measured by any observer anywhere.
So any observer measuring the motion of a light being in their vicinity will
measure of course the speed of light because that's the universal property.
We're talking about a relativistic theory.
And, put these two together, you find these spaces exhibit the kind of
cosmological red-shift that we already discovered underlies the Hubble
discovery. In other words, light that is emitted
from some position in space, at some time t, and is emitted as observed by an
observer at that event with wavelength lambda emission, will then be observed at
certainly some later time, obviously because emission always precedes
absorption or observation, and the ratio between the observed wavelength and the
emitted wave length is the ratio of the scale factors.
In other words, it is the property of the equations of electromagnetism, and we'll
see an example of this much later of how this works out.
That as a light beam travels through space time, if space time is, if space
itself is being scaled by some factor, than the wavelengths of light scale by
exactly the same factor. And this gives rise to the cosmological,
redshift. If a is increasing, then, the wavelength
at which you observe something at a later time, is larger than the wavelength that
which it was emitted. This gives you the cosmological redshift.
Note, this is not a doppler shift. this is the doppler, the, the, the, this
is the, redshift is observed, by observers and emitters and absorbers that
are at rest, if there are perculiar velocities, there will be a doppler shift
on top of this, but these are observers that are as far as I can discuss, address
this. They said at fixed points in this space, and it is the space, if you will,
that is growing. and so it does not require any motion,
and I don't want to think about it as a doppler shift.
And in particular, it is never a blueshift unless spaces shrink.
it is indeed a gravitational effect, but it is a little bit different from the
gravitational redshift we saw in the Pound-Rebka experiment.
Because in the Pound-Rebka experiment, again we had a redshift when light was
propagating up, but a blueshift when light was propagating down.
In this case in some sense, in an expanding universe, all light is
propagating up. There is only a redshift, in a
contracting universe. All light is propagating down, there is
only a blueshift. But if we're both in the same say
expanding universe. and A sees B's light redshifted, then B
will see A's light redshifted, because in both case it takes positive time for the
light to get from A to B or to get from A to B, or to get from B to A.
So does our observation of the Hubble expansion fit into this framework? Well
we have our scale factor, which describes the scale of the universe at time T.
It's a function of T, and for times near to the time T0, which by convention,
remember, is the present, I can approximate this by linear function.
I can approximate anything by linear function and I will call, so it's 1 when
t is equal to t0. This is the first order deviation. We're
going to skate very close to calculus this week.
I'll try to be as clear as I can when we get close to there and I'll designate the
coefficient by the letter H naught, and again this is expected to be valid in
some kind of Newton approximation when this coefficient is small.
I'll justify calling it H naught in a moment.
In this case, by our cosmological redshift understanding, that, if light is
emitted at some time before t 0 and observed, by convention, when I observe
something, I'll be observing it at the present time t0 unless I indicate
otherwise. And so the wavelength at which light is observed divided by the
wavelength at which it is emitted is the ratio of the scale factors. Here should
be A of t0, but a of t0 by convention is 1.
And, what is this? This is one over the scale factor at t emission. So it's 1
over 1 plus h0 times t emission minus t0. But, because this number is small, I can
use Newton's approximation. This is, 1 plus h0 times this inverse.
This is, with Newton's approximation, the same as 1-H0(Tem-T0) and 1-H0(Tem-T0) is
the same as 1+H0(T0-Tem). So this is the expression I'm going to
use it's valid. When H0 times t minus t0 is a small
number, and so to this order I can set 1 plus z equal to that, in other words, z
equals H0 times t0 minus t emission This is remember, a positive number t emission
proceeds t0. So, this is the red shift.
On the other hand, to the same order it'll turn out that the distance light
traveled is the speed of light times the time it is traveling.
This will not always be true because, the universe is changing.
We'll see how that gets corrected. And so, comparing this expression to that
expression, I find that indeed I obtained the Hubble Relation as a low order Newton
type approximation. That small distances and times, small
relative to what? To the amount, to the distance light could have traveled since
the Big Bang, times small relative to the time since the Big Bang.
And that those distances, and therefore, redshifts, small compared
to one, and for under those circumstances, I find the Hubble,
condition. And so, the Hubble constant is the
coefficient of this expansion of the scale factor.
That's the Hubble H0, because it's the Hubble constant today.
The Hubble constant need not, in fact, be a constant. At any other time,
I will define the Hubble constant, erasing all my scribbles, by if you have
a time t*. And for times t not to far from t*,
the Hubble, the skill factor is given by the skill constant at t* times 1 plus
some number times t minus t* , then I would designate this number as the Hubble
constant at time t*. So now imagine, if you will, that this
expression is in fact correct. It's not an approximation, it's an exact equation,
as we might have mistakenly thought last week.
Then what does this tell me about the Hubble Constant? Well you'd think the
Hubble Constant's not changing. Look, this equation always holds.
Yes, but remember the Hubble Constant is the coefficient of the fractional change
in a. So take some later time, and notice that
in a given time interval after that, for every second past, t*, the scale factor
increases by the same amount. But if t* is later than the present, it's
later than t0, then, a, by that time will have grown. And so, the Hubble constant,
which is, the, rate of relative growth of the scale
factor will in fact have decreased. So if you want to understand constant
expansion, to mean this equation where the scale factor is a linear function of
t, then that in fact means a decreasing Hubble constant.
In particular, we have no reason to assume the Hubble constant does not
depend on time and understanding it's time dependance will be something we'll
spend some work on, to find out what it does in our real
work. So, in our real universe.
So this is how Hubble fits into Robertson-Walker as a first approximation
to the actual cosmological redshift. we said that one of the issues in a
universe where distances are changing is what do you mean by distance?
So we will have 3, at least, different ways to define the distance between 2
points. One is, remember we have these co-moving
points and so we label each point on, say the sphere by some name, x and then the
distances are, the distances between the x's, and then
at previous times we have the same sphere with the same points except we considered
all sizes smaller, and in the future we'll have the same sphere with the same
points, but all sizes are bigger. And so the distance alone, the canonical
sphere, is D0. This is the coordinate distance, we call
it. you might think of that,
as the distance right now, because right now the scale factor is 1.
So if you want to define something that measures the actual distance between 2
objects right now, and in particular, will reduce for objects that are near
enough for me to make a measurement. For my right hand and my left hand, if
you ask what's the distance between my right hand and my left hand, if you apply
this crazy Friedman's, Robertson-Walker coordinate system, then this will be
measured by the distance in coordinates between the coordinates of this finger
and the coordinates of that finger, evaluated at the scale factor of 1, which
is, corresponds to right now. Okay.
But it's very rare that we want distances between objects that are very near.
And, in general, the coordinate distance between those 2 things is not going to be
a physical observable, because you cannot have time to lay a ruler, a large
fraction of the size of the universe, before the universe changes on you.
So what can you actually measure? Well, one thing we measure that is associated
to distance, is we measure the small angle formula.
We love our small angle formula. So, imagine that somewhere out in the
universe is a galaxy say, and the galaxy has size D, and the galaxy is at
coordinate distance, D0 for me. So imagine that we're in flat
space, in a flat 2-dimensional universe, so this is the universe at present, and
here is the galaxy and the galaxy is at a distance D0.
OK, now suppose that I want to make a small angle approximation and measure the
apparent angular size of the galaxy. Everything works fine.
Remember, light moves in straight lines in this coordinate plane.
that was our assertion and that is the facts about Robertson-Walker.
But however, when I look at this galaxy, remember, I am looking at this galaxy as
it was a long time ago. And so in fact, the distance that I am
looking at, if you will, is not the size of the galaxy itself, but in fact, the
size of the galaxy as it looks in today's coordinates.
So in other words, that galaxy is sitting at a particular point in the plane.
This is a measurement made in today's plane.
however, the two sides of the galaxy in today's plane the galaxy of course
doesn't expand because it's gravitationally bound.
We'll talk about that. But were the, did a, had I put two points
on the 2 sides of the galaxy, they would have been much bigger today, than they
were then. Therefore the distance between the two
sides of the galaxy in 2 day's coordinates is the size of the galaxy
times the ratio of today's scale factor. But the scale factor at the time of
emission, and if I measure our redshift z for the
light, then the ratio of the scale factors is 1 plus z.
In other words, everything looks, bigger in an expanding universe, because it
looks as big as it would have been, would have looked, had its 2 edges expanded
with the universe. that gives me this expression for the,
small angle formula. I define an angular size distance by,
angular size distance is the distance that you'd put in a small angle formula,
if you wanted it to make sense. If D over D were alpha over 1 radian,
over 206265 arc seconds. And then, from this expression, we see
that the angular size distance is related to coordinate distance or physical
distance today by the expression, take that, divide 1+z, where z is the redshift
at which you observe the light from the object whose angular size you are
measuring. Of course for nearby objects, z is 0.
This is the same old small angle formula. You don't have to change your
understanding of the dimensions of the moon but for distant objects, this is
what the expansion of the universe tells you. You measure an angle, that is the
angle that you would have seen were the two sides of the objects freely moving
and freely falling and expanding. So this is one notion of distance.
There's another equation in which we use distance a lot. One was the small angle
formula. The other of course is the brightness equation.
B is L over 4 pi D squared. is it, which D should we plug in, if we
have an object of no luminosity, and we measure it's brightness, to what D does
that core, th equation correspond. In other words, for what definition of D
does this equation hold? Is it D0, is it Da? It turns out, it's none of the above.
Its something that we call luminosity distance and lets compute the luminosity
distance. So what really happens is that well, 4 pi
is to radians 4 pi. but if you have an object that's at a
distance, that's at a distance D0 in coordinates,
then of course you find D0 squared here. However, remember this is the energy that
the objects emits per second. Now all of those photons that the object
is emitting are getting red shifted. That means every photon that gets emitted
I am observing with an energy that is less than it was emitted with.
And so that means that the factor is the ratio of the scale factors, so that means
that brightness is less than you would have expected at that distance, by a
factor of 1 plus z because of the redshift.
Further more if the object is emitting say, one photon a second and I'm
observing those photons that coming in redshifted.
They're also separated by more than a second, because the redshift means that
all times are dilated in the past. So, there is a, the photons are coming
less frequently, and each carries less energy.
That gives me a total of 2 factors of 1+z.
And again, nothing you can do in a slightly more rigorous way by a
relativistic calculation but this is exactly the logic that applies so that
the luminosity distance is given by the distance that you put in the denominator
that relates brightness to luminosity. It has two factors of 1+Z in it, and so I
have two different definitions of distance that agree.
Of course they're both d0 at small distance but the luminosity distance is
d0 times 1 plus z, because its square is d0 times 1 plus z squared.
And the angular the size distance is z0 over 1 plus z.
this means objects appear dimmer because of redshift than they would, and this
means that objects appear bigger because of redshift than they otherwise would.
And so, the bottom line is since we can't really measure this, but you can
sometimes measure both this and this. They are related by 1+Z^2, a relative
factor, and this sometimes you can measure.
You can use that to verify that we actually observe, live, in a
Robertson-Walker universe. Furthermore, there's another property
that is okay. z, tells you what happens to spectral
lines emitted by a star. a spectral line, emitted from a star at a
particular distance, will be redshifted by the redshift appropriate to that
distance given the distance redshift relation.
What happens to the continuum blocked by the radiation of a star? Does a star, of
course it will all be reddened, so the light that the star emits in blue might
appear as yellow. The light that appears as yellow might
appear as red. What kind of a weird spectrum is it?
This is beautiful. It turns out, that the Black body, the Planck Black body
spectrum has the wonderful property, that if you take every wavelength, and
multiply it by a factor, so stretch out all the waves by some factor a, it's
still a Black body spectrum. But it's a Black body spectrum
corresponding of course to a different temperature, and you wouldn't be
surprised because of Veen's law, that lambda max is 0.0029 meters divided
by the temperature in Kelvin. That if all wavelengths get stretched,
then the temperature, all wavelengths get larger. The temperature decreases by the
same factor as the factor by which wavelengths increase.
And indeed a redshifted Black body spectrum is a Black body spectrum with a
temperature that is decreased by a factor, by the same factor by which
wavelengths are stretched. This will play a role.
So, a star viewed at large redshift, looks like a star but a redder star.
And you have to be careful when you're doing color magnitude diagrams of
clusters, if they're far enough that the cosmological red shift is not trivial,
you will find that this is a meaningful correction and you'll have to take it
into account. Now, one more thing about the kinematics
of relativity. We talked about the red shift.
That tells you that our clocks run slow. That was the 2nd factor of 1 + z, the
coordinate factor velocity of light. I said light moves relative to every
observer with speed c. This does not mean it moves with speed c
as observed in our coordinates which means.
light beams moving in the dim distant past move with velocity C relative to the
observers back then. Those observers have slow clocks and
small distances. It turns out that in terms of the
positions today of the observers that observe the light a long time ago, the
coordinate velocity of light is actually not the speed of light but it does move
along geodesics, along straight lines.
And then, what about things that are not light? Well, do they move along straight
lines? We observe that they do and indeed, even along long distances it
turns out they do. Massive objects, just like as the case
for light, they move along geodesics of space. So they move in straight lines or
whatever the straight lines are in your curved space.
And it turns out that if the objects are freely falling, in other words, if
they're only moving under the influence of the gravity, not something orbiting a
star. I mean they only move under the influence
of the generalized gravity of this homogeneous space, the curvature of this
homogeneous space, not the local fluctuations that are a little galaxy
clusters, or averaging over all that. And object that moves only under the
generalized gravity of all of space, moves through the peculiar velocity, that
decreases, as space expands. In other words, the peculiar velocity of
a galaxy we observe, means that in the past, it was moving twice as fast when
the distances between galaxies were half what they are now.
So, if we observe galaxies to be moving at all, it means in the past they were
moving quite a bit faster. Remember, most of the peculiar velocities
we observe are actually not freely falling velocities.
They have to do with the orbital motion of a galaxy within its cluster or
something, but if you look at the relative motions of distant large
clusters that have nothing to do with gravitational orbital motion,
then they will have moved faster in the past, then they are moving now.
And this is, the analog of studying kinetics.
We have described Robertson-Walker, geometry.
The geometry of, of expanding or changing in sale, size, homogenous isotropic
space. And now, what we need to do is figure
out, what do Einstein's equations tell us? if
you have some distribution of energy and matter what does that do to the time of
evolution of space? So the curvature of space, the density of matter, and the
evolution of the scale factor, are all related by Einstein's equation and that's
the next thing we're going to, turn to. one, another thing to note is that all of
these solutions to Einstein's equation have the property that they have a Big
Bang, in the sense that at some time the scale factor becomes 0 when you solve
Einstein's equation. Or if you wanted, some time all the distances become 0, but
that could be a question of how you measure distances.
Remember, distance is an ambiguous thing. What's the unambiguous thing? At some
time there's a singularity all curvatures diverge.
they diverge either very far enough into the past, that's what we call a big bang,
in some you can construct solutions that don't have a singularity in the past,
but then they have the singularity in the future.
If you want to be creative you can construct, solutions that have
singularities in the past and the future, but there are not solutions without
singularities, so we're going to have to address this issue of the Big Bang at
some point.