Theory of relativity. In theory of space, time, and gravity
provides the theoretical framework for modern cosmology.
Here, I will assume for the purpose of this class that you actually know
something about this and here is just a quick refresher.
The special theory of relatively, which came first, postulates that all inertial
and accelerated frames of references must be equivalent.
And that has a number of important consequences.
It does unify space and time in a novel fashion.
It. postulates the speed of light, is the
maximum speed that can be had in a space time or rather, light moves at the
maximum possible speed. It predicts Lorentz contraction and time
dilation, those are two important concepts and if
you don't know what they are, you really need to look them up.
And, of course, it predicts the equivalence of mass and energy,
the famous EMC^2 = mc^2 formula, as well as a number of other effects.
Here is the front page of Einstein's original special relativity paper from
1905. The special theory of relativity is
limited in its application, because it talks only about unaccelerated
frames of references. What happens with accelerated ones?
Well, this is where general theory of relativity is.
Einstein finished that in 1915 and that was an even more important change in our
view of spacetime. Here, it's assumed that all frames of
references are equivalent, and through equivalence principle, the gravitational
mass and inertia mass are the same. A one sentence summary of what general
relativity says is that the presence of mass or energy, which, remember are
equivalent, determine the local geometry of space.
And local geometry of space determines where would the mass and energy move.
The two have to be computed in a consistent fashion and that is what
general relativity is all about. So general relativity introduced
curvature of space, which also then immediately had
predictions of light bending in gravitational field and so on.
Here is the front page of Einstein's general theory of relativity paper.
However, even before this was published, he already had some of these ideas,
including gravitational bending of light. The equivalence principle postulates that
gravitational and inertial mass are the same.
In addition to that, there is Mach's principle which postulates that
gravitational interaction of all masses is what creates inertia.
In an empty universe, there will be no inertia, there will be no mass, which is
sort of obvious. But Einstein argued that gravity is an
inertial force and there is a famous experiment with elevators and rockets.
There are two important consequences of the equivalence principle.
First, that light should be blue shifted coming in to gravitational potential
wells and redshift coming out of the gravitational potential wells and the
light path would be curved in, space in general.
The presence of mass will introduce local curvature and light moves to what
[INAUDIBLE] is the shortest line. Einstein was already aware of the
possibility gravitational bending of light. And in 1911, he wrote to George
Ellery Hale, famous astronomer, asking him if this can be observed.
Well, apparently, it wasn't possible at the time.
However, it was actually possible to do it during
solar eclipses and there is a famous experiment where Eddington and
collaborators measured positions of stars around solar discs during the maximum
eclipse and then compared them to sunny somewhere, else, else in the sky.
This completely confirmed Einsteins prediction of bending of light by about
almost twoarc arc seconds near the edge of the solar disc and this is why
Einstein really became famous. So let's talk a little bit about geometry
of spacetime. In general, space can be curved.
It's hard to draw four dimensional space time, so we use, use these
two-dimensional analogies embedded in three--dimensional space.
Positive curvature is like curvature of a sphere and that for response physical
closed universe. the curvature will depend on the amount
of mass and energy is present. If, it's exact right critical amount it
will be flat geometry of familiar space. If it's less than that, it'll be
hyperbolic geometry which is also open in existence to infinity.
So how do we quantity geometry of space? The most general approach is to define a
metric, which is a formula that gives the distance between any two points and it's
usually expressed in form of curvature like here, that is the sum of products of
increments in each of the two coordinates multiplied together, sum over all of them
and some coefficients multiply them. So, these coefficients are the metric
answer and its indices in four-dimensional spacetime run from zero
through three, zero is the, the time, one, two, and three are usual spatial XYZ
dimensions. This is a matrix 4x4 or tensor, if you're
familiar with that and for, for plain familiar geometry.
The coefficients are one on diagonal, its the sum of squares in increments in three
spacial coordinates and all of the diagonal limits is zero.
So this is the simple measure of distance in 3D space,
but in general, it can be different and values of these coefficients can be
derived from general theory of relativity.
So, if we know what the metric is, then the geometry completely specified by it.
In the special relativlistic space, which is still flat in Euclidean.
We have to introduce the time element, and, for reasons that we don't need to go
into, it is expressed as square of the time increment minus the spatial
increment of, what that [INAUDIBLE] to be coordinates.
But the very general case, in general activity for homogeneous and isotropic
universe, is the Robertson-Walker metric. Remember that we introduced homogeneity
and isotropy as means of simplifying general relativity.
Homogeneous universe means the same everywhere, and isotropic in all
directions, and therefore, only the radial co, coordinate would matter.
So now, Minkowksi metric from above still has the same old time element and then
has a spatial element which is multiplied by a function r(t), which is called a
scale factor. And the rest of it is expression that is
universally true for different, different values of spatial curvature.
So, Robertson-Walker Metric is expressed in polar coordinates,
this is useful because only radial direction matters.
And, it can be also written in this form where the function S, depending on the
different curvature, is sign, hyperbolic sign for the radius itself.
Remember that, that whole spatial increment can be changing in time, in
principle and this is what the scale factor is all about.
Sometimes it's denoted as R of the [INAUDIBLE] a(t),
I'll be using both of those. You just have to pay attention.
Basically, that's the ratio of any given distance at some time to what it was
relative to the zero point reference, like if you start with an expanding
universe with distance between two galaxies of a gigaparsec, then sometime
later, it will be more than a gigaparsec. So this is just a thumbnail sketch of
what general relativity is all about. Next time, we will talk a little bit more
how it's applied in cosmology.