Okay. So, I tried to make GR 
comprehensible and in words we didn't 
have the equations that are the meat of 
the subject available. 
And the question at this point is, okay, 
have you made all this effort? Is it 
worth it? 
What do we get for it? 
And gravitation is what drives astronomy, 
we know that. So, we expect GR to be 
important in astronomical circumstances, 
and indeed it is. 
And the first check of Einstein's theory, 
performed by Einstein, immediately in 
1916 when he figured out the general 
theory has to do with the orbit of 
Mercury. 
Remember, that what we saw was that 
Newtonian 
results are with a -GmM/R potential 
energy are modified by general relativity 
adding this 1 - v^2/c^2 term. 
And the fastest moving planet in the 
solar system, of course, is Mercury. 
And so you'd, might expect to see this 
effect. 
And what does this do? 
Well, any deviation from -GmM/R will ruin 
this beautiful property of Kepler orbits 
which is that they are closed. 
In other worlds, Keplerian orbit is this 
red ellipse over here in which every 
period the planet returns to exactly the 
same point, and the orbit is exactly 
periodic. 
This is not a property of orbits in 
general, and indeed, it's not a property 
of Mercury's orbit. 
Mercury orbits the sun in an ellipse, but 
that ellipse is precessing like this blue 
ellipse so that 
Mercury does not return after a full 
orbit to exactly the same point, it 
actually deviates likely. 
Now, the deviation is very slight, this 
is highly exaggerated. 
In fact, the deviation had been measured 
long before Einstein think about a one 
and a half degrees per century. 
So imagine, the axis of this blue ellipse 
rotating one and a half degrees per 
century. 
And most of this is due to what perturbs 
the pure Newtonian potential in which 
Mercury orbits. 
Well, most of the deviation is to do, of 
course, with the gravitational influences 
of Jupiter and to smaller extent other 
planets. 
And to an even smaller extent, the fact 
that the sun is not exactly a sphere, 
it's slightly oblate. 
It's fat around the middle due to it's 
rotation. 
All of these effects were in the precinct 
of Newtonian physics, had been taken into 
account and tallied carefully famously by 
[UNKNOWN] in 1859. And he came up with a 
result that was almost a degree and a 
half, per century, 
but was short by 43 arc seconds per 
century. 
This gives you a sense of the precision 
with which the calculations were done. 
And what Einstein does is he computes the 
effect of this term on the procession of 
the [UNKNOWN] of Mercury. 
And putting in the parameters for 
Mercury, he gets exactly 43 arc seconds. 
And so 
some part of the failure of Mercury's 
orbit to be closed is due to general 
relativistic corrections, and this is the 
first validated prediction. If you want 
the first validation of his theory 
clearly given this v^2/c^2 term when you 
have close binary stars like the Algol 
system, two things with masses of order 
in the mass of the sun orbiting each 
other with a an orbit that is an eighth 
of the orbit of Mercury. 
Now, the velocities are suitably higher, 
and there general relativistic effects 
are going to be more important. 
So, whenever we're dealing with orbits 
and using Newtonian physics, we were 
making approximations which we will by 
and large continue to make, by the way. 
We will not be doing, but, but we know 
now that we were missing something. 
since then, precision tests of Einstein's 
theory have been done in various waves. 
Famously, I just have to mention this, 
by the gravity Probe B Mission which 
recently finished a measurement of the 
deviations from Newtonian predictions due 
to the curved geometry around Earth. 
As well as the tiny effect of Earth's 
rotation turns out to change the 
gravitational behavior in the vicinity of 
Earth due to general relativistic 
corrections. 
And both of those were validated 
brilliantly by the, the gravity Probe B 
mission. 
We won't get into too many of the details 
of this. 
There's this famous second validation of 
Einstein's theory. 
Remember, that one of the predictions of 
general relativity is that gravity acts 
on light. 
We found that through the equivalence 
principle, 
you can argue rather easily through the 
mass energy equivalence that gravity has 
to have an effect on light. 
This settles an old dispute 
there was always the question on the one 
hand, 
the force applied by gravity is GMm/R^2 
and light beam has a vanishing little m 
so the force should be zero. 
On the other hand, the acceleration due 
to gravity is GM/R^2, and here the little 
m is gone. 
And so, would light be accelerated in a 
similar way? 
Would light or would it not be affected 
by gravity was an old debate. 
GR settles it, and in fact 
you can make explicit calculation, and 
this is a very important calculation for 
us later, of the degree to which, 
so, so what this means, in fact, is that 
a light beam traveling near a massive 
object will like any massive thing be 
deflected so that it's straight line 
motion in empty space will be curved. 
And, I've drawn the curve clumsily, but 
the net result is that the light beam 
will be deflected. 
And the closer to the planet, to the 
star, the closer to the gravitating 
source, the light beam travels, the more 
it grazes it the larger this deflection 
angle, of the light beam will be. 
The deflection angle is related to the 
mass and the distance by this formula 
which shows you that it's clearly 
relativistic but it's to do with light so 
that's to be expected. 
And the effect of this is that if you are 
studying, if you are observing a star 
which is lined up, let's say, with the 
sun. 
This is a difficult thing to do. 
But if you could imagine viewing a star 
that if you viewed it correctly would 
look as though it were right behind the 
sun, 
then that star would actually appear to 
you because of the deflection of the 
light to be in the wrong part of the sky. 
So you can measure this deflection all 
you need to do is look up at the, a star 
right near the sun. 
But you can't see a star near the sun 
except during a total solar eclipse. 
Fortunately, three years after Einstein 
publishes his paper, there is a total 
solar eclipse in 1919. 
This is the middle of World War I. 
Sir Arthur Eddington, who realizes the 
significance of Einstein's result, 
organizes a big expedition to various 
places in the Southern Hemisphere where 
the eclipse will be visible. And they 
measure, 
here's an, a picture from Eddington's 
original expedition of the total solar 
eclipse. 
This is, of course, a negative. 
You can see the dark being the light 
cromoline, cromosphere and corona that 
they could photograph around the sun and 
they do take plates and notice measure 
the positions of stars that are very near 
the sun during the eclipse. 
And indeed, they are 
deviating from their positions by 
precisely the amount predicted by 
Einstein. 
There were doubts about the precision of 
the measurement. 
It's since been reproduced in many ways 
and at high precision, and its deflection 
of light by massive objects has become a 
topic of research in itself. 
It goes under the main gravitational 
lensing, 
and it's a very important tool in modern 
astronomy. And this famous photo of the 
Abell Cluster is 
one of the the best examples I know of, 
this is the Hubble photo. 
And what we see here in the foreground of 
the picture, we're looking deep into 
space. 
This is the Hubble shot. And these yellow 
objects are cluster of galaxies. 
Now, this weird red shaped thing over on 
the right hand side, 
I hope you can see it. 
this is a galaxy which, in fact, is many, 
many, many, many megap, 
many, many, many, light years behind this 
cluster of galaxies that we're looking 
at. 
And the reason this galaxy looks so weird 
is that, the light coming from the galaxy 
has been deflected by the mass of the 
galaxy cluster in the foreground. 
And we see a deformed image of the galaxy 
spread out, this is not a perfect 
spherical lens. The distribution of mass 
here is somewhat interesting. 
And in fact, this this, this spread out 
thing, if you look at it carefully in a 
magnified, in a larger version of the 
image, you'll see that it is in fact two 
distinct images of the same galaxy both 
lens by this. 
And then once you realize how to look for 
them, you find more lensed background 
objects here, and here, they show this 
sort of streakiness like images through 
bad optics. 
And so 
we have in this image, picture, multiple 
images often of the same background 
galaxy as seen through the lensing of 
this galaxy cluster. 
This has been used by astronomers for 
many purposes. 
one is you have a lens. 
This means that the light coming from 
behind this cluster, if you, things are 
lined up correctly, will be focused and 
intensified. 
And the dimmest, most distant objects 
we've been able to image have been imaged 
through gravitational lenses. 
So, you can basically use this as another 
lens in your telescope. 
even more excitingly this expression that 
I wrote, theta is G, 4GM/rc^2. 
this equation characterizes the 
deflection of light by the mass that is 
deflecting it and the distance at which R 
here is the 
what I called B in the previous picture. 
The distance of closest approach of the 
light beam to the mass. 
you can imagine adding up the deflections 
of a given beam of light due to all of 
the sources of mass in this cluster and 
figuring out how light will be deflected. 
You have here, in the image, the lensed 
image of the galaxy, 
a lot of information on how different 
light beams were deflected. 
You could put them together to learn, in 
fact, about the lens, 
about the mass distribution in this 
cluster. 
And this is very important. 
it has been used, for example, to detect 
exoplanets. 
When an exoplanet passes in front of its 
star, 
the lensing by the planet intensifies the 
light of the star. 
And that light curve peak that is 
characteristic of what's called micro 
lensing events, has been used to detect 
exto planets, which are otherwise too dim 
to see. 
So, you're learning about the lens. 
Another famous example about, of learning 
about the lens is we'll talk 
in the last week or next week about dark 
matter, and one of the methods we use to 
detect dark matter. 
Dark matter does not produce light in any 
wavelength, and does not interact in any 
way other than gravitationally, you know. 
So we need to measure its gravitational 
field. 
One of the ways we measure its 
gravitational field is by observing the 
lensing of light and we can reconstruct 
the mass distribution, and I'll show you 
a brilliant example of that next week. 
So, gravitational lensing is a field of 
study all of its own and its very 
important to an astronomy and is, of 
course intrinsically relativistic. 
And then there's the beautiful case of 
the binary pulsar PSR1913+16.16. 
So, this is a pulsar in the constellation 
Aquila about 6,400 parsecs from Earth. 
It's a millisecond pulsar, we didn't get 
to talk about millisecond pulsar as much. 
But sometimes in binary systems pulsar, 
neutron stars speed up as they accrete 
matter from their companion. 
So that rather than slowing down their 
period for awhile, they speed up. 
And the period of this particular pulsar 
is 59 milliseconds, that's a very rapid, 
rotation rate. 
And, we notice when housian Hulse and 
Taylor careful measurements of the pulses 
from the timing of the pulses from this 
pulsar. 
They found that some of the pulses were 
delayed and some were advanced and there 
was a periodic pulse delay with a period 
of about eight hours. 
Turns out after further study that the 
reason for this is that this pulsar is 
actually a member of a binary system, 
where both members are neutron stars of 
masses about one and a half solar masses. 
And the semi-major axis is 2.8 
astronomical units, 
that's correct there. 
The semi-major axis is about three solar 
radii. 
So, I have two solar mass objects, but 
because they're neutron stars with radii 
of only ten kilometers or so. 
Orbiting each other at a radius about 
three times the sun's radius. 
Moreover, this system is special because 
the elliptical orbits that these objects 
follow are very eccentric. And so, the 
distance of nearest approach at 
periastron, they are only at ten percent 
more than a solar radius away from each 
other. 
They get very near, and then they move 
off to about 4.8 or something solar radii 
at their farthest. 
And this means we have very 
relativilistic velocities for sure when 
we have two solar mass objects at a 
distance of the solar radius from each 
other in orbit. 
And so, this is a great lab to study GR. 
And what do Hulse and Taylor extract from 
this study of GR in this beautiful lab? 
Well, first of all the pulse delay 
structure is interesting. 
It exhibits the usual Doppler effect. 
Pulses are delayed when the pulsar is 
moving away. 
Of course, we don't know if both neutron 
stars are pulsars and once just doesn't 
happen to aim at us or maybe the other 
one does not have jets, we do not know 
this. 
but we only see pulses from one of the 
neutron stars. 
There are other systems that are even 
richer, where both members are pulsars, 
but this is the famous first one. 
And so the pulse display exhibits the 
Doppler effect. 
We see that when the pulsar is retreating 
from us, the pulses are seperated by a 
longer time. But that does not explain 
all of the, pulse delay. 
I will post a link to the, to, to, review 
by Taylor of what it was they did. 
The calculation is very complicated and 
take into account many, many effects, but 
at the end of the day, Doppler Effect 
does not explain everything. 
It turns out that the pulses are more 
delayed near periostron than near 
apostron when the two when the pulsar is 
very near its partner. 
The pulses are delayed by more and the 
reason for this is a gravitational 
redshift. 
The pulses are delayed because the pulsar 
is spinning at it's accustomed rate and 
it's free. 
But as we see it from far away, when it's 
low down in the bottom floor close to 
it's partner, there is an additional 
gravitational redshift and an additional 
delay in the pulses. 
Furthermore, they can measure the 
procession of the perihelion by, because 
they can locate where in these eight hour 
periods periastron lies. 
And the perihelion procession agrees 
completely with GR predictions. 
So they have a whole bunch of 
measurements that in this binary pulsar 
system, 
reproduce brilliantly the predictions of 
GR but it gets much better than this. 
You may recall that a long time ago, I 
talked about quantum mechanics and the 
problem of stability of atoms. 
And I said something like. 
well, if you have a nucleus here, and you 
have an electron that is orbiting it, 
that's an accelerating charge. 
And that should radiate electromagnetic 
radiation. 
And that would mean it's losing energy. 
And the electron would spiral down, and 
an atom wouldn't last. 
How are atoms stable? 
Well, now that we have a field theory of 
gravity, you can ask the same question 
about the Earth orbiting the sun. 
The Earth is orbiting the sun, 
the mass is accelerating. 
Remember, that means that there is a 
ripple in the gravitational field of the 
Earth-Sun system. 
That ripple is broadcast out as something 
called gravitational waves. 
And those carry off energy Shouldn't the 
Earth be spiraling down towards the sun? 
Well, yes, but only very slowly. 
Gravity is a much weaker force than 
electromagnetism. 
And therefore the Earth is in the Earth's 
orbit is spiraling down due to gravity. 
But that effect is not only negligible, 
but completely impossible to measure. 
On the other hand, if you have a system 
that is accelerating as violently as two 
neutron stars a solar radius from each 
other there the emission of gravitational 
waves carries off a non-trivial fraction 
of the energy. 
And what do you expect? 
Well, you expect the two neutron stars as 
they lose energy to spiral down towards 
each other. 
Which means, since they're in 
gravitational orbits, 
the closer they are, the faster they will 
orbit. 
You expect the period, that 7.8 hours, to 
be decreasing with time. 
the pulsar was discovered in the 70s' and 
has been studied since. 
And what you see here is a plot of the 
decline in the period of this binary 
pulsar over the past 30 years plotted 
against a theoretical prediction from GR 
of the rate at which the system will lose 
energy to gravitational waves. 
And so, this is not just evidence for 
general relativity as well as a beautiful 
confirmation of our understanding of the 
binary pulsar. 
But, it's also telling us that this 
system is emitting gravitational waves, 
we'll talk about that in a second. And 
that gravitational energy travels through 
space in the form of waves similar, 
analogous to electromagnetic waves. what 
will be the end of this system? 
Well, as the pulsars, as the two neutron 
stars spiral closer and closer to each 
other, they're accelerating more and more 
violently. The motion becomes faster, the 
rate of energy loss becomes higher. 
Hence, the decay and the period becomes 
more and more precipitously, precipitous. 
And within as little as 250 million 
years, they will actually come close 
enough to violently collide, and 
presumably merge to form well, a bigger 
neutron star if most of the material is 
ejected in the collision or possibly a 
black hole, the subject of our next clip. 
So, there's these gravitational waves 
being emitted. 
Can we see those? Well, the answer is 
almost. 
There are a collection of gravitational 
wave detectors like this one, this is 
LIGO, Laser Interferometer Gravitational 
Wave Observatory and what this is is two 
three miles laser tubes. a gravitational 
wave passing through will cause the large 
and well insulated masses at the tips of 
these to move by about the size of an 
atomic nucleus. 
And because, again, a very sensitive 
interferometry, 
this machine is supposed to be able to 
detect this sort of fluctuation in the 
relative length of the two arms. 
And it has not detected, as far as I 
know, gravitational radiation. 
It's at the end of its calibration run. 
And sensitivities and noise detection and 
background reduction are continually 
being improved. 
And 
maybe any day we will hear the discovery 
of gravitational waves and it won't be 
from the binary, Hulse-Taylor binary 
pulsar because the Hulse-Taylor binary 
pulsar does not emit nearly intense 
enough gravitational waves for this 
detector to detect them. 
Although, it is capable of detecting 
waves, there's a detector in Louisiana, 
one of the two detectors. 
This one I think is in Washington. 
The detector in Louisiana detects waves 
breaking on the Gulf Coast which they 
have to subtract. But it won't be 
sensitive enough to detect the 
Hulse-Taylor pulsar. 
But if something very violent occurs, 
like the merger of neutron stars, 
those last three orbits before a merger 
are sufficiently violent and sufficiently 
energetic. 
That enough energy will be emitted in the 
gravitational radiation that we think we 
will be able to observe such events. 
This will be the second time in this 
class that we're talking about 
non-electromagnetic telescopes. 
We talked about neutrino astronomy 
opening up the field of gravitational 
wave astronomy, which is what this 
detector is hoping to do. 
Would give us a whole new window on the 
universe and one focused on the most 
dramatic and cataclysmic events. 
So, we're all hoping that they do detect 
gravitational waves.