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.