BRENDAN HASSETT:
Hello, everyone.
My name is Brendan Hassett.
I'm a professor of mathematics,
here, at Brown University.
And I'm also the
director of ICERM.
ICERM is the Institute for
Computational Experimental
Research in Mathematics.
We're a mathematics
research institute
supported by the National
Science Foundation, Simons
Foundation, Microsoft
Research, Sloan Foundation,
and other funders.
And basically, our mission
is to bring mathematicians,
physicists, and
computational scientists
together to discuss
problems of common interest
and make progress on
their research questions.
Our special focus is the
interaction between mathematics
and computation.
So we're interested in how
mathematical theorems can
translate into algorithms, new
computational techniques that
can find applications in
industry and in other fields.
We're also interested in
how computational tools
and experiments can generate
discoveries in mathematics.
In many cases, you find
mathematical truths
by looking at
computational examples.
And so that feedback
between math and computation
is what really drives
ICERM's mission.
So we're very pleased to
have Scott Field here.
So Scott field is
actually a student of one
of the founders of
ICERM, Jan Hestaven.
So he did his doctoral
work at Brown.
He won the Houghton
Prize for the top thesis
and in theoretical
physics at Brown.
He went on to work in the
Joint Space Sciences Institute,
in Maryland and
worked in Cornell.
And now, he's an assistant
professor at UMass Dartmouth.
So his work in the
mathematics department
focuses on using techniques
from scientific computing,
Bayesian inference, to try
to draw information out
of data sets that come
from instruments like LIGO.
LIGO, you may have
heard, is this huge tool
to try to hear the sound
of collapsing black holes.
I will probably avoid trying
to say anything more than that.
And so we're actually
very pleased at ICERM
to have a program,
here, in fall of 2020,
co-led by Professor
Field, to discuss
some of the mathematical
tools and techniques that
are needed to translate the
data from these machines
to understanding about what
happens millions and billions
of light years away.
And so we're really
proud that mathematics
has a role in these
important discoveries.
And to some extent, this
is a computational feat
as much as it is an
experimental feat.
Being able to interpret
what the instrument tells
us is a big thing.
And so hopefully, mathematicians
can continue to contribute.
So let me yield the
floor to Professor Field.
Thank you very much.
[APPLAUSE]
SCOTT FIELD: Thank you for
the invitation to speak.
And thank you all for coming.
So as the title says,
we'll be talking
about discovering both black
holes and gravitational waves.
But mostly, the focus
won't be on the observing
and the instrumentation as much
as the mathematical background
that permits
gravitational waves even
to exist in the first place.
And also, the simulation
tools that we need in order
to, basically, solve these
equations on computers in order
to know what sort of
signals to look for.
So this is the rough schematic--
the outline of the talk.
I'm going to begin
by highlighting
some recent developments in
gravitational wave science.
Some of these you may have
already heard a little bit
about.
They've been in the news and
in popular science discussions.
Most of the talk will be spent
on this second and fourth
bullet point, here.
We're going to talk
about what gravitationals
are, both the physical mechanism
and the intuition behind them.
But also, the
mathematical framework
and some of the key developments
in the theory that even permits
Einstein equations to have these
sort of wave-like solutions.
Very briefly, we'll
talk about how
to detect gravitational waves.
But again, I'm not
an experimentalist.
So I wouldn't do
this part justice.
So I'm going to only give the
necessary bits and pieces,
in order to sort of understand
roughly how the detector works.
And then we'll return to how
to simulate gravitational waves
using large computers,
supercomputers, and how this
actually enables discovery.
There will be three
acronyms that will be used.
I don't want to lose
anyone at the outset.
If you see any of these terms--
GW, that will stand
for gravitational wave.
LIGO, that's the Laser
Interferometer Gravitational
Wave Observatory.
And GR is general relativity.
So these terms will be
appearing over and over again
throughout the talk.
And also, what's the approach?
I'm going to try to discuss the
historical angle of this story.
So it's a 100 year project from
1915, when Einstein first wrote
down and proposed the
theory of relativity,
to the detection of
gravitational waves.
And there's a lot
of interesting work
that was done by
hundreds of researchers.
And I'm not a historian.
So I couldn't do
this all justice.
So here are these references.
If you're interested in
learning a little bit more
about the history, you can come
back to this at some point.
So I thought I would
start with a movie.
Everyone likes movies.
This will just give
some idea of what
you should have in the
back of your mind as sort
of the setup of the problem.
So the story begins down
here, at the bottom right hand
corner.
We have two black holes.
So if you don't know
what a black hole is,
briefly, it's the
end result of a star.
After the star has
run out of fuel,
it'll collapse in on itself.
And if the star is
heavy enough, it'll
collapse down into what is
effectively a singularity.
It's a breaking point
of space and time.
What makes it black
is-- well, there's
an event horizon,
which is to say
that if you were to go into
the black hole-- hopefully,
you don't actually
want to do this.
But if you were, and you had
a flashlight or a candle,
and you had this light
on, you could see it.
And other people in the
black hole could see it.
But none of this light would
leave the event horizon.
It could never make
it to some observer
outside of the black hole.
So this animation, down here,
is actually a little bit more
than an animation.
This was run on supercomputers.
And using codes and
techniques developed-- well,
actually, it's very small here.
This little emblem is
simulating extreme space-time
collaboration.
They develop a lot of
the computational tools
for doing binary black
hole simulations.
And they use a sophisticated
ray tracing program
to basically show
what it would look
like if you had a star field.
So I believe this is
probably the Milky
Way of the galactic plane.
If you had two black
holes rotating,
this is what you
would see if you were
looking at them straight on.
So if they're
rotating like this,
and you're looking at
them from overhead.
So that's what's
generating these waves.
And in fact, a large
portion of this talk
will be describing what that
generation mechanism is.
But now, we're just
giving the general setup.
So if you imagine these
two black holes are
orbiting in some other galaxy,
they're emitting these waves.
The waves travel over galactic
distances, in some cases,
hundreds of thousands
of light years away.
And eventually, if they're
lucky, they'll reach Earth.
And they'll pass on by.
And there's a couple of
things going on here.
So this is obviously exaggerated
to scale, in order to see it.
We would be in big trouble
if this is actually
what the effect of
the gravitational wave
was on planet Earth.
And also, they're not green.
The waves are not green.
You can't see them.
If you could, you could just
use an optical telescope.
So they're invisible.
And they don't have
this sort of effect.
But if we had two
detectors, say one over here
in Livingston, another
one at Hanford,
Washington-- and
these detectors,
which we'll just
talk a little bit
about later, feel the effects
of the gravitational waves
as they pass through.
So just like the Earth-- you
see the Earth oscillating.
These gravitational
wave detectors
will oscillate as the
wave passes through.
That's, roughly speaking,
how one measures or observes
gravitational waves.
So I mentioned this was a 100
year long research project.
In 1915, Albert Einstein
proposed the theory
of general relativity as a way
to explain some inconsistencies
in the Newtonian
theory of gravity.
And for the purposes
of our story,
there is really, after
the original birth
of general relativity,
three main landmarks.
You can see here,
by the timestamps,
that it was about 40 to 50
years before mathematicians
and scientists really understood
what a gravitational wave was.
And there are really
two key contributions,
one by Yvonne Choquet-Bruhat.
If there are any French speaking
members of the audience,
I apologize.
And the other by Felix Pirani.
We'll come back to both of them
a little bit later in the talk.
They really laid the
theoretical foundation
for gravitational waves.
And it took about
another 50 years
before computers were able to
simulate gravitational waves.
So once we know that Einstein's
equations will actually
permit these
wave-like solutions,
it makes sense to
try to solve them.
Now, the equations
are too complicated
to sit down with
a paper and pencil
and try to solve them
like you might otherwise
be tempted to do.
So we outsource all of the
hard work to a computer.
And this will be the second
major theme of the talk--
really, how to do this and
some of the breakthroughs
that were required to enable it.
And then 100 years
later, right on schedule,
gravitational waves
were observed by LIGO.
And we'll also quickly
describe the detector.
So we're in the highlight stage.
What was so exciting
about gravitational waves?
Well, they were
predicted as far back
as 1916, which we'll
talk a little bit about
later in the talk.
And on September 14,
2015, a gravitational wave
passed through the Earth.
And the detectors happened
to be on at this time.
The detectors are
not always operating.
In fact, they're off-line
most of the time.
Scientists are going
in there and making
them a little bit
better, doing a lot
of checking and calibration.
But at this point,
it was actually
during an engineering run.
So it wasn't really
even set up to be
recording gravitational waves.
They basically had
turned it on just
to see if the thing was working.
In fact, one of the lead PIs
of the project, Ray Weiss,
who went on to win
the Nobel Prize--
as the story goes, he
was actually very upset
that they turned on the
detectors at this time.
He thought they weren't ready.
And if they had
listened to him, they
would not have seen this signal.
So what we're
looking at here, you
remember the previous slide?
There were two triangles
on the Earth-- one
triangle in Louisiana,
another one in Washington.
These are where the two
detectors are located.
And both of them are
on at the same time.
This allows you to get
a coincident study.
So if you see a gravitational
wave in one detector,
you have some confidence
that it's a real signal,
because you see it in
the other detector.
So over here, this is the signal
that was seen in Washington.
You can see here, the
x-axis is in time.
So this is a signal, a time
series just like I'm speaking,
or if you turn on the
radio in your car.
These wave-like signals could
be measured with a microphone.
And you could look at and
ask, what does the signal look
like as a function of time?
In this case, here, it's not
really the air that's waving.
It is space-time itself.
And these detectors
are set up to measure
the change in distance at
some little, local area
of the Earth.
And that's what's given
here, on the y-axis.
So they measure it,
typically, in Strain,
which is, in some sense, a
relative change in path length,
delta L over L.
We'll come talk about
that a little bit later.
But for our purposes now,
this is really the signal.
And you can see here that
Washington and Louisiana are
on different parts of the map.
So the signal that
comes in at Washington
is not the same one that
will come in at Louisiana.
There's going to
be a time offset.
There may even be a phase
offset to these waves.
And once you account
for all of that
and put them on
top of each other,
you can see they line up
just as you would expect.
So this gives even
more confidence
this is a real signal.
You could imagine somebody
could just come by
and accidentally
kick the detector,
and that would
make it oscillate.
In fact, when they
originally turned these
on for the first
observing run, even things
like the motion of
airplanes-- so airplanes
go by hundreds of miles away.
There's air disturbances.
They can hit the detector.
These are sensitive enough that
they will even pick up on that.
In fact, there are thousands
of environmental channels
that they monitor.
So for example, if there's
an earthquake or a lightning
strike somewhere in
the Midwest, this
will also make the
detector oscillate.
And they have to
control for this.
But if you do all of
your due diligence,
and you're very careful,
you can subtract all of that
and get a nice, clean
gravitational wave signal.
Once you've measured the signal,
observed it with your detector,
it would be interesting to
know what physical setup--
what were the masses
of the black hole, say?
And where in the
sky was it located?
This, broadly speaking, is a
parameter estimation problem.
And since you can't
solve these equations
with a paper and a
pencil, you really
need to do a computer
simulation to figure out
what is the signal
that you would
expect to see if the
masses of the black hole
are such and such.
And that's also
what is shown here.
So up here, this is a little
cartoon of the three phases.
So I mentioned there
are two black holes.
They're rotating.
You may say, why two?
And actually, it turns out
most stars are binary stars.
Most of these large
astrophysical, compact bodies--
there's usually two of them.
So we have these two black
holes that are rotating.
They're emitting waves.
And eventually, they merge.
And they form one
larger black hole.
And that's exactly what
the computer finds.
So this is not just a guess
or maybe a hypothesis.
This is what Einstein's
equations say will happen.
And you can then ask, well,
if you run your computer
code, what signal do you get?
And that is the red curve.
And it's right on top of this
reconstructed template, which
is some model agnostic
filtering of the data.
So you won't know what the
masses of the black holes
are that cause the signal.
But you can do this
model agnostic filtering.
And you can get this
gray line, here.
This particular simulation will
take weeks on a supercomputer.
And it relies on lots
of software development,
but also advanced algorithms
and advanced mathematical tools,
which we'll come back to
talk about in just a second.
Once gravitational
waves were discovered,
the mainstream media
really picked up on this.
And it was all over the place.
It's very rare to see The
New York Times front cover--
this is not the
science section--
the front cover of
The New York Times
describing a
scientific discovery.
In fact, there
should be more that.
But this is usually buried
on the back of the last page.
So the fact that it's the front
cover is quite remarkable.
If you're really
paying attention,
and you have great
eyesight, you'll
notice that says
February of 2016.
And this is September of 2015.
Well, there were many
months in between where
the scientists of LIGO had
to check the data to make
sure it was a real signal.
It's not just about someone
coming by and accidentally
kicking the detector.
There's other things
that can happen.
So for example, people
were very worried
that the signal looked too good.
Maybe there was a hacker
that infiltrated the software
and injected the
signal, artificially.
That was a reasonable
thing that people
thought could have happened.
And so they had to hire
a lot of IT experts
to go over log files, make
sure that didn't happen.
The other thing
that could happen
is there are three people in the
organization that can basically
artificially inject
a signal just
to see what the data
analysis codes would do.
And they won't tell the
rest of the collaboration
they've done it until a paper
is ready to be published on it.
So this is kind of like
a fire alarm dry run.
Yes.
And the first time this
happened, the collaborations
all met at Caltech.
And they sat down,
and they said, OK,
is this a real signal?
And they opened the envelope,
and it said no, fake.
This one wasn't a fire drill.
This was the real deal.
But that's something
that they had
to let the process play out.
If you were riding a New
York City subway at the time,
you would also see that there
were even advertisements.
Apparently, it's easier to
detect a gravitational wave
than it is to find a New York
City apartment with a closet
space.
So this was really
hitting the mainstream.
It was on the top
of people's minds.
I'll point out here,
this is Interstellar,
which actually came out a little
bit before gravitational waves.
Maybe it was just well-timed.
And here, we have
Kip Thorne, which
is another one of the founding
members of LIGO, who went on
to win a Nobel Prize, showing
Jessica Chastain, who is one
of the actresses of the movie--
basically, teaching her a
little bit about general
relativity on the chalkboard.
So these were the
three founding members.
Ray Weiss, he was mostly
responsible for building
and designing the detector.
Barry Barish, who was also
an experimentalist and helped
form the collaboration
and set up the structure.
And Kip Thorne, who was sort
of the in-house theorist that
predicted the signals and also
provided some data analysis
expertise.
Now, this organization
has thousands of people
spanning many, many years.
So this award is
really also on behalf
of the entire collaboration,
which is something that they
certainly pointed out.
And we'll certainly try to talk
about some of the key players
in this besides
these three, which
I'm sure you've all heard of.
We'll come back to them.
So now, before Kip won his Nobel
Prize, we knew him beforehand.
2016-- so this was actually
even before I was at UMass
Dartmouth--
he came and visited.
And he did a special
Physics of Interstellar
at UMass Dartmouth.
And here is Bob Fisher
and Richard Price,
who are both faculty members.
Gaurav Khanna, who is a
faculty member of physics,
and Sigal Gottlieb, who is a
member of the math department--
they're both directors
of the CSCVR,
which is an acronym for--
so I said there wouldn't be any
acronyms besides those three.
I guess I lied.
This is the Center for
Scientific Computing
and Visualization Research.
And Kip came and gave
a nice presentation
about the physics
behind Interstellar,
which is one of the few
Hollywood movies that
is faithful to the physics.
Usually, if you saw Event
Horizon or something,
a ship goes into the black hole.
And it comes out, and
it's evil or something.
Or it's a vacuum cleaner, and
everything gets sucked in.
They really did a careful
job to try to stay faithful
to the physics.
So part of this talk is about
discovering black holes.
And this is really
the main thing
that I had in mind to
talk about in that regard.
So black holes are black.
So by the very nature,
they don't emit light.
You can't point a telescope
into the sky and see them.
So you have to see
them some other way.
And gravitational
waves is that other way
that you see black holes.
And this is sort of a census.
So we're mostly not going
to be worried about stuff
down here or in this area.
The black holes are all up here.
So the way to interpret
this figure is as follows.
Notice that there's a
scale on the y-axis.
This is the mass of the
black hole in solar masses.
So the sun has one solar mass.
So if there's a black hole
up here, that's 80 times
the mass of our sun.
And in order for it to be a
black hole, all of this mass
has to be condensed down
into a small region.
And in fact, if you were to take
a sun that's 80 times as large
as ours and turn it
into a black hole,
it would basically fit
in the state of Maine.
So that is a very small
place for a lot of matter
to be concentrated.
Now, the other
thing that's really
interesting about this figure
is that before LIGO, we
didn't know anything
about this region.
We had no idea that there
were even two black holes that
could be in a binary orbit.
And we certainly didn't
know that black holes could
be as heavy as 80 solar masses.
In fact, I would
say that was even
considered a little
unorthodox to suggest
we'd be seeing a lot of heavy
mass black hole systems.
Yet that's exactly
what nature gave us.
And that's why we
build detectors,
because we don't really
know what we're going
to see until we start looking.
One other thing to point
out-- so I won't really talk
too much about neutron stars.
But I will point out that LIGO
hasn't seen just black holes.
They've also seen merging
binary neutron stars, which is
interesting for many reasons.
Also, what is
interesting here is
you'll notice that these
two neutron stars merge.
And they create a slightly
more massive system.
And the question
might be, what's
the final end state of this?
It's expected to be a black
hole, because it's a bit too
heavy to be a neutron star.
But there's a lot of
interesting micro-physics
that goes on into
describing these problems.
So this is, without a
doubt, an impressive,
I would say, almost unfathomable
experimental feat that you can
measure gravitational waves.
But it will require these
two things, which is really
the focus of our talk--
mathematical and
computational breakthroughs.
So over here, we
have John Wheeler,
who actually won't play
a big role in our story.
But he coined the term
wormhole, black hole.
He trained a whole generation
of gravitational physicists
that had an amazing impact.
For example, Kip Thorne
was one of his students.
He also pioneered the
use of colored chalk.
As you can see here, it's
a very colorful blackboard.
Now, we always use whiteboards.
But back then, they
had blackboards.
And over here, this
is a supercomputer.
So you may not know
what a supercomputer is.
You can imagine you have
a really high end desktop.
There are 16 cores
on your desktop.
That would be a pretty good one.
If you put thousands of those in
a room and connected them all,
you would have a supercomputer.
So Blue Waters is one of
the largest, I would say,
supercomputers that you don't
need government clearance
to get access to.
And probably most of
the binary black hole
simulations run on the planet
have come through Blue Waters.
There have been thousands of
binary black hole simulations
performed on Blue Waters.
And you really need these
large supercomputers
in order to perform these
kinds of calculations.
So moving on a
little bit, first,
a little bit of the theory.
So again, really can't
discuss the entire history
in the development
of the theory.
But I want to give a sense of
what are gravitational waves.
And why should we believe
that they exist without having
to measure them first?
So what does the theory say?
The story really begins
well before 1915.
We can see here, 1687.
That's when Isaac Newton first
wrote down a theory of gravity.
He was the first one to
have a theory of gravity.
And he had a very simple
equation, which many of you
have probably seen if you have
taken high school physics,
even.
If you have two
massive objects--
this could be the sun and the
Earth, the Earth and the moon,
anything that has mass--
and you could measure the
mass, Newton's equations
would tell you the force that
they exert on one another.
So here, we have a
larger massive ball.
And one is being pulled by M2.
So that's what these little
force vectors are saying.
And so if you know
the masses, and you
know the distance
between the two objects,
you can basically
compute the force.
So there's an arbitrary
constant capital G.
That's Newton's constant, which
the theory doesn't specify.
You have to go and measure it.
And actually, that's one
of the hardest things
to measure in nature.
In fact, if you plot the
accuracy of G versus time,
it's actually getting worse.
You don't really know it.
But a couple of decades ago, we
knew it to maybe four digits.
Now, it's probably about
three or something.
It's a very hard
quantity to measure.
But if you knew
what its value was,
you can use Newton's
equations to predict this.
So there is nothing for
these waves to be waving in.
So this is an instantaneous
force between two objects.
And nothing can be waving.
And in fact, this idea
of an instantaneous force
is what led Einstein to
develop the general theory
of relativity.
He refused to believe
that you could
have two things interacting with
one another instantaneously.
There had to be something else.
There had to be a mechanism.
So what he proposed--
the intuition is
that gravity is not
a force in the sense of
pushing and pulling on objects.
But instead, it's really
a warping of space-time.
And this is sometimes
visualized in a cartoon.
So we have a cartoon like this.
Imagine you're over here.
You're far away from
the planet or the moon.
So this is the Earth.
And this is the moon.
You can see that these
squares are undistorted.
If you've heard the terms
Euclidean or Cartesian-like
coordinate system,
you could imagine,
at least if you wanted
to make an experiment,
if you were located
at this corner,
you could walk up
to the next corner.
And you could have a
little meter stick.
And you can measure it
maybe being one meter.
You can measure the
angles of this square,
and it should come out to
be 90 times four, so 360.
But as you get closer to
these massive objects,
the grid gets distorted.
The distances change.
The angles change.
The geometry of
space is changing,
and it's due to
the fact that these
are very massive
objects that are
warping the space and the time.
So this is a complete
rewriting of what gravity is.
And so the main takeaway
here is that gravity
is not an instantaneous
force, but a manifestation
of the bending of
space and time.
So one example, imagine you're
enjoying a beautiful day
in an academic quad.
And there's some students.
And it's spring.
And here's a clock tower.
And all of a sudden,
a black hole appears.
So if you see this,
you should run.
This is not what you
should be wanting
to see on the academic quad.
But this is exactly how it
would look if the black hole was
in front of your face.
And I say exactly, this is
not an artist's rendition.
Again, these
supercomputers were used
to do ray tracing of
how the light comes
to you from the picture
that used to be here.
So there's a lot of
interesting features.
You can see that, one, things
that are on the outskirt
gets warped.
Let's say we're standing right
in the middle of the picture.
So right here,
we're looking at it.
The clock tower has
some light that's
bouncing as a straight line
path, right into our eyes.
Now, it has to go around
the black hole, which
is warping the distances.
So it tries to take a
straight line, but it can't.
So it kind of has the
effect of bending.
It's an optical effect due to
the fact that space is bending.
It bends things
on the outskirts.
The other thing that's
really interesting,
if you have a really
good eye, you'll
notice that the clock tower
also appears over here.
So these are the light rays
that would have otherwise went
over there to maybe your friends
on the other side of the quad.
The light rays get wrapped
around the black hole
and come back to you.
So now, you see this image.
And in fact, if
we could zoom in,
there would be infinitely
many images of clock towers,
due to the fact that
the light is basically
getting tied up and spun around
and then shoots back out.
So this was also developed
by the SXS Collaboration.
Andy, who was a grad student at
the time, did this simulation.
So what are the equations
that describe this?
So far, it's been
very intuitive.
You can write down
the equations.
Of course, you have to if
you want to solve them.
It's Einstein's equation.
And on the left hand
side, this is basically
the part of the
equation that describes
the curving of space and time.
And on the right, that's
where all the matter is.
So if you were to tell
me the mass of the sun,
I would plug that into the
equation on the right hand
side.
I would have to solve this
equation for the geometry
of space and time.
So that's the solution
to this equation.
Now, this looks like
a simple equation.
It's just R times one half
RG equals this other T,
on the right hand side.
But notice that these funny
Greek letters mu and nu.
And we don't really
say what R is.
If you were to
unpack all of this,
you would find that this
is a differential equation.
So this is a non-technical talk.
So if you're not familiar
with differential equations,
don't tune out entirely.
We're not going to be talking
about differential equations
that much.
But it's a
differential equation.
And we would have to solve it.
We can, using computers.
And the solution
to this equation
tells us all we need to know
about how the geometry of space
and time behave.
So for example, the
gravitational lensing
experiment we just did is the
solution to this equation,
if you have a black hole
right in front of you.
If a gravitational wave
passes by the Earth,
even though you
probably can't measure
these changes in
distance, literally,
the distance from you to Boston,
if you're driving to Boston,
is changing.
And another thing, if
you're working at ICERM,
you may interested to know,
you're aging a little bit more
quickly than the people who are
working down at Hemingway's.
The reason is--
sorry to tell you--
we're in a gravitational field.
And the closer we get to
the surface of the Earth,
the deeper we are into that
gravitational potential.
And it turns out that clocks
tick at different rates.
So for example, the best place
to be would be in outer space.
The clock would be in
a deeper kind of well.
It'd be very close
to a black hole.
So according to the
people in Hemingway,
they're looking at us.
So our heart rate maybe is some
kind of measurement device.
Every minute, your heart
maybe beats 80 times.
And if you're working
down at the bottom floor,
and you look up, you
would see our hearts
beating a little bit faster.
Now, this is effectively
unmeasurable.
And you won't really
appreciably live longer
if you go work on
the first floor.
But nevertheless, this is an
experimentally verified thing
that has happened, both
verified in the lab--
and one of the first
experiments is they took clocks
and, basically, airplanes
flew them around the Earth.
And then they would
measure the time difference
when it came back to the Earth.
And they found
that the prediction
of general relativity
agreed perfectly
with the time offset
of these two clocks.
So the difference in
times is pretty small,
but it can be measured.
And one place this is
practically relevant
is GPS systems.
So these are satellites
in outer space
that are very, very
accurate timing devices.
So every so often,
the GPS satellites
effectively have to be reset.
Because the clocks
in the satellites
go out of sync with the
clocks on the earth.
So that's another consequence
of the Einstein equations.
So if you were to
unpack these equations,
it would look like this.
And this is just awful.
This is why the computer
has to solve it.
In fact, this is a
very special case,
where there's
spherical symmetry.
So this is about as
idealized of a system
as you can possibly imagine.
Nothing really gets easier.
If you have a problem
with spherical symmetry,
that's great.
That's a great way
to solve the problem.
These are what the
equations look like.
So notice here, I guess,
there's a G. Remember,
before there was a G. And
these primes are derivatives.
So these are
differential equations.
But we really don't want
to use paper and pencil
to solve those.
Maybe stepping back
a bit, we may just
want to ask some big picture
questions about the equations.
If you didn't have
spherical symmetry,
and you wrote them all out
using first order derivatives,
there would be 52
equations to solve.
Each equation would
have hundreds of terms
on the right hand side.
And if you work in differential
equations, these are nonlinear.
So there's some weird
coupling between them.
And really, we need
computers to solve them.
But it's not even clear,
I would say, at the outset
that gravitational waves exist.
So in the sense that if you
were to look at those equations,
there's no way to know
from first principles
that there are
wave-like solutions that
would reach detectors.
In fact, it's not even obvious
you can solve the equations.
So that's actually
part of the story.
So these are really the two big
pieces that we need to answer.
Do solutions exist?
That's certainly not obvious.
And the equations
are hard to solve.
So are there
gravitational waves?
And notice, here,
that this really
wasn't settled until the
'50s or even the '60s.
So this took about
40 or 50 years.
So existent solutions.
So why does this matter?
If there aren't
solutions, there's
no reason for using a
computer to try to find them.
So it's very useful to
know if a solution exists.
Now, just stepping back a
bit, what does that even mean?
So if you were
taking a math exam,
and someone gave you
this equation to solve,
either they're
trying to trick you,
or they are not really paying
attention when they wrote down
the problem.
So solve for x.
If you subtract 8 from 12
and 5x from 7x, you get this.
Then you cancel the two x's.
And you get 4 equals 5.
So what this means
is there is no value
of x for which this
equation is true.
So that is to say
it cannot be solved.
In the case of the
Einstein equations,
it's a little bit more
complicated, of course.
And you can't just work it
out with paper and pencil.
But you can try to apply
theoretical techniques
to understand the
solvability of the equations.
And for our purposes, the
main contributor to this
is Yvonne Choquet-Bruhat.
Again, sorry if you're a
French speaker in the audience.
She was the first
person to show that you
could solve the
Einstein equations
under general conditions.
And this happened in 1952.
And if we look back
on this timeline,
here, in 1947, she was
a graduate student.
And that was her task.
That's a pretty big task to
give a graduate student--
to prove that you can solve
the Einstein equations.
But she did it.
And remarkably, she did
it in just five years.
And as you can see here, going
back even a little bit further,
I was reading her historical
account of this process.
She was inspired by
mathematicians Friedrichs,
Lewy, and Sobolev, who
wrote, in the 1930s,
a series of papers that
provided the analytical tools
for solvability.
Used in a different context,
but she borrowed these tools
and applied them to
the Einstein equations.
And here, Frederichs and not
these other people, here--
Frederichs was one of the
founding members of the Courant
Institute at NYU, which is
an internationally regarded
applied math institute.
And he lived in New
Rochelle with Courant
and other mathematicians.
They would regularly have
picnics and barbecues.
So here they are at New
Rochelle, in their backyard,
presumably discussing
mathematics.
And those were some of the
tools that came out and allowed
her to prove that you can,
indeed, solve the Einstein
equations under
general conditions.
Another important person
was Robert Geroch,
who, along with Yvonne,
extended her initial 1952
result to effectively allow
these solutions to exist
for all time.
So in this 1952
paper, she proved
that these solutions existed
over very short timescales.
And the 1969 result was
an extension of that.
So that was a key
landmark discovery
in the theoretical
development of the equation.
So yes, it does make sense
to put them on a computer,
because you can
solve the equations.
And do gravitational
waves exist?
That is, again, not clear.
And you'd have a very hard time
convincing Congress to give you
billions of dollars to
build a detector if there's
nothing to detect.
So you have to make sure
that they actually exist.
And this story actually
begins in 1916.
One of the first things
that Einstein recognized
is the possibility of
gravitational waves.
But he immediately
dismissed them
as just artifacts
of the equations.
These are un-physical things.
So again, what does this mean
that you can trust a solution?
What does that really mean?
So one example might
be if someone told you
you're living in a
bedroom that's a square,
and the dimensions of
the walls are 7 feet.
So it's a 7 foot by
7 foot square room.
If someone didn't tell you
that, they just told you
that the area of the
bedroom is 49 feet,
you would have to work backwards
to discover how big of a room
is it that I'm actually
getting into, here.
So mathematically, what
this would look like,
if x was the length
of the wall--
well, x squared equals 49.
And so as the
mathematical condition,
what is the value of x that
satisfies this equation?
Well, it could be 7
feet or minus 7 feet.
That could be the size of the
room that you're living in.
But of course, we
would basically
discard the minus 7 feet
as an un-physical solution.
And we would ignore it.
A very similar thing
is going on here,
where there are these
solutions that we're not
really sure whether
or not we can trust
that they're gravitational
waves or just
a mathematical artifact.
And these can be never
observed in practice.
So 1934, Einstein was
still thinking about this.
He was very sure, very adamant
that they did not exist.
In fact, he wrote this paper
with Rosen, who-- also, there's
this Einstein-Rosen
Wormhole Solution.
That's different
from this paper.
They wrote this paper.
At this point, it
was just a pre-print.
And actually, the
paper has been lost.
But there was a
letter from Einstein
to Max Born, who is a
quantum physicist, where
he says, "I arrive at
the interesting result
that gravitational
waves do not exist."
And the title of the paper is
Do Gravitational Waves Exist?
And the general rule of thumb
is that if the title has
a question mark in
it, the answer is no.
So even though the
paper has been lost,
we can bet that
the answer was no.
And this letter seems
to confirm that.
At the time, it's sort of
an interesting development
of the peer review system.
So scientists write a paper.
You submit it to a journal.
Back then, the expectation would
just be that it got published.
Now, it has to go through a
peer review process, where
outside experts
look at the paper
and make sure you're not
saying anything that crazy.
In 1934, Physical
Review just started
to implement the
peer review system.
And Einstein's
paper was rejected.
He was very upset
about this and said
he would never publish
in Physical Review again.
And he didn't.
And the referee
report came back.
Einstein didn't like that.
But sure enough, he submitted
it to another journal.
And the title had changed
too On Gravitational Waves.
And the result was basically
summarized by this.
So he sent a letter to the
editor describing the changes.
And he says here, "the
second part of the article
was altered by me as we had
misinterpreted the results.
And I want to thank my
colleague, Professor
Robertson."
So Robertson was
not on the paper,
even though by
today's standards,
he probably should
have been on the paper.
But at this point, Einstein
just revised everything
based on this referee
report and resubmitted it
with a new title and
a new conclusion.
Now, they exist.
But that's actually not
quite the end of the story.
There was still a lot
of vigorous debate.
And it wasn't really
until, I would say,
this very landmark
conference, called the Chapel
Hill Conference, in 1957.
So this is a collection
of physicists.
So there aren't going to
be very formal proofs.
So this is more of a proof
by discussion, where there
was arguing and discussing
about the measurability
of gravitational waves.
And really, it was
these four people--
Felix Pirani, we already met.
Richard Feynman, Hermann
Bondi, and Joseph Weber.
Weber actually was inspired by
the talks of the other three.
But Felix Pirani-- he was a
new researcher at the time.
In fact, he may have
even been a post-doc.
And he proposed a
thought experiment
where you could effectively
measure the gravitational waves
by setting up what is
effectively two beads.
If you have two beads, and
you let the waves pass by,
he proposed that
they would oscillate.
And by the rate of
oscillation and the deviation
between the beads, you
can infer everything
you would need to
know about the wave.
So that was his
thought experiment.
At the time, that
was a breakthrough.
And that's actually
the basis for LIGO.
That's exactly
how LIGO operates.
Instead of beads,
there are mirrors.
This was extended a bit by
Feynman and Bondi, who actually
get most of the credit.
But they imagined that instead
of these beads just floating
in free space, they're
attached to a rod.
And they have some friction.
So when they move,
they dissipate heat.
And the fact that you have
a wave turning into heat
means that it must be physical.
Because energy just can't
be created out of thin air.
So this was a
mechanism by which they
reasoned at this conference
that gravitational waves really
do exist.
Joseph Weber.
He actually went off and
built a detector right
after leaving this conference
called the Weber Bar.
And the Weber Bar didn't really
see any gravitational waves.
But it was the first
attempt to detect them.
And so I'm told, he
was the inspiration
for Kip Thorne wanting to go
and detect gravitational waves.
They were on some hike, I
think, in the Swiss Mountains
or something.
And Joseph Weber wouldn't
stop talking about detecting
gravitational waves.
And that was one
of the inspirations
for Kip Thorne's quest
for actually finding them.
So basically, where
have we come to?
Well, there is a 50 year period
where the work of many people--
I've just summarized
the key players,
but there are
many, many people--
prove that you can actually
solve Einstein's equations
under fairly general conditions.
And gravitational waves are one
particularly important feature
of these solutions.
So what does that look like?
We're going to play an
animation with the sound off.
So this is Brian Greene
going to explain it.
But he won't be talking.
I'll be talking.
Right now, there's no black
holes, no neutron stars.
This is just empty space.
There's no matter anywhere.
We have our Cartesian
grid like we had before.
So again, this is totally
undistorted space-time.
And now we plop two
neutron stars-- just
appear out of nowhere.
And you can see that
what they're doing
is they're orbiting one another.
And just as you would imagine
in a bathtub or something,
stirring up the water, or
throwing a rock into a pool,
waves will be emitted.
So what's waving is not water.
It's space itself--
the geometry of space.
And these waves will propagate
over very large distances.
And over here, this is where our
detectors are located on earth.
So this is another way
of visualizing what
we saw in the very first slide.
And now that we know a little
bit about gravitational waves,
we can believe this
animation more.
And we have some understanding
that these solutions even exist
and that it's reasonable to ask
what do the waves look like,
if you knew what the mass
of these two stars were.
So we know that the
waves exist and that you
can solve the equations.
How to actually detect
gravitational waves--
this will be a pretty
short part of the talk.
But I just wanted to give
some sense of the detection
mechanism.
So the best generators
of gravitational waves
are very massive
objects-- massive stars,
massive black holes.
And that's given by
this animation, here.
That's typically what generates
gravitational waves the best.
And here's one
real world example.
I've taken some liberties, one
being that you don't really
know exactly where
the gravitational wave
signal came from.
You know roughly where
it is on the sky.
So you can't localize
it this well.
But one such gravitational
wave detection in 2017,
let's say it was coming
from this galaxy, here.
And if you haven't
seen these images,
you may think these are stars.
These little specks here
are actually other galaxies.
And this is a chunk
of the universe.
So here's a
gravitational wave being
generated by some other galaxy.
And this particular one traveled
about 130 million light years.
So it took a very long time.
And eventually, it
reached us on earth.
And scientists will detect it.
And then once you
have the signal,
that's not really enough.
You want to know what are the
masses of the black holes.
Maybe you want to ask what are
the dimensions of space-time.
Actually, it turns
out that's somewhat
encoded into the signal.
And maybe you want to
know about populations.
How many black holes are there?
What are the typical masses?
So these are all some
of the science questions
you can answer with your data.
So we're not really
putting black holes
at the top of the planet, here.
They're located at these
locations throughout the earth.
So there's these two, here.
LIGO Hanford and Livingston.
And there's one in Germany.
That's called GEO600.
There's one in
Italy called Virgo.
And there's a few
that are planned
or either coming online soon.
And all of these
detectors look the same.
They are basically this L shape.
It's called an interferometer.
And you can imagine
this L as being
a corner of one of those boxes.
So remember, a lot
of these figures,
when we showed the
warping of space-time,
there's a whole bunch of boxes.
This L is basically
one corner of that box.
And in order to build
these interferometers--
you can see here,
these are trees--
and they had to
cut a lot of them
down to get this
very long cavity.
We'll talk about
that in a second.
But this is kind
of an aerial view
of what the detector
actually looks like.
So I'm going to
play a short movie.
It's about a minute.
I didn't think I could explain
it as well as this little clip.
But this is an explanation
by Ray Weiss and also
the narrator, and
sort of an animation
about how these
waves are detected.
AUDIENCE: To measure this
stretching and squeezing,
Ray turned to a device
called an interferometer.
A laser beam is split
and sent down a pair
of long, perpendicular tubes,
each precisely the same length.
The two beams bounce off mirrors
and recombine back at the base.
The light waves come back
lined up in such a way
that they cancel each other out.
RAY WEISS: And you add them
together, you get nothing.
You get a 0-- a big fat 0.
No light gets detected
at the photo detector.
AUDIENCE: But when a
gravity wave comes along,
it distorts space and
changes the distance
between the mirrors.
One arm becomes a little longer.
The other, a little shorter.
An instant later, they switch.
This back and forth
stretching and squeezing
happens over and over
until the wave has passed.
As the distances change,
so does the alignment
between the peaks and valleys of
the two returning light waves.
And the light waves no
longer cancel each other out
when added together in
the recombined beam.
Now, some light does
reach the detector
with an intensity that
varies as the distance
between the mirrors varies.
Measure that intensity, and
you're measuring gravity waves.
SCOTT FIELD: That's roughly
how the detector works.
And the key operative
word there is
distances between the mirrors.
So these mirrors are located
about four kilometers apart.
They are mirrors that are
just basically hanging there.
They're pendulums.
And this light is
bouncing back and forth.
And they're very
accurate meter sticks.
They measure the distance
between these two measures
extremely accurately.
So if there's any change in
the pathway between the two
mirrors, this will show up
as an interference pattern.
And this is the cartoon before.
If we imagine we have
a detector right here,
this is the one in Livingston.
Basically, if you
had a bird's eye view
of what the detector is
doing when a wave passes by,
this is roughly
what we would see.
Of course, we actually
measure the output
of the interferometer, which
is the interference pattern.
But this is what the arms
of the mirrors are doing.
So this is really where
the magic comes in.
So what are these
distances involved?
The mirrors themselves are
located four kilometers apart.
And the gravitational
waves are extremely weak.
This comes down to the fact
that Newton's constant G
is a very small number.
And if we ask what is a
typical gravitational wave
signal, what kind of
displacements can we expect?
It's basically 10 to
the minus 18 kilometers.
That's extremely small.
Just for the sake
of comparison, it's
smaller than the
size of a proton.
So you're trying to
measure distances changing
on the order of
less than the size
of a proton over a four
kilometer distance.
So I find that remarkable
that the experimentalists
can actually get to
this level of accuracy.
I'll point out that it's not
entirely just the detector.
The pipelines that detect
gravitational waves
are fairly complicated
and account
for the properties of the
noise, which is non Gaussian.
There's a lot of
glitchy behavior.
So this all has to be
built in and verified.
And there's an entire team
of people that work on this.
So the role of
computational models
is basically
summarized over here.
So the signal's incredibly weak.
And we can't do a good
job of pulling out
the signal, unless we know
what we're looking for.
So one analogy is
you're driving in a car.
And the car is really noisy.
Maybe the muffler hasn't
been fixed in a while.
You can't really hear the radio
is on, but you know it's on.
You know there's
a song coming out.
And if you were
able to search over
the entire catalog of all
songs ever produced by humans,
you could do a data analysis
technique known as correlation
to figure out if this particular
song is buried in the noise.
That's effectively
what's going on here.
But you need to know
what the signals are
and what you're
looking for in order
to actually make this work.
So this is really where
computation comes in.
Because equations
are hard to solve
with the paper and pencil.
So we need a
computer to do this.
A very brief history of
computational relativity.
What is computational
relativity?
It's, effectively, the
use of computer codes,
mathematical algorithms,
and techniques in order
to solve Einstein's equations.
The basic setup is given
by this diagram, here.
So this is the very canonical
standard binary black hole
setup.
You have two black holes.
You can think of them as,
effectively, spinning tops.
They have masses.
And they also have spin.
And the spins have directions.
So maybe they're spinning
like this or like this.
And they orbit one another.
So if you tell the computer what
the masses of the black holes
are and what their
spins are, in principle,
if the algorithm is good, and
the computer is large enough,
you can solve the equations.
That's kind of what we're after.
In fact, back in the '50s,
also at this Chapel Hill
Conference-- the one from 1957--
there was a comment made
that what's the big deal.
These are just
differential equations.
People solve these
all the time--
people who are working
in weather prediction.
If you're working in
aerodynamics-- so for example,
if you want to know
how to optimize
the shape of an airplane or
a car so it's very efficient,
these are all partial
differential equations.
And people have no
problem solving these.
Why are the Einstein
equations so difficult?
Well, first thing is
it took about 50 years.
So the first thought
that you could actually
put these on a
computer, and that's
a reasonable thing to do--
again, the 1957 conference.
This was suggested
by Bryce Dewitt,
who is a famous mathematical
physicist, and Charles Misner.
And this is basically
summarized in this quote.
At least, to my knowledge,
this is the first time someone
proposed using a computer
for this problem.
They said, "first we assume you
have a computing machine better
than anything we have now.
Many programmers and" the
keyword, a lot of money,
"you can look at a pretty
nice solution of the Einstein
equations if you have
the initial conditions,"
this paragraph goes on to say.
They were basically
laying down a challenge
to the community of
physicists and mathematicians.
Try to solve these
on a computer.
And in 2005, that was
the first time that
was actually done successfully.
So a brief history of
some of the landmarks
along this 50 year journey.
The first real, honest
attempt was in 1964.
And this was done by Susan
Hahn and Richard Lindquist.
So Susan Hahn was a
student of Peter Lax, who
is a pretty prominent, famous
applied mathematician, who
also works at the
Courant Institute at NYU.
And his student, Susan Hahn,
was working in New York for IBM.
She had access to
supercomputers.
At the time, no one else
had access to supercomputers
unless you worked in a lab or
the company that built them.
So in this case, she
worked for the company
that built the supercomputers.
And she met Richard
Lindquist at a conference.
And he was lamenting about
needing access to a big machine
to run these simulations.
So she said, sure, you
can use my machine.
That's fine.
They coded it all up.
They threw it on the
computer, and it crashed.
I'll talk a little bit about
what that means, in a second.
But basically, they got
nothing useful out of it.
And just to give you some
historical perspective,
this machine cost $3 million.
And that was in the 1960s.
And it gives one megaflop
of computing power.
My Samsung Galaxy--
I downloaded something
called LINPACK,
which is a way that people
benchmark supercomputers.
They have an app for this.
And I benchmarked my iPhone.
How quickly can it do matrix
vector multiplication?
It's about 700 to
800 mega flops.
So my phone is 700 to
800 times more powerful
than this entire
room of computers.
And this cost $3 million.
And my phone, I basically got
it for a couple hundred bucks.
So they were up against
pretty formidable odds.
So maybe one approach to
solving these equations
is we'll get a bigger computer.
And we can solve them.
That's a reasonable
thing to decide.
And this is sort of the
approach that Larry Smarr took.
So in 1982, Peter Lax-- the same
Peter Lax that mentored Susan
Hahn--
wrote this report
that was very critical
of the state of affairs
in the United States,
as far as computing access goes.
So if you were a scientist
working in the United States,
you could either run
your code in Germany,
or you had to have
special clearance at a lab
or knew someone like Susan Hahn
that had access to a computer.
That was pretty much the
only ways to run your code.
So Larry Smarr
submitted this proposal
to the NSF, which is the
National Science Foundation.
They fund scientific endeavors.
Usually, the NSF
asks for a proposal.
You write it.
You send it in.
You wait months, and
months, and more months.
And you maybe get
something back.
And they'll fund a
little portion of it.
In this case, he just wrote it.
He said we need a lot of money.
We need to build a network
of supercomputing centers.
Let's do this.
And they funded it.
To my knowledge, this
is the only, at least
it was the first,
unsolicited proposal.
And it led to what, now, we
would probably call it Exceed.
This is a network of
supercomputing centers
across the United States.
At the time, these were
the main institutes.
These were the founding members.
And it wasn't called
Exceed at the time.
And because of Larry
Smarr's involvement,
many of the first
codes that were run
were numerical
relativity codes, trying
to solve this problem
in general relativity.
There was really no success.
We had bigger
computers, more access.
Nothing really was working out.
So basically,
something is wrong.
And the problem is that
there's now an urgency.
In 1993, Congress has funded
the construction of LIGO.
And right now, we
don't really know
what the signals are going
to be, or what to look for.
So this is becoming a
very urgent problem.
So concurrently, the NSF founded
something called the Grand
Challenge, which was
basically offering
lots of money and resources
to numerical relativists
and mathematicians to develop
tools to solve Einstein's
equations on a computer.
That was the goal.
So what is the problem?
If anyone has used
a computer, you've
probably encountered the
blue screen of death.
Basically, your computer
just stops working.
And you don't know why.
In this case, it's more of
an operating system issue.
So usually, there's
a bug in the code.
The operating system
doesn't know what to do.
The computer locks up and just
refuses to do anything else.
This is very similar to
what happens when you're
trying to solve a
partial differential
equation on a computer, but
you do it in the wrong way.
Loosely speaking, the
errors that you're
trying to make
small sort of grow
without bound
until the algorithm
doesn't know what to do.
And it gives up.
And the computer crashes.
People would say
the code crashes.
So the Grand Challenge was about
trying to fix this problem.
Why is the code
crashing when we try
to solve Einstein's
equations on a computer?
And these were some of the
founding members of the Grand
Challenge.
And they all signed their names.
They bet Kip Thorne.
I think, as the bet
goes, Kip Thorne
bet the numerical
relativity community
that LIGO would detect
gravitational waves before they
could simulate them
on their computers.
That was the bet.
And thanks to France Pretorius
and many people before him,
Kip lost this bet.
Basically, they were able to
simulate gravitational waves
before they were detected.
And the big breakthrough
came in 2005.
And here's France Pretorius,
here, drinking some coffee.
As the story goes, this
basically came out of nowhere.
He was working on
new techniques,
exploring these new avenues
that were originally proposed
by applied mathematicians.
And at this very famous
conference in Banff,
which is a mathematical
Institute in Canada,
he announced that
he had successfully
solved the Einstein equations
for two black holes.
They orbited.
You can see here, they
didn't orbit that much, maybe
about one cycle.
But it was enough to get
the gravitational waves
and to see what the
signal would look like.
So this was a huge
milestone for the simulation
of gravitational
waves on a computer.
Today, many groups now know.
In hindsight, it all looks easy.
In hindsight, oh, of
course, we would solve it
with this Kreiss-Oliger
dissipation.
And of course, we would have,
in the more technical speak,
a strongly hyperbolic well-posed
formulation of the Einstein
equation to throw
in the computer.
But that was not
obvious before 2005.
So now, many people have
codes to solve the Einstein
equations.
The one I'm most familiar with
is by the SXS Collaboration.
They've developed something
called the Spectral Einstein
Code.
It's a very accurate code for
binary black hole systems.
This is an example of what
one of these numerical grids
look like.
Basically, you have
two black holes, here.
And you basically refine
it with lots and lots
of computational resources.
We're not going to
really get into that.
We're just going
to look at a movie.
This is the output of the
code, which we can now
do, today, in maybe a week or
two weeks on a supercomputer.
And we can see there's many
things happening, here.
So here are the two black
holes orbiting one another.
You can see that
they're slowly getting
closer together,
because you can see
the tracks are in-spiraling.
That's because the waves
are emitting energy.
And that energy loss is
bringing in the orbital radius.
The gravitational wave
signal that LIGO would see
is given here.
So this moving vertical
bar is effectively saying,
what would the signal be if
we could move it backwards
to the location of the source?
There's also some interesting
color coding going on, here.
So remember, I said
clocks tick more slowly
in a deep gravitational field.
The blue area is where clocks
are ticking at the normal rate.
So if you're a close to
the black hole, in here,
your clock is ticking
a little bit slower.
And these arrows are
effectively saying, where
is the gravitational force?
So if you were to just
spontaneously appear
in this area, you would
have followed the arrows
into the black hole.
So some of the key contributions
to gravitational wave science
that these
breakthroughs enabled--
well, now, we can explore the
binary black hole parameter
space.
So there's masses and spins.
And the solutions are
parametrized by these numbers.
Now, we can compute
them and use them
to process our data to figure
out what sort of signals
were observed and what are the
masses of the black holes that
emitted these signals.
And effectively,
especially if you're
going to use high
precision science--
for example, there are
many alternative theories
of gravity.
All of them make
different predictions
about what the gravitational
wave signal may look like.
If you're going to do
high precision science,
you have to solve Einstein's
equations very accurately.
And these are more
than just simulations.
These are really solving
the equations themselves.
If you don't like the level
of accuracy you're getting,
you can just wait a
little bit longer,
and you'll get a more
accurate solution.
So I'm pretty much
running out of time.
I have a couple of minutes.
I wanted to point out maybe a
few pieces of future directions
and ongoing work, but sort of
biased towards UMass Dartmouth.
And at least as far
as gravitational waves
are concerned, there's a
couple of pressing demands.
The one demand is that the
detectors are getting better.
They're getting
more accurate, which
means we're going
to see more signals.
We're going to see these
signals with higher
signal to noise
ratio, which means we
can do more accurate science.
This is going to place demands
on the numerical relativity
codes.
In fact, a recent
preliminary result
shows that even the
best codes we have today
are insufficient for processing
data for future detectors.
And it may not be
enough to just throw
more dollars at the problem,
get bigger computers.
We may actually need
better techniques.
Also, sort of encoded in
these longer durations,
there is a planned
space-based gravitational wave
detector called LISA.
This should be coming online
in about 10 to 15 years.
And this will track
two binary black holes
over years and years of
their orbital process.
This means that the
signals are very long.
And in order to simulate
this on a computer,
you would need a
computer that really just
hasn't been built yet.
So even the simulations
we currently have,
maybe it does 20 orbits.
And that's really pushing
the frontier of the problem.
To do hundreds of
thousands of orbits,
this is just not possible
with current computing power.
So at UMass Dartmouth,
we have the Center
for Scientific Computing
and Visualization Research.
And we're working
on many things.
But as far as it relates
to gravitational wave
physics, and gravity
theory, and astrophysics,
there's a couple of
different distinct groups.
There are many PhD students,
master students, and undergrads
working on a variety of topics.
So just a couple
of examples of what
people are doing on campus--
Bob Fisher with some of his grad
students, Rahul and Gabriel.
They do simulations of
white dwarf explosions
using this flash code.
And they use enormous
supercomputers
that are located in Texas and
throughout the country in order
to do these simulations.
In fact, as you add
more physics in,
the problem gets more
difficult. So in some sense,
black holes are easy.
There's no matter.
They're very simple objects.
If you start adding
in matter fields,
things really get out
of control quickly.
And there's a lot of issues
with solving the equations that
really require advanced
techniques and very
powerful computers.
Gaurav Khanna-- he works
on, effectively, the types
of problems that LISA will see.
So these are called extreme
mass ratio binaries,
where you have a
super-massive black hole
about a million times larger
than the sun and a very
small black hole orbiting it.
So this is the one that's
orbiting for many years.
And you need to simulate the
signal over many, many years
of its evolution.
You can't really do this
on a traditional computer.
Or you could, but you'd
have to wait too long.
So he's been very good
at getting novel HPC
solutions into the problem.
So one example here is he
built a supercomputer out
of PlayStations.
PlayStations, it turns out,
are great supercomputers.
In fact, they're
sold undervalue.
Because all the markup is
actually in selling the games.
So if you can get a whole
bunch of PlayStations
and tie them together, you
have the computing power
of what might
otherwise be a very
large, national supercomputer.
The problem is
they get very hot.
So you have to cool them off.
So he put them in a meat
cooler and plugged it
into the electricity
of the campus.
And that cools down
these supercomputers,
so they won't
overheat and burn out.
And a lot of his codes
are run on these machines.
One of the things
that I'm involved in
is building fast
computational models.
So we have these numerical
relativity simulations.
They take days or weeks.
So they can't be directly
used in the analysis
of gravitational waves.
They would just be too slow.
But what you can do
is if you simulate
enough binary black
hole configurations,
you can build a model
directly from the data
that is quite
accurate and faithful
to the underlying physics.
And grad students--
Nur is extending this
to the extreme mass ratio case.
And Furoz, who I couldn't
find his picture,
is using these models
to ask questions
about how well can
you resolve the spins
of individual black holes.
And some additional projects
that we're exploring--
Dwyer has been working on using
convolution neural networks
for classifying
gravitational waves.
Kim and Owen have been
working on accurate models
for gravitational
wave propagation.
They propagate over, as
we just saw, hundreds
of millions of light years.
And to accurately resolve
the propagation effects,
you sometimes need very, I would
say, high accurate methods.
Otherwise, you can
get de-phasing.
And a class of methods which
are near and dear to my heart,
I think largely because I was
a grad student here, at Brown.
And Jan Hestaven,
my advisor, was
one of the pioneers in
discontinuous Galerkin methods,
along with Chi-Wang
Shu and others.
And we started looking
at exploring these,
back when I was a grad student,
for the extreme mass ratio
binary problem.
And Ed McClain, who is a grad
student of the PhD program,
is working on extending these
to more complicated systems.
And Akash Kardum,
who just graduated,
was working on exploring
some of the techniques
in the context of hydrodynamics.
He's now working at Microway
building supercomputing
clusters with GPUs.
I think I'm about an hour in.
So I think that's probably
enough of me talking.
So I'll just leave this up,
if you have any questions.
[APPLAUSE]
BRENDAN HASSETT:
Thank you very much.
Are there any questions?
Yeah?
AUDIENCE: There's
an intuition-- why
did the waves stop traveling?
They measure their time.
Black holes are rotating.
Why couldn't you
count the waves?
SCOTT FIELD: So they do.
In fact, if you imagine
a thought experiment
where they're
infinitely far apart,
you would see an
infinitely long signal.
But at some point, they
get closer and closer.
And they merge into
a final black hole.
And at that point, the
wave effectively turns off.
AUDIENCE: But the sensors,
from what you said,
measured some movement.
But like [INAUDIBLE] but
then the waves stop coming?
SCOTT FIELD: Yes, yes.
Because at some point
in the distant past,
in the other galaxy,
the two black holes,
which were distinct
objects, merge
into one larger black hole.
So the waves shut off.
So the detector will
no longer see them.
BRENDAN HASSETT: Yes.
AUDIENCE: [INAUDIBLE]
SCOTT FIELD: Sorry, I couldn't--
AUDIENCE: Earthquakes
or abnormal temperatures
are not going to
affect the detector?
SCOTT FIELD: Oh, so the
properties of the detector.
I know very little about
the experimental design
and effects of the detector.
I am told that they have to
cool it down to very close to 0
Kelvin, I believe.
Basically, thermal vibrations
will make the mirrors move.
So they need to isolate those.
But there are
literally thousands
of sources of possible ways
that the mirrors can move.
And that's an entire
enterprise in and of itself
that I am not that
familiar with.
But I do know that, yes, they
have thought of, basically,
thermal vibrations as a possible
noise source for the mirrors.
BRENDAN HASSETT: Yeah?
AUDIENCE: The problem where
multiple gravitational waves
are coming at different
angles are interfering
with the measurement?
Or are they all just the same?
SCOTT FIELD: Yeah,
that's a good question.
In some sense, the fact that the
detector, as accurate as it is,
is not so accurate
that it can see
the entire symphony of black
holes occurring everywhere.
So you get very distinct
signals coming in.
The future LISA
gravitational wave detector
is going to be so accurate
that, effectively, all
of the gravitational
waves-- you're
kind of seeing them at once.
That is almost like
the noise source.
It's very hard to resolve
individual signals,
because there are so many
other signals that are mixing
and lying on top of each other.
It is true that at any given
instant in time, like right
now, there are many,
many gravitational wave
signals coming from mergers
happening all over the galaxy,
passing through.
But it just so happens
that the detector,
given the frequency band
that it's sensitive in
and the level of
accuracy it can see,
it only seems very
strong signals only
over a certain region
of frequencies.
But yeah, this going to become
a problem as the detectors
get more and more accurate.
It's one of the
frontier challenges,
especially for the
space-based detector.
AUDIENCE: Are there
other things [INAUDIBLE]
which could absorb
those waves [INAUDIBLE]??
SCOTT FIELD: Yeah, so
the waves are so weak,
because basically
Newton's constant--
the way that gravity couples
to other massive objects.
They effectively
travel undistorted
through the universe.
Or at least that's
the approximation.
Of course, though, if they
pass through other galaxies--
and these are very
massive objects,
so there could be some
scattering effects.
But to my knowledge, this really
hasn't been an issue, yet.
Again, coming back to
the space-based LISA.
LISA is so accurate that all
of these things really matter.
There's been a renewed interest
to re-examine this question
in the context of
environmental effects.
So if you have a
super-massive black hole
and a small
perturbing one, there
could be other nearby perturbing
objects or dust clouds.
And taken as a whole, this could
really corrupt your ability
to understand the signal.
So far, this hasn't
been an issue.
But in principle, there's no
reason why it couldn't be.
But right now, we're
kind of saved by the fact
that, yes, these
waves don't really
couple to other massive objects
because of the smallness
of the constants.
BRENDAN HASSETT: Go ahead.
AUDIENCE: Is there any
limit to [INAUDIBLE]
like in a very
massive black hole.
SCOTT FIELD: Yeah, so--
AUDIENCE: If it's
like a huge wave.
SCOTT FIELD: If the black
hole was really big,
or if it was really
close to you, yes.
The wave would be quite strong.
In fact, the dominant
behavior of the wave--
and not just gravity waves.
All waves, kind of like 1
over R. So if you're here,
and you get half the
distance, the wave amplitude
would go up by a factor of two.
And that's also
true for the masses.
If you double the
masses, roughly speaking,
the amplitude will double.
Currently, most of the
black holes we've seen
are roughly 10 to 30
solar masses, 40 maybe.
AUDIENCE: Two
colliding galaxies.
SCOTT FIELD: Yes.
You have two colliding.
So like we're on a collision
course with Andromeda.
And we will eventually
collide with them.
And we have a
super-massive black hole
at the center of our galaxy,
Sagittarius A. There's
one in the center of Andromeda.
They will eventually
in-spiral and merge.
And we will be in this
galaxy, because we're
going along for the ride.
And so yes.
That will be a giant
gravitational wave
that all of the detectors
should have no problem seeing.
[LAUGHTER]
Sorry?
AUDIENCE: The
elasticity of space-time
is limited [INAUDIBLE].
SCOTT FIELD: Oh.
Yes.
There's an extent to which it
maybe cannot stretch beyond it.
So this comes down to the
constants, I would think.
Yes.
BRENDAN HASSETT: One
more question, please.
Yes?
AUDIENCE: So as the
black holes orbit,
do they seem to be
losing kinetic energy?
SCOTT FIELD: Yeah.
AUDIENCE: So this
is what makes them--
because some distant
object can be moved
with respect to each other?
That represents some
heating that is physical.
What's the medium that's
carrying energy through space?
SCOTT FIELD: Yeah, so I'm going
to try to answer two questions.
I'll try to get two
answers into one.
The losing of the
energy-- it's exactly true
that, while they're
in-spiraling,
the wave energy
that's being lost
translates into kinetic
energy being lost.
So they in-spiral.
At some point, they also merge.
And a giant burst of
gravitational waves come out.
So for example, if you have a
solar mass 10 and a solar mass
10 black hole system, the
final object will not be 20.
There'll be some deficit.
And since E equal MC Squared, if
you have a little bit of mass,
you have a lot of energy.
And so during this
merger process,
a lot of gravitational
wave mass is lost.
It's converted as wave energy.
I think, actually,
during the merger event,
it's brighter than all of the
stars in the universe combined,
at that instant in time.
What was the second one?
The original question?
AUDIENCE: It would just
be what moves space-time--
SCOTT FIELD: Oh, yes.
AUDIENCE: --do you think
that's a carrier of energy?
SCOTT FIELD: So it is true.
Yes, it is a carrier.
So that was basically what the
physicists and mathematicians
were wrestling with
for roughly 50 years.
It's not obvious at all that
that should be the case.
But it turns out that,
yes, space and time itself
is a transmitter of energy as
time dependent fluctuations
in the gravitational field.
But it's not obvious.
It's kind of a fact that I
just take for granted, now.
But if I really try to
wrap my head around it,
it's kind of a mysterious
thing, to be honest.
BRENDAN HASSETT:
Thank you very much.
[APPLAUSE]
