GRAVITY WAVES

[spectre]

http://www.ligo.caltech.edu/
http://www.ligo.org/

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RIPPLES IN SPACE-TIME
http://space.newscientist.com/article/dn13579-gravitational-wave-detectors-to-get-major-upgrade.html
Gravitational wave detectors to get major upgrade
BY David Shiga / 02 April 2008

The LIGO project to detect gravitational waves has been given the
green light to begin a major upgrade of its detectors. When the
upgrade is completed in 2014, the project may be sensitive enough to
detect gravitational waves - which have yet to be observed - as often
as once a week. Gravitational waves are ripples in the fabric of space
predicted by Einstein's general theory of relativity. They are
triggered by the motion of massive objects. "[With the upgrade],
either we'll see a signal or Einstein's general theory of relatively
will be wrong," says LIGO director Jay Marx of Caltech in Pasadena,
US.

The ability to listen to gravitational waves would also open up a
completely new window for astronomers to observe the universe,
allowing them to witness violent events like the collision of pairs of
black holes or neutron stars, and even hear the primeval groaning of
the universe as it expanded during its earliest moments. LIGO (Laser
Interferometer Gravitational-wave Observatories) uses two
gravitational wave detectors in the US - one in Livingston, Louisiana,
and the other at the Hanford nuclear facility near Richland,
Washington. Using lasers, the detectors look for slight changes in the
length of tunnels several kilometres long that would occur with the
passage of a gravitational wave.

Powerful lasers

Although LIGO is the world's most sensitive gravitational wave
project, scientists estimate that currently it has a chance of only a
few percent per year of detecting a source for the waves. Scientists
just have to wait and hope that a violent enough event will occur
close enough to the Earth to be noticeable. To improve the situation,
scientists have been planning a major upgrade called Advanced LIGO.
Now, the project has been given approval to begin the upgrades, which
should be finished in 2014. The US National Science Foundation's
governing board gave the go-ahead for the $205 million upgrade at a
meeting on 27 March.

The upgrade will involve replacing the existing 10-watt lasers with
180-watt versions, among other improvements. Put together, the
improvements mean Advanced LIGO will be 10 times more sensitive in the
frequency range it currently monitors. It will also be able to detect
waves at much lower frequencies, down to 10 Hz, compared to its
current lower limit of around 40 Hz. The improvements mean Advanced
LIGO will be able to detect sources 10 times farther from Earth than
it can now, increasing the volume of space it will probe by a factor
of 1000. "With Advanced LIGO, we think we'll be seeing gravitational
waves from sources maybe once a week," Marx told New Scientist.
"Advanced LIGO really opens the door to a new form of astronomy."

-

DETAIL
http://www.ligo.caltech.edu/advLIGO/scripts/summary.shtml
http://mr.caltech.edu/media/Press_Releases/PR13123.html
http://www.ligo.caltech.edu/docs/M/M060056-08/M060056-08.pdf
http://www.space.com/scienceastronomy/astronomy/LIGO_works_991115.html
http://www.space.com/scienceastronomy/astronomy/ligo_open_991116.html
http://www.spacedaily.com/reports/MU_Physicist_Defends_Einstein_Theory_And_Speed_Of_Gravity_Measurement_999.html
http://science.slashdot.org/article.pl?sid=08/01/23/041227
http://www.math.columbia.edu/~woit/wordpress/?p=574
http://geo600.aei.mpg.de/

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MEANWHILE
http://www.wired.com/techbiz/it/news/2007/10/ps3_supercomputer
Astrophysicist Replaces Supercomputer with Eight PlayStation 3s
By Bryan Gardiner Email 10.17.07 | 12:00 AM

Suffering from its exorbitant price point and a dearth of titles,
Sony's PlayStation 3 isn't exactly the most popular gaming platform on
the block. But while the console flounders in the commercial space,
the PS3 may be finding a new calling in the realm of science and
research. Right now, a cluster of eight interlinked PS3s is busy
solving a celestial mystery involving gravitational waves and what
happens when a super-massive black hole, about a million times the
mass of our own sun, swallows up a star. As the architect of this
research, Dr. Gaurav Khanna is employing his so-called "gravity grid"
of PS3s to help measure these theoretical gravity waves -- ripples in
space-time that travel at the speed of light -- that Einstein's Theory
of Relativity predicted would emerge when such an event takes place.

It turns out that the PS3 is ideal for doing precisely the kind of
heavy computational lifting Khanna requires for his project, and the
fact that it's a relatively open platform makes programming scientific
applications feasible. "The interest in the PS3 really was for two
main reasons," explains Khanna, an assistant professor at the
University of Massachusetts, Dartmouth who specializes in
computational astrophysics. "One of those is that Sony did this
remarkable thing of making the PS3 an open platform, so you can in
fact run Linux on it and it doesn't control what you do."

He also says that the console's Cell processor, co-developed by Sony,
IBM and Toshiba, can deliver massive amounts of power, comparable even
to that of a supercomputer -- if you know how to optimize code and
have a few extra consoles lying around that you can string together.
"The PS3/Linux combination offers a very attractive cost-performance
solution whether the PS3s are distributed (like Sony and Stanford's
Folding@home initiative) or clustered together (like Khanna's), says
Sony's senior development manager of research and development, Noam
Rimon.

According to Rimon, the Cell processor was designed as a parallel
processing device, so he's not all that surprised the research
community has embraced it. "It has a general purpose processor, as
well as eight additional processing cores, each of which has two
processing pipelines and can process multiple numbers, all at the same
time," Rimon says. This is precisely what Khanna needed. Prior to
obtaining his PS3s, Khanna relied on grants from the National Science
Foundation (NSF) to use various supercomputing sites spread across the
United States "Typically I'd use a couple hundred processors -- going
up to 500 -- to do these same types of things."

However, each of those supercomputer runs cost Khanna as much as
$5,000 in grant money. Eight 60 GB PS3s would cost just $3,200, by
contrast, but Khanna figured he would have a hard time convincing the
NSF to give him a grant to buy game consoles, even if the overall
price tag was lower. So after tweaking his code this past summer so
that it could take advantage of the Cell's unique architecture, Khanna
set about petitioning Sony for some help in the form of free PS3s.
"Once I was able to get to the point that I had this kind of
performance from a single PS3, I think that's when Sony started paying
attention," Khanna says of his optimized code. Khanna says that his
gravity grid has been up and running for a little over a month now and
that, crudely speaking, his eight consoles are equal to about 200 of
the supercomputing nodes he used to rely on. "Basically, it's almost
like a replacement," he says. "I don't have to use that supercomputer
anymore, which is a good thing."

"For the same amount of money -- well, I didn't pay for it, but even
if you look into the amount of funding that would go into buying
something like eight PS3s -- for the same amount of money I can do
these runs indefinitely." The point of the simulations Khanna and his
team at UMass are running on the cluster is to see if gravitational
waves, which have been postulated for almost 100 years but have never
been observed, are strong enough that we could actually observe them
one day. Indeed, with NASA and other agencies building some very big
gravitational wave observatories with the sensitivity to be able to
detect these waves, Khanna's sees his work as complementary to such
endeavors. Khanna expects to publish the results of his research in
the next few months. So while PS3 owners continue to wait for a fuller
range of PS3 titles and low prices, at least they'll have some reading
material to pass the time.

.

CONTACT
Gaurav Khanna, Ph. D.
http://www.umassd.edu/engineering/phy/people/faculty/gkhanna/welcome.cfm
email : gkh...@umassd.edu

SEE ALSO
http://xxx.lanl.gov/
http://xxx.lanl.gov/find/gr-qc/1/au:+Khanna_G/0/1/0/all/0/1
http://gravity.phy.umassd.edu/
http://gravity.phy.umassd.edu/qc.html

.

BUILD YER OWN
http://hpc.sourceforge.net/
http://gravity.phy.umassd.edu/ps3.html

"The Sony PlayStation 3 has a number of unique features that make it
particularly suited for scientific computation. To start with, the PS3
is an open platform, which essentially means that one can run a
different system software on it, for example, PowerPC Linux. Next, it
has a revolutionary processor called the Cell processor which was
developed by Sony, IBM and Toshiba. This processor has a main CPU,
called the PPU and several (six for the PS3) special compute engines,
called SPUs available for raw computation. Moreover, each SPU performs
vector operations, which implies that it can compute on multiple data,
in a single step. Finally, its incredibly low cost makes it very
attractive as a scientific computing node, that is part of a cluster.
In fact, its highly plausible that the raw computing power per dollar
that the PS3 offers, is significantly higher than anything else on the
market today!

Thanks to a very generous, partial donation by Sony, we have a sixteen
PS3 cluster in our department, which we call PS3 Gravity Grid. Check
out some pictures of the cluster here: 1) the PS3's arrive; 2) the
rack arrives; 3) front view of the cluster; 4) side view of the
cluster. We are using "stock" PS3s for this cluster, with no hardware
modifications. They are networked together using an inexpensive
netgear gigabit switch. For Linux installation, there are several
guides available on the internet. For YDL Linux, consider using the
guide by Terrasoft Solutions. For Fedora Core 5/6, I found this guide
particularly useful. For deploying a parallel job on this cluster, we
use a code that implements a standard domain decomposition approach,
based on message-passing (MPI). There are more details available on
our code below. For compiling, we use GCC and also IBM's XL compilers
for the Cell, that are available as part of IBM's Cell SDK. These are
available from IBM's alphaworks site. The MPI distribution that we are
using is the recently released, OpenMPI distribution for PowerPC
Linux.
Projects

* Binary Black Hole Coalescence using Perturbation Theory (GK)
This project broadly deals with estimating properties of the
gravity waves produced by the merger of two black holes. Gravitational
waves are "ripples" in space-time that travel at the speed of light.
These were theoretically predicted by Einstein's general relativity,
but have never been directly observed. Currently, there is an
extensive search being performed for these waves by the newly
constructed NSF LIGO laboratory and various other such observatories
in Europe and Asia. The ESA and NASA also have a mission planned in
the near future, the LISA mission, that will also be attempting to
detect these waves. To learn more about these waves and the recent
attempts to observe them, please visit the LISA mission website.
http://lisa.jpl.nasa.gov/

The evolution code for the extreme-mass-ratio limit of this
problem (referred to as EMRI) is essentially like an inhomogeneous
wave-equation solver which includes a very complicated source-term.
The source-term describes how the smaller black hole (or star) affects
the space-time of the larger one. Because of the computational
complexity of the source-term, it is often the most numerically
intensive part of the whole evolution. On the PS3's Cell processor, it
is precisely this part of the computation that is farmed out to six
SPUs. This approach essentially eliminates the entire time spent on
the source computation and yields a speed up of over a factor of five
over a PPU-only computation. It should be noted that the context of
this computation is double-precision floating point operations. In
single-precision, the speed-up is significantly higher.

Overall, a single PS3 performs better than the highest-end
desktops available and compares to as many as 25 nodes of an IBM Blue
Gene supercomputer. And there is still tremendous scope left for
extracting more performance through further optimization. Furthermore,
we distribute the entire computational domain across the sixteen PS3s
using MPI (message passing) parallelization. This enables the entire
cluster to run together, harmoniously, working on the computation in
an efficient way. Each PS3 works on its part of the domain and
communicates the appropriate data to the others, as needed.
http://www.open-mpi.org/

-

TECTONICS
http://www.math.columbia.edu/~woit/wordpress/?p=563
http://www.nytimes.com/imagepages/2006/05/01/science/20060502_HOLE_GRAPHIC.html
http://www.nytimes.com/2006/05/02/science/space/02hole.html
Black Holes Collide, and Gravity Quivers
BY Kenneth Chang / May 2, 2006

In the most precise effort yet to detect gravitational waves -- the
quiverings of space-time predicted by Einstein's theory of general
relativity -- the National Science Foundation in the late 1990's carved
two large V's, one in the barren landscape of central Washington
State, the other among the pines outside Baton Rouge, La. The tunnels
are part of the Laser Interferometer Gravitational-Wave Observatory,
known as LIGO. If something astronomically violent, like a collision
of two black holes, shakes the fabric of the universe within 300
million light-years of Earth, an expanse that encompasses several
thousand galaxies, LIGO should see the resulting gravitational
ripples.

The observatory is sensitive enough to detect a change of less than
one ten-quadrillionth of an inch, or about a thousandth of the
diameter of a proton, in the length of the 2.5-mile-long tunnels.
After several years of testing and fine-tuning -- special dampers had
to be installed at the Louisiana site to counteract vibrations
generated when nearby loggers cut down trees, for instance -- the
observatory began full operation in November. The centers cost nearly
$300 million to build and $30 million a year to operate.

The data so far, reported last week at a meeting of the American
Physical Society in Dallas, contain nothing of interest. In fact,
scientists would not be surprised if the initial run of the experiment
over the next year or so found nothing at all. "I would still sleep
well about general relativity," said Peter R. Saulson, a physics
professor at Syracuse and an observatory spokesman. Jay Marx, LIGO's
executive director, estimated that the chance of success was "25
percent, if nature's kind."

General relativity, formulated 90 years ago by Einstein to explain the
properties of space and time, fits well with measurements of gravity
in and around the solar system. But predictions about what happens
around black holes and other places where gravity is extremely strong
remain largely untested. One of the predictions is that in such
conditions, sizable gravitational waves will be produced.

With new research, scientists have a better idea of what LIGO should
look for. Researchers led by Joan M. Centrella, chief of the
Gravitational Astrophysics Laboratory at NASA's Goddard Space Flight
Center, announced last month that they had succeeded in calculating
the shape of the gravitational waves that should result when two black
holes, orbiting one another, merge. "This is not something made up
like in a science fiction movie," Dr. Centrella said in a news
conference announcing the findings. "Rather, we have confidence that
these results are the real deal, that we have the true gravitational
wave fingerprint predicted by Einstein for the black hole merger."

The equations of general relativity can be easily written down but are
notoriously hard to solve. Astrophysicists were able to simulate the
head-on collision of two black holes three decades ago, but computing
the paths of orbiting black holes and their violent merger proved much
harder. "This has been a holy grail type of quest for the last 30
years," Dr. Centrella said.

Dr. Centrella's simulations still contain some simplifications that do
not reflect attributes of actual black hole pairs: the two black holes
have the same mass, and neither is spinning. The calculations
predicted, for example, that 4 percent of the mass of the black holes
should be converted into gravitational waves. "That's a very important
number," Dr. Saulson said. "That tells us that these gravitational
waves are going to be about as strong as we hoped they could be." He
added, "And that's got those of us working on the detectors very
excited, making it seem more likely we'll bump into something."

Einstein's theory of general relativity changed the idea of gravity
from a simple force dragging apples from a tree to a puzzle of
geometry. Imagine a rubber sheet pulled taut horizontally and then
tossing a bowling ball and a tennis ball onto it. The heavier bowling
ball sinks deeper, and the tennis ball will move toward the bowling
ball not because of a direct attraction between the two, but because
the tennis ball rolls into the depression around the bowling ball.

In this two-dimensional analogy of space-time, one can also imagine a
sudden collision of objects creating ripples that skitter across the
sheet. Those are the gravitational waves LIGO hopes to detect. At each
site, a laser beam generated at the base of the V is split in two and
shot through tunnels buried along each 2.5-mile-long arm. The light
bounces back and forth in the two tunnels. When a gravitational wave
speeds past, it should stretch and shrink the distance that the laser
beams travel, causing the laser light to flicker into a detector at
the base of the V.

Because the instruments are susceptible to tiny disturbances, only
signals seen by both LIGO detectors, nearly 2,000 miles apart, would
likely be convincing to scientists. The skepticism about whether LIGO
will actually spot gravitational waves comes not from questions about
general relativity -- "People would be incredibly surprised if it
wasn't right," Dr. Marx said -- but uncertainty about how often events
that create gravitational waves occur in the universe.

Pairs of orbiting black holes should be the end result of star systems
consisting of two massive stars. Over time, the black holes would
spiral inward and eventually collide. Astronomers can see plenty of
pairs of massive stars twirling in the sky, but they cannot be sure
that they ultimately collapse into pairs of black holes. Because
astrophysicists do not fully understand how stars age, "There are
multiple factors of uncertainty," said Vassiliki Kalogera, a professor
of physics and astronomy at Northwestern University. "We don't know
that binary black holes exist."

At the optimistic end, her calculations suggest that LIGO could detect
up to 10 black hole mergers a year. But the calculations are still
uncertain by a factor of 100, which means that at the pessimistic end,
the rate of detectable black hole mergers may be just one every 50
years or so. A more common event is the merger of neutron stars, the
dense, burned-out cores left over by some exploding stars. The most
convincing evidence so far for gravitational waves was the observation
in 1974 by two Princeton physicists, Joseph H. Taylor and his student
Russell A. Hulse. They saw a pair of pulsating neutron stars spiraling
inward toward each other. The amount of energy lost in the decaying
orbits turned out to match the amount of energy expected to be emitted
in gravitational waves.

However, the gravitational waves produced by orbiting neutron stars
are too weak to be detected by LIGO. And even when the neutron stars
slam into each other, the cataclysm is not nearly as violent as the
merger of black holes, so a neutron star collision would have to occur
much closer in order for LIGO to see it. Dr. Kalogera's calculations
suggest that the observatory will see a neutron star merger once every
seven or eight years, at best. For LIGO to detect gravitational waves
routinely, the instruments will need a proposed $200 million upgrade,
which includes more powerful lasers, to increase their sensitivity by
a factor of 10, Dr. Marx said.

Astronomers hope that LIGO and its successors, as well as similar
detectors in Europe and Japan, will become a new type of telescope. If
the detection of gravitational waves becomes common, astronomers
should be able to deduce many physical properties of black holes and
neutron stars. They may also find that such objects are more common in
certain types of galaxies. The upgraded observatory may also be able
to detect gravitational waves produced by exploding stars or even
reverberations of the Big Bang 13.6 billion years ago.

Sometime in the next decade, NASA and the European Space Agency hope
to launch a space-based gravitational wave detector called the Laser
Interferometer Space Antenna, or LISA. Consisting of three satellites
flying around the sun in the formation of an equilateral triangle 3.1
million miles apart, LISA would be able to detect gravitational waves
with much larger wavelengths, like those produced when mega-black
holes at the center of galaxies merge. For now, the scientists await
their first gravitational wave. "We are all hoping we are lucky," said
Gabriela González, a physics professor at Louisiana State and a LIGO
scientist. "Even if we are not, we will know more about nature."

-

http://science.msfc.nasa.gov/headlines/y2008/19mar_grits.htm
http://science.nasa.gov/headlines/y2007/11oct_undularbore.htm
http://science.msfc.nasa.gov/headlines/y2008/images/grits/tama_gw_070506.mpeg
http://science.nasa.gov/headlines/y2007/images/undularbore/saylorville_timelapse.mov

http://science.msfc.nasa.gov/headlines/y2008/images/grits/radar.gif
http://www.terradaily.com/reports/Yale_Group_To_Study_Atmospheric_Tsunamis.html
http://mesonet.agron.iastate.edu/cool/
http://vortex.nsstc.uah.edu/mips/

"The gravity waves of this story should not be confused with the
gravitational waves of astrophysics. One is an ordinary wave of water
or air; the other is a ripple in the fabric of spacetime itself."

CAUSING TORNADOES
http://www.terradaily.com/reports/Gravity_Waves_Make_Tornados_999.html
Gravity Waves Make Tornadoes / Mar 20, 2008

Did you know that there's a new breakfast food that helps
meteorologists predict severe storms? Down South they call it "GrITs."
GrITs stands for Gravity wave Interactions with Tornadoes. "It's a
computer model I developed to study how atmospheric gravity waves
interact with severe storms," says research meteorologist Tim Coleman
of the National Space Science and Technology Center in Huntsville,
Alabama. According to Coleman, wave-storm interactions are very
important. If a gravity wave hits a rotating thunderstorm, it can
sometimes spin that storm up into a tornado.

What is an atmospheric gravity wave? Coleman explains: "They are
similar to waves on the surface of the ocean, but they roll through
the air instead of the water. Gravity is what keeps them going. If you
push water up and then it plops back down, it creates waves. It's the
same with air." Coleman left his job as a TV weather anchor in
Birmingham to work on his Ph.D. in Atmospheric Science at the
University of Alabama in Huntsville. "I'm having fun," he says, but
his smile and enthusiasm already gave that away. "You can see gravity
waves everywhere," he continues. "When I drove in to work this
morning, I saw some waves in the clouds. I even think about wave
dynamics on the water when I go fishing now."

Gravity waves get started when an impulse disturbs the atmosphere. An
impulse could be, for instance, a wind shear, a thunderstorm updraft,
or a sudden change in the jet stream. Gravity waves go billowing out
from these disturbances like ripples around a rock thrown in a pond.
When a gravity wave bears down on a rotating thunderstorm, it
compresses the storm. This, in turn, causes the storm to spin faster.
To understand why, Coleman describes an ice skater spinning with her
arms held straight out. "Her spin increases when she pulls her arms
inward." Ditto for spinning storms: When they are compressed by
gravity waves, they spin faster to conserve angular momentum. "There
is also wind shear in a gravity wave, and the storm can take that wind
shear and tilt it and make even more spin. All of these factors may
increase storm rotation, making it more powerful and more likely to
produce a tornado."

"We've also seen at least one case of a tornado already on the ground
(in Birmingham, Alabama, on April 8, 1998) which may have become more
intense as it interacted with a gravity wave." Coleman also points out
that gravity waves sometimes come in sets, and with each passing wave,
sometimes the tornado or rotating storm will grow stronger. Tim and
his boss, Dr. Kevin Knupp, are beginning the process of training
National Weather Service and TV meteorologists to look for gravity
waves in real-time, and to use the theories behind the GrITs model to
modify forecasts accordingly. Who would have thought grits could
predict bad weather? "Just us meteorologists in Alabama," laughs
Coleman. Seriously, though, Gravity wave Interactions with Tornadoes
could be the next big thing in severe storm forecasting.

-

CONTACT
Sergei Kopeikin
http://web.missouri.edu/~umcasphyswww/people/SergeiKopeikin.html
email : kope...@missouri.edu

CLOCKING JUPITER'S GRAVITY
http://www.spacedaily.com/reports/MU_Physicist_Defends_Einstein_Theory_And_Speed_Of_Gravity_Measurement_999.html
MU Physicist Defends Einstein's Theory And Speed Of Gravity
Measurement / Oct 04, 2007

Scientists have attempted to disprove Albert Einstein's theory of
general relativity for the better part of a century. After testing and
confirming Einstein's prediction in 2002 that gravity moves at the
speed of light, a professor at the University of Missouri-Columbia has
spent the past five years defending the result, as well as his own
innovative experimental techniques for measuring the speed of
propagation of the tiny ripples of space-time known as gravitational
waves.

Sergei Kopeikin, associate professor of physics and astronomy in the
College of Arts and Science, believes that his latest article,
"Gravimagnetism, causality, and aberration of gravity in the
gravitational light-ray deflection experiments" published along with
Edward Fomalont from the National Radio Astronomical Observatory,
arrives at a consensus in the continuing debate that has divided the
scientific community. An experiment conducted by Fomalont and Kopeikin
five years ago found that the gravity force of Jupiter and light
travel at the same speed, which validates Einstein's suggestion that
gravity and electromagnetic field properties, are governed by the same
principle of special relativity with a single fundamental speed.

In observing the gravitational deflection of light caused by motion of
Jupiter in space, Kopeikin concluded that mass currents cause non-
stationary gravimagnetic fields to form in accordance with Einstein's
point of view. The research paper that discusses the gravimagnetic
field appears in the October edition of Journal of General Relativity
and Gravitation. Einstein believed that in order to measure any
property of gravity, one has to use test particles. "By observing the
motion of the particles under influence of the gravity force, one can
then extract properties of the gravitational field," Kopeikin said.
"Particles without mass - such as photons - are particularly useful
because they always propagate with constant speed of light
irrespectively of the reference frame used for observations."

The property of gravity tested in the experiment with Jupiter also is
called causality. Causality denotes the relationship between one event
(cause) and another event (effect), which is the consequence (result)
of the first. In the case of the speed of gravity experiment, the
cause is the event of the gravitational perturbation of photon by
Jupiter, and the effect is the event of detection of this
gravitational perturbation by an observer. The two events are
separated by a certain interval of time which can be measured as
Jupiter moves, and compared with an independently-measured interval of
time taken by photon to propagate from Jupiter to the observer. The
experiment found that two intervals of time for gravity and light
coincide up to 20 percent. Therefore, the gravitational field cannot
act faster than light propagates."

Other physicists argue that the Fomalont-Kopeikin experiment measured
nothing else but the speed of light. "This point of view stems from
the belief that the time-dependent perturbation of the gravitational
field of a uniformly moving Jupiter is too small to detect," Kopeikin
said. "However, our research article clearly demonstrates that this
belief is based on insufficient mathematical exploration of the rich
nature of the Einstein field equations and a misunderstanding of the
physical laws of interaction of light and gravity in curved space-
time."