Very weak gravity waves produced when a massive body is disturbed or accelerated. The phenomenon is predicted by the general theory of relativity, but not yet observed with certainty.
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In physics, a gravitational wave is a fluctuation in the curvature of spacetime which propagates as a wave. Gravitational radiation results when gravitational waves are emitted from some moving object or system of objects. Important examples of systems which emit gravitational waves are binary star systems, where the two stars in the binary are white dwarfs, neutron stars, or black holes.
Although gravitational radiation has not yet been directly detected, it has been indirectly shown to exist.
(Gravitational waves are sometimes called gravity waves, but this term should be reserved for a completely different kind of wave encountered in hydrodynamics.)
Introduction
In Einstein's theory of general relativity, the force of gravity is due to curvature of spacetime. These ripples are gravitational waves.
The simplest example of a strong source of gravitational waves is a spinning neutron star with a small mountain on its surface.
As these waves pass a distant observer, that observer will find spacetime distorted in a very particular way. Distances between objects will increase and decrease rhythmically as the wave
passes. Any gravitational waves expected to be seen on Earth will be quite small;
By measuring these waves, astrophysicists hope to learn about systems they could not observe with more traditional means such as optical telescopes, radio telescopes, etc. Gravitational waves
can penetrate regions that the more familiar waves cannot, providing us with a view of black holes and other mysterious objects in the distant Universe. Using precise measurements of
gravitational waves in this way will also allow us to test the theory of general relativity more thoroughly.
The effects of a passing gravitational wave
Imagine a perfectly flat region of spacetime, with a group of motionless test particles lying in a plane. Then, a weak gravitational wave arrives, passing through the particles along a line perpendicular to the plane of the particles.
Like other waves, there are a few useful numbers describing a gravitational wave:
The frequency, wavelength, and speed of a gravitational wave are related by the equation
just like the equation for a light wave. Factoring in the speed of light (c = 3 x 108 meters per second) the wavelength of the waves would be roughly
or about 50 times the width of the Earth.
In the example just discussed, we actually assume something special about the wave. Polarization of a gravitational wave is just like polarization of a light wave, except that the polarizations
of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, if we had a "cross"-polarized gravitational wave, , the effect on the test particles would be basically the
same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, two linearly polarized gravitational waves can be combined in just the right way to give a
circularly polarized wave. Gravitational waves are polarized because of the nature of their sources. The polarization of a wave actually depends on the angle from the source, as we will see in
the next section.
Sources of gravitational waves
In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically symmetric (like a spinning, expanding or contracting sphere) or cylindrically symmetric (like a spinning disk).
Some more detailed examples:
More technically, the second time derivative of the quadrupole moment (or the l-th time derivative of the l-th multipole moment) of an isolated system's stress-energy tensor must
be nonzero in order for it to emit gravitational radiation.
Power radiated by the Earth-Sun system
We imagine a simple system of two masses — such as the Earth-Sun system — moving slowly compared to the speed of light. That is, the system will give off gravitational waves. The power given off (radiated) by this system is:
where the negative sign means that power is being given off by the system, rather than received. In this case, the power is:
Thus the total power radiated by the Earth-Sun system in the form of gravitational waves is about 300 Watts (i.e.
Wave amplitudes from the Earth-Sun system
We can also think in terms of the actual amplitude of the wave. c / Ω), the two polarizations of the wave will be
Here, we use the angular velocity of a simple Keplerian system. We also see that the frequency of the wave given off is ν = 2Ω / 2π = Ω / π. If we put in numbers for the Earth-Sun system, we
find
In this case, the minimum distance to find waves is r = 1light year, so typical amplitudes will be .
Radiation from other sources
Although the waves from the Earth-Sun system are minuscule, astronomers can point to other sources for which the radiation should be substantial.
The information about the orbit can be used to predict just how much energy (and angular momentum) should be given off in the form of gravitational waves. In 1993, Russell Hulse and Joe Taylor
were awarded the Nobel Prize in Physics for this experiment, which was the first experimental evidence for gravitational waves.
Inspirals are very important sources of gravitational waves. Any time two compact objects (white dwarfs, neutron stars, or black holes) come close to each other, they send out intense
gravitational waves. As the objects come closer and closer (R becomes smaller and smaller), the gravitational waves become more and more intense. At some point these waves should become
so intense that they can be directly detected by their effect on objects on the Earth.
The only difficulty is that systems like the Hulse-Taylor binary are so far away. The amplitude of waves given off by the Hulse-Taylor binary as seen on Earth would be roughly .
Gravitational wave detectors
Though the Hulse-Taylor observations were very important, they were only indirect evidence for gravitational waves. A more interesting observation would be a direct measurement of the effect of a passing gravitational wave. Not only would a direct measurement of gravitational waves rule out other possible (however unlikely) reasons for changes to the orbit of an inspiraling system, it would also provide us more information on the system. Perhaps more importantly, such a detection could give us information about things we can't see with radio or light waves — such as black holes.
The great challenge of this type of detection, though, is the extraordinarily small effect the waves would produce on a detector. The amplitude of any wave will fall off as the inverse of the
distance from the source (the 1 / r term in the formulas for h above). Thus, even waves from extreme systems like merging binary black holes die out to very small amplitude by the
time they reach the Earth. Astrophysicists expect that some gravitational waves passing the Earth may be as large , but generally no bigger.
A simple device to detect this motion is called a Weber bar — a large, solid piece of metal with electronics attached to detect any vibrations. This type of instrument was the first type of
gravitational wave detector. The idea is to wait for a passing gravitational wave to "ring up" a bar at its resonant frequency, which would basically amplify the wave naturally. Unfortunately,
Weber bars are not sensitive enough to detect anything but extremely powerful gravitational waves. The most sensitive is LIGO — the Laser Interferometer Gravitational Wave Observatory. A
passing gravitational wave will then slightly stretch one arm as it shortens the other.
Even with such long arms, a gravitational wave will only change the distance between the ends of the arms by about 10 − 17 meters at most. LIGO's should be able to detect gravitational waves as
small as , but needs to wait until a gravitational wave with at least that amplitude passes. Passing cars and trains, falling logs, and even waves crashing on the shore hundreds of miles away
are all very significant sources of noise in real interferometers.
There are other prospects such as MiniGRAIL, a spherical gravitational wave antenna based at Leiden University. Some scientists even want to use the moon as a giant gravitational wave detector.
The moon should be somewhat pliable to the contortions caused by gravitational waves, and the hope is that the motion of the moon caused by these waves will be detectable, much like the motion
of a Weber bar. Supernovae and neutron star or black hole mergers should have larger amplitudes and be more interesting, but their waves will be more complicated. The waves given off by a
spinning, bumpy neutron star would be "monochromatic" — like a pure tone in music.
The Einstein@Home project is a distributed computing project similar to SETI@home intended to detect this type of simple gravitational wave. Searches for gravitational waves from other types of
systems require large supercomputers running for long periods.
Astrophysics and gravitational waves
Unsolved problems in physics: Is our universe filled with gravitational radiation from the big bang?Scientists are eager to directly measure gravitational waves from astronomical sources, as they can probe phenomena that are difficult or impossible to study with electromagnetic radiation. For instance, although a black hole emits no visible radiation in the way that a regular star does, gravitational waves can be emitted when an object falls into a black hole, or when two black holes collide. If the inspiraling mass is significantly smaller than the central black hole, the emitted gravitational waves may, at least in some circumstances, allow physicists to directly probe the spacetime geometry around the event horizon (such observations are a primary goal of the LISA mission). Also, because gravitational waves are disturbances of spacetime itself, objects opaque to light are often transparent to gravitational radiation. In particular, gravitational waves could propagate while the universe was still opaque to light (i.e., at times before recombination). In this way, gravitational waves could help reveal information about the very structure of the universe.
In contrast to electromagnetic radiation, it is not fully understood what difference the presence of gravitational radiation would make for the workings of the universe. More promising is the
hope to detect waves emitted by sources on astronomic size scales, such as:
By directly studying the details of gravitational radiation given off by these systems, astronomers could potentially learn much which they would not be able to learn from electromagnetic
radiation.
Energy, momentum, and angular momentum carried by gravitational waves
Waves familiar from other areas of physics — such as water waves, sound waves, and electromagnetic waves — are able to carry energy, momentum, and angular momentum. By carrying these away from a source, waves are able to rob that source of its energy, momentum, or angular momentum.
In much the same way, a gravitational wave can carry off energy, momentum, and angular momentum, robbing them from its source. The waves can also carry off linear momentum, which will leave the
binary with a net velocity relative to its inital velocity — just like the usual third law of motion. This curvature is related to the stress-energy tensor — Tμν — by the key equation
where GN is Newton's gravitational constant, and c is the speed of light.
With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are just wave equations. This is just a wave equation for the field with a source, despite the
fact that the source involves terms quadratic in the field itself.
Linear approximation
The equations above are valid everywhere — near a black hole, for instance. That is, Einstein's equations become
If we are interested in the field far from a source, however, we can treat the source as a point source; everywhere else, the stress-energy tensor would be zero, so
Now, this is the usual homogeneous wave equation — one for each component of . For a wave moving away from a point source, the radiated part (meaning the part that dies off as 1 / r far from the source) can always be written in the form A(t − r,θ,φ) / r, where A is just some function. Now, if we further assume that the source is positioned at r = 0, the general solution to the wave equation in spherical coordinates is
where we now see the origin of the two polarizations.
Relation to the source
If we know the details of a source — for instance, the parameters of the orbit of a binary — we can relate the source's motion to the gravitational radiation observed far away. With the relation
we can write the solution in terms of the tensorial Green's function for the d'Alembertian operator:
Though it is possible to expand the Green's function in tensor spherical harmonics, it is easier to simply use the form
where the positive and negative signs correspond to ingoing and outgoing solutions, respectively. Generally, we are interested in the outgoing solutions, so
If the source is confined to a small region very far away, to an excellent approximation we have:
where .
Now, because we will eventually only be interested in the spatial components of this equation (time components can be set to zero with a coordinate transformation), and we are integrating this
quantity — presumably over a region of which there is no boundary — we can put this in a different form. Ignoring divergences with the help of Stokes' theorem and an empty boundary, we can see
that
Inserting this into the above equation, we arrive at
Finally, because we have chosen to work in coordinates for which , we know that . With a few simple manipulations, we can use this to prove that
With this relation, the expression for the radiated field is
In the linear case, τ00 = ρ, the density of mass-energy. Explicitly, if the masses of the two objects are M1 and M2, and the positions are and , then
We can use this expression to do the integral above:
Using mass-centered coordinates, and assuming a circular binary, this is
where .
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