Gravitational radiation

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In physics, a gravitational wave is a fluctuation in the curvature of space-time which propagates as a wave. Just as electromagnetic radiation is the collection of electromagnetic waves emanating from some source or sources, gravitational radiation is the collection of gravitational waves emanating from some source or sources.

Gravitational waves should not be confused with gravity waves in hydrodynamics.

Contents

Overview

In Einstein's theory of General Relativity, gravity is the curvature of space-time. We fall down because space-time is curved. The mass of the Earth bends space-time around it causing this gravitational attraction. Without this space-time curvature, we would merely float through space in a straight line motion.

Along with this curvature and gravitational attraction, Einstein's theory also permits space-time to fluctuate in a wave-like manner. This is similar, but not completely analogous, to the situation in electromagnetism where electromagnetic waves were first derived from Maxwell's equations.

The gravitational waves are oscillations of space-time itself and yet they also travel through space-time. Compare this to sound waves which are an oscillation of the air and yet travel through the air.

Gravitational waves travel at the speed of light and they are of a different character than sound or light waves. Sound waves are longitudinal, spin-0 pressure waves. Spin-0 means that sound is rather ordinary. Light waves are transverse, spin-1 electromagnetic waves. Spin-1 means that light has polarization. But gravitational waves are transverse spin-2 waves, something entirely different from any other known wave.

It is important to note that gravitational waves are unique to Einsein's theory of gravity and were not predicted by Newton's theory of gravity. Thus they are a new prediction of General Relativity but they have not been confirmed by observation, perhaps because of inherent difficulties.

In electromagnetism, certain motions of charged particles, like electrons, will radiate electromagnetic waves. Analogously in gravity, certain motions of mass or energy will radiate gravitational waves. In electromagnetism, electromagnetic waves can be observed as individual particles called photons. But in gravity, no such particle counterpart to the gravitational wave has ever been observed. If it were to be discovered, it would be called the graviton. Although neither gravitational waves nor gravitons have been detected, for various technical reasons it could be the case that gravitational waves exist and gravitons do not.

The Nature of Gravitational Waves

Gravity waves are fluctuations in space-time. They would alter the relative distance between objects. To detect them you would look for relative motion between objects induced by the waves.

Gravitational waves are transverse. If a gravitational wave were traveling straight toward you, the oscillations would be perpendicular to that: up, down, left, and right.

Gravitational waves have spin-2 polarization. They will cause expansion in one direction and contraction in the perpendicular. Overall size will remain unchanged. This will also oscillate back and forth. So if a gravitational wave were traveling straight toward you, you might see expansion up and down with contraction left and right. Slightly later in time as the wave traveled more, the left and right direction would expand and the up and down direction would contract ... back and forth the oscillations would continue until the wave traveled past you.

Sources of Gravitational Waves

Gravitational waves are caused by certain motions of mass or energy. The type of motion required is different from electromagnetism where any accelerating charge will radiate electromagnetic radiation. In gravity, an object's quadropole moment must be changing in time for it to radiate. This means that a single moving mass (monopole) will not radiate. A pair of orbiting masses with no intrinsic spin (dipole) will not radiate either. You either need a pair of orbiting masses where one mass has a spin that isn't aligned with the plane of orbit or you need three masses orbiting in different planes. A changing monopole moment will not radiate due to the conservation of translational energy-momentum. The changing dipole moment will not radiate due to the conservation of angular momentum. But the changing quadropole moment will radiate gravitational waves. The system will lose energy due to this process. Much as a speeding charge decelerates as it radiates, the gravitational system will spin down as it radiates.

Examples

  • Our solar system will not radiate very much gravitational radiation because the planets are mostly on the same plane of orbit around the sun and their intrinsic spinning is mostly aligned to that plane.
  • If two spinning blackholes were to collide and their spinning was not parallel, then they would emit an enormous amount of gravitational radiation and lose energy in the process.


Detection

In 2005 it was announced that observations of the binary pulsar PSR J0737-3039 appeared to confirm predictions of general relativity with respect to energy emitted by gravitational waves, with the system's orbit observed to shrink 7 mm per day. This further confirms predictions made by Russell Alan Hulse and Joseph Hooton Taylor Jr. observing binary pulsar PSR B1913+16, for the discovery and analysis of which they were awarded the Nobel Prize in Physics in 1993.

But to directly detect gravitational waves you would have to look for any motion they cause. Typically you would place 2 test masses very far apart in one direction to look for the expansion and contraction oscillations. You would also place another 2 test masses perpendicular to the initial. This setup is called a Webber bar and would detect a gravitational wave coming at it. The test masses need to be placed at least kilometers apart. And the motion to be detected would be very slight.

One reason for the lack of direct detection so far is that the gravitational waves that we expect to be produced in nature are very weak, so that the signals for gravitational waves, if they exist, are buried under noise generated from other sources. Reportedly, ordinary terrestrial sources would be undetectable, despite their closeness, because of the great relative weakness of the gravitational force.

Gravitational radiation has not been directly observed, although there are a number of existing and proposed experiments such as LIGO that intend to do so. A number of teams are working on making more sensitive and selective gravitational wave detectors and analysing their results. A commonly used technique to reduce the effects of noise is to use coincidence detection to filter out events that do not register on both detectors. There are two common types of detectors used in these experiments:

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.

In November 2002, a team of Italian researchers at the Instituto Nazionale di Fisica Nucleare and the University of Rome La Sapienza produced an analysis of their experimental results that may be further indirect evidence of the existence of gravitational waves. Their paper, entitled "Study of the coincidences between the gravitational wave detectors EXPLORER and NAUTILUS in 2001", is based on a statistical analysis of the results from their detectors which shows that the number of coincident detections is greatest when both of their detectors are pointing into the center of our Milky Way galaxy.

Einstein@Home

Bruce Allen of UWM's LIGO Scientific Collaboration (LSC) group is leading the development of the Einstein@Home project, developed to search data for signals coming from selected, extremely dense, rapidly rotating stars observed from LIGO in the US and the GEO 600 gravitational wave observatory in Germany . Such sources are believed to be either quark stars or neutron stars; a subclass of these stars are already observed by conventional means and are known as pulsars, electromagnetic wave-emitting celestial bodies. If some of these stars are not quite near-perfectly spherical, they should emit gravitational waves, which LIGO and GEO 600 may begin to detect.

Einstein@Home is a small part of the LSC scientific program. It has been set up and released as a distributed computing project similar to SETI@home. That is, it relies on computer time donated by private computer users to process data generated by LIGO's and GEO 600's search for gravity waves.


Prospects

Unsolved problems in physics: Is our universe filled with gravitational radiation from the big bang? From astrophysical sources, such as inspiralling neutron stars? What can this tell us about quantum gravity and general relativity?

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 so weak (and thus difficult to detect), objects opaque to light are often transparent to gravitational radiation. In particular, gravitational waves could propagate while the universe was still opqaue 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. A sufficiently strong sea of primordial gravitational radiation, with an energy density exceeding that of the big bang electromagnetic radiation by a few orders of magnitude, would shorten the life of the universe, violating existing data that show it is at least 13 billion years old. More promising is the hope to detect waves emitted by sources on astronomic size scales, such as:

Derivation

Perturbation off Flat Space-time

Consider that the full metric <math>g</math> is nearly the flat metric <math>\eta</math> plus some small perturbation <math>h</math>.

<math>g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}</math>

The Einstein equation in vacuum is

<math>R_{\mu \nu} = \mathbf{0}</math>

Where <math>R</math> is the Ricci curvature. We will expand <math>R</math> in perturbatively in powers of <math>h</math>.

<math>R_{\mu \nu} = \mathbf{0} + \delta R_{\mu \nu} + \delta^2 R_{\mu \nu} + \cdots</math>

The zeroth order term can only be a function of the flat metric and therefore is identically zero. As the perturbation is to be small, we will solve only for the first order term and ignore all higher orders.

<math>\delta R_{\mu \nu} = \mathbf{0}</math>

Where <math>\delta R_{\mu \nu}</math> is the deviation from the flat (and thus zero) Ricci curvature that depends linearly on the perturbation <math>h</math>.

Now we need the formula for the Ricci curvature.

<math>R_{\mu \nu} = \partial_{\alpha} \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \Gamma_{\mu \alpha}^\alpha + \Gamma_{\mu \nu}^\alpha \Gamma_{\alpha \beta}^\beta - \Gamma_{\mu \beta}^\alpha \Gamma_{\nu \alpha}^\beta</math>

Where <math>\Gamma</math> are the Christoffel symbols and <math>\partial_{\alpha}</math> is shorthand for <math>\frac{\partial}{\partial x^{\alpha}}</math>. Only first two terms which are linear in <math>\Gamma</math> will contribute to the first order correction.

<math>\delta R_{\mu \nu} = \partial_{\alpha} \delta \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \delta \Gamma_{\mu \alpha}^\alpha</math>

Next we need the formula for the Christoffel symbols.

<math>\Gamma^\alpha_{\mu \nu} = \frac{1}{2} g^{\alpha \gamma} \left( \partial_{\nu} g_{\gamma \mu} + \partial_{\mu} g_{\gamma \nu} - \partial_{\gamma} g_{\mu \nu} \right)</math>

Seeing as the flat metric is constant, the only first order terms will involve derivatives of the perturbation.

<math>\delta \Gamma^\alpha_{\mu \nu} = \frac{1}{2} \eta^{\alpha \gamma} \left( \partial_{\nu} h_{\gamma \mu} + \partial_{\mu} h_{\gamma \nu} - \partial_{\gamma} h_{\mu \nu} \right)</math>

The linearized Einstien equation now becomes

<math>\delta R_{\mu \nu} = \frac{1}{2} \left( \Box^2 h_{\mu \nu} + \partial_\alpha V_\beta + \partial_\beta V_\alpha \right)</math>

Where <math>V_\alpha</math> substitutes the expression <math>\partial_\beta h_\alpha^\beta - \frac{1}{2} \partial_\alpha h_\beta^\beta</math> and <math>\Box^2 = \partial_t^2 - \nabla^2</math> is the d'Alembertian or 4-Laplacian. Raising and lowering indices can be tricky. To first order you only use the flat metric. Also note the inverse metric has a negative perturbation plus higher order terms.

Next we choose a particular coordinate system where <math>V_\alpha</math> is identically zero. Some proof is necessary to make sure this is possible, but it is. We are left with a wave equation and our gauge condition.

<math>\Box^2 h_{\mu \nu} = \mathbf{0}</math>
<math>\partial_\beta h_\alpha^\beta = \frac{1}{2} \partial_\alpha h_\beta^\beta</math>

From experience with simpler wave equations we can guess the general form of the solution.

<math>h_{\mu \nu} = A_{\mu \nu} e^{\imath k \cdot x}</math>

Where <math>k \cdot k = 0</math> is a null vector. The wave equation is now satisfied, but what choices of <math>A</math> will satisfy the gauge condition we used.

<math>A_\alpha^\beta \partial_\beta e^{\imath k \cdot x} = A_\beta^\beta \partial_\alpha e^{\imath k \cdot x}</math>
<math>A_\alpha^\beta k_\beta = A_\beta^\beta k_\alpha</math>

If we don't want transformations to disturb our choice of gauge, then we better make the wave traceless, <math>A_\beta^\beta = 0</math>, and transverse, <math>A_\alpha^\beta k_\beta = 0</math>.

For a wave traveling in the <math>z</math> direction, <math>k = (1,0,0,1)</math>, the perturbation will take the following form.

<math> h_{\mu \nu} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & a & b & 0\\ 0 & b & -a & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} e^{\imath k \cdot x} </math>

Thus the oscillations are transverse spacial distortions. The wave is called spin-2 because there are 2 different polarizations <math>a,b</math>. Light only has one! <math>a</math> is called the plus polarization and <math>b</math> is called the cross polarization.

Trivia

Perturbative versus Exact

Gravitational waves differ markedly from electromagnetic waves in that electromagnetic waves can be derived exactly from Maxwell's equations. However gravitational waves, as a linear, spin-2 wave, as they are often thought of are only perturbations to certain space-time geometries. In other words, classically there are always linear, spin-1 E&M waves, but there are not always (or even usually) linear, spin-2 gravitational waves. There are still wave-like fluctuations, but in general things are nonlinear, as is often the case in General Relativity. This is one of the reasons there may be no graviton.

The thing that is analagous to electromagnetic radiation is the Weyl curvature, not the linear, spin-2 wave.

Gravitational waves transmit energy

Within parts of the scientific community there was initially some confusion as to if gravitational waves could transmit energy like electromagnetic waves can. This confusion came from the fact that gravitational waves have no local energy density - no contribution to the stress-energy tensor. Unlike Newtonian gravity, Einstein gravity is not a force theory. Gravity is not a force in General Relativity, it is geometry. Therefore the field was thought to not contain energy, like a gravitational potential would. But the field can most certainly carry energy as it can do mechanical work at a distance. And this has been proven using stress-energy psuedo tensors that transport energy as well as seeing how radiation can carry energy out to infinity.

See also

External links

References

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