Black Holes and Gravitational Waves: a brief introduction
Gravitational waves are oscillations in the gravitational field that propagate throughout the universe at the speed of light. They are analogous to the more familiar electromagnetic (EM) waves, which are oscillations of the electric and magnetic fields. While EM waves are produced by the motions of magnets or electric charges, gravitational waves are produced by the motions of masses. The gravitational waves produced in normal situations (such as when you shake your fist back and forth) are far too weak to ever be detectable. However, there are astronomical sources that can produce detectable gravitational waves.
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These sources usually involve
compact objects: black holes, neutron stars, or white dwarfs. Compact objects are produced after stars expend all of their fuel, so that the pressure from nuclear fusion in their cores can no longer support their own self-gravitational attraction. When this happens the stars
collapse — they shrink in radius until quantum-mechanical forms of pressure prevent further collapse, forming either a white dwarf (about the size of the Earth) or a neutron star (about 10 km in radius) with a mass close to the Sun's mass. When the most massive stars collapse, they instead form
black holes: the gravitational field produced by compressing so much mass into so small a volume becomes strong enough to prevent light from escaping its "surface".
All of the material of the star is compressed to a nearly infinitesimally small volume at the center of the black hole. The resulting super-strong gravitational field has
closed off a region of space from the rest of the universe (for a solar-mass black hole, this region is about 6 km across). The black hole itself is not made of any physical material — it is simply a region in space and time that is separated from the rest of the universe by a one-way membrane (the
event horizon) through which anything can enter, but nothing can escape.
While the collapse process itself produces gravitational waves, the strongest waves are actually produced from two compact objects that are orbiting each other in a binary system. (Most stars are in binaries since their initial formation). The orbital motion of the two compact objects produces gravitational waves, just as moving your finger through still water produces water waves. The binary responds by shrinking, orbiting faster, and producing even stronger waves. Eventually the compact objects get close enough to merge into a single object (a black hole or neutron star), producing an enormous burst of gravitational waves in the process. This process is so powerful, that the luminosity of a black hole collision is greater than the combined electromagnetic luminosity of all the stars in the universe combined. (Even more interestingly, this statement is true regardless of the mass of the black holes.)
Unfortunately, even the strongest astronomical sources of gravitational waves are very difficult to detect. However, a recent experiment called
LIGO (Laser Interferometer Gravitational-wave Observatory), along with its international partners, will likely detect gravitational waves in the next few years. To understand how gravitational-wave detection works, it is important to understand how a passing gravitational wave affects matter. While an electromagnetic wave causes an oscillatory displacement of charged particles perpendicular to the waves' motion, a passing gravitational wave causes an oscillatory ellipsoidal stretching and squeezing of matter (also perpendicular to the wave's motion; see the diagram at right).
In a gravitational-wave detector like LIGO, the distance to two large mirrors (each oriented in a 4-km-long "L" shape) is monitored with lasers to detect the passing wave. Remarkably, even for the strongest sources, the gravitational wave will move the mirrors by less 1/1000 the size of a proton — an exceedingly small distance! But the magic of laser interferometry allows such miniscule distances to be measured.
In addition to LIGO and the other ground-based detectors, there are also plans for a joint NASA -
ESA mission called
LISA (Laser Interferometer Space Antenna). This involves three spacecraft (separated by 5 million km) that will use interferometry to measure their relative separations. If funded, LISA will detect an incredible number and variety of gravitational wave sources.
Research Motivations
Because the gravitational-wave signals from even the strongest astronomical sources are very weak compared to the noise in the detector, it is necessary to have an accurate model (or template) for the gravitational-wave signal we are trying to measure. Accurate templates allow us to find a weak signal in the noise-dominated detector output. The main theoretical difficulty lies in generating these templates. This usually involves solving Einstein's equations, a complicated system of nonlinear, partial differential equations that describes the gravitational interactions between objects in the general theory of relativity.
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The primary source of interest for LIGO and other detectors are compact-object binaries. In Newton's theory of gravity (which is described by a simple, linear partial-differential-equation) the orbital motion of a binary (the so-called two-body problem) has a class of simple, exact solutions: the two masses move on circles, ellipses, hyperbolas, or parabolas. But when gravity is very strong, Newton's theory is inaccurate, and the complexities of Einstein's equations do not allow for a simple analytic solution. We therefore have to use either (i) perturbation methods, or (ii) numerical computation to solve for the motion of the binary and the resulting gravitational waves. Perturbation methods roughly fall into two categories: (a) post-Newtonian theory, where we series-expand the solutions of Einstein's equations in a weak-gravity and slow-motion approximation;
and (b) black-hole perturbation theory, where we assume that a small mass causes a perturbation to the known exact solution for a single black hole. The numerical approach —
numerical relativity — involves discretizing Einstein's equations on a grid of points and using powerful computers to solve the resulting equations.
This describes the main mathematical problem that one needs to solve in order to best detect gravitational waves. In the context of this problem, some of the questions that interest me include:
- How well can we develop perturbative solutions to the 2-body problem? How far can we push these solutions into the strong-gravity region?
- What are some of the interesting nonlinear features that arise in the solution of the relativistic 2-body problem, and can we gain an analytic understanding of these features? Also, are these features observable?
- How can numerical and perturbative solutions best be combined to provide reliable gravitational-wave templates for LIGO and LISA?
In addition to the above questions, I am also interested in a variety astrophysical and fundamental-physics questions that can only be answered with gravitational-wave observations:
- What is the merger rate, number density, and distribution of masses and spins of the various types of compact object binaries?
- What can we learn about the central engines of gamma-ray-bursts?
- Can neutron-star mergers constrain the equation-of-state of nuclear matter?
- Can we constrain models of binary stellar evolution and globular cluster dynamics?
- How well will LIGO or LISA be able to test general relativity, constrain alternative theories of gravity, and map the spacetime geometry around a black hole?
- How large a kick do black holes receive after they merge, and what are the consequences of such kicks? Do black holes commonly get ejected from galaxies?
- How well can gravitational-wave observations constrain the nature of dark energy?
Research Projects: overview
Below are some brief descriptions of some of the projects that I've worked on. Most of these concern the coalescence of binary compact objects, a key source for LIGO and LISA. To better understand the context of these projects, it is important to understand the three phases of coalescence:
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First is the
inspiral phase, in which the two compact objects slowly orbit around each other. As the binary system radiates gravitational waves, the orbit shrinks and the orbital speed increases, causing more gravitational waves to be emitted. The two masses (which at this stage can be modeled as point-particles) slowly inspiral until they reach the
plunge/merger phase. At this point, the orbital dynamics becomes unstable (the inspiral ceases to be "slow") and the two objects rapidly plunge together, forming a single object and releasing a strong burst of gravitational waves in the process. For two black holes, this process is relatively "clean" and is analogous to two water droplets that merge into one. But if one or both objects are a neutron star or white dwarf, the plunge/merger process is more "messy": tidal forces and the collision process itself can cause the star to be disrupted, producing partially ejected material that can form a disk around the system. The result of the merger is a single object (usually a black hole or massive neutron star) that is highly "distorted." In the final
ringdown phase, these distortions radiate gravitational waves until the merger remnant settles down to a more "quiet" state. This is analogous to the way a struck bell damps its vibrational excitations by radiating sound waves.
Different calculational techniques are used to compute the gravitational waveform for each of these phases. If the binaries are of comparable mass, post-Newtonian expansions are used to compute the inspiral waveforms, black hole perturbation theory is used to compute the ringdown, and numerical relativity is used to compute the plunge/merger (as well as the late-inspiral and early-ringdown). For extreme-mass-ratio systems (eg., a solar-mass black hole orbiting a supermassive black hole), black hole perturbation theory can — in principle — be used to describe all three phases of coalescence. However, this can only be done in the context of certain additional approximations.
With this picture of the coalescence process in mind, you can read on to learn about some specific projects that I have worked on.
◊ gravitational wave memory
In the usual picture of a gravitational wave signal, the waveform amplitude starts small at early times, grows to some maximum value, and then decays back to zero (see, eg., the black hole cartoon picture in the previous section). But some sources have what's called gravitational-wave memory: the gravitational wave signal does not decay back to zero amplitude, but instead asymptotes to some non-zero value at late-times.
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A simple example of a source with memory is a binary on a hyperbolic orbit (i.e., two masses in an unbound orbit that gravitationally scatter off of each other; see the diagram on the right). In this case the asymmetry between the ingoing and outgoing velocity of the particle leads to the memory. Another example of a source with memory is a supernova explosion that asymmetrically ejects matter or neutrinos.
When a gravitational wave "with memory" passes through an idealized gravitational-wave detector (like the free-floating ring of particles in the section
above), it causes the detector to be permanently distorted. That is, while a normal gravitational-wave will return an initially circular ring of particles back to its initial shape, a wave with memory will leave the ring in an elliptical shape even after the wave has passed.
Gravitational waves with memory were originally proposed in the 1970's, but they were generally assumed to be unimportant. It was not expected that sources with memory would be common, and in any case, the low-frequency memory effect is difficult to detect with devices like
LIGO which are mostly sensitive at high frequencies (10 to a few hundred hertz). However, as my recent research has emphasized, the gravitational-wave memory effect
is common, its magnitude is not small, and it can be detectable with
LISA and pulsar-timing arrays.
The memory sources discussed above are all examples of
linear memory. But there is another
nonlinear form of memory (sometimes called the
Christodoulou memory) that is present in all gravitational-wave sources. This nonlinear memory (which was discovered, but only briefly studied, in the 1990s) arises from the fact that gravitational waves themselves produce gravitational waves. This means that
every gravitational wave source is a source with nonlinear memory. For example, the nonlinear memory for a binary black hole merger is pictured on the right, and you can see that instead of waves damping back to their zero-value during the ringdown, they instead approach some constant non-zero value.
While our understanding of the nonlinear memory effect was previously quite limited, recent developments in post-Newtonian theory and numerical relativity have advanced our understanding of the memory. More specifically, my research has focused on (i) computing post-Newtonian memory corrections to the gravitational waveforms during the inspiral phase; (ii) modeling the memory during the merger and ringdown using input from numerical relativity simulations; and (iii) assessing the detectability of the memory effect.
You can learn more about the specifics of my work on the memory effect by checking out the talks or papers below:
- Talk at the 2009 East Coast Gravity Meeting: The nonlinear gravitational wave memory in binary black hole mergers [pdf]
- Conference proceeding:Gravitational-wave memory revisited: memory from the merger and recoil of binary black holes, Marc Favata, Proceedings of the 7th International LISA Symposium, J. Phys. Conf. Ser. 154, 012043 (2009); arXiv:0811.3451 [astro-ph] [local pdf]
- Paper published in Astrophysical Review Letters: Nonlinear gravitational-wave memory from binary black hole mergers, Marc Favata, Astrophys. J. Letters, 696, L159, (2009); arXiv:0902.3660 [astro-ph.SR] [local pdf]
- Paper published in Physical Review D: Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries, Marc Favata, Phys. Rev. D, 73, 104005, (2009); arXiv:0812.0069 [gr-qc] [local pdf]
- Several other memory-related papers coming "soon"
◊ radiation recoil: how black holes get their kicks
As discussed above, gravitational waves carry away energy from a binary system, causing the orbital separation to shrink (decreasing the orbital energy of the binary). In addition to energy, these waves also carry away linear momentum. This loss of linear momentum imparts a corresponding change to the momentum of the binary, causing a "recoil" of the center-of-mass of the binary. This effect is similar to the recoil of a gun after it is fired. 
Amazingly, when two black holes merge the resulting recoil can be large enough to eject the remnant hole from its galaxy or star cluster.
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One can picture the radiation recoil effect as resulting from the asymmetric "beaming" of the gravitational waves emitted by each orbiting black hole. If the black holes were identical, they would each eject gravitational waves in equal amounts but in opposing directions, causing a cancelation of the linear momentum emitted. But any asymmetries in the system (such as different masses or spins) cause a net radiated momentum in some direction, and a corresponding recoil in the opposite direction. Because black holes play important roles in the dynamics and structure of galaxies and star clusters, it is important to calculate the kicks that merging black holes receive so that we can determine if they will be ejected from their environments.
One of the main parts of my PhD thesis was to estimate the size of these kick velocities. Previous work on this subject was slim and only considered the momentum emitted during the inspiral phase for non-spinning black holes. However, most of the momentum is radiated during the plunge/merger/ringdown phases and is expected to be more important for spinning black holes.
My thesis involved the first computations that considered the recoil for realistic plunging orbits about spinning black holes. At that time, these were the best recoil estimates available. In collaboration with
Scott Hughes,
Daniel Holz,
David Merritt, and
Milos Milosavljevic, we also performed the first detailed study of the astrophysical consequences of recoiling black holes. These projects were the subject of a cover story in
Astronomy magazine.
Because our recoil calculations relied heavily on perturbation methods, our estimates for the kick velocity were uncertain. In the years since our study, numerical relativity has finally (after 30 years of trying!) been able to compute the merger of two black holes without approximations. (You can see some movies of these simulations
here.) These results confirmed the range of kick velocities (as well as the spin scalings) that we predicted. However, the numerical relativity estimates are much more accurate and have even predicted much larger recoils than we thought possible.
In addition to my thesis work, I am also involved in some other projects related to the recoil effect. One study involves gaining a better analytic understanding of the recoil effect. Another concerns the effects of recoil on the detected gravitational-wave signal.
You can learn more about my work from the following links:
- Popular science article on my thesis work in Astronomy magazine: [local pdf]
- Press release on this work: [web link] [local pdf]
- Talk summarizing recent progress in computing kick velocities (and, more briefly, some recent work), at the 2007 Theoretical Astrophysics in Southern California meeting: [slides, pdf]
- Slides from my PhD defense related to the recoil effect: [slides, pdf]
- Talk on the Gravitational Radiation Rocket Effect at GR17, Dublin, July 2004: (n.b., some symbols did not convert correctly) [slides, pdf]
- Poster presentation on the recoil effect given at The Astrophysics of Gravitational-Wave Sources meeting at Univ. of Maryland, April 2003: [slides, pdf]
- Paper on recoil calculations, published in Astrophysical Journal Letters: How black holes get their kicks: gravitational radiation recoil revisited, M. Favata, S. A. Hughes, & D. E. Holz, Astrophys. J. Letters, 607, L5, (2004); astro-ph/0402056 [local pdf]
- Paper on the astrophysical consequences of the recoil effect, published in Astrophysical Journal Letters: Consequences of radiation recoil, D. Merritt, M. Milosavljevic, M. Favata, S. A. Hughes, & D. E. Holz, Astrophys. J. Letters, 607, L9, (2004); astro-ph/0402057 [local pdf]
- Conference proceedings that summarizes our work on the recoil effect: How black holes get their
kicks: radiation recoil in binary black hole mergers, S. A. Hughes, M. Favata, & D. E. Holz, Contribution to proceedings of the conference on "Growing Black Holes," held in Garching, Germany on June 21-25, 2004, edited by A. Merloni, S. Nayakshin, and R. Sunyaev, Springer-Verlag series of "ESO Astrophysics Symposia;" astro-ph/0408492 [local pdf]
◊ eccentric binary inspiral
When scientists at LIGO and other ground-based detectors search for gravitational waves from inspiraling compact binaries, the search templates they use usually assume that the binaries' orbits are circular rather than eccentric. This is generally considered to be a good approximation because gravitational-wave emission tends to make binaries more circular — so binaries formed in the standard way are likely to have negligible eccentricity when they enter the LIGO frequency band. However, there are a variety of non-standard scenarios through which the binary eccentricity might be non-negligible. For this reason, it is important to also account for the effects of eccentricity when modeling gravitational-wave sources. Furthermore, eccentricity effects are even more important for gravitational-wave sources that will be observed by LISA.
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There are several well-developed formalisms for calculating the gravitational-wave signal from eccentric binaries. When relativistic effects are included, this formalism can be somewhat complicated and the resulting equations usually have to be solved numerically. One of the projects I have worked on involves simplifying this formalism to produce fast, easy-to-use analytic waveform templates that include relativistic effects but assume that the eccentricity is small (which is a good approximation for many sources). You can learn more about the details from the following links:
◊ extreme-mass-ratio inspirals
An "extreme-mass-ratio inspiral" (or EMRI ) refers to a binary system consisting of a solar-mass star or compact object (most likely a stellar-mass black hole) in a close orbit around a supermassive black hole. Such systems are likely to exist at the centers of most galaxies and one of the key gravitational-wave sources for LISA. EMRIs are especially interesting because the orbit of the compact object (and the corresponding gravitational waves that it emits) acts as a probe of the spacetime geometry of the larger black hole. Measuring the waves from such systems will thus allow us to test how accurately the black holes we observe in nature agree with the simple and precise mathematical solution provided by Einstein's equations.
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...descriptions of some EMRI projects are coming soon. As are links to some related papers and talks:
- coming soon...
- coming soon...
◊ gravitomagnetic Love numbers & tidal crushing of neutron stars
When two gravitating bodies are widely separated, their interactions and dynamics can be described by assuming that they are two point particles. But when the objects are "close", we need to worry about finite-size effects: For example, because one side of the Earth is closer to the Moon than the other, the Moon's gravity pulls more strongly on the closer side than on the opposing side. This causes a stretching or tidal distortion of the Earth's oceans that results in the twice daily high and low tides (the Earth's surface and interior is also stretched, but not as much). For inspiraling neutron stars and white dwarfs, these tidal interactions can also be important.
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In Newtonian gravity, the amount by which a star or planet is tidally distorted is directly proportional to the strength of the external gravitational tidal field that is causing the distortion. The proportionality constant depends on the body's material: a stiff, compact body is hard to distort, while a soft, fluffy object distorts easily. This constant of proportionality is called the
tidal Love number (named after
Augustus Edward Hough Love).
In relativistic gravity there are quantities that reduce to the Newtonian tidal forces and Love numbers. However general relativity also introduces additional kinds of tidal interactions that are not present in Newtonian gravity.
For example the motion of material bodies produces a new kind of gravitational interaction called a
gravitomagnetic field. This is analogous to the magnetic field produced by the motion of an electric charge. Just as a spherical body placed in an ordinary gravitational field develops an ellipsoidal tidal distortion, I showed that a body placed in a gravitomagnetic tidal field develops internal velocity currents. The strength of these tidally-induced currents is proportional to the external gravitomagnetic tidal field as well as a new quantity that I termed the
gravitomagnetic Love number (which is a constant that depends on the material properties of the star).
These new gravitomagnetic Love numbers were introduced in the context of a study of the
binary-induced collapse instability. This refers to a strange effect that was seen in computer simulations of binary neutron stars performed by James Wilson (now
deceased),
Grant Mathews, and
Pedro Marronetti (Wilson and Mathews were among the earliest pioneers in simulating binary neutron stars). Their simulations predicted that the tidal field of one neutron star could cause its binary companion to collapse to a black hole before the stars plunge together (i.e., when the stars are still in the "inspiral phase"). This "crushing" or binary-induced collapse effect was surprising because Newtonian tidal fields tend to make the stars
less likely to collapse. If such an instability really operated in neutron star binaries, it would have significant implications for their detectability with LIGO.
This binary-induced-collapse result was disputed by several authors and led to the so-called
Wilson-Mathews-Marronetti controversy (you can read about the sociology of this controversy in
this article by Daniel Kennefick). My thesis adviser
Eanna Flanagan eventually found a mistake in one of the equations used by Wilson & Mathews, and, once corrected, they found that their crushing effect was reduced in size but did not entirely disappear. Other research groups have since performed more accurate simulations of binary neutron stars; none have found a crushing effect.
In my work on this subject, I reviewed the various arguments for and against the crushing effect and concluded that it likely results from poor resolution in Wilson and Mathews' simulations. Another part of my analysis investigated if such a crushing effect is possible even in principle. The answer turns out to be
yes, although this effect is usually smaller than the Newtonian tidal-stabilization effect. Only if an especially fast circulation pattern exists in the star can the crushing effect dominate.
You can read more about the details of this work from the following links:
- Selected slides from my PhD defense that discuss the star-crushing controversy and the gravitomagnetic tidal crushing mechanism: [slides, pdf]
- Talk at the 2005 April APS meeting: Are neutron stars crushed? Gravitomagnetic tidal fields as a mechanism for binary induced collapse, [slides, pdf]
- Talk at the 2002 April APS meeting, on an early version of the above work: [slides, html]
- Paper in Phys. Rev. D that introduces gravitomagnetic Love numbers and discusses the gravitomagnetic tidal crushing mechanism: Are neutron stars crushed? Gravitomagnetic tidal forces
as a mechanism for binary-induced collapse, Marc Favata, Phys. Rev. D, 73, 104005, (2006); astro-ph/0510668 [local pdf]
◊ tidal energy transfer in general relativity
One of the projects I worked on as an undergraduate concerned the description of tidal interactions in general relativity. This was part of a SURF summer research project with Kip Thorne. The project was motivated by Thorne's analysis of tidal interactions related to the Wilson-Mathews-Marronetti controversy discussed above; but the main problem can be phrased in a more general context: unlike some other forms of energy, gravitational energy is known to be non-localizable — in some local region one can always transform to a freely falling frame in which the gravitational-field vanishes. How does this non-localizability affect tidal interactions, which clearly can do mechanical work on a system? (For a dramatic example of tidal work, check out the volcanos on Jupiter's moon, Io.)
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It turns out that the nonlocalizability of gravity does
not affect the tidal work that is performed in a system (i.e., the work that can be turned into the mechanical or thermal energy of a planet or moon); but it can affect the non-invariant tidal interaction energy between two bodies. The analysis I performed essentially consisted of computing the tidal energy transfer for different schemes of localizing gravitational energy in general relativity. The resulting tidal-energy transfer rate always contained the same invariant piece (the tidal work) plus a non-invariant interaction-energy piece.
For further details, see the following links:
Last updated on November 3, 2009
