By Prof. L. Sriramkumar
On February 11, 2016, the two Laser Interferometer Gravitational-wave Observatories (LIGO) at Hanford and Livingston, U.S.A. [1], announced the detection of Gravitational Waves (GWs) from two merging black holes. [2] The GWs are an important prediction of Einstein’s theory of General Relativity (GR), a relativistic theory of gravitation which is expected to describe the gravitational field at the largest scales and at the greatest strengths. In what follows, I shall first outline the theories and experiments that led to the formulation of GR. Thereafter, I shall describe GR itself briefly before going on to discuss the concept of GWs, their sources and the recent detection by LIGO. I shall conclude by briefly sketching as to how the detection of GWs opens up a new window to the universe.
PART I: FROM NEWTONIAN GRAVITY TO GENERAL RELATIVITY
Newtonian gravity and the motion of Mercury
The Newtonian law of gravitation which describes the force between masses is well known. According to the law, two masses attract each other with a force that is directly proportional to the products of the masses and inversely proportional to the square of the distance between them. The Newtonian theory has been enormously successful in explaining a variety of terrestrial phenomena from falling apples to long range projectiles, and celestial phenomena such as the motion of planets as encoded in the Kepler’s laws. [3] According to the Kepler’s laws, the planets are expected to move around the Sun along ellipses indefinitely. However, observations of the planet Mercury over the course of the nineteenth century revealed that, instead of being confined to an ellipse, it followed the trajectory of a slowly precessing ellipse, where its axis of rotation constantly changed. The extent of the precession of the perihelion of Mercury (i.e. the advancement of the point of its closest approach to the Sun) had remained unexplained despite detailed efforts to take into account other effects such as the influence of other planets and the oblateness of the Sun. We shall revisit this issue in the light of GR in due course.
At this stage, it should be noted that, in Newtonian theory, the gravitational force acts instantaneously, i.e. without any delay.
Speed of light and the theory of special relativity
There exist two threads that lead to the formulation of GR. One is the Newtonian law of gravitation, which we described above and the other is the theory of Special Relativity (SR). SR had owed its origins to the Galilean principle of relativity (according to which the Newton’s second law of motion is valid in any inertial frame, i.e. frames that move with constant velocities) and the nature of Electro-Magnetic (EM) waves. The theory describing electricity and magnetism was developed during the course of the nineteenth century and culminated in its comprehensive description by Maxwell in terms of equations which carry his name. During the latter half of the nineteenth century, as the concept of EM waves (i.e. light) emerged, it was thought that light requires a medium for propagation, just as sound waves do. In order to explain their propagation, an all pervading ether was invoked to play the role of the medium. If that was indeed the case, light rays propagating in mutually perpendicular directions as Earth travels through ether will have different speeds. While the velocity of light in the direction of motion of the Earth will be modified (according to the conventional Newtonian law of addition of velocities, viz. that the velocity of a person moving in a train with respect to the station is the sum of the velocities of the train and the velocity of the person with respect to the train) by the ether wind generated due to its motion, the velocity of light in the perpendicular direction will not be affected.
This issue was examined by Michelson and Morley using an interferometer towards the closing years of the nineteenth century. An interferometer is a rather sensitive instrument that consists of two arms of equal length along which light is made to propagate back and forth before they interfere with each other at the end. The difference in speeds along the two perpendicular directions as Earth moves through ether will lead to a shift in the interference pattern. However, Michelson and Morley did not observe any such shift.
The result implied that the speed of light is unaffected by the motion of the Earth. This, in turn, suggested that the concept of ether is not supported by the experiments. As we shall see later, the Michelson-Morley interferometer plays a primary role in the detection of GWs.
The null result of the Michelson and Morley experiment led Einstein to formulate SR in 1905, based on two postulates. The first postulate simply states that the physical laws have the same form in any inertial frame (which is a generalization of the Galilean principle of relativity that had applied to Newton’s second law of motion). The second postulate states that, in contrast to the conventional Newtonian law of addition of velocities, the speed of light is the same in all frames of reference. These postulates lead to a variety of consequences such as the relative nature of the concept of simultaneity (which was an absolute concept in Newtonian physics), Lorentz contraction (i.e. moving rods appear shorter) and time dilation (i.e. moving clocks run slower). These phenomena are regularly encountered in experiments involving elementary particles moving at relativistic velocities and the validity of SR has been tested and confirmed in a multitude of situations over the last century. It should be mentioned that the equations of motion in SR reduce to the Newton’s laws of motion in the non-relativistic limit, i.e. when the velocities involved are much smaller than the speed of light. [4]
GR and dynamical spacetimes
Time is an absolute concept in Newtonian mechanics. It does not depend on either the location or motion of the observer.
But, in SR, the transformations of the coordinates from one inertial frame to another (called the Lorentz transformations) mix space and time.Hence, it proves to be convenient to merge space and time into a single entity called spacetime.
In SR, the spacetime is fixed (referred to as the Minkowski or flat spacetime) and it basically provides a background arena in which various phenomena take place. As we mentioned above, the Newtonian gravitational force acts instantaneously, which is, evidently, inconsistent with SR, according to which the velocity of light is the maximum speed of propagation.
During the period of 1905 to 1915, Einstein worked towards constructing a theory of gravity that is consistent with SR. An important step in the process was the recognition of the so-called principle of equivalence. In fact, the principle is often stated in different ways and we shall consider two versions of the principle to construct our arguments. The first of the versions corresponds to the fact that the motion of test particles in a gravitational field proves to be independent of their masses or their composition (as Galileo is supposed to have famously illustrated by dropping different objects from the leaning tower of Pisa). Secondly, there seems to be complete equivalence between an inertial frame and a freely falling frame in a uniform gravitational field, say, in an elevator that is in free fall above the Earth. In other words, no experiments carried within these frames can distinguish one from the other. For example, an apple dropped will float freely in both these frames. However, this equivalence breaks down as the size of the freely falling frame is increased. In a wider elevator, dropped apples will begin to move towards each other because of the tidal forces arising due to the curvature of the Earth. Such arguments had led Einstein to recognize that the gravitational field manifests itself as the curvature of spacetime.
The geometry of spacetime characterizes the gravitational field. Geometry is described by the metric, i.e. the quantity that determines the distances between events in spacetime. Einstein realized that the presence of matter curves spacetime, and the curved spacetime, in turn, influences the motion of test particles.
These ideas are best described by the following quote, attributed to Wheeler: matter tells spacetime how to curve, and spacetime tells matter how to move. A popular analogy that helps visualize the above two points is the behaviour of a rubber sheet under the weight of a large mass that is placed on it. In this analogy, the sheet corresponds to spacetime. It bends or curves due to the large mass. The bent sheet in turn determines the trajectory of test particles (i.e. masses much smaller than the original one).
GR, formulated in 1915, describes the dynamics of spacetimes through the Einstein’s equations, which relate the spacetime dynamics to the matter content. [3],[5]
Tests of GR
Since its formulation, GR has been tested in a variety of situations. To begin with, let us reconsider the behaviour of the planet Mercury in the context of GR. Einstein himself had investigated the issue and had found that GR indeed predicts the precession of the perihelion of Mercury. Moreover, the extent of the precession predicted by GR is found to match the observations very well. There exist two other phenomena that GR had predicted (which, along with the precession of the perihelion of Mercury are referred to as the classic tests of GR): bending and red-shifting of light due to strong gravitational fields. [3],[5] Since atomic clocks run based on the frequency of atomic transitions, the latter implies that clocks will run slower in a stronger gravitational field (referred to as gravitational time dilation). The phenomenon of gravitational lensing (i.e. bending and focusing of light by the gravitational field) is regularly observed by astronomers and gravitational red-shift was experimentally established in an exquisite experiment by Pound and Rebka about half-a-century ago. In fact, both gravitational and relativistic time dilation (that we had mentioned earlier) need to be accounted for in the Global Positioning System (GPS), if it is to meet the high accuracy desired for, say, military applications.
GR is not only required to describe gravitational fields at its greatest strengths, it is also needed on the largest scales, wherein Newtonian gravity ceases to be valid. For this reason, GR is essential to study cosmology, i.e. the physics of the universe as a whole. With the advent of increasingly accurate observational data over the last couple of decades, it is often said that we have entered an era of precision cosmology. These data have helped us converge on a standard model of cosmology.
It ought to be highlighted here that GR has proved to be remarkably consistent with virtually all the astrophysical and cosmological observations to date. [3],[5]
PART II: GWS AND THEIR DETECTION
GWs, properties and sources
An important prediction of GR are GWs. It was predicted by Einstein himself in 1916, very soon after the formulation of GR. However, there had remained a lack of clarity about their physical reality for almost half-a-century. It was only in the late 1950’s that it was recognized that GWs carry energy and hence they can be detected. [6] GWs are fundamentally small disturbances (ripples, as the title of this article suggests) in the fabric of spacetime, akin to ripples on the surface of water. GWs have some similarity to EM waves. They too travel at the speed of light. Also, as in the case of EM waves, they are transverse in nature, i.e. the disturbances occur in a direction perpendicular to their direction of propagation. Moreover, they are characterized by two degrees of polarization as EM waves are. However, the fundamental forms of polarization of GWs are somewhat different and are referred to as plus and cross polarizations. A GW impinging on a ring of masses will set them in oscillatory motion. The nomenclatures for the two types of polarization are due to the manner in which the ring of masses are affected by the polarized wave. [3],[5]
In electromagnetism as well as gravitation, radiation is emitted by a system of charges and masses which are in motion.
It is possible to express the radiation emitted by the system in terms of the so-called moments of the charge and mass distribution. These moments characterize the strength and shape of the distribution. They constitute a hierarchy involving increasingly higher powers of the distances of the location of charges or masses from a given origin. For instance, the leading moment of a system, viz. the monopole moment, does not involve the distances at all and essentially reflects the total charge or mass of a distribution. While the next two moments, i.e. the dipole and the quadrupole moments, involve the first and the second power of the distances of the charges or masses from the origin. In the case of electromagnetism, a varying monopole is not possible due to charge conservation. It is the time-dependent dipole moment resulting in a non-zero second time derivative of a charge distribution that leads to the dominant contribution to the radiation from the system. One such simple system is an accelerating charge. In the context of gravitation, while the conservation of mass forbids radiation from a monopole, conservation of linear and angular momenta rule out radiation by a dipole. Therefore, gravitational radiation primarily arises due to a time-dependent quadrupole moment (leading to a nonvanishing second time derivative). [3],[5]
As it is merely a matter of a suitably varying quadrupole moment, clearly, there can exist a variety of sources of GWs. In fact, even simply waving our hands can generate them! However, it turns out that, generating GWs of sufficient strength that can be detected requires rather strong gravitational fields and relativistic velocities, only found in astrophysical situations. Strong sources of GWs include: rotating highly dense and compact objects known as neutron stars, exploding supernovae, binary neutron stars or black holes that are spiraling in towards each other, supermassive binary black holes at the centre of galaxies and quantum fluctuations in the early universe.
LIGO
The frequency of the GWs will depend on the internal frequency of the source that generates them. At the lowest range of frequencies are the GWs produced in the early universe (of about 10−16 hertz) and those generated by in-spiralling compact binaries correspond to the highest frequencies (of about 102 hertz). The former leaves its imprints as anisotropies in the cosmic microwave background (an ubiquitous thermal distribution of photons that we are immersed in, which is a vestige of the hot early universe). The latter is expected to be detected by the so-called interferometric detectors, which we shall now describe.
LIGO, or Laser Interferometer Gravitational-Wave Observatory, is basically a very large Michelson interferometer, with massive mirrors (weighing about 40 kilograms) that are suspended freely.
The principle of the detection of the GWs using an interferometer is quite simple. The freely suspended mirrors will be disturbed by an incoming GW, leading to a difference in the path length between the two arms and therefore a shift in the fringes. However, the amplitude of the incoming GW, even from quite a strong source, proves to be extremely small – it will at most move the mirrors by a fraction of the size of a proton! Therein lies the challenge. In order to improve the sensitivity of the interferometers, their arms are made very long. The two arms of LIGO are actually four kilometer long optical (so-called Fabry-Perot) cavities. Also, in order to increase the path length, the beams of light are made to travel back and forth as many as 280 times (leading to a path length of 1120 kilometers) before they interfere. (In comparison, the path length in the original Michelson and Morley experiment was 11 meters.) Actually, the source of light is a highly focused beam of sufficiently powerful laser. The laser light is constantly recycled and boosted so that it does not disperse or dissipate even after many trips back and forth. The interferometer is designed such that, in the absence of any disturbance, the laser beams travelling in the two arms arrive at a photodetector exactly out of phase, resulting in destructive interference or no signal. [7] This makes it easier to detect a GW when it arrives, as it will lead to an increase in the brightness. It should be stressed that the characteristics of a variety of other noises, such as the seismic noise, thermal noise and the noise due to fluctuations in the photon number, have to be understood in detail and accounted for before the signal from GWs can be decoded.
The concept of interferometers as GW detectors was originally suggested in the 1960’s. The concept was developed over the next two decades and the two LIGO facilities at Hanford and Livingston were completed in the 1990’s. They were in operation for about a decade until 2010. Over the last few years, LIGO was completely overhauled. The refurbished detector, referred to as advanced LIGO (aLIGO) [8], with three times the sensitivity of LIGO, began operation in 2015. [9]
Recent observations by LIGO
On September 14, 2015, the two LIGO detectors measured a transient signal of GWs lasting for about half-a-second. [10] The signals had arrived at Hanford about 7 milliseconds later than at Livingston, which almost exactly corresponds to the time taken by the GWs to travel the distance. The signals at the two observatories were very similar in nature. [11] The peak dimensionless amplitude of the signal (i.e. the strain induced by the GWs on the mirrors) was about 10−21 and the frequency of the signal had steadily increased from about 35 to 250 hertz. These amplitude and frequencies had fallen exactly within the range of sensitivity of LIGO. [12]
At this stage, to interpret the signals detected, we need to digress in order to understand their possible origin. Even as the GW detectors were being planned and constructed, there was a constant theoretical effort during the last two to three decades to arrive at the characteristics of GWs emanating from various sources. One source of GWs that has been investigated extensively are two compact objects (say, neutron stars or black holes) that are orbiting each other in their mutual gravitational field. When the binary system evolves, they lose energy due to the emission of GWs. As a result, their orbits keep shrinking and their orbital periods keep constantly reducing. When the binaries are spiralling towards each other in such a fashion, the amplitude as well as the frequency of the GW emitted by the system increases.
The resulting waveform is very similar to the short shrill sounds that birds often make and for this reason it is referred to as a chirp. The amplitude and the frequency of the chirp carries information about the nature of the objects that constitute the binary system.
The waveform observed by LIGO had corresponded to a chirp. [10], [11], [12] It had suggested that the GWs had originated from inspiralling black holes of masses about 36 and 29 times the mass of the Sun. Associated numerical modelling suggests that the two black holes had merged together into a single black hole of mass about 62 times the mass of the Sun. In fact, the detected signal had also contained the last throes (referred to as the quasi-normal modes) of the merged black hole, before it reached a quiescent stage. Energy corresponding to about 4 solar masses was radiated away as GWs. Amazingly, the corresponding luminosity proves to be much larger than the luminosity of all the stars in the universe put together!
It should be pointed out that the two black holes would have been approaching each other for eons (billions of years), constantly emitting GWs. It is only towards the very end of their inspiral that the GWs emitted are sufficiently strong for our detectors to pick up their signals.
As the signal had begun to be detected, the black holes were about 350 kilometers apart and were moving at about one-third the velocity of light. In less than half-a-second, they had approached each other at velocities close to the speed of light and had merged into a single black hole. [13] The event has been estimated to have occurred about 1.3 billion light years away.
There were many firsts with the LIGO observations. This is indeed the first direct detection of GWs. (It should be stressed that this is the first direct detection. Observations of the famous Hulse-Taylor binary pulsar had pointed out to GWs indirectly. [14]) It is also the first direct detection of black holes, not to mention a black hole binary. Moreover, this is the first experimental test of GR in the strong gravitational regime. All the analyses carried out until now point to the fact that GR is remarkably consistent with the observations. [15]
A new window to the universe
Our ability to observe the universe in regions of the EM spectrum beyond the optical domain has revealed many facets of the universe that were hitherto hidden from us. In a similar fashion, the detection of GWs opens up a new window to the universe. As is often clarified, the two LIGO facilities are not just detectors, but actually observatories. They have been designed to carry out GW astronomy. The detection of GWs from the merging black holes (the event is now referred to as GW150914) has ushered in a new era (remarkably, in the centennial year of GR, 2015) and the expectation is that aLIGO’s sensitivity is high enough to be able to observe at least a few such events every year. [16] The information carried by GWs is expected to be complementary to the information encoded in the EM waves and, say, neutrinos, if they can be detected. Such a multi-messenger astronomy is expected to lead to a far superior understanding of the physics of the universe.
Role of India
It is noteworthy that many Indian scientists, working from different institutions, have been a part of the LIGO scientific collaboration.
There have been major contributions by the Indian scientists on the theoretical, numerical and analysis fronts.
With a LIGO facility planned in India [17], we can look forward to Indian scientists playing a significant role in GW astronomy over the next decade.
Timeline of the main events leading to the detection of GWs
| Year/Period | Event |
| 1887 | Michelson-Morley experiment |
| 1905 | Special Relativity proposed by Einstein |
| 1905-15 | Formulation of General Relativity by Einstein |
| 1916 | Prediction of Gravitational Waves (GWs) by Einstein |
| 1955-70 | Understanding the nature of GWs |
| 1974 | Discovery of Hulse-Taylor binary pulsar |
| 1984 | LIGO founded |
| 1999 | LIGO inauguration ceremony |
| 2006 | LIGO design sensitivity achieved |
| 2011-14 | aLIGO installation and testing |
| 2015 | Detection of GWs by aLIGO |
References and Links
[1] The LIGO web-pages are located here
[2] Videos of the announcement can be found here
[3] For a discussion of gravity and GR aimed at a general reader, see, for instance, B. Schutz, Gravity from the Ground Up (Cambridge University Press, Cambridge, England, 2003).
[4] For undergraduate level texts on SR, see, for example, A. P. French, Special Relativity (W. W. Norton, New York, 1968) and E. F. Taylor and J. A. Wheeler, Spacetime Physics: Introduction to Special Relativity (W. H. Freeman, San Francisco, 1992).
[5] For an undergraduate level text on GR, see, for instance, J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity (Pearson Education, Singapore, 2003).
[6] For a historical account, see, D. Kennefick, Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves (Princeton University Press, New Jersey, 2007).
[7] An animation that explains the concept of the interferometer and illustrates how the interferometer is expected to respond to an incoming GW can be found here
[8] The aLIGO web-pages are located here
[9] A detailed time-line of LIGO can be found at here
[10] The announcement of the detection of the gravitational waves and the technical details of the associated analyses appeared in: The LIGO Scientific Collaboration and The Virgo Collaboration, Physical Review Letters 116, 061102 (2016). For a more popular account, see E. Berti, Physics 9, 17 (2016).
[11] The signals detected by the two observatories and a comparison of the two observations can be found, for instance, on this page
[12] In an interesting animation, the GWs from two inspiralling black holes have been converted to sound waves. To ‘hear’ the ‘sound’ from inspiralling black holes, see this.
[13] For a simulation of two inspiralling black holes and the resulting GW waveform, see the video here.
[14] In this context, see, for instance, J. M. Weisberg, J. H. Taylor and L. A. Fowler, Scientific American 245, 74 (1981).
[15] For detailed technical analyses, see, The LIGO Scientific Collaboration and the Virgo Collaboration, arXiv:1602.03841[gr-qc].
[16] The LIGO Scientific Collaboration and The Virgo Collaboration, Physical Review Letters 116, 131103 (2016)
[17] ] See this page. Also, see the recent article T. Souradeep, Resonance 21, 225 (2016).
The author, Prof. L. Sriramkumar, is a faculty member in the Department of Physics at IIT Madras. T5E is grateful to him for this lucid and insightful article.
