# Gravitational Waves

**Gravitational waves**are propagating disturbances in the curvature of spacetime, caused by some of the most violent and energetic physical processes in the universe. A common analogy in general relativity considers gravity to be caused by masses warping a rubber sheet; smaller masses tend to fall into the indentations caused by larger masses, representing the attractive effects of gravity. In this analogy, gravitational waves are "ripples" in the rubber sheet propagating outwards like waves on the surface of water.

Unlike in Newtonian gravity, where gravity acts instantaneously, according to special relativity all interactions including gravity cannot propagate faster than the speed of light. As part of his theory of general relativity, Einstein predicted in 1916 that gravity propagates as a wave, mediated by a massless particle called a **graviton** that travels at the speed of light. These particles comprise gravitational waves in the same way that photons comprise light waves.

The mathematics of gravitational waves and their detection is difficult, requiring the solution of a simplified version of Einstein's equations, applications of Fourier analysis, and extensive knowledge of quantum mechanics. Regardless, physicists have been able to predict to remarkable accuracy observable quantities such as the power radiated in gravitational waves from an inspiraling binary system of stars or black holes as well as phase shifts in laser interferometers designed to detect gravitational waves. On February 11, 2016, the LIGO Scientific Collaboration and Virgo Collaboration confirmed that they had directly observed on September 14, 2015 a gravitational wave signal from the merger of two previously mutually orbiting black holes, using laser interferometry techniques. On October 3, 2017, the 2017 Nobel Prize in Physics was awarded to the three physicists principally involved in the conception and development of LIGO: Rainer Weiss, Barry C. Barish, and Kip S. Thorne [3].

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## What are Gravitational Waves?

To say that gravitational waves are propagating disturbances in the curvature of spacetime means that gravitational waves affect the distances measured between objects as they pass through them. Just like light waves, gravitational waves have different polarizations. These polarizations are characterized by how gravitational waves with a certain polarization act on a circular ring of masses. When a gravitational wave with a certain polarization passes through the ring of masses, it distorts the locations of the masses in a characteristic pattern. The two polarizations of gravitational waves are referred to as "\(+\)" and "\(\times\)" polarizations, corresponding to the two characteristic patterns of oscillation over time of a circular ring of masses as gravitational waves with these polarizations pass through them:

The relative change in measured distance between two masses as a gravitational wave passes through them is referred to as the **strain**.

A particularly strong source of gravitational waves is inspiraling binary systems. These are two stars or black holes orbiting around their common center of mass. As they shed gravitational waves (and thus energy) throughout their orbit, the radius of orbit decreases slowly until the stars/black holes collide. If two stars/black holes each of mass \(M\) are mutually orbiting at angular frequency \(\Omega\) and radius \(R\) to their center of rotation, the magnitude of strains at a distance \(r\) from the binary system oscillates as a function of time \(t\), roughly [4]:

\[h = \frac{8GM}{rc^4} \Omega^2 R^2 \cos \left(2\Omega \Big(t - \frac{r}{c} \Big) \right),\]

where \(G\) is Newton's gravitational constant and \(c\) is the speed of light. The magnitude of the strains caused by gravitational waves thus falls off inversely proportional to the distance from the source. Since most binary systems of stars/black holes are far away from Earth, the strongest signals for gravitational waves will come from either very massive or very rapidly rotating binary systems. Even so, the strains caused by gravitational waves from most systems are on a scale of roughly \(h = 10^{-26}\). This means that if two masses are separated by a distance of \(L\), the change in separation \(\Delta L\) caused by the passing gravitational wave satisfies

\[\frac{\Delta L}{L} = h \approx 10^{-26}.\]

Note that the strain is unitless since both the numerator and denominator of the left-hand side have units of length. Even the strongest known gravitational wave signals cause strains of order only \(h \approx 10^{-20}\). As a result, it is very difficult to directly measure gravitational wave signals, since few experimental apparatuses can detect such minute strains.

The radius of a proton is about \(10^{-15}\) meters. If a \(4 \text{ km}\) arm of the Advanced LIGO apparatus is displaced by just the radius of a proton, what strain does this correspond to? (What does this say about the detectability of gravitational waves?)

In addition to causing the oscillation of a system of free masses, gravitational waves also cause a so-called **memory effect**, in which a free system of masses through which gravitational waves have passed is permanently displaced from its initial configuration. Recent work in high-energy physics has investigated the existence of memory effects for the other fundamental forces of nature as well.

Since direct signals of gravitational waves are miniscule, for decades the only evidence for the existence of gravitational waves was indirect. Rather than directly observing strains caused by the gravitational waves, astronomers were able to observe in 1974 a decrease in the period of rotation of a binary star system over time due to energy loss to gravitational waves. This system, known as PSR B1913+16 or the **Hulse-Taylor binary** after its discoverers, consists of a neutron star and a pulsar (radiating neutron star). Neutron stars are very dense and small, so the period of rotation of this system was very small. A pulsar emits radiation in a beam which can only be observed when pointing towards Earth, thus providing a highly accurate clock with which to measure the period of rotation. According to general relativity, the power radiated by such a binary system is [4]

\[P = -\frac25 \frac{G^4 M^5}{c^5 R^5},\]

where \(M\) is the mass of each star (assuming they are approximately equal), \(R\) is the distance of each star to the center of rotation, and \(G\) is Newton's gravitational constant. This power loss causes the binary system to inspiral at a faster and faster rate, since once the system begins to inspiral faster it emits gravitational waves faster, which in turn causes faster inspiral. Below is plotted the data from the Hulse-Taylor system for change in the period of rotation over time, compared to the prediction from the theory of gravitational waves. The agreement is remarkable:

Two black holes in a mutually orbiting binary system both have mass \(M\) and orbit at radius \(R\) to their center of rotation.

What is the magnitude of the rate of change of their radius (the rate at which they inspiral due to energy loss to gravitational radiation)?

**Assumptions and Details:**

- The power radiated in gravitational waves by an inspiraling binary is \(P = -\dfrac25 \dfrac{G^4 M^5}{c^5 R^5}.\)
- Apart from the expression for gravitational wave radiation, assume Newtonian mechanics applies.

Suppose that an inspiraling black hole binary system is at radius \(R_0\) at time \(t=0\) and that the radius of the binary changes as

\[\frac{dR}{dt} = -\frac{\kappa}{R^3}\]

for some constant \(\kappa\). What is the period shift in the rotation of the binary as a function of time, noting that the frequency of rotation in a Hulse-Taylor binary is

\[\Omega = \left(\frac{GM}{4R^3}\right)^{1/2}?\]

## Operation of Advanced LIGO

Although the strains caused by gravitational waves are miniscule, they can be measured directly using techniques of laser interferometry. This is the technique used by the Advanced Laser Interferometer Gravitational-wave Observatory (Advanced LIGO), which directly detected gravitational waves on September 14, 2015 from a binary black hole merger now called GW150914.

The Advanced LIGO apparatus consists of two Michelson interferometers, located in Hanford, Washington, USA and Livington, Louisiana, USA, which are sensitive to strains of order down to \(h\approx 10^{-22}\). Two detectors are necessary because a single detector would not be able to detect the position of a gravitational wave source and in order to provide additional verification of an apparent signal. Each detector consists of two arms of four kilometers in length. Light from a \(1064 \text{ nm}\) Nd:YAG laser enters a beam splitter, which splits the beam into each arm of the detector. In each arm, the laser light is amplified to a power of \(100 \text{ kW}\) by bouncing between two mirrors ("test masses") located at either end of the arm. A passing gravitational wave changes the distance between the mirrors by a factor given by the strain of the gravitational wave.

The detector arms are highly insulated from both seismic noise and thermal noise, since any minute vibrations due to Earth's tectonic activity or thermal fluctuations would vastly overwhelm gravitational wave signals. The mirrors between which the laser bounces in each arm are suspended on vibration-isolating stages and kept in extreme vacuum conditions at temperatures around a millionth of one Kelvin, at such a low energy that the mirrors themselves are near their quantum ground state [7].

A difficult effect to control for is the quantum fluctuation in the number of photons in the lasers in the cavity. This is a problem because light exerts a pressure on objects when it reflects from them; therefore, uncertainty in photon number translates into an uncertain/fluctuating amount of pressure on the mirrors in the cavity which could overwhelm the displacements on the mirrors due to gravitational waves. In lasers, the fluctuation in photon number scales as the square root of the number of photons. Although this fact makes it seem like using a less energetic laser would reduce this problem, there exists a number-phase uncertainty relation which means that the phase of the light would become more uncertain as the number of photons became more certain, which would also pose a problem for measuring strains, as discussed below.

The changing displacement between the mirrors in each arm is measured by taking advantage of the fact that light has a phase. If the distance between mirrors increases by a tiny amount, the laser light in the arm will travel a little extra piece of a wavelength before bouncing off of the mirrors and thus will shift its phase angle slightly. In Advanced LIGO, the laser light in each cavity is repeatedly cycled from end to end of each arm, picking up a small phase accumulation each time. This ultimately amplifies any gravitational wave strain by three hundred times [6].

Finally, the beams are recombined at the beam splitter. As a result of the accumulated phase shift, the wavelike nature of light will cause an interference pattern on a photodetector. From this interference pattern, the displacement of the mirrors and thus the strain of the gravitational waves can be measured.

If the laser light used in Advanced LIGO is \(1064 \text{ nm}\), an arm of the detector is \(4 \text{ km}\) long, and the GW150914 system causes strains \(h = 10^{-21}\), what is the phase shift in laser light traversing a detector arm at maximum displacement versus no displacement, in radians?

\(\)

See Gravitational Waves.

Below is a plot of the theoretically expected signal for the strain in the Advanced LIGO detector throughout the process of a black hole merger. As the black holes inspiral rapidly right before combining, there is a rapid oscillatory burst that rings down to steady state.

Finally, the data experimentally observed at Advanced LIGO is presented below. As illustrated, not only does the data agree well with the model, but in fact the sound of the black hole merger can be directly heard in a characteristic "chirping" noise as the binary black hole system released a massive burst of gravitational waves in its final instants before merging.

The signal demonstrates quite remarkable and beautiful agreement with the theoretical model computed numerically from the equations of general relativity. The black holes involved in the merger were computed to have masses \(29^{+4}_{-4}\) and \(36^{+5}_{-4}\) times the mass of the sun, with the final merged black hole mass of \(62^{+4}_{-4}\) times the mass of the sun. The burst of gravitational waves observed at LIGO therefore contained three solar masses worth of energy, keeping in mind Einstein's conversion between mass and energy \(E=mc^2\). The signal strength had a statistical significance of \(5.1\,\sigma\), meaning that the probability such a signal is a false positive is \(0.000036\%\). As quoted from Advanced LIGO's paper, "this was the first direct detection of gravitational waves and the first observation of a binary black hole merger" [6].

## Mathematics of Gravitational Waves in General Relativity

In general relativity, the way measured distances change as a function of space and time is encoded by a mathematical object called a metric. To understand the mathematics of gravitational waves, it is essential to understand what a metric is and how it is represented. Here is a short summary: a metric is a matrix, the components of which describe the factor by which measured distances are changing. In general, the components of the metric are functions of space and time. Gravitational waves will turn out to be perturbations to the flat metric of spacetime that obey the wave equation.

Explicitly, gravitational waves are represented as perturbations to the flat Minkowski metric, i.e. the metric is written as

\[g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu},\]

where \(\eta_{\mu \nu}\) is the Minkowski metric \(\text{diag} (-1,1,1,1)\) and \(h_{\mu \nu}\) is a small perturbation, i.e. each component is of magnitude much smaller than one. The invariant interval for the full metric \(g_{\mu \nu}\) can be written as follows [4]:

\[ds^2 = -(1+2\Phi)dt^2 + w_i \big(dt dx^i + dx^i dt\big) + \big[(1-2\Psi)\delta_{ij} + 2s_{ij}\big] dx^i dx^j,\]

where \(\Phi\), \(w_i\), \(\Psi\), and \(s_{ij}\) describe the different possible kinds of perturbations. The perturbation \(s_{ij}\) is a traceless symmetric tensor which will turn out to represent gravitational waves or their constituent particle, the **graviton**. Because \(s_{ij}\) is a matrix, it is often called the **tensor perturbation** to the metric; it is traceless because its trace has been separated out into the \(\delta_{ij}\) term above. It is an interesting problem to compute the degrees of freedom of \(s_{ij}\), as in the problem below, because this is how the gravitational modes of a string are identified in string theory.

In a particularly nice coordinate system called **transverse gauge**, plugging this metric into Einstein's equations gives all of the perturbations to be zero except for \(s_{ij}\), which satisfies the wave equation

\[\Box s_{ij} = 0 \iff \frac{1}{c^2} \frac{\partial^2 s_{ij}}{\partial t^2} = \frac{\partial^2 s_{ij}}{\partial x^2} + \frac{\partial^2 s_{ij}}{\partial y^2} + \frac{\partial^2 s_{ij}}{\partial z^2}.\]

Since all perturbations except \(s_{ij}\) are zero in this gauge, it is common to just work with \(h_{\mu \nu}\) directly, which just looks like the matrix

\[h_{\mu \nu} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 &&&\\0&&2s_{ij} & \\ 0 &&& \end{pmatrix}.\]

The components of \(h_{\mu \nu}\), i.e. the components of \(s_{ij}\), are exactly the strains \(h\) mentioned above, but in different directions for each component. The "\(+\)" and "\(\times\)" polarizations discussed above refer to the two possible actions of these strains on geodesics of particles moving in this perturbed metric. Using this formalism and more extensive mathematics in general relativity, the formulas quoted above for the magnitudes of strains as a function of space and time and power radiated by a binary black hole system can be computed.

## References

[1] Image from https://en.wikipedia.org/wiki/Gravitational_wave under Creative Commons licensing for reuse and modification.

[2] K. McLin, C. Peruta, and L. Cominsky. *Direct Observation of Gravitational Waves: Educator's Guide*. SSU Education and Public Outreach Group, Sonoma State University, Rohnert Park, CA. https://dcc.ligo.org/public/0123/P1600015/004/LIGOEdGuide_Final.pdf

[3] "The Nobel Prize in Physics 2017". Nobelprize.org. Nobel Media AB 2014. Web. 4 Oct 2017. <http://www.nobelprize.org/nobel_prizes/physics/laureates/2017/>

[4] Carroll, Sean. *Spacetime and Geometry: An Introduction to General Relativity*. San Francisco: Pearson, 2004.

[5] Weisberg, J.M.; Taylor, J.H. (July 2005). *The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis*. Written at Aspen, Colorado, USA. In F.A. Rasio; I.H. Stairs. Binary Radio Pulsars. ASP Conference Series. San Francisco: Astronomical Society of the Pacific. p. 25.

[6] B.P. Abbott, et al. *Observation of Gravitational Waves from a Binary Black Hole Merger*. PRL **116**, 061102 (2016).

[7] B.P. Abbott, et al. *Observation of a kilogram-scale oscillator near its quantum ground state*. New Journal of Physics, Volume 11, July 2009. http://iopscience.iop.org/article/10.1088/1367-2630/11/7/073032/meta;jsessionid=DC52784888F8C04DC86C16069DE35826.c2.iopscience.cld.iop.org

**Cite as:**Gravitational Waves.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/gravitational-waves/