Introduction to gravitational wave astronomy

Soon after proposing the general theory of relativity (GR), Einstein showed, by linearizing the field equations, that the theory implies the existence of gravitational waves (GWs) Einstein1916 ; Einstein1918 , and obtained what has become known as the quadrupole formula. However, the concept of a GW actually predates GR. Newtonian gravitation theory can be expressed as an elliptic equation with the gravitational field changing instantaneously if the source changes. Since special relativity had established the speed of light as a universal speed limit, this suggests that the gravitational theory should be expressed as a hyperbolic system, so implying the existence of GWs. During the early years, i.e. from about 1920 to 1960, there was uncertainty about the nature of GWs. It was not clear whether GWs carry energy since GW energy at a given event in spacetime cannot be defined; it was not clear whether the quadrupole formul could be applied to gravitationally bound systems; and it was argued that the full nonlinear theory of GR does not permit GW solutions. For a detailed discussion of these issues, see Kennefick:1997kb .

The mathematical theory of GWs became well-established in the 1960s. The above issues were resolved by expressing the Einstein equations of the full nonlinear theory of GR in a suitable coordinate system Bondi62 ; Sachs62 , and also by the “shortwave approximation” averaging procedure Isaacson68 . Once it was clear that the emission of GWs causes a loss of energy in the emitting system, the dynamics of an orbiting binary and the resulting inspiral could be calculated using linearized theory and the quadrupole approximation Peters:1963ux ; Peters:1964 .

In this period, there were also important developments in the experimental area. The theory of stellar evolution had long indicated that the end-state of a massive star would be a neutron star or a black hole. However, the first observation of a neutron star was of a pulsar in the Crab nebula in 1969, and the first evidence of a massive ($\approx 15M_{\odot}$) compact object that could only be a black hole was Cygnus X-1 in 1971. The confirmation of the existence of these compact objects was important: their inspiral and merger would be both powerful and in the frequency range $\approx 10$Hz to $\approx 1000$Hz which would be suitable for a terrestrial detector. The first attempt to detect gravitational waves was reported in 1969 using a bar detector Weber:1969bz . This paper actually reported the detection of a number of GW events, although subsequently it became clear that these events were not astrophysical but rather the result of experimental error. In 1979, observations of the binary pulsar system PSR 1913+16 showed that the orbit was inspiralling at a rate consistent with GW emission in GR, thus providing the first experimental evidence for the existence of GWs Taylor79 .

The most powerful GW sources, and therefore those most likley to be detected, are the merger of two compact objects, but this case does not satisfy the conditions of the quadrupole approximation and there is a need to go beyond linearized theory. This can be achieved by means of a series expansion, adding terms of quadratic, cubic and higher orders to the linearized expressions. This approach was considered in the early years of GR Lorentz17 ; Lorentz37 , and in the 1980s formalized as the Post-Newtonian method. For the actual merger, a numerical simulation of the full Einstein equations is needed, normally with the spacetime foliated into a sequence of spacelike hypersurfaces  Arnowitt62 . In 1977, the GWs from a head-on collision of two Schwarzschild black holes were computed  Smarr77 , but it was only in 2005 that codes were able to make a stable evolution of the physically realistic problem of the inspiral and merger of two black holes  Pretorius:2005gq . Since then, many other groups have successfully evolved black hole spacetimes, as well as neutron star mergers and supernovae, often using a combination of Post-Newtonian and numerical methods.

The possibility of using laser interferometry to detect GWs was suggested in the 1960s, and simple prototypes were constructed at that time. In 1980, the US National Science Foundation provided funding for the construction of certain prototypes, as well as for a study of the technical issues and the costs of building an interferometer with arms several km in length. This eventually led to the construction of LIGO (Laser Interferometry Gravitational-Wave Observatory) facilities at Hanford and Livingstone, USA, with 4km arms; scientific studies commenced in 2002. Also at this time, the much smaller detector GEO600 (600m arms) started operation, and in 2007 Virgo in Italy, which has 3km arms and is a project of the European Gravitational Observatory consortium, made its first science run. These instruments underwent a number of upgrades to improve the sensitivity, and the first direct detection of GWs was made on 14 September 2015 by LIGO Hanford and Livingstone Abbott2016a .

This chapter provides an overview of GW astronomy, and describes aspects of the basic theory of GWs that form the foundation of the field. It provides the background material that underpins the specialized chapters in this handbook. The discussion is mainly within the context of the Einstein equations linearized about Minkowski spacetime. This theory is sufficiently straightforward that it can be presented within a single chapter, yet is also widely applicable. The magnitude of GWs is dimensionless, and those propagating past the Earth that have been detected can be characterized as being ${\mathcal{O}}(10^{-22})$; thus linearized theory certainly applies. The later phase of compact object inspiral is driven by GW emission; this process lasts millions or billions of years and, apart from the final seconds or minutes, is accurately described by linearized theory. Even though linearized theory does not provide an accurate waveform for the actual merger of two compact objects, it remains useful as it provides a simple guide in analytic form.

The theory of linearizing Einstein’s equations about Minkowski spacetime is developed in Sec. 2. A key issue is the use of gauge transformations (which, in this context, are coordinate transformations that are almost the identity transformation) to simplify the equations to a form that is manifestly the wave equation. The simplest solution to the wave equation is a plane wave, i.e. in Cartesian coordinates $(t,x,y,z)$ a function $f(t,z)$ that satisfies the wave equation; this solution well describes GWs in the solar system produced by a source many Mpc away. We construct the plane wave solution, and make further gauge transformations to the Transverse Traceless, or TT, gauge. In this way, the 10 components of a symmetric tensor are reduced to 2 independent components or polarizations, usually denoted as $h_{+},h_{\times}$. Finally, this section investigates the effect of a plane GW on a system of test particles. In the TT gauge, the coordinates of the particles do not change, but that does not mean that there is no movement: a coordinate independent quantity such as the proper distance between two of the particles does vary with time. It is also shown that the Riemann tensor has non-zero components, so the spacetime is not flat.

The generation of GWs by the motion of matter is discussed in Sec. 3, and the quadrupole formula is derived. As an example, the formula is applied to the astrophysically important problem of a binary comprising two masses in circular orbit around each other. The two polarization modes, $h_{+},h_{\times}$, are evaluated with respect to spherical polar coordinates. We next discuss the energy, as well as the linear and angular momenta, carried away by GWs. These effects are beyond the scope of linearized theory, being quadratic in the perturbations, and the calculations are only outlined with much detail omitted. This section also summarizes methods that are used for situations where the quadrupole formula is inadequate: Post-Newtonian approximations, numerical relativity, the theory of quasinormal modes of a black hole, and Extreme Mass Ratio Inspirals (EMRIs).

Now that we know what GWs carry away from a system, conservation laws are used in Sec. 4 to find the evolution of an orbiting circular binary. The orbital diameter and wave period slowly decrease, and we find an expression for the time to coalescence. We also find a relation between the chirp mass, which is a function of the individual masses of the binary, and an expression involving the wave frequency and its rate of change; this means that the chirp mass is entirely determined by observational GW data. It is also shown that GW emission causes an eccentric orbit to circularize, so the focus on circular orbits is for reasons of physics rather than for mathematical convenience.

Sec. 5 provides an outline of various existing and planned GW detection facilities, including terrestrial laser interferometers, satellite systems, and pulsar timing. We indicate the frequency range in which a detector is sensitive, and some actual or expected astrophysical sources. Then, for the LIGO and Virgo network, we outline the process of determining basic source parameters of an observed GW event representing a compact object inspiral and merger. Some contributions that GW observations have already made to physics, cosmology and astrophysics are outlined.

The chapter ends with a Conclusion, Sec. 6.

The essential idea is to consider spacetimes that comprise small perturbations about Minkowski spacetime. More precisely, the metric tensor $g_{\alpha\beta}$ is written

$$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}\,,$$

(1)

where $\eta_{\alpha\beta}=$ is the metric of Minkowski spacetime which in Cartesian $(t,x,y,z)$ coordinates is diag$(-1,1,1,1)$, and $h_{\alpha\beta}$ is a small perturbation. In the linearized approximation, terms of order ${\mathcal{O}}\left(\left(h_{\alpha\beta}\right)^{2}\right)$ are neglected. The contravariant metric is written as $g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}$, and the identities $\delta^{\alpha}_{\gamma}=g^{\alpha\beta}g_{\beta\gamma}=\eta^{\alpha\beta}\eta_{\beta\gamma}$ imply that

$$h^{\alpha\beta}=\eta^{\alpha\gamma}\eta^{\beta\delta}h_{\gamma\delta}\,.$$

(2)

Thus, the indices of quantities of order ${\mathcal{O}}(h_{\alpha\beta})$ are raised and lowered using the background metric $\eta_{\alpha\beta}$ rather than the full metric $g_{\alpha\beta}$. The first step towards constructing the Einstein equations is to determine the metric connection

$$\Gamma^{\mu}_{\ \alpha\beta}=\frac{1}{2}\eta^{\mu\nu}(\partial_{\beta}h_{\nu\alpha}+\partial_{\alpha}h_{\nu\beta}-\partial_{\nu}h_{\alpha\beta})=\frac{1}{2}(\partial_{\beta}h^{\mu}_{\ \ \alpha}+\partial_{\alpha}h^{\mu}_{\ \ \beta}-\partial^{\mu}h_{\alpha\beta})\,.$$

(3)

Then the Ricci tensor is

$$R_{\mu\nu}=\partial_{\alpha}\Gamma^{\alpha}_{\ \mu\nu}-\partial_{\nu}\Gamma^{\alpha}_{\ \mu\alpha}=\frac{1}{2}(\partial_{\alpha}\partial_{\nu}h^{\ \;\alpha}_{\mu}+\partial_{\alpha}\partial_{\mu}h^{\ \;\alpha}_{\nu}-\partial_{\alpha}\partial^{\alpha}h_{\mu\nu}-\partial_{\mu}\partial_{\nu}h)\,,$$

(4)

where $h=h^{\alpha}_{\alpha}=\eta^{\alpha\beta}h_{\alpha\beta}$. The Ricci scalar is thus

$$R=R^{\mu}_{\mu}=\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}-\partial_{\alpha}\partial^{\alpha}h\,,$$

(5)

and therefore Einstein’s equations $G_{\mu\nu}=R_{\mu\nu}-Rg_{\mu\nu}/2=8\pi T_{\mu\nu}$ are

	$\displaystyle\frac{1}{2}$	$\displaystyle\left(\partial^{\alpha}\partial_{\nu}h_{\mu\alpha}+\partial^{\alpha}\partial_{\mu}h_{\nu\alpha}-\partial_{\alpha}\partial^{\alpha}h_{\mu\nu}-\partial_{\mu}\partial_{\nu}h-\eta_{\mu\nu}(\partial^{\alpha}\partial^{\beta}h_{\alpha\beta}-\partial^{\alpha}\partial_{\alpha}h)\right)$
		$\displaystyle=8\pi T_{\mu\nu}\,.$		(6)

Eq. (2.1) may be somewhat simplified by changing the metric perturbation $h_{\alpha\beta}$ to its trace reversed form, that is

$$\bar{h}_{\alpha\beta}=h_{\alpha\beta}-\frac{1}{2}\eta_{\alpha\beta}h\mbox{ so that }\bar{h}=-h\mbox{ and }{h}_{\alpha\beta}=\bar{h}_{\alpha\beta}-\frac{1}{2}\eta_{\alpha\beta}\bar{h}\,.$$

(7)

Einstein’s equations are then

$$G_{\mu\nu}=-\frac{1}{2}\left(\partial_{\alpha}\partial^{\alpha}\bar{h}_{\mu\nu}+\eta_{\mu\nu}\partial^{\alpha}\partial^{\beta}\bar{h}_{\alpha\beta}-\partial_{\nu}\partial^{\alpha}\bar{h}_{\mu\alpha}-\partial_{\mu}\partial^{\alpha}\bar{h}_{\nu\alpha}\right)=8\pi T_{\mu\nu}.$$

(8)

The first term $\partial_{\alpha}\partial^{\alpha}\bar{h}_{\mu\nu}$ is the wave operator applied to $\bar{h}_{\mu\nu}$, and the problem would reduce to a standard wave equation if the other terms could be made to disappear. This can be achieved on applying the Lorentz gauge condition

$$\partial^{\alpha}\bar{h}_{\alpha\beta}=0\,,$$

(9)

in Eq. (8) to obtain

$$-\frac{1}{2}\Box\bar{h}_{\mu\nu}=8\pi T_{\mu\nu}\,,$$

(10)

where the operator $\Box=\partial_{\alpha}\partial^{\alpha}$, and clearly in vacuum $\Box\bar{h}_{\mu\nu}=0$. Imposition of the Lorentz gauge condition is a constraint on the coordinates being used, and is not a restriction on the geometry of the spacetime. This matter is discussed further in the next section.

A gauge transformation is a coordinate transformation of the form

$$x^{\mbox{{\tiny(NEW)}}\alpha}=x^{\mbox{{\tiny(OLD)}}\alpha}+\xi^{\alpha}(x^{\mbox{{\tiny(OLD)}}\beta})$$

(11)

where $\partial_{\beta}\xi^{\alpha}$ is of the same order of smallness as $h_{\alpha\beta}$; terms of order ${\mathcal{O}}(h_{\alpha\beta})^{2}$, ${\mathcal{O}}(h_{\alpha\beta}\partial_{\beta}\xi^{\alpha})$, ${\mathcal{O}}(\partial_{\beta}\xi^{\alpha})^{2}$ are neglected. Applying the coordinate transformation (11) to the metric (1) gives

$$h^{\mbox{{\tiny(NEW)}}}_{\alpha\beta}=h^{\mbox{{\tiny(OLD)}}}_{\alpha\beta}-\partial_{\beta}\xi_{\alpha}-\partial_{\alpha}\xi_{\beta}\,,$$

(12)

and then the change to trace-reversed form leads to

$$\bar{h}^{\mbox{{\tiny(NEW)}}}_{\alpha\beta}=\bar{h}^{\mbox{{\tiny(OLD)}}}_{\alpha\beta}-\partial_{\beta}\xi_{\alpha}-\partial_{\alpha}\xi_{\beta}+\eta_{\alpha\beta}\partial_{\mu}\xi^{\mu}\,.$$

(13)

Requiring $\bar{h}^{\mbox{{\tiny(NEW)}}}_{\alpha\beta}$ to satisfy the Lorentz gauge condition then implies

$$\Box\xi_{\beta}=\partial^{\alpha}\bar{h}^{\mbox{{\tiny(OLD)}}}_{\alpha\beta}\,.$$

(14)

The right hand side of Eq. (14) is regarded as given, and then Eq. (14) comprises separate wave equations for each of the four unknowns $\xi_{0},\cdots,\xi_{3}$. The existence of solutions to these four wave equations follows from the theory of partial differential equations. Note however that the solutions are not unique, and we are free to write

$$\xi^{\alpha}=\xi^{(0)\alpha}+\zeta^{\alpha}\,,$$

(15)

where $\xi^{(0)\alpha}$ is a solution to Eq. (14), and where $\zeta^{\alpha}$ satisfies $\Box\zeta^{\alpha}=0$.

The possibility to make gauge transformations means that a metric that appears to be wavelike may not represent a GW. For example, suppose that the spacetime is Minkowski (and so does not contain gravitational waves) with $\bar{h}^{\mbox{{\tiny(OLD)}}}_{\alpha\beta}=0$, and that $\xi_{\alpha}=(0,0,0,\epsilon\cos(t-x))$. Then

$$\bar{h}_{\alpha\beta}=\left(\begin{array}[]{cccc}0&0&0&\epsilon\sin(t-x)\\ 0&0&0&-\epsilon\sin(t-x)\\ 0&0&0&0\\ \epsilon\sin(t-x)&-\epsilon\sin(t-x)&0&0\end{array}\right)\,.$$

(16)

A simple test to determine whether or not a vacuum spacetime is Minkowski in unusual coordinates, is to evaluate the Riemann tensor: $R^{\alpha}_{\phantom{\alpha}\beta\gamma\delta}=0$ if and only if the spacetime is Minkowskian.

Let us now make the ansatz

$$\bar{h}^{\alpha\beta}=\Re\left(A^{\alpha\beta}\exp(ik_{\alpha}x^{\alpha})\right)\,,$$

(17)

where the $k_{\alpha}$ are real constants and the $A_{\alpha\beta}$ are complex constants; note that a general wave may be expressed as a Fourier sum of terms with the above form. Substituting into the vacuum field equations $\Box\bar{h}_{\alpha\beta}=0$ leads to $\eta^{\alpha\beta}k_{\alpha}k_{\beta}=0$, so that $k_{\alpha}$ is a null vector, and usually $k_{0}$ is written as $\omega$. Since $A^{\alpha\beta}$ is symmetric, it has 10 independent components. Now, the Lorentz gauge condition $\partial^{\beta}\bar{h}_{\alpha\beta}=0$ implies 4 conditions

$$A^{\alpha\beta}k_{\beta}=0\,,$$

(18)

reducing the number of independent components of $A^{\alpha\beta}$ from 10 to 6. Then a gauge transformation can be made where

$$\xi^{\alpha}=\Re(B^{\alpha}\exp(ik_{\mu}x^{\mu}))$$

(19)

with $B^{\alpha}$ constant; since $\xi^{\alpha}$ satisfies the identity $\Box\xi^{\alpha}=0$, the condition $\partial_{\beta}\bar{h}^{\alpha\beta}=0$ is not affected. The 4 constants $B^{\alpha}$ are used to reduce the independent components of $A^{\alpha\beta}$ from 6 to 2.

In order to apply the above and construct in a transparent way an explicit form of $A^{\alpha\beta}$ with only two independent components, we align the coordinates with the direction of propagation of the wave. Let the wave be propagating in the $z-$direction with frequency $\omega/(2\pi)$ so that $k_{\alpha}=(\omega,0,0,-\omega)$. Then the conditions (18) imply that $A^{\alpha 0}=A^{\alpha 3}$, so we can write

$$A^{\alpha\beta}=\left(\begin{array}[]{cccc}A^{00}&A^{01}&A^{02}&A^{00}\\ A^{01}&A^{11}&A^{12}&A^{01}\\ A^{02}&A^{12}&A^{22}&A^{02}\\ A^{00}&A^{01}&A^{02}&A^{00}\end{array}\right)\,,$$

(20)

which has six independent components $A^{00},A^{01},A^{02},A^{11},A^{12},A^{22}$. Differentiating the gauge transformation (19) gives $\partial^{\alpha}\zeta^{\beta}=\Re(iB^{\alpha}k^{\beta}\exp(ik_{\mu}x^{\mu}))$, so that from Eq. (13),

$$\bar{h}^{\prime\alpha\beta}=\Re\left((A^{\alpha\beta}-iB^{\alpha}k^{\beta}-iB^{\beta}k^{\alpha}+\eta^{\alpha\beta}iB^{\mu}k_{\mu})\exp(ik_{\mu}x^{\mu})\right)\,.$$

(21)

Now, $A^{\alpha\beta},B^{\alpha},(k^{\prime\alpha}-k^{\alpha}),(x^{\prime\alpha}-x^{\alpha})$ are small quantities, and so the product of any pair is negligible. Thus we can write $\bar{h}^{\prime\alpha\beta}=\Re\left((A^{\prime\alpha\beta}\exp(ik_{\mu}x^{\mu})\right)$, and then

$$A^{\prime\alpha\beta}=A^{\alpha\beta}-iB^{\alpha}k^{\beta}-iB^{\beta}k^{\alpha}+\eta^{\alpha\beta}iB^{\mu}k_{\mu}\,.$$

(22)

	$\displaystyle A^{\prime 00}$	$\displaystyle=A^{00}+i\omega(B^{0}+B^{3})\,,\qquad A^{\prime 01}=A^{01}+i\omega B^{1}\,,\qquad A^{\prime 02}=A^{02}+i\omega B^{2}$
	$\displaystyle A^{\prime 11}$	$\displaystyle=A^{11}+i\omega(B^{3}-B^{0})\,,\qquad A^{\prime 22}=A^{22}+i\omega(B^{3}-B^{0})\,,\qquad A^{\prime 12}=A^{12}\,.$		(23)

The constants $B^{\alpha}$ may be chosen so as to simplify $A^{\prime\alpha\beta}$ in any desired way, and common practice is to use them to set

$$A^{\prime 00}=A^{\prime 01}=A^{\prime 03}=0\,,\;\;A^{\prime 22}=-A^{\prime 11}\,,$$

(24)

so that $A^{\prime\alpha\beta}$ is

$$A^{\prime\alpha\beta}=\left(\begin{array}[]{cccc}0&0&0&0\\ 0&A^{\prime 11}&A^{\prime 12}&0\\ 0&A^{\prime 12}&-A^{\prime 11}&0\\ 0&0&0&0\end{array}\right)\,.$$

(25)

The resulting metric $\bar{h}_{\alpha\beta}$ is transverse: the wave travels in the $z$-direction, and its effects are in the $x$- and $y$-directions. More generally, the transverse property is expressed as

$$\bar{h}^{\prime}_{\alpha\beta}k^{\alpha}=0\,.$$

(26)

Further the metric is traceless, i.e.

$$\bar{h}^{\prime\alpha}_{\alpha}=\bar{h}^{\prime}=0\,,$$

(27)

so that the metric and its trace-reverse form are the same, $\bar{h}^{\prime}_{\alpha\beta}=h^{\prime}_{\alpha\beta}$. A metric satisfying Eqs. (26) and (27) is said to be in the Transverse Traceless or TT gauge. From now on, we will drop the ^′ to denote a quantity in this gauge, and in circumstances where there is a need to differentiate the gauge of quantities, we will use the superfix ^TT to denote a quantity in the TT gauge.

The GW derived above has two independent components $h^{11},h^{12}$. However, there is no essential difference between the components, as can be seen by making a coordinate transformation which, geometrically, corresponds to a rotation of the $(x,y)$ axes about the $z$-axis through an angle $\theta=\pi/4$. More precisely, $(t,x,y,z)\rightarrow((t^{\prime},x^{\prime},y^{\prime},z^{\prime})$ where

$$x^{\prime}=x\cos\theta+y\sin\theta\,,\qquad y^{\prime}=x\sin\theta-y\cos\theta\,,$$

(28)

and here $\cos\theta=\sin\theta=\sqrt{2}/2$. Application to the metric $g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}$ leads to

$$h^{\prime 11}=-h^{\prime 22}=h^{12}\,,\qquad h^{\prime 12}=h^{\prime 21}=-h^{11}\,.$$

(29)

The two independent components of a GW, $h^{11},h^{12}$, are usually denoted by $h_{+},h_{\times}$ respectively and are called the two polarization states of the GW. There is an analogy to electromagnetic waves which are also transverse to the direction of propagation and which also have polarization states, although in this case the angle between the two states is $\pi/2$ rather than $\pi/4$. As shown above, the distinction between $h_{+},h_{\times}$ is observer dependent, but in some circumstances the polarization can be described in an observer-independent way. We write

$$h_{+}=\Re(A_{+}\exp\left(i\omega(t-z))\right)\,,\;\;h_{\times}=\Re(A_{\times}\exp\left(i\omega(t-z))\right)\,.$$

(30)

Now, $A_{+},A_{\times}$ are complex constants (i.e., independent of the coordinates $x^{\alpha}$), and consider the ratio $A_{\times}/A_{+}=R_{A}$ which must also be constant. There are two interesting special cases: (1) Suppose that the ratio $R_{A}$ has zero imaginary part, then $h_{\times}/h_{+}$ is constant, and so we may change coordinates by rotating the $(x,y)$ axes about the $z$-axis through some angle $\theta$ to achieve $h_{\times}=0$; in this case the GW is said to be linearly polarised. (2) Suppose that $R_{A}=\pm i$, then from the identity $\cos^{2}s+\sin^{2}s=1$, it follows that the magnitude of the GW $\sqrt{h_{+}^{2}+h_{\times}^{2}}$ is constant, and the wave is said to be circularly polarised. The physical properties leading to these names are discussed in the next sub-section.

We now consider the physical effects of a GW in the TT gauge as described in the previous sub-section. Since $h_{\alpha 0}=0$, it follows from Eq. (3) that $\Gamma^{\alpha}_{00}=0$. The geodesic equation is

$$\frac{d^{2}x^{\alpha}}{d\tau^{2}}=-\Gamma^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\,,$$

(31)

where $\tau$ is the proper time; if a particle is initially at rest, ${dx^{\mu}}/{d\tau}=(1,0,0,0)$ so that ${d^{2}x^{\alpha}/}{d\tau^{2}}=0$ and the particle remains at rest. But this is just a consequence of the coordinates being used, and does not mean that the GW has no physical effect. In order to analyze this further, we use the equation of geodesic deviation. Consider two nearby particles, $A$ and $B$, each with 4-velocity $U^{\alpha}$, and let $c^{\alpha}$ be a vector connecting $A$ and $B$ and orthogonal to $U^{\alpha}$; then

$$\frac{d^{2}c^{\alpha}}{d\tau^{2}}+\Gamma^{\alpha}_{\beta\gamma}U^{\beta}U^{\gamma}=-R^{\alpha}_{\phantom{\alpha}\beta\gamma\delta}U^{\beta}U^{\delta}c^{\gamma}\,.$$

(32)

Now suppose that we use proper coordinates with $A$ located at the origin; then $U^{\alpha}=(1,0,0,0)$, $c^{\alpha}=(0,c^{i})=(0,x_{B}^{i})$ (with $i=1,2,3$ denoting a spatial index), the metric is

$$ds^{2}=\left(\eta_{\alpha\beta}+{\mathcal{O}}(|x^{i}|^{2})\right)dx^{\alpha}dx^{\beta}\,,$$

(33)

$\Gamma^{\alpha}_{\beta\gamma}=0$, $\tau=t$, and Eq. (32) simplifies to

$$\frac{d^{2}x_{B}^{i}}{dt^{2}}=-R^{i}_{\phantom{i}0j0}x_{B}^{j}\,.$$

(34)

It is important to note that $R^{\alpha}_{\phantom{\alpha}\beta\gamma\delta}$ is gauge-invariant, that is if we make a gauge transformation given by Eq. (11), then in the new gauge $R^{\mbox{{\tiny(NEW)}}\alpha}_{\phantom{\mbox{{\tiny(NEW)}}\alpha}\beta\gamma\delta}=R^{\mbox{{\tiny(OLD)}}\alpha}_{\phantom{\mbox{{\tiny(OLD)}}\alpha}\beta\gamma\delta}$. This result follows since $R^{\alpha}_{\phantom{\alpha}\beta\gamma\delta}={\mathcal{O}}(h_{\alpha\beta})$, and the coordinate transformation matrix is

$$\frac{\partial x^{\mbox{{\tiny(NEW)}}\alpha}}{\partial x^{\mbox{{\tiny(OLD)}}\beta}}=\delta^{\alpha}_{\beta}+{\mathcal{O}}(h_{\alpha\beta})\,,$$

(35)

so to ${\mathcal{O}}(h_{\alpha\beta})$ only the $\delta^{\alpha}_{\beta}$ is used in the coordinate transformation. Thus we may use the TT gauge to evaluate the Riemann tensor in Eq. (34), and we find

$$R^{1}_{\phantom{1}010}=-R^{2}_{\phantom{2}020}=-\frac{1}{2}\partial^{2}_{t}h_{+}\,,\;\;R^{1}_{\phantom{1}020}=R^{2}_{\phantom{2}010}=-\frac{1}{2}\partial^{2}_{t}h_{\times}\,,$$

(36)

with the remaining components of the form $R^{i}_{\phantom{i}0j0}$ being zero. Then Eq. (34) simplifies to

$$\frac{d^{2}x_{B}^{i}}{dt^{2}}=\left(\frac{1}{2}\partial^{2}_{t}h_{+}x_{B}^{1}+\frac{1}{2}\partial^{2}_{t}h_{\times}x_{B}^{2},\frac{1}{2}\partial^{2}_{t}h_{\times}x_{B}^{1}-\frac{1}{2}\partial^{2}_{t}h_{+}x_{B}^{2},0\right)\,.$$

(37)

We now write $x^{i}_{B}(t)=x^{i}_{B0}+\epsilon^{i}(t)$ with $\epsilon^{i}$ of order ${\mathcal{O}}(h)$, and replace $x^{i}_{B}$ by $x^{i}_{B0}$ on the right hand side of Eq. (37) (as terms of order ${\mathcal{O}}(h^{2})$ are ignorable) so that Eq. (37) can be integrated to give

$$\epsilon^{i}=\left(\frac{1}{2}h_{+}x_{B0}^{1}+\frac{1}{2}h_{\times}x_{B0}^{2},\frac{1}{2}h_{\times}x_{B0}^{1}-\frac{1}{2}h_{+}x_{B0}^{2},0\right)\,.$$

(38)

In order to facilitate the interpretation of Eq. (38), let us suppose that the $(x^{1},x^{2})$ axes have been rotated so that $x^{2}_{B0}=0$, use Eq. (30), and suppose that the origin of the spacetime coordinates is such that $A_{+}$ is real. Then in the case of a linearly polarised wave

$$\epsilon^{i}=\frac{1}{2}x_{B0}^{1}A_{+}\cos\omega t\left(1,R_{A},0\right)\,,$$

(39)

so that $B$ moves in a straight line; and in the case of a circularly polarized wave

$$\epsilon^{i}=\frac{1}{2}x_{B0}^{1}A_{+}\left(\cos\omega t,\sin\omega t,0\right)\,,$$

(40)

so that $B$ moves in a circle. The effect of a GW, for both linear and circular polarization states is illustrated in Fig. 1.

The preceeding analysis brings out two important general points. Firstly, since $R^{\alpha}_{\phantom{\alpha}\beta\gamma\delta}\neq 0$, the spacetime is not Minkoskian. Secondly, suppose that the particles $A$ and $B$ are massive and are connected by a damping system; then the relative motion and acceleration will mean that work is done on the damping system and therefore its temperature must increase. These considerations show that GWs are real physical effects that contain energy that must have originated in some source and which can, in principle, be extracted from the GWs.

From standard wave equation theory, the solution to Eq. (10) representing an outgoing wave is

$$\bar{h}^{\mu\nu}(t,x^{i})=4\int\frac{T^{\mu\nu}(t-|x^{i}-x^{\prime i}|,x^{\prime j})}{|x^{i}-x^{\prime i}|}d^{3}x^{\prime}\,,$$

(41)

where $(t,x^{i})$ is the event at which the metric perturbation $\bar{h}^{\mu\nu}$ is evaluated, $x^{\prime i}$ represents a spatial point inside the source, and $|x^{i}-x^{\prime i}|$ is the Euclidean distance between $x^{i}$ and $x^{\prime i}$. Suppose now that the source is matter localized near the origin in a region $r=|x^{i}|<r_{0}$, and we are evaluating the metric perturbations where $r={\mathcal{O}}(r_{E})$ with $r_{0}\ll r_{E}$; then Eq. (41) simplifies to

$$\bar{h}^{\mu\nu}(t,x^{i})=\frac{4}{r}\int T^{\mu\nu}(t-r,x^{\prime j})d^{3}x^{\prime}\,.$$

(42)

The discussion in Sec. 2.3 showed that the 10 components of $\bar{h}^{\mu\nu}$ have only 6 independent components, so it is sufficient to determine the spatial components $\bar{h}^{ij}$. In the linearized approximation, the conservation condition $\nabla_{\alpha}T^{0\alpha}=0$ simplifies to $\partial_{t}T^{00}+\partial_{i}T^{0i}=0$, and applying this relation twice gives

$$\partial_{i}\partial_{j}T^{ij}=\partial^{2}_{t}T^{00}\,.$$

(43)

Using the above together with the rules of basic calculus gives

	$\displaystyle\partial^{2}_{t}(T^{00}x^{i}x^{j})=$	$\displaystyle(\partial_{k}\partial_{\ell}T^{k\ell})x^{i}x^{j}$
	$\displaystyle=$	$\displaystyle\partial_{k}\partial_{\ell}(T^{k\ell}x^{i}x^{j})-2\partial_{k}(T^{ki}x^{j}+T^{kj}x^{i})+2T^{ij}\,,$		(44)

from which it follows that

$$\int T^{ij}d^{3}x=\frac{1}{2}\partial^{2}_{t}\int T^{00}x^{i}x^{j}d^{3}x\,,$$

(45)

since the remaining terms $\partial_{k}\left[\partial_{\ell}(T^{k\ell}x^{i}x^{j})-2(T^{ki}x^{j}+T^{kj}x^{i})\right]$ can be transformed by the divergence theorem into a surface integral over the boundary of the volume of integration where $T^{\alpha\beta}=0$.

As well as the gravitational field being weak with $h_{\mu\nu}\ll 1$, we now assume that throughout the source region the relative speed of the matter flow is much less than that of light. Then $T^{00}$ can be replaced by the matter density $\rho$ so that Eq. (41) becomes

$$\bar{h}^{ij}(t,x^{i})=\frac{2}{r}\frac{d^{2}I^{ij}(t-r)}{dt^{2}}\,,$$

(46)

where the 3-dimensional tensor

$$I^{ij}(t)=\int\rho(t,x^{\prime k})x^{\prime i}x^{\prime j}d^{3}x$$

(47)

is the second moment, also called the quadrupole moment, of the mass distribution. In the case that the matter distribution is treated as a system of $N$ particles,

$$I^{ij}(t)=\sum_{a=1}^{N}M_{a}x^{\prime i}_{a}x^{\prime j}_{a}\,.$$

(48)

The energy associated with gravitational waves is a second-order effect, and so its justification goes beyond the linearized approximation considered so far. Here, we outline the calculation, but with many details omitted Isaacson68 . Eq. (1) is amended to

$$g_{\alpha\beta}=g^{[B]}_{\alpha\beta}+h^{[1]}_{\alpha\beta}+h^{[2]}_{\alpha\beta}\,,$$

(57)

where: $h^{[1]}_{\alpha\beta}$ is $h_{\alpha\beta}$ in Eq. (1) and is the first-order perturbation; $h^{[2]}_{\alpha\beta}$ is the second-order perturbation; and $g^{[B]}_{\alpha\beta}$ is the background metric, and is a little different to the Minkowski metric $\eta_{\alpha\beta}$ because it includes the response of the background geometry to the first-order perturbations. Evaluation of the Ricci tensor for the metric of Eq. (57) gives a similar expansion

$$R_{\alpha\beta}=R^{[B]}_{\alpha\beta}(g^{[B]})+R^{[1]}_{\alpha\beta}(h^{[1]})+R^{[2]}_{\alpha\beta}(h^{[1]})+R^{[1]}_{\alpha\beta}(h^{[2]})\,,$$

(58)

where $R^{[1]}_{\alpha\beta}(h^{[1]})$ is of order ${\mathcal{O}}\left(h^{[1]}\right)$, and the last two terms are of order ${\mathcal{O}}\left(\left(h^{[1]}\right)^{2}\right)$. Then solving Eq. (58) to first-order in $h^{[1]}$ gives $R^{[1]}_{\alpha\beta}(h^{[1]})=0$, from which the linearized theory discussed in the preceeding sections of this chapter is obtained.

The next step is to split the second-order part of $R_{\alpha\beta}$ into a part that varies on scales much larger than the wavelength of the GWs, and a remainder that contains the local fluctuations. Defining $<x>$ to be the average of the quantity $x$ over a suitable region (in a sense made precise in Isaacson68 ), then

	$\displaystyle R^{[B]}_{\alpha\beta}(g^{[B]})+<R^{[2]}_{\alpha\beta}(h^{[1]})>$	$\displaystyle=0\,,$		(59)
	$\displaystyle R^{[1]}_{\alpha\beta}(h^{[2]})+R^{[2]}_{\alpha\beta}(h^{[1]})-<R^{[2]}_{\alpha\beta}(h^{[1]})>$	$\displaystyle=0\,.$		(60)

We can then define

$$T^{[GW]}_{\alpha\beta}=-\frac{1}{8\pi}\left(<R^{[2]}_{\alpha\beta}(h^{[1]})>-\frac{1}{2}g^{[B]}_{\alpha\beta}<R^{[2]}(h^{[1]})>\right)\,,$$

(61)

since, from Eq. (59), the Einstein tensor for the background metric can be written as

$$G^{[B]}_{\alpha\beta}=R^{[B]}_{\alpha\beta}-\frac{1}{2}g^{[B]}_{\alpha\beta}R^{[B]}=-\left(<R^{[2]}_{\alpha\beta}(h^{[1]})>-\frac{1}{2}g^{[B]}_{\alpha\beta}<R^{[2]}(h^{[1]})>\right)\,.$$

(62)

Evaluation of $T^{[GW]}_{\alpha\beta}$ leads to a simple form when $h^{[1]}_{\alpha\beta}$ satisfies the Lorentz and TT gauge conditions

$$T^{[GW]}_{\alpha\beta}=\frac{1}{32\pi}<(\partial_{\alpha}h_{\mu\nu})(\partial_{\beta}h^{\mu\nu})>\,.$$

(63)

In the case of GWs moving radially outwards from a source which is far away, we can write

$$T^{[GW]}_{00}=-T^{[GW]}_{01}=T^{[GW]}_{11}=\frac{1}{32\pi}<(\partial_{t}h_{\mu\nu})(\partial_{t}h^{\mu\nu})>=\frac{1}{16\pi}<(\partial_{t}h_{+})^{2}+(\partial_{t}h_{\times})^{2}>\,.$$

(64)

The stress-energy tensor $T^{[GW]}_{\alpha\beta}$ can be written in the form $\rho k_{\alpha}k_{\beta}$ with $k_{\alpha}$ being the null vector $(-1,1,0,0)$, and $\rho$ being the right hand side of Eq. (64); thus $T^{[GW]}_{\alpha\beta}$ can be interpreted as being sourced by dust comprising massless particles moving radially outwards at the speed of light. The total power crossing the 2-surface $S$ at $r=$constant, $t=$ constant is then

$$L^{[GW]}=\int_{S}T^{[GW]01}r^{2}d\Omega=\frac{r^{2}}{16\pi}\int_{S}(\partial_{t}h_{+})^{2}+(\partial_{t}h_{\times})^{2}d\Omega\,,$$

(65)

with the averaging $<>$ removed since $h_{+},h_{\times}$ vary slowly in the angular directions. For example, in the case of a circular binary with $h_{+},h_{\times}$ given by Eq. (56),

$$L^{[GW]}=\frac{32}{5}\mu^{2}r_{0}^{4}\omega^{6}\,.$$

(66)