Linearized General Relativity and Gravitational Waves

Abstract

Gravitational waves are the analog of radio waves in electromagnetic theory. They were first predicted soon after the advent of general relativity theory, and after about a century of theoretical research and decades of experimental work they have been finally detected. In this chapter we develop the theory of the production, the propagation, and the detection of gravitational waves. Gravitational waves provide an entirely new observational window on the universe; the mergers of black holes and neutron stars are the sources of the waves so far observed.

Appendices

Appendix 1: Solutions for Retarded Potentials

Equations involving the d’Alembertian operator and a source, such as (11.18), are ubiquitous in physics. As such, anyone who has studied electromagnetism is familiar with them (Jackson 1999). This Appendix is essentially a short reminder of the solutions and their meaning. We consider an equation that relates a field $\psi$ via the d’Alembertian operator to some source $f$ according to

$$ \square^{2} \psi \left( {\vec{x},t} \right) = 4\pi f\left( {\vec{x},t} \right),\quad \square^{2} \equiv \eta^{\alpha \beta } \frac{\partial }{{\partial x^{\alpha } }}\frac{\partial }{{\partial x^{\beta } }} = \frac{{\partial^{2} }}{{\partial t^{2} }} - \nabla^{2} . $$

(11.78)

We will not give a rigorous derivation of the relevant solutions but instead a convincing heuristic discussion.

The time independent case is very familiar; it is the same as Coulomb’s law of electrostatics; for a unit point source at the origin, $\vec{x} = 0 \,{\text{and}}\, r = 0,$ the solution is

$$ \psi \left( {\vec{x}} \right) = - \frac{1}{r},\quad r = \left| {\vec{x}} \right|. $$

(11.79)

For a localized distribution of the source $f$ we may superpose a continuum of such point solutions and get a more general solution

$$ \psi \left( {\vec{x}} \right) = - \int \frac{1}{r}f\left( {\vec{x}'} \right){\text{d}}^{3} x' ,\quad r = \left| {\vec{x} - \vec{x}^{'} } \right|. $$

(11.80)

Such superposition is a key element in linear theories that allows relative ease of solution.

For the time dependent case we proceed in a similar way. We first look for a solution for a point source that only exists for an instant, that is the source is a delta function in space and time

$$ f\left( {\vec{x},t} \right) = \delta \left( {t - t^{\prime}} \right)\delta^{3} \left( {\vec{x} - \vec{x}^{\prime} } \right) . $$

(11.81)

The solution of (11.78) for such a point source is called the Green’s function, and can be written

$$ G\left( {\vec{x},t:\vec{x}^{'} ,t^{\prime}} \right) = \frac{1}{r} \delta \left( {t - (t^{'} + r/c)} \right) ,\quad r = \left| {\vec{x} - \vec{x}^{'} } \right|. $$

(11.82)

Here the time $t$ is the time $t'$ at the source plus the travel time to the field point. Thus a virtual point source localized in spacetime produces the same field as in the static case (11.79) but at a later time due to propagation of the effect at velocity $c$. A superposition of such fields gives a general solution formed by an integral, analogous to what we did in the static case. That is

$$ \begin{aligned} \psi \left( {\vec{x},t} \right) & = \int \frac{1}{r}f\left( {\vec{x}^{{\prime }} ,t^{{\prime }} } \right)\delta \left( {t - (t^{{\prime }} + r/c)} \right)~{\text{d}}t\,{\text{d}}^{3} x^{{\prime }} \\ & = \int \frac{1}{r}f\left( {\vec{x}^{'} ,t_{{{\text{ret}}}} } \right){\text{d}}^{3} x^{{\prime }} ,\quad r = \left| {\vec{x} - \vec{x}^{{\prime }} } \right|,\quad t_{{{\text{ret}}}} = ~t - r/c. \\ \end{aligned} $$

(11.83)

The quantity $t_{\text{ret}}$ is called the retarded time for obvious reasons, and the above solution is called the retarded solution.

We actually have the option of choosing the opposite sign in the last equation to give an advanced time $t_{\text{adv}} = t + r/c$ and an advanced solution. It appears however that nature has chosen the retarded time: this is called causality and is usually taken as a general principle of physics.

There are two further simplifications one may often make for many sources, including masses radiating gravitational waves and charges radiating electromagnetic waves. First, if the size of the source $L$ is much smaller than the distance $r$ then the factor $1/r$ may be removed from the integral. Second, if the time delay for light traveling across the source, $L/c$, is negligible compared to the characteristic time of change for the source, call it 1/ $\omega_{\text{ch}}$, then the waves from each source element are in phase and the integral may be done for a single time. Then (11.83) is

$$ \psi \left( {\vec{x},t} \right) = \frac{1}{r}\int f\left( {\vec{x}^{'} ,t_{\text{ret}} } \right){\text{d}}^{3} x',\quad r = \left| {\vec{x}} \right|,\quad t_{\text{ret}} = t - r/c,\quad L\omega _{{{\text{ch}}}} \ll c. $$

(11.84)

This may be called the small source approximation.

In this Appendix we have given the Green’s function or point source solution (11.82) without a rigorous derivation. Instead we chose to show the result intuitively but convincingly. For the interested reader a derivation can be obtained in a straight-forward way using integrals in the complex plane, as is done in many texts on electricity and magnetism (Jackson 1999).

Appendix 2: Electromagnetic Plane Waves

We provide here a brief sketch of the solution of Maxwell’s equations for plane electromagnetic waves to demonstrate how similar the mathematics is to the gravitational wave mathematics in Sect. 11.3. The theory of electromagnetic waves can be nicely expressed in terms of the 4-vector potential $A_{\mu }$. It obeys the wave equation and a Lorentz gauge condition, by choice,

$${{A_{\mu}}^{,\lambda}}_{,\lambda} = \square^{2} A_{\mu } = 0, $$

(11.85a)

$$ {A^{\nu }}_{,\nu } = 0. $$

(11.85b)

For plane waves the solution of the wave equation may be expressed as an arbitrary smooth function $\ell \left( U \right)$; here the wave vector is denoted as $k_{\beta }$ and the quantity $U = k_{\beta } x^{\beta }$. This is easily shown by substitution, and holds for a null wave vector,

$$ A_{\mu } = \epsilon_{\mu } \text{ }\ell \left( {k_{\beta } x^{\beta } } \right),\quad k_{\beta } k^{\beta } = 0. $$

(11.86)

It remains to determine the polarization vector $\epsilon_{\nu }$, which must be consistent with the Lorentz condition (11.82). With the solution in (11.86) this condition is

$$ \epsilon^{\beta } k_{\beta } = 0. $$

(11.87)

If we align the z axis along the space part of the null vector we may take $U = ct - z$

or some constant multiple as in Sect. 11.3. We may choose the polarization to be along either the $x$ or $y$ direction and thus obtain the solution in terms of either components or unit vectors as

$$ A_{\nu } = \left( {0, A_{1} \left( U \right), A_{2} \left( U \right), 0 } \right) = A_{1} \left( U \right)\hat{\varvec{e}}_{1} + A_{2} \left( U \right)\hat{\varvec{e}}_{2} . $$

(11.88)

The picture we thereby obtain is that the polarization vector has no time component and points along either the $x$ or $y$ direction, perpendicular to the propagation direction z of the wave. The wave is therefore called transverse.

A transformation to what we will call a tilde gauge is defined in terms of a scalar function $\varphi$ as

$$ \widetilde{A}_{\nu } = A_{\nu } + \varphi_{,\nu } . $$

(11.89)

This does not change the antisymmetric Maxwell electromagnetic field tensor, which is related to the vector potential by

$$ F_{\mu \nu } = A_{\mu ,\nu } - A_{\nu ,\mu } . $$

(11.90)

Moreover, such a gauge change can take us between various choices of the polarization.

By using a gauge transformation we can put any solution of the equations  (11.85) into the transverse form (11.88). In terms of $U = k_{\beta } x^{\beta }$ the solution and Lorentz gauge condition are

$$ A^{\mu } = A^{\mu } \left( U \right),\quad {A^{\nu }}_{ ,\nu } = \frac{{\partial A^{\nu } }}{{\partial x^{\nu } }} = \frac{\partial U}{{\partial x^{\nu } }}\frac{{\partial A^{\nu } }}{\partial U} = {A^{\nu} }_{ ,U} k_{\nu } = 0. $$

(11.91)

Here the notation ${,}U$ denotes differentiation with respect to the argument $U$. For the gauge function $\varphi$ we choose some function of $U$ to be determined; because of (11.89) the function $\varphi$ must be a solution of the wave equation,

$$ \square^{2} \varphi = {\varphi_{,\lambda}}^{,\lambda } = 0. $$

(11.92)

In the tilde system the vector potential and its divergence are then

$$ \tilde{A}_{\nu } = A_{\nu } + \varphi_{,\nu } ,\quad {{\tilde{A}^{ ,\nu }}}_{\ \ ,\nu } = {A^{ \nu }}_{ ,\nu } + {\varphi^{ ,\nu }}_{ ,\nu } = {A^{ \nu }}_{ ,\nu } = 0. $$

(11.93)

Thus if we begin in a Lorentz gauge and make a transformation with a solution of the wave equation we remain in the Lorentz gauge; this is convenient and elegant.

Most importantly we can choose $\varphi$ so that in the tilde gauge $\tilde{A}^{0} = \tilde{A}^{3} = 0$. For simplicity we align the vector $k_{\nu }$ along the time and z axis, $k_{\nu } = \left( {1, 0, 0, - 1} \right),$ so that $U = ct - z$. Then the Lorentz gauge condition is

$$ {A^{\nu }}_{ ,U} k_{\nu } = {A^{0}}_{ ,U} - {A^{3}}_{ ,U} = 0,\quad A^{0} = A^{3} ,\quad A_{0} = - A_{3} . $$

(11.94)

Here the last step follows from integration, since all components are functions of only $U$, and any constant would be irrelevant. Exactly the same Formula holds in the tilde gauge since the Lorentz condition holds there also. Finally, in the tilde gauge we may then force the 0 and 3 components to be zero according to (11.93) by choosing

$$ \begin{aligned} & \tilde{A}_{0} = A_{0} + \varphi_{,0} = 0,\quad \varphi_{,0} = - A_{0} ,\quad \varphi_{,U} = - A_{0} \\ & \tilde{A}_{3} = A_{3} + \varphi_{,3} = 0,\quad \varphi_{,3} = - A_{3} ,\quad \varphi_{,U} = A_{3} . \\ \end{aligned} $$

(11.95)

The two expressions for the $U$ derivative of $\varphi$ are same according to (11.94). Thus we can integrate to give $\varphi$ as a function of $U$, with an irrelevant constant. Thereby the vector potential has only 1,2 components in the tilde system. Note also that the 1,2 components of the vector potential are not changed by the gauge transformation.

From the discussion of gravitational waves in the text and the above comments on electromagnetic waves the following mathematical analogies are apparent:

$$ A_{\mu } \leftrightarrow h_{\alpha \beta } \quad {\text{vector}}\,{\text{potential,}}\,{\text{metric}}\,{\text{perturbation}} $$

(11.96)

$$ \epsilon_{\mu } \leftrightarrow \epsilon_{\alpha \beta } \quad {\text{polarization}}\,{\text{vector}}, {\text{metric}}\,{\text{polarization}} $$

$$ \begin{aligned} A^{\prime}_{\nu } & = A_{\nu } + \varphi _{{,\nu }} \quad \leftrightarrow \quad h_{{\alpha \beta }}^{\prime } = h_{{\alpha \beta }} + (f_{{\alpha ,\beta }} + f_{{\beta ,\alpha }} ) \\ & {\text{gauge}}\,{\text{transformation,~}}\,{\text{small}}\,{\text{coordinate}}\,{\text{transformation}} \\ \end{aligned} $$

$$ F_{\mu \nu } \leftrightarrow R_{\alpha \beta \gamma \delta } \quad \text{physical electromagnetic, gravitational tidal fields} $$

These analogies are rather elegant and simple. However this does not mean that electromagnetism and gravity are in any sense the same thing with a few indices altered. Einstein spent many of his later years trying to establish a deep physical connection between gravity and electromagnetism and did not succeed.

Appendix 3: Electromagnetic Wave Sources

As in Appendix 1 we give a brief sketch of the solution of Maxwell’s equations with a source to demonstrate the similarity to the gravitational wave mathematics in Sect. 11.5. In terms of the 4-vector potential $A_{\mu }$ the equations to be solved are

$$ {{A_{ \mu}}^{ ,\lambda }}_{, \lambda} = \square^{2} A_{\mu } = J_{\mu } , $$

(11.97a)

$$ {A^{\nu} }_{ ,\nu } = 0 . $$

(11.97b)

Notice that these are consistent with the conservation of charge relation ${J^{\mu }}_{ ,\mu } = 0$. As we discuss in Appendix 1 the retarded solution is

$$ A\left( {\vec{x},t} \right)^{\mu } = - \left( {\frac{1}{4\pi }} \right)\int \frac{1}{r}J\left( {\vec{x}^{'} ,t_{ret} } \right)^{\mu } {\text{d}}^{3} x^{\prime}. $$

(11.98)

Far from the source we expect such a wave to approach a plane wave, and we know from our discussion of plane waves in Appendix 2 that for such a wave there is a gauge in which the 0 and 3 components of the field vanish, so we may focus on the 1,2 components of the field. Moreover, if we assume the small source approximation we may remove the $1/r$ factor from the integral and evaluate the integral at a single retarded time to obtain

$$\begin{aligned} & A\left( {\vec{x},t} \right)^{k} = - \left( {\frac{1}{4\pi }} \right)\frac{1}{r}\int J\left( {\vec{x}^{'} ,t_{\text{ret}} } \right)^{k} {\text{d}}^{3} x^{\prime},\\ & r = \left| {\vec{x}} \right|,\quad t_{\text{ret}} = t - r/c,\quad L\omega_{\text{ch}} \ll c . \\ \end{aligned}$$

(11.99)

For a “sanity check” note that the zeroth component of this expression is Coulomb’s law involving the total charge. Thus the problem reduces to finding integrals over the space components of the current.

The integral on the right side of the solution (11.99) may be simplified with the use of the conservation of current relation, which states that the divergence of the current is zero. That is

$$ {J^{\mu }}_{ ,\mu } = 0,\quad {J^{0}}_{ ,0} = - {J^{i}}_{ ,i} . $$

(11.100)

From this we see that

$$ \frac{\partial }{\partial t}\int J^{0} x^{{{\prime }k}} {\text{d}}^{3} x^{{\prime }} = \int {J^{0}}_{ ,0} x^{{{\prime }k}} {\text{d}}^{3} x^{{\prime }} = - \int {J^{i}}_{ ,i} x^{{{\prime }k}} {\text{d}}^{3} x^{{\prime }} = \int J^{k} {\text{d}}^{3} x^{{\prime }} . $$

(11.101)

Thus the integral reduces to the time derivative of an integral involving the charge density $J^{0}$ that we will call a dipole integral.

$$ A\left( {\vec{x},t} \right)^{k} = - \left( {\frac{1}{4\pi }} \right)\frac{1}{r}\frac{\partial }{\partial t}\left( {\int J^{0} x^{{{\prime }k}} {\text{d}}^{3} x^{\prime}} \right). $$

(11.102)

This result lets us calculate some simple and interesting cases of radiation. For example consider a point charge $q$ oscillating along the $x$ axis with amplitude $L$ at frequency $\omega$. Then the charge density function is

$$ J^{0} = q\delta \left( {x^{{\prime }} - L\cos \omega t} \right)\delta \left( {y^{{\prime }} } \right)\delta \left( {z^{{\prime }} } \right) , $$

(11.103)

and the field from (11.102) is

$$ A\left( {\vec{x},t} \right)^{1} = \left( {\frac{q}{4\pi }} \right)\frac{L\omega }{r}\sin \omega t. $$

(11.104)

This is a reasonable model for a short dipole antenna. Note that the corresponding electric field is the derivative of this vector potential and thus is proportional to the square of the frequency.

Exercises

• 11.1 Write out the approximate metric in (11.22) for a source which has monopole and quadrupole moments. What of a dipole moment? What of higher moments? Where might this equation be useful?

• 11.2 Do the two functions in the traceless transverse gauge solution (11.28) need to be related to each other? Can you imagine a source in which the elements of the metric have different time dependence?

• 11.3. Design a simple gravitational wave detector using springs and masses. Design one consisting of elastic rods. Equations (11.47) and (11.48) should be a help.

• 11.4 The current official definition of physical distance is that of light travel time. Consider then the line element (11.29). Light moves on a null line, ${\text{d}}s = 0,$ at constant physical velocity $c$, so in the $x$ direction it obeys $c{\text{d}}t = \left( {1 - h_{11} /2} \right){\text{d}}x.$ The definition of distance thus means the relation between coordinate distance and physical distance is ${\text{d}}\ell = c {\text{d}}t = \left( {1 - h_{11} /2} \right){\text{d}}x.$ This gives justification for the relation (11.43) for test bodies in a gravitational wave. Now use this to analyze the operation of an interferometer wave detector that is not of negligible size compared to the wavelength of the gravitational wave. This analysis is relevant for very large machines such as LISA.

• 11.5 We studied in Example 11.2 an orbiting pair of equal mass bodies in a plane perpendicular to the line toward earth; see Fig. 11.3. Work out the metric field if the line to earth is at an angle $\theta$ from the perpendicular, and thereby show that the $+$ polarized wave picks up a factor of $\left( {1 + { \cos }^{2} \theta } \right)/2$ and the $\times$ polarized wave picks up a factor of $\cos \theta$.

  Thus verify (11.67).

• 11.6 The amplitude solution (11.64) has a general order of magnitude form involving a characteristic velocity $v_{\text{ch}}$ and we may write it as

  $$ h\sim \left( {\frac{GM}{{c^{2} r}}} \right)\left( {\frac{{v_{\text{ch}}^{2} }}{{c^{2} }}} \right) = \left( {\frac{m}{r}} \right)\left( {\frac{{v_{\text{ch}} }}{c}} \right)^{2} . $$

  Can you show heuristically that this should be roughly true for a fairly general source? See also Exercise 11.16.

• 11.7 In the text we discussed as sources of gravitational waves orbiting black holes and neutron stars. Can you think of any other possibly interesting astronomical sources?

• 11.8 In Sect. 11.2 on Newtonian limits we neglected some velocity dependent effects in the geodesic motion of particles and also velocity dependent effects on the sources of gravity. Such effects are interesting, although usually very small in the real world. They are called gravitomagnetic effects or sometimes “frame dragging” effects; they have been observed in the orbits of satellites and on the precession of an orbiting gyroscope. Work out the effects for the field produced by a spinning body and on the motion of bodies; see Adler (2000).

• 11.9 Verify Kepler’s law expressed in (11.69) for the orbiting system of Example 11.3.

• 11.10 Verify that the wave metric may be written in terms of the chirp time as in (11.70). What is the approximate chirp time in seconds for a pair of orbiting solar mass black holes?

• 11.11. It is fairly straight-forward to derive the energy density in a gravitational wave using linearized general relativity, but a bit tedious and lengthy. We may take a shortcut and obtain the result heuristically using the analogy with electromagnetic waves and dimensional analysis. The energy density in an electric field is well-known by physics students to be proportional to $\vec{E}^{2}$ and for an electromagnetic wave it is thus proportional to $\left( {\dot{A}} \right)^{2}$. Use the analogy between electromagnetism and linearized general relativity discussed in Appendix 2 and (11.96) to see that the analogous expression for the gravitational wave should have the form

  $$ \rho \propto \left\langle {(\dot{h}_{{11}} )^{2} + (\dot{h}_{{12}} )^{2} } \right\rangle . $$

  The angle brackets in this expression indicate that the quantity is to be averaged over a wavelength or so. Think a bit about why the averaging should occur and see Schutz (2009).

• 11.12 Next use dimensional analysis to see that a factor of $c^{2} /G$ should be included in the expression for the energy density in the above exercise. The energy density thus becomes

  $$ \rho = \frac{1}{{16\pi }}\frac{{c^{2} }}{G}\left\langle {(\dot{h}_{{11}} )^{2} + (\dot{h}_{{12}} )^{2} } \right\rangle $$

  where the numerical factor $1/16\pi$ must be gotten from a more detailed analysis such as Schutz (2009).

• 11.13 Verify the expression (11.72) for the total energy of the orbiting system according to classical mechanics. Then use (11.70) and (11.71) to calculate the wave energy in a thin spherical shell of thickness ${\text{cdt}}$. Balance this energy with the energy the orbiting system must lose in a time ${\text{d}}t$ to verify (11.73).

• 11.14 Verify (11.74) which gives the frequency of the waves as a function of time.

• 11.15 Take the size of the orbiting bodies in Example 11.3 to be nonzero and calculate a more realistic coalescence time than given by (11.74). Also calculate the maximum frequency due to the finite size.

• 11.16 We noted in the text that the production and detection of gravitational waves in a laboratory is not likely in the foreseeable future. Use the order of magnitude relation in Exercise 11.6 to show this; you need only estimate the maximum mass and velocity one might hope to achieve in a terrestrial laboratory setting.