LOG#237. GW music.

The spectrum of gravitational waves!!!!! Purely gravitational wave music!

The blog post today will cover two topics from elementary viewpoints: falling into a non-rotating black hole and gravitational wave “music”, i.e., gravitational wave formulae! It is a hard equilibrium just to fall into the BH singularity and to be spaghettified, but you will be awarded with gravitational wave physics at the second act! Happy???

Non-rotating black holes are called Schwarzschild black holes, or Schwarzschild-Tangherlini, since the latter generalized the black hole metric into extra space-like dimensions in 1963. I will keep mathematics as simple as possible, but that will introduce some imprecisions that, I wish, experts in the field will forgive.

A classical particle owns a lagrangian

$\mathcal{L}=-m\sqrt{-g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu}$

It implies that

$\dfrac{\partial L}{\partial \dot{t}}=-\dfrac{m^2 g_{tt}\cdot \dot{t}}{L}=\mbox{constant}=-E$

Thus

(1) $\begin{eqnarray*} \dot{t}=\dfrac{ L E}{m^2g_{tt}}\\ \dfrac{ds^2}{d\tau^2}=-1\\ L=-m\rightarrow \dot{t}=-\dfrac{E^2}{mg_{tt}}\end{eqnarray*}$

and then

$g_{tt}\dot{t}^2+g_{rr}\dot{r}^2=-1\rightarrow \dot{r}^2=-\left(1-\dfrac{2GM}{r}\right)+\dfrac{E^2}{m^2}$

Firstly, let us consider a test particle at rest from an infinite distance from the black hole event horizon, with $E=m$ , and thus set up

$\dot{r}^2=\dfrac{2GM}{r}$

Note that the above result is essentially the classical escape velocity. Secondly, make a trip towards the singularity, starting from $r=r_0$ (at $\tau=0$ ). Supposing its distance from $\tau=0$ is $r$ you get

$\dfrac{dr}{d\tau}=-\sqrt{\dfrac{2GM}{r}}$

and now proceed to integrate the above equation with the proper above mentioned limits

$\int_{r_0}^{R}r^{1/2}dr=-\sqrt{2GM}\int_0^\tau d\tau$

Remark: until the singularity from the boundary event horizon radius, you get a distance $r=R_S$ , until the initial point, you have a distance $r=r_0$ , until the final point you have $r=R$ from the initial point but $r=r_0-R$ from the singularity at the center.

After integration, and reintroducing $c$ , you will obtain that

(2) $\begin{equation*}\tau_{BH}^f=\dfrac{2}{3R_S^{1/2}c}\left(r_0^{3/2}-R^{3/2}\right)\end{equation*}$

If $r_0=R_S$ and $R=0$ for the time we reach the singularity, then

$\tau=\dfrac{2R_S}{3c}=\dfrac{4GM}{3c^3}$

This calculation can be performed for a $D$ -dimensional black hole with $d$ space-like dimensions. If you define the higher dimensional version of the Schwarzschild radius $r_s=r_S(D)$ the analogue integration is about

(3) $\begin{equation*}\tau_{BH}^f=\dfrac{(D-2)}{(D-1)R_S^{(D-3)/2}c}\left(r_0^{(D-1)/2}-R^{(D-1)/2}\right)\end{equation*}$

and similarly

$\tau(XD)=\dfrac{(D-2)r_S(D)}{(D-1)c}$

A variation of this time (but you will be likely death before the singularity arrives), can be done if you use the same geodesic equation above but with initial $E=0$ (or a big test mass so $E<<m$ ), then the integral is lightly different and you have to be careful to evaluate it. I will calculate it only in the usual $D=4$ spacetime to compare it with the previous result:

$\tau_f=\dfrac{1}{c}\int_0^{R_s}\dfrac{1}{\sqrt{\dfrac{2GM}{r}-1}}dr$

The result is

$\boxed{\tau_f=\dfrac{\pi}{2}\left(\dfrac{R_s}{c}\right)=\dfrac{\pi GM}{c^3}}$

The difference is not big, since $\tau_f=3\pi\tau/4$ , so we can be sure that we will be pushed into the singularity in about that order of time. Of course, the caveats are the life of the passenger and that quantum gravity should be taken into account at some point in the interior of the black hole. Maybe even before, according to the firewall paradigm.

What is next? Small problem: what is the density for a spherical exoplanet, moon or compact object to stay weightlessness in the equator surface? For 3d space, we can equalize

$F_g=F_c\leftrightarrow \dfrac{GMm}{r^2}=m\dfrac{v^2}{r}$

and plugging $v=2\pi r/T$ and $V=4\pi r^3/3$ , $\rho=M/V$ , then we will obtain

$\dfrac{4\pi G r^3\rho}{3r^2}=\dfrac{4\pi^2 r^2}{rT^2}$

and thus

$\boxed{\rho=\dfrac{3\pi}{GT^2}=\dfrac{3\pi f^2}{G}}$

$\boxed{f=\dfrac{\omega}{2\pi}=\sqrt{\dfrac{G\rho}{3\pi}}}$

By the other hand, our Universe is mysterious. Just like when we have different types of gravitational lensing (strong lensing, weak lensing and microlensing),via e.g. a simple equation

$\theta_E=\dfrac{4GM}{rc^2}=\dfrac{2R_S}{r}=\dfrac{D_S}{r}$

the study of the gravitational waves just began. Perhaps, we will have new tools to test even the black hole entropy of our almost de Sitter Universe. The dS Universe entropy is given by

$S_{dS}=\dfrac{A_{dS}}{4L_p^2}=\dfrac{\pi}{L_p^2H_\Lambda^2}$

For Keplerian orbits, the same argument than the previous one will help us to find the gravitational wave frequencies. The orbital frequency for quasi-circular keplerian orbits will be

$f_K=\dfrac{1}{2\pi}\sqrt{\dfrac{GM}{r^3}}$

The gravitational wave frequency in Einstein theory is twice the orbital frequency, i.e., $f_{GW}=2f_K$ , and then

$\boxed{f_{GW}=2f_{K}=\dfrac{1}{\pi}\sqrt{\dfrac{GM}{r^3}}}$

or, in terms of density,

$\boxed{f_{GW}=2f_{K}=\dfrac{1}{\pi}\sqrt{\dfrac{4\pi G\rho}{3}}\sim\sqrt{G\rho}}$

We want to compute this quantity from some reference scale, e.g., solar mass scale. Then, since $R_s=2GM/c^2$ and $M_\odot$ are the Schwarzschild radius and the solar mass respectively, by ratios we can calculate

$f_ {GW}=\dfrac{\sqrt{G}}{\pi}\left[\left(\dfrac{M}{R^3}\right)\left(\dfrac{M_\odot}{M_\odot}\right)\left(\dfrac{R_S^3}{R_S^3}\right)\right]^{1/2}$

$f_ {GW}=\dfrac{c^3}{2\sqrt{2}\pi GM_\odot}\left(\dfrac{M_\odot}{M}\right)^{\frac{3}{2}}\left(\dfrac{M}{M_\odot}\right)^{\frac{1}{2}}\left(\dfrac{R_S}{R}\right)^{\frac{3}{2}}$

and then

$\boxed{f_ {GW}=\dfrac{c^3}{2\sqrt{2}\pi GM_\odot}\left(\dfrac{M_\odot}{M}\right)\left(\dfrac{R_S}{R}\right)^{\frac{3}{2}}}$

Therefore, the dominant gravitational wave frequency for quasicircular keplerian orbits reads off as

$\boxed{f_{GW}\approx 2.29\cdot 10^4\left(\dfrac{M_\odot}{M}\right)\left(\dfrac{R_S}{R}\right)^{\frac{3}{2}}\mbox{Hz}\simeq 23kHz\left(\dfrac{M_\odot}{M}\right)\left(\dfrac{R_S}{R}\right)^{\frac{3}{2}}}$

For periods

$\boxed{T_{GW}=\dfrac{1}{f_{GW}}=43.7\mu s\left(\dfrac{M_\odot}{M}\right)\left(\dfrac{R_S}{R}\right)^{\frac{3}{2}}}$

In the end, unless quantum gravity changes the rules, any gravitational bounded system will decay gravitationally! The time for the gravitational coalescence of two orbiting bodies can also be computed from General Relativity for any (even eccentric) orbit. If initially the two bodies, with masses $M_1, M_2$ have a separation $a$ and eccentricity $e$ , then the time till coalescence will be

$t_{GW}^c=\dfrac{5}{256}\dfrac{c^5a^4f(e)}{G^3 M_1M_2(M_1+M_2)}$

withe the eccentricity function

$f(e)=\dfrac{(1-e^2)^{7/2}}{1+\frac{73}{24}e^2+\frac{37}{96}e^4}$

Since $a$ is the major semi-axis, the apoastron and periastron centers will be $R=a(1+e)$ , $r_p=a(1-e)$ . Also, define the chirp mass

$\mathcal{M}_c=\dfrac{(M_1M_2)^{3/5}}{(M_1+M_2)^{1/5}}$

Then, the time to coalescence is

$\boxed{t_{GW,c}\approx 10^5Gyr\left(\dfrac{1}{AU}\right)^4\left(\dfrac{10M_\odot}{M_1}\right)\left(\dfrac{10M_\odot}{M_2}\right)\left(\dfrac{20M_\odot}{(M_1+M_2)}\right)\left(1-e^2\right)^{7/2}}$

while the peak of the gravitational wave frequency (music) will be

$\boxed{f_{GW,peak}=0.35mHz\left(\dfrac{M_{BHB}}{30M_\odot}\right)^{1/2}\left(\dfrac{a}{0.01AU}\right)^{-3/2}\dfrac{(1+e)^{1.1954}}{(1-e^2)^{1.5}}}$

See, e.g., Wen 2003, or Antonini et al. 2014. to check the above formulae. By the other hand, merger anatomy is generally complex and hard with current tools. There are phases: inspiral, plunge, merger and coalescence. The final phase is “balding hair” of the black hole (in the case of binary black hole mergers). After plunge and merger phase end, there is also a new phase called “ring-down”. The final product is a Kerr black hole in general. At least from the current knowledge of gravity, general relativity and black hole theory. Kerr black hole is perturbed and the so-called quasinormal modes are emitted. For quadruple QNM (quasinormal mode), from the paper PRD 34, 384 (1986), you can get the values

$\dfrac{\omega_{QNM}}{2\pi}=f_{QNM}=32kHz\left(\dfrac{M_\odot}{M}\right)\left(1-0.63(1-j)^{0.3}\right)$

and

$\tau_{QNM}=20\mu s\left(\dfrac{M}{M_\odot}\right)\dfrac{(1-j)^{-0.45}}{1-0.63(1-j)^{0.3}}$

where $j=a/M_f$ is the Kerr parameter for the final state. In this field, there are also zoom-whirls. Zoom-whirl orbits are perturbations of unstable circular orbits that exist within the inner stable circular orbits (ISCO). The number of whirls $n$ is related to perturbation magnitude $\delta r$ and instability exponent $\gamma$ via $e^n\propto \vert\delta r\vert^{-\gamma}$ . Orbit taxonomy and classification of Kerr-like black hole orbits are also possible. It is generally defined a rational number $q=\omega+\nu/z$ , where $\omega$ is the number of whirls, and $z$ is the number of leaves that make up the zooms, while $\nu$ is the sequence in which the leaves are traced out ( $\nu/z<1$ ). Any non-closed orbit is arbitrarily close to some periodic orbit. If $a=J/M_fc$ , then $j=a/M_f=J/M_f^2c$ .

There are two more places where gravitational waves are important. Firstly, for the so-called Kozai-Lidov resonances in ternary systems. When a binary system is perturbed by a third body, the latter can induce some periodic oscillations in the libration of the orbital ellipse and a variation in the orbital eccentricity. The typical Kozai-Lidov oscillations can be calculated to have a period

$T_{KL}=\dfrac{2T_0^2}{3\pi T}\left(1-e_0^2\right)^{3/2}\left(\dfrac{m_1+m_2+m_0}{m_0}\right)$

where $T$ is the period of the triplet’s inner orbit (keplerian), $T_0$ is the period of the triple’s outer orbit (keplerian).

Finally, we have the so-called stochastic gravitational wave background, a buzz in GW caused by non-resolved GW sources. This is different from the so-called gravitational or graviton relic background (the cosmic analogue of the cosmic microwave background). Just as the current CMB has a temperature $T_\gamma\sim 3K$ , and the cosmological neutrino background has a temperature $T_\nu\leq 2K$ , the relic graviton background is expected to be $T_g\leq 1K$ . For the stochastic background, however, we define the energy density

$\Omega_{GW}(f)=\dfrac{1}{\rho_c}\left(\dfrac{ d\rho_{GW}}{d\ln f}\right)$

and the gravitational wave cosmological stochastic density to be

$\rho=\dfrac{c^2}{32\pi G}\left<\dot{h}_{ab}\dot{h}^{ab}\right>$

Therefore, the strain spectrum and the strain scale are respectively

$\boxed{S(f)=\dfrac{3H_0^2}{10\pi^2}\dfrac{\Omega_{GW}(f)}{f^3}}$

$\boxed{h(f)=6.3\cdot 10^{-22}\sqrt{\Omega_{GW}(f)}\left(\dfrac{100Hz}{f}\right)^{3/2} Hz^{-1/2}}$

Where do all of those equations come from? Consider a small perturbation of the spacetime metric

$\delta \eta_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$

The Einstein’s Field Equations read off

$R_{\mu\nu}+\dfrac{1}{2}g_{\mu\nu}=\dfrac{8 \pi G}{c^4}T_{\mu\nu}$

In vacuum $T_{\mu\nu}=0$ implies that $\square^2\Psi\equiv \square\Psi$ can be used to compute

$\partial_\nu h^\nu_\mu(x)-\dfrac{1}{2}\partial_\mu h^\nu_\nu=0$

and with

$\square=\dfrac{1}{c^2}\dfrac{\partial^2}{\partial t^2}-\nabla^2$

it yields

$\square \overline{h}_{\mu\nu}=0$

and plus sources

$\square \overline{h}_{\mu\nu}=-\dfrac{16 \pi G }{c^4}T_{\mu\nu}$

The simplest solution to a wave-equation is the plane wave

$h_{\mu\nu}(x)=\begin{pmatrix} h_+ & h_\times\\ h_\times & -h_+\end{pmatrix}\exp \left(i(kz-\omega t)\right)$

With many simplifications, the solution is rewritte as the variation of a quadrupole moment

$h^{ij}(t,x)\sim \dfrac{2G}{c^4 r}\dfrac{d^2}{dt^2}\left(I^{ij}(t-r/c)\right)$

There, the distance to the source is $r$ and the quadrupole moment is delayed or computed in the so-called retarded time. This is easier than the entropic uncertainty principle $H(x)+H(p)\geq \ln (e\pi)$ for sure. Gravitational waves from binaries, during the inspiral phase, is just elementary physics from these considerations. First, assum edge on observation ( $\theta=i=\pi/2$ ) such as $\cos\theta=04$ . Then, you can check that ( $t_r=r-r/c$ is the retarded time):

$h_+=h_+(t,\theta,\psi,r)=\dfrac{4G\mu}{c^4r}\omega^2_Ka^2\dfrac{1+\cos^2\theta}{2}\cos\left(2\omega_Kt_r+\psi\right)$

$h_\times=h_\times (t,\theta,\psi,r)=\dfrac{4G\mu \omega^2_Ka^2}{c^4r}\cos\theta\sin\left(2\omega_Kt_r+\psi\right)$

Now, define

$h=\sqrt{h_+^2+h_\times^2}=\dfrac{4G\mu \omega_K^2a^2}{c^4r}\sqrt{\dfrac{1+\cos^2\theta}{4}+\cos^2\theta}$

and then, with the previous conditions, deduce that

$h=\dfrac{4G\mu\omega_K^2a^2}{c^4r}$

where $\mu=m_1m_2/(m_1+m_2)$ , and with the aid of Kepler third law you get

$h=\dfrac{4G^2}{c^4}\left(\dfrac{\mu M}{ra}\right)=\dfrac{4G^2m_1m_2}{c^4ra}=\dfrac{4}{c^4 }\left(\mathcal{M}G\right)^{5/3}\left(\dfrac{\omega_{GW}}{2}\right)^{2/3}$

with $M=m_1+m_2$ . Now, resume the formulae

(4) $\begin{eqnarray*}\mathcal{M}_c=\dfrac{(m_1m_2)^{3/5}}{M^{1/5}}\\ M=m_1+m_2\\ \mu=\dfrac{m_1m_2}{M}\\ \eta=\dfrac{m_1m_1}{M^2}\\ T_K=\dfrac{2\pi}{\omega_K}=\left[\dfrac{4\pi^2a^3}{GM}\right]^{1/2}\\ a=\dfrac{GM^{1/3}}{\omega^{2/3}_K}\\ \omega_{ISCO}=\dfrac{2c^3}{6^{3/2}GM}\\ \omega_{GW}=2\omega_K\\ a_{\rm ISCO} = 3\times\frac{2G(m_1+m_2)}{c^2}\end{eqnarray*}$

Furthermore,

$h = \dfrac{4 G^{5/3}}{c^4} \dfrac{m_1m_2}{(m_1+m_2)^{1/3}}\omega_{ orb}^{2/3} = \dfrac{4 G^{5/3}}{c^4} m_{ chirp}^{5/3}\omega_{orb}^{2/3}$

$h\propto \left[\frac{m_{\rm chirp}}{{\rm M}_\odot}\right]^{5/3} \left[\frac{P_{\rm b}}{{\rm hours}}\right]^{-2/3} \left[\frac{r}{{\rm kpc}}\right]^{-1}$

Moreover, we can calculate the gravitational wave power radiated in the GW emission as

$\begin{aligned}P &= \frac{{\rm d}E_{\rm orb}}{{\rm d} t} = -\frac{{\rm d}}{{\rm d} t}\left[\frac{Gm_1m_2}{2a}\right] = \frac{Gm_1m_2}{2}\frac{1}{a^2}\frac{{\rm d}a}{{\rm d}t}=\\&=\frac{32}{5}\frac{G^4}{c^5}\frac{1}{a^5}(m_1m_2)^2(m_1+m_2)\quad\text{from the quadrupole formula}\\&=\frac{32}{5}\frac{c^5}{G}\left[\frac{Gm_{\rm chirp}\omega_{\rm GW}}{2c^3}\right]^{10/3}\end{aligned}$

and the time varying angular frequency will be

$\frac{{\rm d}\omega_{\rm GW}}{{\rm d}t} = \omega_{\rm GW}^{11/3}m_{\rm chirp}^{5/3}$

from which

$\omega_{\rm gw} = \left[\frac{64}{5\times2^{2/3}}\right]^{-3/8}\left[\frac{Gm_{\rm chirp}}{c^3}\right]^{-5/8}t_{\rm GW}^{-3/8}$

and for quasicircular orbits, you get the previously mentioned value

$t_{\rm GW}\sim\frac{5}{256}\frac{c^5}{G^3}\frac{a^4}{(m_1m_2)(m_1+m_2)}$

while the general formula in the literature is

$t_{\rm GW} = \frac{5}{256} \frac{c^5}{G^3}\frac{a^4(1-e^2)^{7/2}}{(m_1m_2)(m_1+m_2)}$

From Peter and Mathews, we also have

(5) $\begin{eqnarray*}\begin{aligned}\langle\frac{{\rm d} E}{{\rm d} t}\rangle &= -\frac{32}{5}\frac{G^4m_1^2m_2^2(m_1+m_2)}{c^5a^5(1-e^2)^{7/2}}\left(1+\frac{73}{24}e^2+\frac{37}{96}e^4\right)\\\langle\frac{{\rm d} L}{{\rm d} t}\rangle &= -\frac{32}{5}\frac{G^{7/2}m_1^2m_2^2(m_1+m_2)^{1/2}}{c^5a^{7/2}(1-e^2)^2}(1+\frac{7}{8}e^2)\label{eq:angularMomentumAverageEmissionRate}\end{aligned}\\ \begin{aligned}\langle\frac{{\rm d} a}{{\rm d} t}\rangle &= -\frac{64}{5}\frac{G^3m_1m_2(m_1+m_2)}{c^5a^3(1-e^2)^{7/2}}\left(1+\frac{73}{24}e^2+\frac{37}{96}e^4\right)\label{eq:smaAverageVariation}\\\langle\frac{{\rm d} e}{{\rm d} t}\rangle &= -\frac{304}{15}e\frac{G^3m_1m_2(m_1+m_2)^3}{c^5a^4(1-e^2)^{5/2}}(1+\frac{121}{304}e^2)\label{eq:eccAverageVariation}\end{aligned}\\ \langle\frac{{\rm d} a}{{\rm d} e}\rangle = \frac{12}{19}\frac{a}{e}\frac{\left[1+\frac{73}{24}e^2 + \frac{36}{96}e^4\right]}{(1-e^2)\left[1+\frac{121}{304}e^2\right]}\\ \langle\frac{{\rm d} a}{{\rm d} t}\rangle = -\frac{64}{5}\frac{G^3m_1^2m_2^2(m_1+m_2)}{c^5a^3} \end{eqnarray*}$

Integrating the semi-axis from $a_0$ to $a$ , you would get

$\label{eq:AOfTZeroEcc} a(t) = (a_0^4-4Ct)^{1/4}$

For the major semi-axis shrinking value we get in the literature being similar

$\label{eq:AOfTAllEcc}a(t) = \left(a_0^4-4C\frac{t}{(1-e^2)^{7/2}}\right)^{1/4}$

and the integration constant

$C=\dfrac{64}{5}\dfrac{G^3m_1^2m_2^2(m_1+m_2)}{c^5}$

The function of radial shrinking in terms of eccentricity reads

$a(e)=\dfrac{\omega e^{12/19}}{1-e^2}\left[1+\dfrac{121}{304}e^2\right]^{870/2299}$

The period decreases as

$\dot{T}=-\dfrac{192G^{5/3}}{5c^5}\left(m_1m_2\right)\left(m_1+m_2\right)^{-1/3}\left(\dfrac{T}{2}\right)^{-5/3}f(e)$

where

$f(e)=\dfrac{1}{(1-e^2)^{7/2}}\left[1+\dfrac{73}{24}e^2+\dfrac{37}{96}e^4\right]$

The power decrease is given by the formula

$\dot{P}=-\dfrac{192\pi}{5c^5}G^{5/3}(m_1+m_2)^{-1/3}m_1m_1\left(\dfrac{P}{2\pi}\right)^{-5/3}f(e)$

where $f(e)$ is, as before,

$f(e)=\dfrac{1}{(1-e^2)^{7/2}}\left(1+\dfrac{73}{24}e^2+\dfrac{37}{96}e^4\right)$

Maggiore’s book on gravitational waves define

$\tau_{c}=t_c-t_0=t_{GW}$

with

$f_{GW}=\dfrac{1}{\tau}\left[\dfrac{5}{256}\dfrac{1}{\tau}\right]^{3/8}\left[\dfrac{G\mathcal{M}_c}{c^3}\right]^{-5/8}=134Hz\left(\dfrac{1.21M_\odot}{\mathcal{M}_c}\right)^{5/8}\left(\dfrac{1s}{\tau}\right)^{3/8}$

$\tau_0=\dfrac{5c^5}{256G^3}\dfrac{a_0^4}{m_1m_2M}$

$\tau\sim 2.18s\left(\dfrac{1.21M_\odot}{\mathcal{M}_c}\right)^{5/3}\left(\dfrac{100Hz}{f_{GW}}\right)^{8/3}$

See you in other blog post!

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LOG#237. GW music.

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