Chapter VII: Tidal response and near-horizon boundary conditions for spin-
7.5 Waveforms and quasi-normal modes of the ECO
Energy Contents of Incoming and Reflected Waves
The reflection coefficient we defined in last subsection is indeed the (square root of) power reflectivity of the gravitational waves on the ECO boundary. To see this, consider a solution tos =−2 Teukolsky equation near the ECO surface. The energy flux down to the surface is given by [187]
dEhole dω =X
`m
ω
64πk(k2+42)(2rH)3|Y`holemω|2, (7.127) while the energy propagating outward from the surface is given by
dErefl dω =X
`m
ω
4πk(k2+42)(2rH)3|Z`mωrefl |2. (7.128) Hereωare all taken as real numbers. See Appendix 7.7 for detailed discussions on the energy flux and the energy conservation law. In the simple case of neglecting
`-`0mixing, incoming energy from the(`-m-ω)-mode will return from the(`mω)- mode, with
dErefl
dω
!
`mω = |R-ω+mΩH|2 dEhole
dω
!
`-m-ω
. (7.129)
This means our reflectivityR indeed acts as an energy reflectivity.
This satisfies the Teukolsky equation with the appropriate source term away from the horizon, the outgoing condition at infinity, but not the ECO boundary condition near the horizon. We will need to add an additional homogeneous solution, which satisfies the outgoing boundary condition at infinity. Recall that for the radial part, we have
R`mω+∞ =
Din`mω∆2e−ikr∗+ D`outmωeikr∗, r → b, r3eiωr∗, r → +∞. Thus we add the following homogeneous solution toΥ(0):
−2Υecho= X
`m
Z dω
2πc`mωR+`mω∞ −2S`mω(θ, φ)e−iωt, (7.132) so that−2Υ(0)+−2Υechois of the form (7.14), also satisfying (7.114). The asymptotic behavior of−2Υechois given by
−2Υecho =
X
`m
Z dω
2πc`mωr3e+iωr∗e−iωt−2S`mω(θ, φ), r∗ →+∞, X
`m
Z dω 2πc`mω
fD`mωin ∆2e−ikr∗+ Dout`mωeikr∗g
e−iωt−2S`mω(θ, φ), r∗ → b∗. (7.133) Here we already see that the amplitudesclmω directly give us theadditionalgrav- itational waves due to the reflecting surface. Identifying term by term between
−2Υ(0) +−2Υechoand Eq. (7.14), we find
Z`mωhole = Z`mωhole(0)+c`mωD`inmω, Z`mωrefl = c`mωD`mωout . (7.134) Applying Eq. (7.112), we obtain
c`mωDout`mω =X
`0
G``0mω f
Z`hole0-m(0)∗-ω +c`∗0-m-ωD`in0-m∗ -ω
g , c∗`-m-ωDout`-m∗-ω =X
`0
G``∗0-m-ω
fZ`hole0mω(0)+c`0mωDin`0mω
g . (7.135)
Here we restrict ourselves to real-valuedωonly. Using the symmetry of the Teukol- sky equation, for real-valuedω, it is straightforward to show that the homogeneous solutions have the symmetry that
Din`mω = D`in-m∗ -ω, D`outmω = Dout`-m∗-ω. (7.136)
We can then write
* ,
δ``0D`mωout −G``0mωD`inmω
−G``0mωDin`mω δ``0D`outmω + -
* ,
c`0mω
c`∗0-m-ω
+ -
= * ,
G``0mω 0 0 G``0mω +
-
* ,
Z`in0-m(0)∗-ω
Z`in0mω(0)
+ - , (7.137) G``0mω ≡ G``∗0-m-ω, (7.138) where the components in all matrices are also block matrices with` and`0repre- senting sections of rows and columns. This will allow us to solve forc`mω, therefore leading to the additional outgoing waves at infinity, i.e. the gravitational-wave echoes.
In the simple case where there is no `-`0 mixing for reflected waves (so that the relation between reflected waves and incoming waves is simply given by Eq. (7.114)), and that
Gˆ∗`-m-ω ≡ Gˆ`mω, (7.139) we can have simpler results. For each harmonic for the Z components (similar for the c components), we can define symmetric and anti-symmetric quadrature amplitudes
Z`holemω(0),S ≡ Z`mωhole(0)+ Z`hole-m-ω(0)∗
√
2 , (7.140)
Z`mωhole(0),A ≡ Z`mωhole(0)− Z`hole-m-ω(0)∗
√
2i . (7.141)
We then have
c`Smω = Gˆ`mω
D`outmω−Gˆ`mωDin`mωZ`mωhole(0),S, (7.142)
c`Amω = − Gˆ`mω
Dout`mω+Gˆ`mωD`mωin Z`holemω(0),A. (7.143)
Here we see that theAquadrature has a reflectivity of−Gˆ`mω, compared with ˆG`mω for the Squadrature. These quadratures correspond to electric- and magnetic-type perturbations.
As it turns out, non-spinning binaries, or those with spins aligned with the orbital angular momentum, only excite the S quadrature — although generically both quadratures are excited — they will have different echoes. In the case when echoes are well-separated in the time domain, the first, third, and other odd echoes, the A
andSwill have transfer functions negative to each other, while for even echoes, they will have the same transfer function.
If we further simplify the problem by demandingc`mω = c`∗-m-ω, Eq. (7.137) gives that
c`mω = Gˆ`mω
Dout`mω −Gˆ`mωD`mωin Z`mωhole(0). (7.144) This expression coincides, for instance, with the one obtained in [21] for the spherically-symmetric spacetime with a reflecting surface. Note that the phase factore−2ik b∗ has been absorbed into our definition of ˆG.
Echoes driven by symmetric source terms
In our reflection model (7.114), as discussed in Ref. [190], the coefficientsZ`hole-m∗-ω∗
and Z`mωhole are related for quasi-circular orbits. For such orbits, one can define a series of frequencies as
ωmk =mΩφ+kΩθ, (7.145)
whereΩφandΩθare two fundamental frequencies defined for periodic motions inφ andθ. Then, for real frequencies, we can decompose the amplitudeZ`mωin according to
Z`mωhole =X
k
Z`mkholeδ(ω−ωmk). (7.146) It is easy to check that for Kerr black holes,
Z`-m-khole∗ = (−1)`+kZ`mkhole. (7.147) That is, if we consider a specific circular orbit, we have the symmetry thatZ`hole-m∗-ω∗is either equal toZ`holemω, or they differ by a minus sign. In this simple case, our reflection model (7.114) does not involve different modes, and the model becomes similar to those reflection models based on Sasaki-Nakamura functions like in Ref. [170].
However, if we consider the full quasi-circular motions, i.e. adding up all orbits, this symmetry no longer exists, and one has to consider the mixing of modes when dealing with the reflecting boundary. For general orbits that are not quasi-circular, the symmetry between Z`hole-m∗-ω∗ andZ`mωhole may not exist.
Now for the symmetric source, where there is no mode mixing, let us consider a solution−2Υ(0)to the Teukolsky equation, which has the following form atr∗ → −∞:
−2Υ(0) = X
`m
Z dω
2π Z`mωhole(0)∆2e−ikr∗−2S`mω(θ, φ)e−iωt. (7.148)
Following the same steps as in the last subsection, it is straightforward to show that the echo solution to the Teukolsky equation at infinity is given by
−2Υecho=X
`m
Z dω
2π Z`mωecho−2S`mω(θ, φ)e−iωt, (7.149) with
Z`mωecho= Gˆ`mω
D`outmω−Gˆ`mωD`mωin Z`mωhole(0), (7.150) where we have chosen the normalization D`mω∞ = 1. The tidal reflectivity can also be directly related to the SN reflectivity as
R`mωSN = (−1)m+1 D`mω
4C`mωf`mωd`mωR∗-ω+mΩ
H . (7.151)
In this simple scenario, the tidal reflectivity is exactly the energy reflectivity for each mode.
Quasi-Normal Modes and Breakdown of Isospectrality
For Quasi-Normal Modes, we setZ to zero, and analytically continue Eq. (7.137) to complexω. The QNM frequencies can be directly solved by setting the determinant of the lhs matrix of Eq. (7.137) to zero, i.e.
det* ,
δ``0Dout`mω −G``0mωDin`mω
−G``∗0-m-ω∗D`mωin δ``0Dout`mω + -
= 0. (7.152)
This will in general cause a mixing between QNMs with different`, and break the isospectrality property of the Kerr spacetime and lead to two distinct QNMs for each(`,m).
Neglecting the`-`0mixing, we can simply write fDout`mωg2
= Gˆ`mωG¯ˆ`mω f
D`inmωg2
, G¯ˆ`mω ≡ Gˆ∗`-m-ω∗. (7.153) In the special case of ˆG`mω =G¯ˆ`mω (which is satisfied by all the reflectivity models discussed in this paper), we note that the ECO’s QNMs split into S and Amodes, withωn`mS andωn`mA satisfying different equations:
D`mωout
S −Gˆ`mωSD`mωin
S = 0, (7.154)
D`mωout
A+Gˆ`mωAD`inmω
A = 0. (7.155)
This still breaks the isospectrality properties of Kerr spacetime. Note that this property has also been found and studied in Ref. [184] with their echo model which describes the ECO as a dissipative fluid. Since modes of the ECO are usually excited collectively, the main signature of the breakdown of isospectrality is still the fact thatSand Aechoes have alternating sign differences in even and odd echoes.