Distortion of Gravitational-Wave Packets Due to their Self-Gravity
Lumdiff_self_k05_tau.tex
Distortion of Gravitational-Wave Packets Due to their Self-Gravity
Bence Kocsis1, 2, ∗ and Abraham Loeb1, †
1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
2Institute of Physics, Eötvös University, Pázmány P. s. 1/A, 1117 Budapest, Hungary
(Dated: October 25, 2018)
When a source emits a gravity-wave (GW) pulse over a short period of time, the leading edge of
the GW signal is redshifted more than the inner boundary of the pulse. The GW pulse is distorted
by the gravitational effect of the self-energy residing in between these shells. We illustrate this
distortion for GW pulses from the final plunge of BH binaries, leading to the evolution of the GW
profile as a function of the radial distance from the source. The distortion depends on the total GW
energy released ǫ and the duration of the emission τ , scaled by the total binary mass M . The effect
should be relevant in finite box simulations where the waveforms are extracted within a radius of
2M . For characteristic emission parameters at the final plunge between binary BHs of arbitrary
spins, this effect could distort the simulated GW templates for LIGO and LISA by a fraction of 10−3.
Accounting for the wave distortion would significantly decrease the waveform extraction errors in
numerical simulations.
PACS numbers:
I. INTRODUCTION
The observation of gravitational waves (GWs) is ex-
pected to open a new window on the universe within the
following decade. First generation GW detectors (In-
LIGO [76], VIRGO [77], TAMA [78], GEO [79]) are al-
ready operating at or close to their design sensitivity lev-
els and the development of the next advanced-sensitivity
GW detectors (Advanced LIGO [80], Advanced Virgo
[81], LCGT [82]) and the space-detector LISA [83] are
well underway. It is now increasingly important to fully
understand the precise characteristics of the GW wave-
forms that we expect to observe.
The most luminous GW sources are expected to be
associated with mergers of BH binaries. The physical
understanding of these sources has greatly improved by
recent breakthroughs in numerical relativity [1, 2, 3]. It
is now finally possible to simulate the merger of a BH bi-
nary, from the initial circular inspiral, through the plunge
to a common surrounding horizon, to the final ringdown,
as the remnant settles down to a quiescent stationary
Kerr-BH. It is widely believed now that existing simu-
lations are sufficiently precise to allow targeted searches
for these waveforms in real data [4]. In fact, it has been
recently shown that the errors are not even limited by
the numerical precision of the simulation (∼ 10−5), but
the GW extraction method itself entails a much larger
uncertainty (∼ 10−3) [5]. In this paper, we demonstrate
that the self-gravity during the propagation of gravita-
tional radiation in the zone of wave extraction of numer-
ical simuations leads to the distortion of the waves, cor-
responding to similar magnitude modifications in typical
cases.
∗Electronic address: bkocsis@cfa.harvard.edu
†Electronic address: aloeb@cfa.harvard.edu
A. Description of the effect
Let us imagine a compact spherically-symmetric con-
figuration of matter (representing the remnant) and a
rapidly expanding sphere of massless particles (represent-
ing the radiation) carrying away some of the initial mass
of the system (Fig. 1). First, let us assume Newtonian
gravity and spherical symmetry. In this case, the var-
ious shells are pulled back only by the gravity of the
mass interior as if it was concentrated to a point mass
at the center of the sphere, and the effect of the outer
enclosing shells exactly cancels out. Thus, the particles
on the outermost shell are always attracted by the total
mass, including the mass of the radiation, but the inner-
most shells experience only the gravity of the remnant.
Therefore, the gravity of the radiation implies that the
innermost shells of radiation will be continuously catch-
ing up to the outer boundary during their journey from
the source to the observer.
Do we expect an analogous effect to exist also for grav-
itational waves in full general relativity? First, let us
consider conventional (i.e. non-gravitational) radiation.
In analogy to the Newtonian gravitational pull, relativis-
tic test particles are slowed down by gravity: the null-
geodesics in a gravitational field experience the so-called
Shapiro delay [6], decreasing the radial coordinate ve-
locity with increasing gravity. Furthermore, according
to Birkhoff’s theorem, the spacetime outside a spheri-
cally symmetric distribution of energy is equivalent to
the spacetime of a point-mass placed at the center of
the sphere, the Schwarzschild metric, and the spacetime
inside a cavity is the free-space Minkowski spacetime.
More generally, the spherically symmetric expansion of
collisionless radiation is a known simple exact solution
of the Einstein equations, the Vaidya metric [7, 8]. This
solution has exactly the same characteristics as the New-
tonian example, whereby various shells react only to the
mass interior to them, i.e. they move on world lines ne-
http://arxiv.org/abs/0704.1149v4
mailto:bkocsis@cfa.harvard.edu
mailto:aloeb@cfa.harvard.edu
Mf ∆τ ′
FIG. 1: A sketch of the effect under consideration. The co-
alescence of two BHs in a binary of total initial mass M0
results in the emission of a burst of gravitational radiation
which carries away a non-negligible ǫ fraction of M0. The
remnant BH mass is Mf . The proper temporal width of the
wave-packet for a hypothetical observer fixed at a radial dis-
tance r is ∆τ . As the packet propagates outwards, it (1)
expands due to gravitational redshift of the initial mass M0
(solid lines), (2) contracts due to the mean self-gravity of the
radiation (dotted line, ∆τ ′), and also (3) distorts its profile
due to the self-gravity of the radiation (not shown). As a re-
sult the inner shells begin to catch up, and the proper time
separation in excess of gravitational redshift from the front
of the burst decreases with distance. Consequently, the net
luminosity of the radiation burst changes with distance.
glecting the exterior shells and the effect of the interior
shells is the same as if they were concentrated to a point
mass at the center.
Next, let us turn to the case of gravitational radiation.
The effect of the self-energy of gravitational radiation can
be accounted for in the first nonlinear-order approxima-
tion of the Einstein field equations by attaching terms
of order h2 to the stress-energy, Tij , considering these
terms as sources in addition to the regular radiation fields
[9, 10, 11, 12, 13, 14, 15]. Here hij is the wave amplitude
which is the correction to the background metric. If the
wavelength of the GW wave-packet is much smaller than
the size of the wave envelope, the evolution of the wave-
packet is determined by the WKB cycle-averaged effec-
tive stress-energy tensor [10, 11], independent of the spe-
cific wave-characteristics of the radiation. In this regime,
we may treat the GW packet as an ensemble of relativis-
tic particles for which our previous arguments apply. In
conclusion, we anticipate that
(i) the wave-envelope will continuously expand due to
the redshift of the initial mass of the binary,
(ii) it will contract due to the self-gravity of the radia-
tion, and
(iii) in analogy to electrodynamics, we expect that the
distortion of the wave envelope would lead to a
continuous adiabatic modification in the GW fre-
quency.
The purpose of this paper, is to quantify these expec-
tations for typical BH merger waveforms using simple
models and to demonstrate that this effect should be
accounted for in relation to numerical simulations and
observed merger waveforms.
B. Related literature
To our knowledge the effect of self-gravitational dis-
tortion of GWs had not been explicitly recognized previ-
ously. We elaborate on the relation of the self-distortion
effect to numerical general relativity, analytical investiga-
tions like the multipolar post-Minkowskian (MPM) and
post-Newtonian (PN) theory, and the studies of the scat-
tering of gravitational radiation in curved spacetimes.
The self-gravitational distortion of GWs is a relatively
small effect on short scales currently accessible to numer-
ical simulations. Current state-of-the-art simulations of
binary BH mergers are restricted to the central strong-
gravity domain near the black holes, and extract gravity
waves from the boundary of this domain. The standard
method of extracting and extrapolating the waveforms to
larger distances, is based on the Regge–Wheeler–Zerilli-
Moncrief perturbation formalism [16, 17, 18]. This is a
linear-order representation of the Einstein field equations
and so it neglects self-energy effects of order h2. Cumula-
tive nonlinear effects like the self-distortion effect should
lead to systematical deformations of the linear waveform
extracted at different radii, which can in principle be dis-
covered by a rigorous convergence test. In fact, nearly
all papers on simulated merger GWs study the conver-
gence behaviour in some detail. However, due to com-
putational limitations, the extraction radius is currently
restricted to r ∼
< 50M , and the extraction has been pre-
formed on only a few, typically 3–4 different radii with
the extrapolation done empirically based on these radii
[1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Recently,
Pazos et al. [5] reported a systematic effect of order 10−3
(for extraction radii r ≤ 80M), which is much larger than
the numerical precision of the simulation ∼ 10−5. This
is roughly the same order of magnitude systematic effect
that we expect for the GW distortion for the given radii
(see below). Note however, that Berti et al. [25] showed
that the convergence behavior is also largely sensitive to
simulation assumptions. On small scales probed by these
simulations, other near-field nonlinearities might also be
of equal importance.
A precise treatment of the waveforms at large radii
is possible by analytical methods, such as the MPM-
expansion introduced by Thorne [14] (see also [13]) and
developed further by Blanchet & Damour [28, 29, 30]. In
Ref. [14], Thorne introduced the concept of “local-wave
zone,” which is outside the dynamical zone of wave gener-
ation, but where nonlinear effects are still important. In
this region the propagation of gravity waves is expressed
in terms of an expansion in powers of hij as an infinite
sum of multipole contributions with rapidly decreasing
amplitude, which needs to be matched to the dynamical
gravitational field generated by the source. The GWs
in the local wave zone are given formally by the MPM
expansion, whose terms correspond to different powers
of the gravitational coupling constant G. In the PN ap-
proach, the dynamical wave generation is calculated an-
alytically in an infinite series in the inverse speed of light
c−2. Matching the PN and MPM expansions in the local
wave zone is a successful method for the calculation for
steady source GWs produced by relatively slow motions,
like the inspiral phase of BH mergers where the distance
between the BHs is large enough to allow an adiabatic
quasicircular orbit at r > 6M . To date, the PN wave-
forms for circular binary inspirals are available to 3.5PN
order for general mass ratios [31] (which is the highest
order that is expected to have a measurable contribution
for circular inspirals by a LISA-type detector [32]) and
5.5PN order for extreme mass ratios and no BH spins [33].
To our best knownledge, the self-gravitational distortion
effect has not been identified in these works. However,
in this paper we show that the radius for a fixed lumi-
nosity shift or a fixed frequency shift is linearly sensitive
to the energy density of the radiation. The luminosity at
the inspiral phase of binaries is less than 1% of the lu-
minosity at the final plunge [26]. Therefore, even if it is
negligible for inspirals, the self-gravitational modulation
of GWs could be significant for the final plunge.
We expect the self-gravitational distortion of GWs to
be consistent with the PN/MPM expansion, and to have
corresponding PN tail counterparts [34]. The tails of
GWs are caused by the scattering of linear waves on
the spacetime curvature generated by the total mass-
energy of the source [35], which is related to the Shapiro
time-delay [6] of the radiation crawling out of the back-
ground gravity of the source. At 2.5PN order beyond the
Newtonian quadrupole formula, GW tails scatter off the
monopole field of the remnant [36, 37, 38]. Furthermore,
above 3PN order, the tails of the tails are produced by
curvature scattering of the tails of the waves themselves,
associated with the cubic nonlinear interaction between
two mass monopole moments and the mass quadrupole of
the source [39]. In this paper we show that the modifica-
tion caused by the spherical self-gravitational distortion
of GWs is to lowest order proportional to the original
waveform (i.e. without this effect) times the energy den-
sity of the waveform. Since the energy density is pro-
portional to the square of the amplitude, this possibly
implies to lowest order a monopole-quadrupole2 type in-
teraction counterpart.
The self-gravitational distortion effect is also related to
the “memory effects” (or “hereditary effects”) of gravi-
tational radiation, since it is a cumulative effect that de-
pends on the full past history of the radiation, as opposed
to regular PN terms which only depend on the instanta-
neous retarded fields. Other known hereditary GW ef-
fects are the tails of GWs [30, 40], the Christodoulou
effect [41, 42], and the GW recoil kick [43].
Finally, nonlinear effects were also examined for the in-
teraction of plane GWs on a free space (i.e. Minkowski)
background [44, 45, 46, 47, 48, 49]. The nonlinear terms
in the scattering problem are found to exactly cancel up
to fourth order, leaving no self-phase-modulation effect
for GWs in vacuum to this order. However, the geome-
try of this case is very different from the one discussed in
this paper where a curved background is initially present.
It is also unclear whether there are self-phase modulation
effects at higher nonlinear orders. Fortunately, our ap-
proach does not face such convergence issues, since we
adopt the exact (i.e. non-perturbative) solution of the
Einstein equations in the spherical WKB approximation
of an expanding radiation shell.
The present paper aims to quantify the self-
gravitational distortion effect for recently compiled
merger waveforms. In § II, we list the main properties of
the waveforms relevant for our study. In § III we present
our analysis in spherical symmetry, and derive the results
for the self-gravitational distortion of signal duration and
the luminosity profile. In § IV we summarize the main
conclusions, and then discuss their implications. Finally,
we discuss the validity of the spherical approximation
and consider the possible effect of the anisotropy in the
Appendix. We use units with G = c = 1 and a metric
signature of (−1, 1, 1, 1).
II. MERGER WAVEFORMS
To illustrate our effect, we adopt a simplified treatment
for the merger GW waveforms. Table I lists the total ra-
diated mass ∆mtot relative to the initial total mass M0
for various encounters between compact objects found in
the literature. While all of these encounters would have a
nonnegligible GW self-gravitational effect, the inspiral–
merger–ringdown events are expected to have the most
prominent event rates for interferometric GW detectors.
Inspiral detection rate estimates are between 0.3−3 yr−1
and several per day for NS/BH mergers for inLIGO and
adLIGO, respectively [50]; 3 and 100 yr−1 for stellar
BH/BH inspirals in globular clusters [51] and in galac-
tic nuclei [52] with adLIGO, respectively; 1–100 yr−1 and
30 yr−1 for supermassive (SMBH) and intermediate mass
BH (IMBH) [53, 54, 55] and SMBH/SMBH inspirals
[56, 57] with LISA, respectively. The dynamic time of the
encounter is proportional to the total mass; consequently
we do not consider extreme mass ratio inspiral–mergers
(EMRI), as the GW luminosity of these sources is much
smaller. Although high velocity stellar BH encounters
can be very bright in GWs, these events are expected to
be rare, less than 1 yr−1 for adLIGO or LISA [58]. In
this analysis, we focus on equal mass inspiral–merger–
ringdown waveforms.
Generally, the waveforms can be expanded in multi-
TABLE I: Total radiated mass relative to the initial total
mass, ǫ, for bright GW encounters. For a detailed comparison
of BH inspiral computations see Ref. [59].
Objects Encountera Spins orien- Refs. ǫ
tationb [%]
NS–BH head-on (0.0,0.0) [60] 0.01
BH–BH head-on (0.0,0.0) [61] 0.05
BH–BH head-on (0.1,0.1) [61] 0.06
NS–BH orbiting (tidalc,0) [62, 63] 0.1
BH–BH head-on (0.2,0.2) [61] 0.12
BH–BH head-on (0.0,0.0) [64] 0.13
BH–BH whirl (0.0,0.0) [24] 0.5–3
BH–BH grazing (0.9,0.7) − ⊥ − [65] 0.9
BH–BH grazing (0.9,0.7) + ⊥ + [65] 1.0
BH–BH grazing (0.0,0.0) [65] 1.2
BH–BH inspiral (0.2,0.2) + ‖ + [20] 1.8
BH–BH inspiral (0.1,0.1) − ‖ − [20] 2.0
BH–BH inspiral (0.2,0.2) − ‖ − [20] 2.0
BH–BH inspiral (0.8,0.8) − ‖ − [22] 2.2
BH–BH inspiral (0.1,0.1) + ‖ + [20] 2.4
BH–BH inspiral (0.0,0.0) [20] 2.5
BH–BH inspiral (0.0,0.0) [3, 21] 3.2
BH–BH inspiral (0.0,0.0) [19, 22] 3.5
BH–BH inspiral (0.0,0.0) [25, 27] 3.7d
BH–BH inspiral (0.1,0.1) + ‖ + [26] 5.2
BH–BH inspiral (0.8,0.8) generale [23] 5–6
BH–BH inspiral (0.8,0.8) + ‖ + [22] 6.7
BH–BH r.whirl (0.0,0.0) [24] 15f
BH–BH r.head-on (0.0,0.0) [66] 16
BH–BH r.head-on (1.0,0.0) ⊥ [67] 17 η
BH–BH whirl (0.0,0.0) [24] 24h
BH–BH r.whirl (0.0,0.0) [24] 100h
aHere “head-on” stands for a direct collision with v ‖ r initially,
“orbiting” stands for the tidal stripping of a NS by a BH in close
orbit, “grazing” stands for inspiral collisions in which the initial
separation is within the final orbit in a merger, “inspiral” is the
complete inspiral–merger–ringdown event, “whirl” stands for parti-
cles approaching from infinity with some impact parameter leading
to a quasi-circular whirl-type orbit before merger, “r.whirl” and
“r.head-on” corresponds to relativistic initial velocities v ≈ 1 at
r ≫ M .
bHere we give the sign of S1 · J , the relationship between S1 and
S2, and the sign of S2 ·J , where J is the orbital angular momentum.
In case of a head-on collision with spins, we give the initial direction
of the spin relative to the separation vector.
cNS tidally locked
dRef [25] provides the results for different mass ratios, m1/m2 =
1–4, and found that ∆m ∝ η2, where η = m1m2/(m1 + m2)
e8 different choices of spin orientations
fIn case the impact parameter is small enough to end up in a
merger.
gCalculated for m1 ≪ m2. (See η above at
hIn case the impact parameters are fine tuned for the binary to
approach the unstable circular orbit.
poles [14]
hµν =
(1 + z)Anµν
eiφn (1)
where φ is the high frequency GW phase which is related
to the instantaneous frequency through f = dφ/dt; Aµν
is a slowly varying envelope describing how the instanta-
neous amplitude changes over the waveform, the index n
labels the various polarizations and multipoles, and dL is
the luminosity distance, and z is the redshift. For binary
inspiral merger waveforms, the (l = 2,m = 2) multipole
(i.e. quadrupole) dominates the waveform [25, 26, 27].
For the sake of simplicity, we restrict our attention to a
single monochromatic wave.
During a single GW cycle, corresponding to a time
interval f−1, the envelope Aµν can be regarded as con-
stant. In the WKB approximation, the energy carried by
the radiation can be calculated as a cycle-averaged quan-
tity. The effective stress-energy tensor of the radiation is
[11, 12]
T µν =
kµkν , (2)
where A2 = AµνAµν is the squared effective amplitude
and kµ = φ,µ is the wave number.
Each infinitesimal volume of the wave can be at-
tributed the mass-energy that it carries according to Eq.
(2). In the spherically symmetric approximation, the to-
tal luminosity (or more precisely the graviton number) is
= 4πr2T tr =
ktkr (3)
With this equation it is possible to obtain the waveform
[A(t), f(t)] within radius r from the luminosity function
L(t).
Based on the waveforms derived by numerical simula-
tions (such as Fig. 25 in Ref. [26]), we adopt the following
simple fit to the effective luminosity
L(t) = L0 ×
[(t1 − t)/t1]
−1.5 if − t0 < t < 0
1 if 0 < t < t1
exp[−(t− t1)/t2] if t > t1
where the intervals t < 0, 0 < t < t1 and t1 > t corre-
spond to the late inspiral/final orbits, the peak luminos-
ity at the plunge, and the ringdown phases, respectively,
−t0 sets the initial time of the calculated profile, t1 rep-
resents the characteristic timescale of the most intensive
part of the radiation, t2 sets the ringdown decay rate, and
L0 is the normalization luminosity. Numerical simula-
tions show [26] that the characteristic frequency of the ra-
diation rapidly increases after the inspiral and saturates
at ω = 2πf ∼ M−10 /2 at t ∼ 0, where M0 is the initial
mass of the source. The characteristic number of wave
cycles during the brightest phase is N = ∆t/f−1 ∼ 3
over a time ∆t = 12πM0. Note that we assume that
the WKB method is applicable to the waveform, imply-
ing that L(t) does not change greatly over a single cycle.
This condition is just marginally satisfied for these wave-
forms.
For a simple analysis, we assume that the total lu-
minosity crossing a sphere at infinity L∞(t) is given
by Eq. (4) with the following parameters: (t0, t1, t2) =
(100, 10, 5)M0, and use ∆t = 12πM0 in reference to av-
erage quantities below. Under the WKB approximation
the luminosity (4) evolves independently of the carrier
frequency f ∼ 1/(4πM0). The L0 normalization is set
using the total radiated mass of the system as a frac-
tion of M0 by ǫ = ∆mtot/M0 for the inspiral-mergers (3)
listed in Table I. For pedagogical comparison purposes
we distinguish the inspiral events only based on the nor-
malization L0, and do not consider the variations in the
shape of the waveform (e.g. ∆t).
III. QUANTITATIVE ESTIMATES IN
SPHERICAL SYMMETRY
To give a first quantitative estimate of the magnitude
of the wave-packet distortion due to self-energy, we start
by computing the propagation of unpolarized radiation
packets in the spherically symmetric Vaidya spacetime.
The possible effect of anisotropies is discussed in the Ap-
pendix.
The Vaidya metric [7] in radiation coordinates
(u, r, θ, φ) is
ds2 = −
2m(u)
du2 − 2drdu + r2dΩ, (5)
where dΩ = dθ2 + sin2 θdφ2. This metric is an exact
(i.e. nonperturbative) solution of Einstein’s equations in
spherical symmetry in the eikonal approximation to a ra-
dial flow of unpolarized radiation. Here u is the retarded
time parameter which is constant along the world lines
of radially outgoing radiation, and m(u) describes the
mass function interior to u. Outside the radiation (i.e.
where m(u) is constant in the space-time), the Vaidya
metric (5) is the Schwarzschild solution in Eddington-
Finkelstein coordinates [12]. For our approximate wave-
forms decribed in § II, m(u) is M0 constant outside, it
is quickly changing within a short range 0 ≤ u ≤ ∆utot
across, and it is Mf = M0 −∆mtot inside the radiation
shell. Here, ∆mtot and ∆utot are set by the simulated
waveforms § II.
For the metric given by Eq. (5) it is straightforward
to derive the convergence of null-geodesics describing the
world lines of radiation shells using Raychaudhuri’s equa-
tion or the equation of geodesic deviation [68]. In either
way, we find that radially outgoing shells of radiation sim-
ply follow the world lines u(r) = constant, implying that
the ∆u coordinate difference between the shells does not
change during the propagation. The physical contrac-
tion of radiation shells can be examined using the proper
time measure between shells and the observed luminosity
profile.
A. Proper time duration
First, we estimate the proper-time duration of the GW
signal along the world-line of a hypothetical observer
crossing the radiation shell. For simplicity, we restrict
to observers at a fixed spacial coordinate (r, θ, φ). Then
we have dr ≡ dθ ≡ dΩ ≡ 0 and Eq. (5) gives
dτ2 = −ds2 =
2m(u)
du2 (6)
leading to dτ =
1− 2m(u)/rdu. Therefore, du can
be interpreted as the infinitesimal proper time difference
between two radiation shells at fixed radius approaching
infinity. Thus, we adopt the notation du ≡ dτ∞, and
similarly ∆u ≡ ∆τ∞ for integrated quantities. Finally,
let us define mi = m(ui) for i ∈ {1, 2} and ∆m = |m2 −
m1|. We set (m1,m2,∆mtot) = (M0,Mf , ǫM0) when
referring to the total GW signal duration.
Integrating between two arbitrary shells of radiation
u2 and u1 gives
2m(u)
du. (7)
After expanding m(u) in a Taylor-series, the integrand
becomes
2u〈m′〉
u2〈m′′〉
+ . . .
. (8)
Substituting in Eq. (7), and using 〈m′〉 = −∆m/∆τ∞
and 〈m′′〉 = 0 to first order, we get
∆τ(m1,m2, r) =
3|m2 −m1|
. (9)
Setting m1 = M0 and expanding in terms of ∆m, we get
r − 2M0
+O(∆m2)
The leading order term can be indentified as the gravi-
tational redshift for constant mass M0, the second term
describes the correction due to the radiation mass. If we
expand also in terms of powers of 1/r, the relative change
in the proper time duration of the signal becomes
∆τ −∆τ∞
+O(r−2,∆m2). (11)
Equations (9–11) describe the self-gravitational distor-
tion between radiation shells in terms of proper time
along world lines of r = const. One can notice that to
leading order this is simply the gravitational redshift for
the average enclosed mass between the shells. However,
since ∆m changes along the wave packet non-trivally for
a fixed radius as a function of time, the modification of
the profile is generally not self-similar, leading to the dis-
tortion of the luminosity profile as a function of radius.
To make this point clearer, we correct for the average
distortion of the signal and calculate the residual distor-
tion of the signal. Let us define
∆τ ′ = (1 + z)∆τ −∆τ∞ (12)
where (1+z) is a time-independent constant representing
the “average gravitational redshift” at a given r, given by
1 + z =
, (13)
〈m〉 ≡ (m1 +m2)/2 is the average mass, and to leading
order
. (14)
Now let us take an arbitrary radiation shell enclos-
ing mass ∆m relative to the shell enclosing mass M0 −
0.5∆mtot, i.e. we set m1 = M0 − 0.5∆mtot and m2 =
m1−∆m in eq. (9). After correcting for the average grav-
itational redshift using Eqs. (12) and (13), the residual
relative distortion to leading order is
+O(r−2,∆m2). (15)
Figure 2 shows the residual distortion using the exact
formula (Eq. (12), thick lines) and the leading order con-
tribution (given by Eq. (15), dotted lines). At the typ-
ical radius used by numerical simulations for waveform
extraction, r = 50M⊙, the primary bulk waveform distor-
tion changes the signal duration by M0/r ∼ 2%, and the
secondary relative waveform distorsion between the front
and the back of the signal is ǫM0/(2r) ∼ 7 × 10
−4 for a
typical BH inspiral–merger with high spins ǫ = 7% (see
§ II). The figure also shows that the higher order effects
beyond 1/r lead to an uncertainty of order 10−3–10−4 for
r = (30–50)M0.
B. Luminosity Profile
Since Eqs. (9) and (11) are applicable to two arbitrary
shells of radiation, we can use them to compute the evo-
lution of an arbitrary initial radiation profile, whereas the
luminosity is simply L = ∆m/∆τ , the total mass-energy
crossing a sphere at radius r within proper time ∆τ .
The profile at infinity is given by m(u) in radiation
coordinates, or ∆τ∞(m), the proper time a shell enclos-
ing mass m arrives at r = R where R → ∞, relative to
the outermost shell of radiation [84]. Here ∆τ∞(m) can
10 100 1000
FIG. 2: The residual self-gravitational distortion of shells af-
ter correcting for the bulk gravitational redshift. The thick
curves show the change in the proper time duration of the
signal at radial distance r from the source between shells en-
closing 7% (top) or 3% (bottom) of the total mass, dotted
lines correspond to the leading order term given by Eq. (15).
The vertical lines highlight the typical radii used in numerical
simulations for GW extraction.
be any monotonically decreasing function, for which the
luminosity at R in Eq. (3) is
L∞(τ) = −
d∆τ∞(m)
, (16)
where the minus sign originates from our definition of m:
the shell labeled by the largest value of m arrives the ear-
liest. The luminosity profile can also be obtained as the
function of time, L∞(τ), using the relationship ∆τ(m).
Conversely, for given L∞(τ), we can compute ∆τ∞(m)
using Eq. (16). The luminosity at some other distance r
can be obtained similarly if given ∆τ(m, r), the arrival
time of mass m relative to the outermost shell at distance
r. This function is given by Eq. (9), substituting the
waveform ∆τ∞(m) for ∆τ∞, and (m1,m2) = (M0,m).
The luminosity profile at r is then
Lr(m) =−
∂∆τ(m, r)
)−1/2
L∞(m)
Therefore, the modification of the profile in Bondi ra-
diative coordinates (m, r) the profile is distorted self-
similarly. However, in terms of the observer proper time
variable, ∆τ(m) =
Lr(m)
−1dm, Lr(τ), the modifica-
tion to the profile will not be self-similar:
Lr(τ) =
′)dτ ′
)−1/2
′)dτ ′
Equation (18) relates the luminosity profile as a function
of proper time at radius r, Lr(τ), to the profile at infinity,
L∞(τ). Comparing Eqs. (17) and (18) shows the advan-
tage of Bondi type radiative coordiantes as opposed to
proper time.
-100 -80 -60 -40 -20 0 20 40
r = 1000Mtot
r = 12.5Mtot
-100 -80 -60 -40 -20 0 20 40
r = 12.5Mtot
r = 30Mtot
r = 80Mtot
FIG. 3: Our fits to the GW luminosity profile for binary BH
inspiral merger simulations as a function of observer proper
time τ and the evolution of the profile at various distances, r.
The profile parameters are given in § II and ǫ = 7%. Top: The
absolute profile (in units of c5/G) is shown for two extremes,
a nearby distance (r = 12.5M0) and far-away distance (r =
103M0). Bottom: The difference between the GW luminosity
profiles at infinity (i.e. r = 103M0) and three cases of smaller
r, in units of peak luminosity at infinity. The peaks of the
profiles are set to τ = 0. The main effect responsible for the
differences seen in this figure is the bulk gravitational redshift.
Figure 3 plots Lr(τ) for our fit to the luminosity profile
at infinity of merging binary BHs L∞(τ) with ǫ = 7% (see
§ II). The top panel shows the absolute profile while the
bottom panel shows the difference between the profile at
some radius r and the profile at infinity, such that the
peak of the profiles are at τ = 0. The bottom panel
is useful to visualize the characteristic evolution of the
profile.
-100 -80 -60 -40 -20 0 20 40
r = 12.5Mtot
r = 30Mtot
r = 80Mtot
-100 -80 -60 -40 -20 0 20 40
r = 12.5Mtot
r = 30Mtot
r = 80Mtot
FIG. 4: The residual self-gravitational distortion to the lumi-
nosity profiles after accounting for the avarage gravitational
redshift, z = M0/r (top) or (M0 − 0.5∆mtot)/r (bottom), re-
spectively. Other parameters are the same as in Fig. 3.
In § III A, we have identified the two main effects re-
sponsible for the convergence rate of the waveform to
be the gravitational redshift corresponding to the aver-
age mass and the self-gravitational effect. Indeed, the
differences visible in Figure 3 are primarily due to the
former. However, correcting for only the average gravi-
tational redshift at each radius r leaves a nonnegligible
systematic error with respect to the true signal. To see
this, we substitute ∆τ∞/(1 + z) given by Eq. (13) into
Eqs. (16) and (17), and refer to the corresponding lumi-
nosity as the average gravitational redshifted luminosity
profile, Lzr(τ). After subtracting from the true profile for
each τ , the residual luminosity distortion is
L′r(τ) = Lr(τ) − L
r(τ), (19)
where we set the reference time again to τ = 0 for the
peak of the luminosity profiles.
Figure 4 shows the residual self-gravitational distortion
L′r(τ) for various radii in units of the peak luminosity at
infinity, L∞,max. Naturally, the definition of the “av-
erage gravitational redshift,” z, used for defining L′r(τ)
makes a difference in the result. The top panel uses only
the initial binary mass 〈m〉 = M0/r in Eq. (14) totally
neglecting the gravity of the radiation shell, while the
bottom panel has 〈m〉 = (M0 − 0.5∆mtot)/r, i.e. the
redshift is chosen to account also for the average gravity
of the radiation shell. In the later case we find a much
quicker convergence for the waveform peak at increasing
radii, but the former choice is more suitable for the early
parts of the waveform corresponding to the late inspiral
waveform. A comparison of Fig. 3 and 4 shows that the
self-gravitational distortion is roughly an order of magni-
tude smaller than the effect of the average gravitational
redshift.
C. Self-Gravitational Coordinate Effects
In the previous sections we have derived the time du-
ration and the luminosity profile of the radiation shell as
it propagates radially outward from the source. We have
assumed that the GW profile at each fixed arial radius r
is parameterized by the proper time τ of a hypothetical
observer fixed at that radius, in particular the luminosity
Lr(τ) was the total mass-energy crossing a sphere at ra-
dius r within infinitesimal proper time dτ . Therefore, the
adopted time-coordinate τ corresponds to a synchronous
gauge at each radius. Since physical observables depend
precisely on proper measures, these coordinates allow a
simple interpretation of the convergence characteristics
of the GW profile at large radii.
Other choices of coordinates would have introduced
additional artificial distortion effects making the conver-
gence characteristics of the waveforms much different.
Consider for instance the “natural” coordinate system
(t, r, θ, φ) in the spherically symmetric case that is chosen
to be Schwarzschild both before and after the GW burst
has arrived with masses M0 and Mf = M0 −∆mtot, re-
spectively, and which changes smoothly in between these
regions. An example of such a coordinate system can be
derived from the Vaidya metric Eq. (5) with the implicit
transformation t ≡ u + r + 2m(u) ln[r − 2m(u)] where
m(u) describes the mass function interior to u (which is
constant along the outgoing radiation world lines). In-
deed, everywhere in the spacetime where dm/du = 0,
these coordinates yield a Schwarzschild metric, and the
Vaidya metric in radiation coordinates (5) is then sim-
ply the Schwarzschild solution in Eddington-Finkelstein
coordinates [12] in these regions. This map covers all
relevant parts of the spacetime including the GW zone.
The world-lines of radiation shells can be shown to follow
, (20)
where the second term is called the Shapiro-time delay
[6] for a particle crawling out of the gravitational poten-
tial of mass mu interior to it. After integration, we find
that to first order the temporal separation of two radi-
ation shells enclosing mass ∆m at radius r evolves to
leading order as ∆t(r) = ∆t(R0)− 2∆m ln(r/R0), where
R0 is an arbitrary initial radius. In these coordinates,
the signal duration contracts uniformly in exponential
distance intervals. Even though the metric is asymptoti-
cally Minkowski (where t approaches τ for r ≫ M0), the
resulting profile evolution is fundamentally different from
∆τ(r) given by Eq. (11)!
The appearance of the logarithmic radial dependence
of the waveform was first realized by Fock [69]. This ef-
fect is specific to the harmonic coordinates and can be
avoided if changing to Bondi type radiative coordinates
[70, 71]. Blanchet & Schäfer [38] have shown that a sim-
ilar logarithmic dependence of the GW tail leads to a
tail-induced amplitude and phase shift (typically of order
10−7) for stationary sources. In contrast, the logarith-
mic radial dependence of the wave contraction for merger
waveforms can be significant for GW merger simulations.
Between r = (20–40)M0, the waveform contracts in ∆t
by a fraction of 5×10−3, which is just of the order of the
current wave extraction precision [5, 26, 27]. This appar-
ent logarithmic contraction effects can be avoided if one
changes to the proper time variable as we have done in
the previous sections. The remaining ∆τ(r) evolution is
however a physical effect.
Both the logarithmic ∆t(r) contraction and the phys-
ical ∆τ(r) evolution (in particular the contribution de-
noted by ∆τ ′(r) above) are consequences of higher order
radiation effects in the Einstein equations beyond the
scope of first order methods such as the Regge–Wheeler–
Zerilli-Moncrief perturbation method used for extrapo-
lating the numerical waveforms to infinity. Therefore
these effects cause the extrapolated waveforms to be
different when extracting GWs from numerical simula-
tions at various radii by standard methods using no self-
gravitational interaction. For a related recent analysis
see Ref. [72].
IV. DISCUSSION
A. Summary
We considered the self-gravitational effect of gravi-
tational radiation on the propagation of GWs from a
compact source. We adopted simple approximations for
the geometry of the radiation, by considering spherical
symmetry on scales comparable to the radial width of
the radiation packet. This approximation appears ade-
quate for the quadrupolar (l = 2,m = 2) radiation pat-
tern around binary BH sources in numerical simulations
[25, 26, 27]. Nevertheless, we use the Appendix to ex-
amine the maximal effects of anisotropy in the opposite
(exaggerated) extreme, when the outgoing radiation is
concentrated into a compact region. We find that irre-
spective of the level of anisotropy, the gravitational radi-
ation is distorted under the influence of its own gravity
as it propagates. Contrary to the standard gravitational
redshift, which is a uniform shift of the waveform, the
self-gravitational effect depends on the intensity and is
predominant only for the most intensive bursts of radi-
ation causing a non-uniform distortion of the waveform.
The self-gravitational distortion depends on distance to
leading order as ∆m/r, and is therefore relevant on scales
rsg/M0 ∼
< ǫ/δ where ǫ is the radiation efficiency and δ is
the desired calculation accuracy. For BH binary merg-
ers simulations ǫ ∼ 7% and δ ∼ 10−5 implying that
rsg ∼
< 7 × 103M0. If the GWs are extracted within this
region, the self-gravitational distortion should be taken
into account.
B. Testing the Effect with Numerical Simulations
Numerical simulations based on the full set of Einstein
equations for binary BH inspirals have not yet reported
evidence for the waveform distortion effect considered
here although they have shown that the waveforms do
not converge within a fractional accuracy of δ ∼ 10−3
[5, 26, 27]. This is because the simulations are restricted
to a limited volume, typically of radii ∼ (80–850)M ,
while the extracted waveforms are typically compared be-
tween r ∼ 20–50M . For a radiation mass ∆m/M0 ∼ 7%,
the primary effect is a shift of the waveform due to a
logarithmic Shapiro time delay of the remnant, a uni-
form gravitational redshift, and the self-gravitational ef-
fect. We have shown that the logarithmic Shapiro de-
lay does not show up if using proper measures to de-
scribe the waveform, and the uniform gravitational red-
shift is accounted for in the linear wave propagation mod-
els. However, the residual self-gravitational effect in the
GW luminosity has a characteristic profile that has to be
subtracted when extrapolating the extracted waveform.
The peak of the effective luminosity distortion reaches
2× 10−3 and 5× 10−4 at r = 30 and 50M0, respectively.
Present-day numerical relativity simulations should al-
ready be capable of directly measuring the relevance of
our effect by artificially amplifying the gravitational radi-
ation found at the extraction radius r ∼ 20M , and start-
ing the simulation with these amplified initial conditions.
For example, for a total GW energy ∆m/M0 = 30%, the
effective luminosity distortion between r = 20M between
20–50M is several percent, which is well within simula-
tion and extraction errors. For consistency, the simu-
lation should confirm that the total energy content of
the radiation does not change. We also expect the initial
ringdown frequency (corresponding to the most energetic
shell) to be smaller than the final ringdown frequency.
In order to avoid errors caused by the self-gravitational
distortion effect up to the desired numerical precision δ ∼
10−5, the waveform extraction radius should be chosen
to be rsg ∼
> 7 × 103M0. Alternatively, if the waveforms
are extracted at smaller radii, the waveforms should be
converted to Bondi type radiative coordinates and then
extrapolated with the scaling 1/r.
C. Observational Implications
The self-gravitational waveform distortion is important
for future observations of BH binary mergers.
1. The waveform distortion is expected to be resolv-
able for the LISA instrument with respect to sim-
ulated waveforms for total BH masses of (104–
109)M⊙. The total signal to noise ratio of merger
waveforms is 104 for LISA observing zc ∼ 1 [4].
The distortion effect modifies the waveform ampli-
tude and frequency by∼ (10−3–10−4) for numerical
waveforms extracted between r = (30–80)M0.
2. The distortion involves a systematic modification
of the waveform which needs to be accounted for
in order to interpret observed merger waveforms
and improve the estimation uncertainty of physical
parameters beyond the uncertainty of the preced-
ing inspiral signal. The signal to noise ratio of the
final BH merger waveform is an order of magni-
tude larger than for the inspiral, implying that the
merger waveform has a potential to greatly reduce
parameter estimation errors. Note that the relative
accuracy using only the inspiral signal with LISA
is expected to be 10−3–10−5 [73, 74] for estimating
the component masses, which is smaller than the
distortion effect.
3. This effect is different from the uncertainties caused
by gravitational lensing [75], in that it is only an
issue concerning the convergence properties of nu-
merical simulations. Lensing causes an error of
several percent on the inferred luminosity distance
(due to the unresolved matter along the line of
sight), and lensing errors increase with the source
distance. In contrast, the self-gravitational effect is
of order 0.1-0.01 percent for numerical simulations
if the waveforms are extracted at 30M–80M and
dies off quickly as 1/r. The wave distortion effect
is of order 10−20 relative to the waveform ampli-
tude for typical astrophysical scales. Therefore, the
wave-contraction effect does not provide any addi-
tional physical parameters for observations.
Acknowledgments
We thank George Rybicky, Irwin Shapiro and Kip
Thorne for enlightening discussions. BK acknowledges
support from a Smithsonian Astrophysical Observatory
Predoctoral Fellowship and from NKTH Öveges József
Fellowship.
APPENDIX A: ANISOTROPY
Our analysis considered only perfectly spherically-
symmetric configurations. The approach was motivated
by the quasi-spherical radiation patterns found around
binary BH sources in numerical simulations [25, 26, 27].
In this Appendix, we would like to examine the sensitiv-
ity of our basic results to deviations from sphericity. To
gauge whether there is any such sensitivity, we analyse
the most extreme case in which the outgoing radiation is
concentrated into a highly compact region.
But first let us define more precisely what we assumed
so far. The derivation presented in § III requires that the
radiation field is “initially locally spherically symmetric”,
so that it is initially described by the Schwarzschild met-
ric locally within some narrow solid angle ∆θ ∼
< ∆r/r be-
fore the GW arrives, where ∆r ∼ c∆t is the radial width
of the wave-packet along its propagation direction. But
since the radiation propagating through this solid angle
is in no causal contact with the radiation field expanding
towards other directions, it cannot distinguish the ac-
tual spacetime from a spherically symmetric one. Note
that the outermost shells of radiation expanding along
different directions are always causally disconnected by
definition, and the interior shells of radiation can only be
affected by the outer shells within ∆θ. Since we examine
the distortion effect on large distances compared to the
width of the burst r ≫ ∆r, spherical symmetry must
only be required within a very narrow angle. This simple
set of considerations implies that if high-order multipoles
have a vanishing contribution at large distances, the re-
sults derived in § III are applicable very generally for
short bursts of radiation, ∆r ≪ r. Indeed, numerical
simulations confirmed that the dominant contribution to
the wave amplitude is given by the (l = 2,m = 2) multi-
pole and higher order terms are suppressed by more than
a factor of magnitude (see references in § II). In the re-
mainder of this Appendix we demonstrate the validity of
this simple conclusion through explicit calculations.
We consider three variations on a toy model to estimate
the effect of anisotropy. We start with the simplest model
and refine this model by adding more details and com-
plexity in the successive models. In each case, we discuss
general implications for the model under consideration.
In all models we consider the extreme opposite regime to
spherical symmetry, namely that the radiation is maxi-
mally clumped into two outgoing BHs L (leading) and T
(trailing) of masses mL and mT , representing the leading
and trailing edges of the radiation, respectively. We as-
sume that L and T are moving in the same direction on
light-like world lines, so that T lies in the causal past of
L, but L is outside the causal past of T throughout their
propagation. We assume that there is also a remnant
Schwarzschild BH R with mass mR. The instantaneous
radial position coordinate of R, T , and L at time t are
0, rT (t), and rL(t). We are interested in obtaining the
world lines of BHs T and L to see how the coordinate
separation ∆r(t) = rL(t) − rT (t) decreases with time as
compared to the spherically symmetric result. In our
first model we neglect the remnant R (setting mR = 0),
assume that L moves with constant velocity vL in free
space, and calculate the trajectory of T in the spacetime
created by L. Subsequently, we will generalize L to move
on a more general world line with a slowly changing ve-
locity vL(t). Finally, we can turn on the remnant R in
addition to L, and include the retardation effect when
calculating the relative motion of T .
We note that the spacetime of BHs moving at the
speed of light have been calculated previously in Ref. [46],
which found that BHs moving in the same direction do
not interact. However, Ref. [46] assumed that the BHs
move in free space and consequently adopted v = 1 for
their velocity. In contrast, the BHs T and L travel on
null-geodesics in the perturbed spacetime which is ini-
tially the Schwarzschild spacetime. This difference gives
rise to a non-trivial interaction between the BHs T and
We compare our results to the spherical case, using the
(t, r) coordinate system defined by Scwarzschild coordi-
nates before and after the radiation shells as described in
§ III C.
1. No remnant mR = 0, constant vL velocity
We start by assuming that L is a BH with constant
velocity vL < 1 in free space, and wish to calculate the
world line of T in this background. Here, we assume that
no remnant is present, and that L and T move along the
same spatial direction, which we denote by x. Thus it is
sufficient to restrict our attention to the two dimensions
(t, x) of the spacetime.
Let us start by deriving the metric. In the coordi-
nate system (t′, x′) comoving with L, the metric is the
Schwarzschild metric ds2 = −(1−φ′)dt′2+(1−φ′)−1dx′2,
where φ′ = 2m′L/|x
′|. Here x′ ≡ 0 corresponds to the BH
L for all t′, and m′L = mL/γ is the rest mass of L, where
mL is the energy carried by L in the original (t, x) co-
ordinates and γ = 1/
1− v2L is the Lorentz factor. To
derive the metric in the (t, x) coordinate system, we ap-
ply the diffeomorphism (t, x) = γ(t′+vLx
′, vLt
′+x′), i.e.
a global Lorentz transformation,
ds2 = −
(1 − φ′)2 − v2L
(1− v2L)(1 − φ
dt2 +
1− v2L(1− φ
(1 − v2L)(1− φ
′(2− φ′)
(1− v2L)(1 − φ
dtdx (A1)
which can be rearranged as
ds2 =
[vL + (1− φ
′)]dt− [1 + vL(1− φ
′)]dx
(1− v2L)(1 − φ
[vL − (1 − φ
′)]dt− [1− vL(1 − φ
′)]dx
(1− v2L)(1 − φ
.(A2)
Here φ′ is to be expressed as the function of the new
coordinates (t, x), i.e. φ′ = γ−2φ where φ = 2mL/|∆x|,
∆x = xL−x, and xL = vLt is the instantaneous position
of the singularity.
The xT (t) null-geodesics describing the world line of
T can be obtained by setting ds2 = 0. Equation (A2)
shows that there are two solutions
[vL + (1− γ
−2φ)]dt − [1 + vL(1− γ
−2φ)]dxT = 0,
[vL − (1− γ
−2φ)]dt − [1− vL(1− γ
−2φ)]dxT = 0.
These differential equations can also be obtained more
simply by finding the null-geodesics in the comoving co-
ordinates (t′, x′) first, and changing to the (t, x) coordi-
nates only in the resulting equation. The null geodesics in
the comoving coordinates are simply dx′T /dt
′ = ±|1−φ′|
(see Eq. 20), and the Lorentz boost coordinate trans-
formation of this differential equation leads instantly to
(A3-A4). Therefore, the two solutions (A3-A4) describe
the null geodesics approaching or receding the moving
BH, respectively.
We would like to find the solution for the T test particle
approaching the source L from behind, namely Eq. (A3)
for an initial condition xT < xL. This first-order dif-
ferential equation can be solved analytically by a linear
substitution. The coordinate velocity vT = dxT (t)/dt
monotonously decreases from 1 to vL as the event horizon
at xLhor(t) = vLt− 2γ
−2mL is approached. In particular
if vL = 1, i.e. the source L has the speed of light in the
free-space background, then the trailing test particle T
will not be delayed at all, vT (t) = 1 for all t. However,
if vL ≪ 1 then T is considerably affected by the Shapiro
delay near the horizon of L.
In concluding the description of this model, let us sum-
marize how the clumpy case compares to the spherically
symmetric case of expanding radiation shells. First re-
call that in the spherically symmetric case, L has no ef-
fect on T throughout the dynamics regardless of vL or
∆r. In the clumpy case, the gravity of L delays the mo-
tion of T . The magnitude of this delay is significant only
if both of two conditions are violated: (a) vL ≈ 1 and
(b) ∆x = xL − xT ≫ 2γ
−2mL. What are the “typi-
cal numbers” for these quantities? Eq. (20) implies that
vL = 1 − 2M0/x (which is also true in the clumpy case,
see § A3), implying that γ−2 ∼ 4M0/r, and for binary
mergers ∆x ∼ 12πM0, mL < M0 − Mf ∼
< 0.06M0, we
find that the two cases are equivalent to (a) x ≫ 2M0
and (b) x ≫ 0.2∆m ∼
> 0.01M0. Quite clearly, these
conditions will not be violated for distances outside the
dynamical regime of strong gravity e.g. x ∼
> R0 = 30M0.
Thus, we expect only very minor modifications relative to
the spherically symmetric case, even in the most clumpy
case. For a quantitative estimate we need to integrate
these modifications over the relevant distances which we
describe next.
2. No remnant, slowly changing vL(t)
Next we consider a slowly changing source velocity
vL(t) for the BH L, continue to neglect a remnant R,
and calculate the motion of T in this spacetime. Since L
is assumed to move on a light-like world-line, it is not ef-
fected by R and only responds to the background created
prior to the production of the bursts. Thus we assume
L moves on the null-geodesic of the background as de-
scribed by (20) with vL = 1− 2mT /xL.
If vL is slowly changing, we can consider vL to be con-
stant during short time intervals with infinitesimal jumps
on their boundaries. We can then find the correspond-
ing world line segments of T by solving the differential
equation (A2) and matching the boundary conditions of
the successive segments by requiring continuity. In the
limit that the length of the constant time intervals ap-
proaches zero, the world line xT (t) at every instant is
given by Eq. (A2) with vL now denoting the instanta-
neous velocity, and φ′ referring to the instantaneous value
of the potential: φ′ = γ−2φ, where φ = 2mL/|∆x| with
∆x = xL − x and xL =
vL(t)dt. Thus,
vL + 1− γ
1 + vL(1− γ−2φ)
2mL(1 − vL)
∆x− 2mLvL(1− vL)
Substituting vL = 1−2mT /xL, the distance between the
clumps of radiation satisfies
4mLmT
xL∆x− 4mLmT
1− 2mTx
Note that xL(t) can be used to express the width of the
packet ∆x as a function of xL. To simplify the result,
we use qL = mL/mT , set the units to the Schwarzschild
radius 2mT = 1, and express (A6) in terms of the loga-
rithmic distance variable, y = ln(xL − 1). Then
= −1 +
xL∆x− qL
1− x−1L
) (A7)
which to first order in 1/xL becomes
= −1 +
. (A8)
Equation (A8) shows that to leading order, the wave-
packet packet contracts linearly in terms of the logarith-
mic distance variable x. In the spherically-symmetric
case, the inner shell T is not influenced by L and so
dxT /dt = 1 instead of Eq. (A5), leading to d∆x/dy =
−1. Therefore the distortion of wave-packets in the max-
imally clumpy case is the same as in the spherically sym-
metric case to leading order. The difference arises in the
next order given by the second term in (A8) describing
how the gravity of L Shapiro-delays the motion of T .
This is typically of order (4 × 6%)/(12π) × x−1L ∼ 10
for xL ∼ 30M0 and gets exponentially smaller for expo-
nentially larger distances.
3. Remnant included, changing vL
The previous model assumed no remnant (i.e. mR =
0), and postulated that T propagated in the spacetime of
L by assuming that the spacetime at T was the spacetime
of L at the same instant. Here we consider a nonzero
mR and account for the retardation of the effect of L as
percieved by T . We assume a slowly changing velocity
and that the gravitational perturbations are sufficiently
small to allow simple superposition to leading order. We
follow a simplified approach with the following essential
assumptions:
1. The initial condition is a spherically symmetric
Schwarzschild spacetime centered at x = 0.
2. Lmoves on a null-geodesic xL(t) of the initial back-
ground metric of R and T , i.e. in a Schwarzschild
metric centered at x = 0 for all t and for a mass
mR +mT . The world-line of L follows (20) accord-
ingly.
3. T moves on a null-geodesic xT (t) of the background
metric of R and L, which we assume to be a sim-
ple superposition gij = ηij + δg
ij + δg
L,ret
ij . Here
δgNij = g
ij − ηij for a given metric, g
ij , where ηij
is the Minkowski metric, gRij is the metric of the
remnant i.e. the Schwarzschild metric centered at
x = 0 for all t with mass mR. The metric g
ij is the
stationary boosted Schwarzschild metric (A1–A2)
with mass mL, velocity vL, centered at xL. The
label ret stands for retardation, which we describe
next separately.
4. We account for retardation by setting ∆tret = t −
tret to be the light-travel time from L to T . For this
we compute the inward propagating null-geodesics
from L to T [i.e. between positions (tret, rL(tret))
and (t, rT (t))], based on the initial backgroundmet-
ric of R and T (i.e. neglecting the gravity of L).
To find the retardation time, we note that the inward
propagating null geodesics satisfies dx/dt = −[1−2(mR+
mT )/x]. Since this is exactly the time-reversed world line
of L, we get xT (t) = xL(t − 2∆tret). Integrating dt/dx
for the world line of L between xL(t− 2∆tret) and xL(t),
2∆tret =
∫ xL(t)
xT (t)
xL − 2mR − 2mT
dxL, (A9)
from which
∆tret =
xL − xT
+ (mR +mT ) ln
xL − 2mR − 2mT
xT − 2mR − 2mT
(A10)
The distance where T percieves L is xretL (t) = xL(t −
∆tret) and the separation is ∆xret ≡ x
L − xT . Substi-
tuting xT as xT = xL −∆x and xT = x
L −∆xret into
(A10), we can find ∆xret for given ∆x and xL. To first
nonvanishing order in 1/xL,
∆xret =
(mR +mT )∆x
. (A11)
The leading order term corresponds to the propagation at
the speed of light in free space, vL = vT = 1. Note that
the correction is proportional to x−2L , which is extremely
small for the physical cases beyond the strong field zone.
Finally, we define the retarded position and velocity
xretL = xL − ∆xret, and v
L = 1 − 2(mR + mT )/x
which can be written in terms of xL and ∆x using Eq.
(A11).
The metric contribution δg
L,ret
ij of L at T at time t, is
the boosted Schwarzschild metric (A1,A2) with instan-
taneous velocity vretL , a singularity at x
L , and distance
∆xret.
Now we can redo the derivation presented in § A2 to
find the motion of T , using the modified spacetime gij
given above. Again we find two solutions for ds2 = 0 rep-
resenting the ingoing and outgoing radiation. Expanding
the outgoing solution in a series in x−1L , we find
clumpy
qR∆x+
2qL(1 + q
(A12)
where qi = mi/(mR+mT ) for i ∈ {R, T, L} and distance
units are chosen to be the Schwarzschild radius 2(mR +
mT ) ≡ 1. In order to get the instantaneous shell width
∆x as a function of logarithmic distance y, we can redo
the manipulations of (A6–A8) for the result (A12). To
first order in 1/xL,
clumpy
= −qT +
qR∆x+
2qL(1 + q
(A13)
In the limit of no remnant qR = 0, we almost recover
the solution derived previously in Eq. (A8). There is a
factor 2 difference, which is the direct consequence of the
retardation of the percieved distance ∆x, which had been
neglected in Eq. (A8).
Equation (A13) should be contrasted to the expansion
of two spherically symmetric shells mT and mL in the
presence of a remnant mR. We may expand the corre-
sponding spherical solution of § III C in a series in 1/xL
to first order:
spherical
= −qT +
. (A14)
The first two terms in Eqs. (A13) and (A14) are iden-
tical. The correction describing the “Shapiro delay” of
contraction in the clumpy case due to the gravity of L is
2qL(1 + q
R)/(xL∆x). Substituting typical physical val-
ues qR = 97%, qT = 3%, qR = 3%, and ∆x0 ∼ 6π ∼ xL0
in units of Schwarzschild radii, we see that the correc-
tion is of order 10−4 initially, and becomes exponentially
smaller at exponentially larger distances. In summary,
even in the most extreme case of clumpiness, the radia-
tion packet propagates to very high precision according
to the spherically symmetric description.
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[84] Throughout this paper we assume that the metric at
r → ∞ is the Minkowski metric, neglecting cosmologi-
cal effects. This is justified by the orders of magnitudes
or radii where the radiation shell distortion is relevant
r < 103M0 (see Fig. 2) is much less than cosmological
scales r ∼
> 1018M0.
