# Can Two Ultrarelativistic Objects
Lose Almost All Their Energy
to Gravitational Radiation?

Don N. Page
Department of Physics
4-183 CCIS
University of Alberta
Edmonton, Alberta T6G 2E1
Canada Internet address<EMAIL_ADDRESS>
(2022 December 7)

###### Abstract

In 2007 Pretorius and Khurana [37] did “speculate that at threshold [at a
critical impact parameter], all of the kinetic energy of the system [two
ultrarelativistic black holes] is converted to gravitational waves, which can
be an arbitrarily large fraction of the total energy.” However, in 2012
Sperhake, Berti, Cardoso, and Pretorius [57] performed numerical calculations
that led them to the contrary conclusion: “An extrapolation of our results to
the limit $\gamma\rightarrow\infty$ suggests that about half of the center-of-
mass energy of the system can be emitted in gravitational radiation, while the
rest must be converted into rest-mass and spin energy.” Here I present
arguments against this latter conclusion and in support of the earlier
speculation that for sufficiently large $\gamma$, all but an arbitrarily small
fraction of the total energy can be radiated away.

##  1 Introduction

LIGO has had enormous success in detecting the gravitational waves from
astrophysical inspiraling black holes that coalesce to form a larger black
hole [1, 2, 3, 4]. In these cases the black holes have been inspiraling long
before their gravitational wave emission is strong enough to be detected by
LIGO, so that well before the final coalescence, the two black holes are
moving with nonrelativistic velocities $v\ll c$ relative to each other and are
gravitationally bound to each other.

Astrophysically, it seems that it would be very rare (and not yet observed)
for two black holes or other macroscopic objects to approach at relativistic
velocities $v\sim c$ before coalescing. However, it is an interesting question
what would happen if this occurred. Indeed, there has been an enormous amount
of work [5–92] calculating what happens for compact objects approaching each
other at arbitrary relative velocities in 4-dimensional spacetime (not
including considerable additional work that has also been done in higher
dimensions that I shall not discuss here). Much of this work has shown
essentially that if the Schwarzschild radius corresponding to the total
center-of-momentum (COM) energy $E$, that is, $R_{S}=2GE/c^{4}=2E$ (where here
and henceforth I shall use units with $G=c=1$), is much larger than the sum
$R$ of the intrinsic linear sizes of the two objects, then the composition of
each object does not matter significantly for how much gravitational wave
energy $\Delta E$ is emitted, which, for fixed total COM energy $E$ instead
depends almost entirely on the impact parameter $b\equiv J/p\equiv\beta E$,
where $J$ is the total angular momentum in the COM and $p$ is the magnitude of
the initial spatial momentum of each object in the COM (equal magnitudes but
opposite directions, since the total spatial momentum is zero in the COM).
Here

$\beta\equiv\frac{b}{E}\equiv\frac{J}{pE}\approx\frac{J}{2E^{2}}=\frac{2J}{R_{S}}$
(1)

is a dimensionless measure of the impact parameter (not to be confused with
$v/c$). The approximation of the last two terms occurs in the limit that the
COM energy $E$ is much greater than the sum $M=m_{1}+m_{2}$ of the rest masses
$m_{1}$ and $m_{2}$ of the two objects, so that then $p\approx E/2$, giving
$b\approx 2J/E$. (Note that I am not using $M$ for the total ADM mass of the
spacetime as many others do, but rather $E$.)

For fixed $E\gg M$ and $R_{S}=2E\gg R$, there is at least one critical impact
parameter $b_{c}=\beta_{c}E=\beta_{c}R_{S}/2$ (with $\beta_{c}$ some
dimensionless number of the order of unity, which Shibata, Okawa, and Yamamoto
[44] have estimated to be about $2.5/v$), that should be independent of $E$,
say with value $\approx 2.5$, in the limit $E/M\rightarrow\infty$ and
$R_{S}/R\rightarrow\infty$, such that for impact parameter $b$ infinitesimally
on one side of $b_{c}$, the two objects will, at least eventually, form a
single black hole, but for $b$ infinitesimally on the other side of $b_{c}$,
the two objects will scatter separately to infinity (though leaving open the
possibility that either of the objects might separately become a black hole
even if it were not one originally, or that both might become black holes that
never merge).

It is perhaps simplest to assume that for fixed $E\gg M$ and $R_{S}=2E\gg R$,
there is a unique critical impact parameter $b_{c}$ such that for all smaller
impact parameters, $b<b_{c}$, the two objects eventually merge and form a
single black hole, but for all larger impact parameters, $b>b_{c}$, the two
objects scatter separately to infinity. However, near the end of this paper I
shall discuss the possibility that this might not be true, and that there
might instead be a larger odd number of critical impact parameters $b_{c}$.
Counting with positive integers $n$, with $n=1$ for the smallest critical
impact parameter $b_{c}=b_{c,1}$ (so that at all lower values of $b$, that is
for all $0\leq b<b_{c,1}$, a single black hole forms), each odd critical
impact parameter, $b_{c,2n-1}$, has, for $b$ infinitesimally smaller than
$b_{c,2n-1}$, the two objects merging to form a single black hole and has, for
$b$ infinitesimally larger than $b_{c,2n-1}$, the two objects scattering
separately to infinity. Conversely, each even critical impact parameter,
$b_{c,2n}$, has, for $b$ infinitesimally smaller than $b_{c,2n}$, the two
objects scattering separately to infinity and has, for $b$ infinitesimally
larger than $b_{c,2n}$, the two objects merging to form a single black hole.

In the following, when I discuss the impact parameter $b_{c}$, if in fact
there are more than one, I shall mean by $b_{c}$ the particular critical
impact parameter at which the largest fraction $f$ of the fixed total energy
$E$ is emitted into gravitational waves of energy $\Delta E=fE$. Similarly,
$\beta_{c}\equiv b_{c}/E$ will denote the dimensionless quantity analogous to
that particular critical impact parameter.

When $b$ is much smaller than $b_{c}=\beta_{c}E$, it appears that the fraction
$f$ of the initial COM energy $E$ that is radiated into gravitational waves of
energy $\Delta E=fE$ is significantly less than unity (approximately $0.14\pm
0.03$ for a head-on collision, $b=0$, when $E\gg M$ [43]), and it seems that
increasing $b$ increases $f=\Delta E/E$. On the other hand, when $b$ is much
larger than $b_{c}=\beta_{c}E$, the two objects have a small scattering angle
and also do not emit a large fraction of their energy into gravitational
radiation, and $\Delta E$, the amount radiated, goes down with increasing $b$
as the scattering angle also decreases. Therefore, there should be some value
in between for $b$, say $b_{m}=\beta_{m}E$, at which the maximum fraction
$f=\Delta E/E$, say $f_{m}$, of the total COM energy $E$ (which is to be kept
fixed when $b$ is varied) is emitted as gravitational wave energy $\Delta E$.
If there are in fact more than one critical impact parameters, it also seems
possible that $f$ does not decrease monotonically from $f_{m}$ as $b$ is moved
in either direction away from $b_{m}$, so that there may be more than one
local maximum for $f$ (perhaps one for each odd critical impact parameter
$b_{c,2n-1}$), but I am taking $f_{m}$ to be the global maximum for $f$. It
seems plausible that $b_{m}$ would be close to the critical impact parameter
$b_{c}$ (so $\beta_{m}$ is close to $\beta_{c}$), especially in the limit
$E/M\rightarrow\infty$ and $R_{S}/R\rightarrow\infty$, but here I shall not
assume that this needs to be true.

Now Conjecture 1, a slight generalization of the 2007 speculation of Pretorius
and Khurana [37] to an arbitrary pair of initial objects of finite size, is
that in the limit $\gamma\equiv E/M\rightarrow\infty$ and
$R_{S}/R\rightarrow\infty$, $f_{m}\rightarrow 1$.

Slightly stronger is Conjecture 2, that at the critical impact parameter
$b_{c}$ that has the largest $f=\Delta E/E$ for finite $E/M$ and finite
$R_{S}/R$, when one takes the limit $\gamma\equiv E/M\rightarrow\infty$ and
$R_{S}/R\rightarrow\infty$, $f=\Delta E/E$ at this $\gamma$-dependent $b_{c}$
approaches 1.

However, in 2012 Sperhake, Berti, Cardoso, and Pretorius (SBCP) [57] performed
numerical calculations that led them to the contrary conclusion: “An
extrapolation of our results to the limit $\gamma\rightarrow\infty$ suggests
that about half of the center-of-mass energy of the system can be emitted in
gravitational radiation, while the rest must be converted into rest-mass and
spin energy.”

Here I wish to revisit the question of what fraction $f=\Delta E/E$ of the
initial COM energy $E$ can be emitted in gravitational wave energy $\Delta E$
in the limit that $\gamma\equiv E/M$ is taken to infinity. Despite the
numerical evidence presented by [57], I shall argue that it is more plausible
that $f$ can be made arbitrarily near 1 by taking $\gamma$ to infinity while
fine tuning the impact parameter $b$ appropriately.

##  2 Ultrarelativistic Collisions of Black Holes

My arguments should generalize to apply for any pair of objects with total
linear size $R$ and with $M$ the sum of their rest masses (here not the total
ADM mass of the spacetime, which is what I shall denote by the total energy
$E$), when one separately boosts the two objects so that the center-of-
momentum (COM) energy $E$ is much greater than both $R$ and $M$ (in units with
$G=c=1$). However, for simplicity and concreteness let me assume that the two
objects are initially Schwarzschild (nonrotating) black holes having positive
rest masses $m_{1}$ and $m_{2}$, with $M=m_{1}+m_{2}$ and $R=2m_{1}+2m_{2}=2M$
if one uses the Schwarzschild radius of each black hole as its linear size.

Then the critical impact parameter that gives the largest $f=\Delta E/E$ out
of all critical impact parameters (if there is more than one) should be

$b_{c}=b_{c}(m_{1},m_{2},E)=\beta_{2}(m_{1}/E,m_{2}/E)E,$ (2)

with the dimensionless function $\beta_{2}(m_{1}/E,m_{2}/E)$, a symmetric
function of its two dimensionless arguments, going to a constant, which I
shall call $\beta_{0}$, in the limit that $m_{1}/E\rightarrow 0$ and
$m_{2}/E\rightarrow 0$ (which is implied by the single limit $M/E\rightarrow
0$):

$\beta_{0}\equiv\lim_{M/E\rightarrow 0}\frac{b_{c}}{E}.$ (3)

For impact parameter $b$ just infinitesimally larger than this particular
critical impact parameter $b_{c}$, the two black holes (each initially much
smaller than the impact parameter, since their Schwarzschild radii, $2m_{1}$
and $2m_{2}$, are much smaller than $b_{c}\sim E$, leaving out the
dimensionless factor $\beta_{2}(m_{1}/E,m_{2}/E)$ that is expected to be of
the order of unity) do not merge into a larger black hole. On the other hand,
for impact parameter $b$ just infinitesimally smaller than the critical impact
parameter $b_{c}$, the two black holes would merge into a larger black hole.

In this example, there are four independent quantities that with $G=c=1$ have
the dimension of mass or length or time, namely $m_{1}$, $m_{2}$, $E$, and
$b$, and hence there are three independent dimensionless ratios, namely
$m_{1}/E$, $m_{2}/E$, and $b/E$. At the critical impact parameter $b_{c}$
given by Eq. (2), there are only two independent dimensionless ratios, namely
$m_{1}/E$ and $m_{2}/E$.

It is also interesting to restrict to an even simpler model in which each
initially Schwarzschild black hole has the same mass, $m_{1}=m_{2}=M/2$, so
that then there are only three independent quantities that have the dimension
of mass or length or time, namely $M$, $E$, and $b$, and hence there are only
two independent dimensionless ratios, namely $\gamma\equiv E/M$ and
$\beta\equiv b/E$ (here not equal to $v/c=\sqrt{1-1/\gamma^{2}}$). At the
particular critical impact parameter $b_{c}$ that gives the maximum $f=\Delta
E/E$ out of all critical impact parameters if there are more than one, which
now has the form

$b_{c}=b_{c}(E,M/2,M/2)=\beta_{1}(\gamma)E,$ (4)

there is only one independent dimensionless ratio, namely $\gamma\equiv E/M$.
The fraction $f=\Delta E/E$ of the initial energy $E$ that is radiated into
gravitational waves with energy $\Delta E$ at the critical impact parameter
would then be a function only of $\gamma$, and the question under dispute is
how $f(\gamma)$ depends on $\gamma$ in the limit that $\gamma$ is taken to
infinity.

##  3 Qualitative Argument for Ultrarelativistic Scattering Near the Critical
Impact Parameter

Here I shall restrict to the simplest model, with two initially incoming
Schwarzschild black holes, each of mass $M/2$, which in the COM have opposite
spatial momenta each of magnitude $p=(M/2)\gamma v$ with equal gamma-factors,
$\gamma=1/\sqrt{1-v^{2}}$, so that the total energy in the COM frame is
$E=M\gamma$, and with total angular momentum $J=bp$ with impact parameter
$b=J/p=2J/(Ev)$ that is near the critical impact parameter
$b_{c}=\beta_{1}(\gamma)E$ with the maximum $f=\Delta E/E$ (out of all
critical impact parameters, if there are more than one, not out of all
possible values of the impact parameter $b$). I shall describe the picture
Pretorius and Khurana [37] painted and then argue that it seems generally
qualitatively correct, despite the opposing arguments of Sperhake, Berti,
Cardoso, and Pretorius (SBCP) [57].

In particular, the idea is that near the critical impact parameter
$b_{c}=\beta_{1}(\gamma)E$, the two black holes, each of size much smaller
than the critical impact parameter $b_{c}$ when $\gamma=E/M\gg 1$ as I shall
always assume, spiral inward while radiating away almost all their initial
energy, until they coalesce at the center to form a black hole that on a
logarithmic scale has a mass closer to the original total rest mass $M$ than
to the initial COM energy $E=M\gamma$. At each time $t$ that is measured by a
static observer in the COM frame at radial infinity, let $r(t)$ be some
measure of the distance of each black hole from the center of momentum, such
as the proper distance along a spacelike geodesic from the center of momentum
that at this center is orthogonal to the worldline of the center of momentum.

With the black holes orbiting around the COM at very nearly the speed of
light, one would expect, in analogy with the quadrupole formula, that the sum
of the rest masses and kinetic energies of the black holes, which I shall call
$E_{h}$ (not including the gravitational potential energy, which would be
expected to be much smaller than $E_{h}$ for the ultrarelativistic motion, and
not including the gravitational radiation energy $E_{GW}\approx E-E_{h}$,
which would carry away significant amounts of energy to cause $E_{h}$ to
decrease) would decrease at a rate per time that would be roughly proportional
to $(E_{h}/r)^{2}$, assuming that the shape of the orbit is nearly a self-
similar equiangular spiral. Dividing this radiation power outflow from the two
black holes by the negative radial velocity $v_{r}=dr/dt$, which would be
expected to have a magnitude 1–2 orders of magnitude smaller than unity but
still be of the general order of unity (i.e., not having any strong dependence
on $\gamma$ when $\gamma\gg 1$), and including an unknown numerical factor of
$\alpha$, one gets (here initially ignoring the effect of the absorption of
gravitational waves by the black holes)

$\frac{dE_{h}}{dr}\sim\frac{1}{\alpha}\frac{E_{h}^{2}}{r^{2}},$ (5)

which leads to a solution that asymptotically, as $r$ decreases far below the
original impact parameter $b$, has the form $E_{h}\sim\alpha r$ for the total
rest mass plus kinetic energy of the two black holes.

Sperhake, Berti, Cardoso, and Pretorius [57] argue that, even in the limit
$\gamma\rightarrow\infty$, a significant fraction of the kinetic energy of the
two black holes is not radiated away but falls into the black holes to
increase their sizes significantly. Here I shall argue that the fraction of
the total energy that falls into the black holes goes to zero in the limit
that one takes $\gamma$ to infinity, so long as at each $\gamma$ one chooses
the impact parameter $b$ to be near the critical impact parameter $b_{c}$
giving the largest $f=\Delta E/E$ out of all critical impact parameters (if
there are more than one). Alternatively, one could choose $b$ to be near
$b_{m}$, the value of $b$ that maximizes $f$ without restricting to critical
impact parameters, though I would expect $b_{m}$ to be near $b_{c}$ for each
value of $\gamma$, at least for $\gamma\gg 1$.

The argument is that during most of the inspiral, when most of the energy is
radiated into gravitational waves, the black holes, initially each of mass
$m=M/2=E/(2\gamma)\ll b_{c}\sim E$ and hence each of Schwarzschild radii
$2m=M=E/\gamma$, are so much smaller than the region of linear size $\sim
r\sim E_{h}$ where the gravitational wave energy is localized near the black
hole orbits before it flows outward, that only a tiny fraction of this energy
will be absorbed by the black holes.

Let me make a crude model to estimate the order of magnitude of the fraction
$F\equiv 1-f$ of the initial total COM energy $E$ that is absorbed by the
black holes, as a function of $\gamma=E/M$ when it is large, and when $b$ is
sufficiently close to either $b_{c}$ or to $b_{m}$ for that $\gamma$. Each
black hole of rest mass $m$ has an effective cross section for absorbing
gravitational radiation that is $m^{2}$ multiplied by some dimensionless
number (depending on the frequency spectrum of the gravitational radiation in
the frame of the black hole and which is $27\pi$ for gravitational radiation
of wavelength very short compared with the Schwarzschild radius $2m$ of the
black hole when it is nonrotating, which is the geometric optics limit in
which null geodesics fall into a Schwarzschild black hole when they have
impact parameter $b<\sqrt{27}m$ [93, 94]). Therefore, the rate of
gravitational wave energy absorbed by each black hole is some dimensionless
number multiplying $m^{2}\rho$, where $\rho$ is the energy density of the
gravitational waves at the location of the black hole. One can crudely
approximate $\rho$ to be the energy in gravitational waves within the distance
$r$ of the COM origin that is the distance each black hole is from that origin
at the time $t$ of absorption, divided by the volume $\sim 4\pi r^{3}/3$
inside that region. Since the gravitational wave energy is produced at a rate
$\approx-dE_{h}/dt\sim dE_{h}/dr$ and is flowing generally outward with an
effective radial velocity near the speed of light ($c=1$), the energy within
the distance $r$ of the COM origin is of the order of $rdE_{h}/dr$, giving an
effective gravitational wave energy density
$\rho\sim(rdE_{h}/dr)/r^{3}=(1/r^{2})dE_{h}/dr$, dropping factors of the order
of unity such as $4\pi/3$ and various velocities in units of the speed of
light.

For the total rate of the increase of $E_{h}$, the time-dependent the rest
mass plus kinetic energy of the two black holes, if we now add the positive
contribution of the absorption of gravitational waves to the negative
contribution from the emission of the gravitational waves, and divide by the
negative $v_{r}=dr/dt$, we get, as an improvement over Eq. (5),

$\frac{dE_{h}}{dr}\sim\frac{1}{\alpha}\frac{E_{h}^{2}}{r^{2}}-\delta\frac{m^{2}}{r^{2}}\frac{dE_{h}}{dr},$
(6)

where I have introduced a second unknown dimensionless factor $\delta$ to take
into account the unknown factors of the order of unity that were dropped in
the previous paragraph. Of course, Eq. (6) can readily be rearranged to give

$\frac{dE_{h}}{dr}\sim\left(1+\delta\frac{m^{2}}{r^{2}}\right)^{-1}\frac{1}{\alpha}\frac{E_{h}^{2}}{r^{2}}.$
(7)

Next, we need to get an approximate differential equation for how $m$ evolves
as $r$ decreases. If each black hole were absorbing radiation whose center-of-
momentum energy were at rest in the COM frame (i.e., if there were no energy
flux in that frame), the mass would increase at a rate that is
$\gamma_{h}\equiv(1/2)E_{h}/m$ times the rate of its energy increase from the
absorption of gravitational waves that was approximated above by
$-(m^{2}/r^{2})(dE_{h}/dt)$, so then

$\frac{dm}{dt}\sim-\frac{1}{2}\frac{E_{h}}{m}\frac{m^{2}}{r^{2}}\frac{dE_{h}}{dt}.$
(8)

Replacing the factor of $1/2$ by a third unknown dimensionless factor of
$\epsilon$ to take into account the uncertainty in the mean effective velocity
of the gravitational waves, rearranging Eq. (8), and combining it with Eq. (7)
gives

$\frac{r}{m}\frac{dm}{dr}\sim-\epsilon\frac{E_{h}}{r}\frac{dE_{h}}{dr}\sim-\left(1+\delta\frac{m^{2}}{r^{2}}\right)^{-1}\frac{\epsilon}{\alpha}\frac{E_{h}^{3}}{r^{3}}.$
(9)

One can then combine Eqs. (7) and (9) to get the rate at which the logarithm
of $m$ increases when the logarithm of $E_{h}$ decreases:

$\frac{d\ln{m}}{d\ln{E_{h}}}\equiv\frac{E_{h}}{m}\frac{dm}{dE_{h}}\sim-\epsilon\frac{E_{h}^{2}}{r^{2}}.$
(10)

One can see from Eq. (7) that when $\delta m^{2}/r^{2}\ll 1$, as $r$ decreases
with time as the black holes spiral closer together, say with initial
conditions $E_{h}\sim E$ at $r\sim r_{0}$,

$E_{h}\sim\frac{\alpha r}{1+\alpha r/E-r/r_{0}},$ (11)

so that as $r$ drops far below $r_{0}$ and $E$, $E_{h}$ approaches $\alpha r$.
Therefore, during this stage of the evolution, when $E_{h}\sim\alpha r$, Eq.
(10) gives

$\frac{d\ln{m}}{d\ln{E_{h}}}\sim-\epsilon\alpha^{2},$ (12)

so

$\frac{m}{m_{0}}\sim\left(\frac{E_{h}}{E}\right)^{-\alpha^{2}\epsilon}\sim\left(\frac{\alpha
r}{E}\right)^{-\alpha^{2}\epsilon},$ (13)

where $m_{0}=M/2=E/(2\gamma)$ is the initial rest mass of each black hole
(with the sum of the two initial black hole rest masses being
$2m_{0}=M=E/\gamma$), and where the initial value of $E_{h}$, the total rest
mass plus kinetic energies of the two black holes, is the constant total COM
energy $E=\gamma M=2\gamma m_{0}$, and where the initial value of $r$ at the
beginning of inspiral is $r_{0}\sim b\sim E$, assuming $b\sim b_{c}=\beta E$
with $\beta\sim\beta_{c}\sim 1$.

The black holes will merge when $m\sim r$, so that

$1\sim\frac{m}{r}=m_{0}\frac{1}{r}\frac{m}{m_{0}}\sim\frac{E}{2\gamma}\frac{\alpha}{E_{h}}\left(\frac{E_{h}}{E}\right)^{-\alpha^{2}\epsilon}=\frac{\alpha}{2\gamma}\left(\frac{E_{h}}{E}\right)^{-1-\alpha^{2}\epsilon}.$
(14)

Therefore, one gets that the fraction $F$ of the initial total COM energy $R$
that is not radiated into gravitational waves is approximately

$F\equiv
1-f\sim\frac{E_{h}}{E}\sim\left(\frac{\alpha}{2\gamma}\right)^{\frac{1}{1+\alpha^{2}\epsilon}}\sim\left(\frac{\alpha}{2\gamma}\right)^{p},$
(15)

with exponent

$p\sim\frac{1}{1+\alpha^{2}\epsilon}.$ (16)

Therefore, for large $\gamma\equiv E/M=E/(2m_{0})$, $F=1-f$ goes to zero as a
positive power of $1/\gamma$, though without numerical calculations the
exponent, $p\sim 1/(1+\alpha^{2}\epsilon)$, which lies in the range between 0
and 1, cannot be predicted precisely.

The numerical coefficient of the power of $1/\gamma$, here crudely estimated
to be $(\alpha/2)^{p}$, would also be expected to have relatively small
corrections from ignoring nonzero values of $\delta$ and $E/r_{0}-\alpha$ and
from also ignoring various departures from the power-law relations assumed
above. However, in any case the evidence seems strong that the fraction of
energy radiated away goes to zero as a positive power of $1/\gamma$ when
$\gamma$ is taken to infinity, with the exponent apparently being positive but
less than one.

##  4 Possible Reasons for the Contrary Conclusion of Sperhake, Berti,
Cardoso, and Pretorius

Now I wish to examine some of the possible reasons for the contrary conclusion
of Sperhake, Berti, Cardoso, and Pretorius [57], “An extrapolation of our
results to the limit $\gamma\rightarrow\infty$ suggests that about half of the
center-of-mass energy of the system can be emitted in gravitational radiation,
while the rest must be converted into rest-mass and spin energy.”

First, the numerical data SBCP used only went up to $\gamma=2.49$, which is
somewhat large compared with unity, but hardly large enough to form definite
conclusions about what happens when $\gamma$ is taken to infinity. For
example, Sperhake, Cardoso, Pretorius, Berti, and González [43] found that a
good fit to their head-on collision data of the fraction, which I am hereby
labeling $f_{0}(\gamma)=\Delta E/E$, of the gravitational wave energy
radiated, $\Delta E$, out of the total energy, $E$, for impact parameter $b=0$
as a function of $\gamma\equiv E/M$ (with my $M$ and $E$, not their $M$ which
is the total ADM mass that is my $E$, and not their $E$ which is my $\Delta
E$) is given by the Zero Frequency Limit (ZFL) formula [14], the right hand
side of Eq. (3) in [43], with the energy cutoff for the ZFL evaluated as
$E_{\infty}=0.14\pm 0.03$, say $E_{\infty}=0.14$ for the numerical values
below, so that Eq. (3) in [43] can be written as

$\displaystyle f_{0}(\gamma)$ $\displaystyle=$ $\displaystyle
0.14\left(1+\frac{1}{2\gamma^{2}}-\frac{(4\gamma^{2}-1)\ln{(\gamma+\sqrt{\gamma^{2}-1})}}{2\gamma^{3}\sqrt{\gamma^{2}-1}}\right)$
(17) $\displaystyle=$ $\displaystyle
0.14\left(\frac{3-v^{2}}{2}-(1-v^{2})(3+v^{2})\frac{1}{4v}\ln{\frac{1+v}{1-v}}\right)$
$\displaystyle=$ $\displaystyle
0.14\sum_{n=2}^{\infty}\frac{8(n-1)v^{2n}}{(2n-3)(2n-1)(2n+1)}$
$\displaystyle=$ $\displaystyle
0.14\left(\frac{8}{15}v^{4}+\frac{16}{105}v^{6}+\frac{8}{105}v^{8}+\frac{32}{693}v^{10}+\frac{40}{1287}v^{12}+\frac{16}{715}v^{14}+\\!\cdots\\!\\!\right)\\!\\!.$

For $\gamma=2.49$, this formula gives a fraction $f_{0}(2.49)=0.077$ that is
only 55% as large as the $\gamma\rightarrow\infty$ limit of 0.14, which does
not seem sufficient for obtaining firm conclusions about the
$\gamma\rightarrow\infty$ limit.

Second, SBCP [57] fit to a formula that, as $1/\gamma\rightarrow 0$,
approaches a constant with a deviation proportional to $1/\gamma$, rather than
to $(1/\gamma)^{p}$ with the smaller exponent $p=1/(1+\alpha^{2}\epsilon)$
that my analysis gives. Therefore, it is perhaps not surprising that they did
not see the slower decrease of $1-f=(E-\Delta E)/E$ to zero for
$\gamma\rightarrow\infty$.

In particular, SBCP fit their numerical estimates for what I am calling
$f(\gamma)=\Delta E/E$, the fraction of the total energy $E$ radiated as
gravitational waves of energy $\Delta E$ at the $\gamma$-dependent critical
impact parameter $b=b_{c}=\beta_{c}E$, with the following formula for what I
shall label as $f_{\mathrm{SBCP}}(\gamma)$, in terms of the fraction
$f_{0}(\gamma)$ at $b=0$:

$f_{\mathrm{SBCP}}(\gamma)=\frac{f_{0}(\gamma)}{\mathcal{R_{\mathrm{SBCP}}}(\gamma)},\
\mathcal{R}_{\mathrm{SBCP}}(\gamma)=0.34(1-1/\gamma).$ (18)

Let me compare different fits to the data given for initially nonrotating
black holes in Table I of [57] for $f(\gamma)=\Delta E/E$, which is the
product of what is labeled in Table I as $K/M=(\gamma-1)/\gamma$ and as
$E_{\mathrm{rad}}/K$, with their $E_{\mathrm{rad}}$ being the same as my
$\Delta E$, the total gravitational wave energy radiated. These data for
$f(\gamma)$ are (here retaining all 6 digits from multiplying the 3-digit
values of $K/M$ and $E_{\mathrm{rad}}/K$ in Table I, leaving rounding to fewer
digits until later)

$f(1.22)=0.160\,921,\ f(1.88)=0.298\,116,\ f(2.49)=0.343\,252.$ (19)

More relevant for my arguments that $f(\gamma)\rightarrow 1$ as
$\gamma\rightarrow\infty$ are the values for

$F(\gamma)\equiv 1-f(\gamma)=M_{f}/E=(\mathrm{final\ rest\ mass\
}M_{f})/(\mathrm{total\ energy\ }E),$ (20)

where $M_{f}$ is the final mass of the black hole that forms for impact
parameter $b$ infinitesimally below the critical impact parameter $b_{c}$.
Table I of [57] then implies

$F(1.22)=0.839\,079,\ F(1.88)=0.701\,884,\ F(2.49)=0.656\,748.$ (21)

The SBCP fitting function $f_{\mathrm{SBCP}}(\gamma)$ of Eqs. (17) and (18)
then gives what I call

$F_{\mathrm{SBCP}}(\gamma)=1-\frac{0.14/0.34}{(1-1/\gamma)}\left(1+\frac{1}{2\gamma^{2}}-\frac{(4\gamma^{2}-1)\ln{(\gamma+\sqrt{\gamma^{2}-1})}}{2\gamma^{3}\sqrt{\gamma^{2}-1}}\right),$
(22)

with values

$\displaystyle F_{\mathrm{SBCP}}(1.22)$ $\displaystyle=$ $\displaystyle 0.854\
=\ 1.018\,F(1.22),$ $\displaystyle F_{\mathrm{SBCP}}(1.88)$ $\displaystyle=$
$\displaystyle 0.676\ =\ 0.963\,F(1.88),$ $\displaystyle
F_{\mathrm{SBCP}}(2.49)$ $\displaystyle=$ $\displaystyle 0.620\ =\
0.944\,F(2.49),$ (23)

which have an rms error relative to the numerical values of $F(\gamma)$ of
about 4.0%.

However, since my Eq. (15) for the asymptotic behavior of $F(\gamma)$ has two
unknown parameters, $\alpha$ and $\epsilon$ (ignoring weak dependencies on the
other unknown dimensionless parameters such as $\delta$ and $E/r_{0}-\alpha$),
for a fair comparison I should replace the one-parameter SBCP fitting function
$f_{\mathrm{SBCP}}(\gamma)=f_{0}(\gamma)/\mathcal{R}_{\mathrm{SBCP}}(\gamma)$,
having its one fitted constant 0.34 in
$\mathcal{R}_{\mathrm{SBCP}}(\gamma)=0.34(1-1/\gamma)$, with a two-parameter
fitting function $f_{2}(\gamma)=f_{0}(\gamma)/\mathcal{R}_{2}(\gamma)$ with
$\mathcal{R}_{2}(\gamma)=a-b/\gamma$, having two fitting parameters, $a$ and
$b$. Then a least-squares fit to the numerical data for $f(\gamma)$ given in
Eq. (19) yields $a=0.389$ and $b=0.406$ after rounding to 3 digits, and hence

$\displaystyle F_{2}(\gamma)$ $\displaystyle=$ $\displaystyle
1-\frac{f_{0}(\gamma)}{(0.389-0.406/\gamma)}$ (24) $\displaystyle=$
$\displaystyle
1-\frac{0.14}{(0.389-0.406/\gamma)}\left(1+\frac{1}{2\gamma^{2}}-\frac{(4\gamma^{2}-1)\ln{(\gamma+\sqrt{\gamma^{2}-1})}}{2\gamma^{3}\sqrt{\gamma^{2}-1}}\right)\\!.$

This fit gives

$\displaystyle F_{2}(1.22)$ $\displaystyle=$ $\displaystyle
0.8408=1.0021\,F(1.22),$ $\displaystyle F_{2}(1.88)$ $\displaystyle=$
$\displaystyle 0.7018=0.9999\,F(1.88),$ $\displaystyle F_{2}(2.49)$
$\displaystyle=$ $\displaystyle 0.6577=1.0014\,F(2.49),$ (25)

which have an rms error relative to the numerical values of $F(\gamma)$ of
only about 0.14%, a remarkably good fit, but perhaps the extreme excellency of
this fit is somewhat spurious because of the numerical uncertainties of the
values of $F(\gamma)$ given in Eq. (21) derived from the SBCP Table I of [57].

In contrast to these fits that have $1/\gamma$ appearing linearly in
$\mathcal{R}_{\mathrm{SBCP}}(\gamma)$ and $\mathcal{R}_{2}(\gamma)$ (though
there is a term asymptotically going as $(\ln{\gamma})/\gamma^{2}$ in
$f_{0}(\gamma)$ given by Eq. (17) from the right hand side of the ZFL Eq. (3)
of [43] that originally came from Eq. (2.20) of [14]), the derivation in
Section 3 above of the large-$\gamma$ asymptotic form given by Eq. (15)
suggests a fit by $F_{p}(\gamma)=A\gamma^{-p}$ with two parameters, $p\sim
1/(1+\alpha^{2}\epsilon)$ and $A\sim(\alpha/2)^{p}$. Then a least-squares fit
of the three numerical values of $\ln{F(\gamma)}$ by $\ln{A}-p\ln{\gamma}$
gives, to three digits each, $A=0.892$ and $p=0.343=(0.7)^{3}$, or

$F_{p}(\gamma)=0.892\,\gamma^{-0.343}.$ (26)

This fit gives $\alpha\sim 1.44$, $\epsilon\sim 0.93$ (though I would not
assign much precision to these estimated values, because of the uncertainty of
the extrapolation to $\gamma\gg 1$).

Then $F_{p}(\gamma)=0.892\,\gamma^{-0.343}$ gives

$\displaystyle F_{p}(1.22)$ $\displaystyle=$ $\displaystyle
0.833=0.993\,F(1.22),$ $\displaystyle F_{p}(1.88)$ $\displaystyle=$
$\displaystyle 0.718=1.023\,F(1.88),$ $\displaystyle F_{p}(2.49)$
$\displaystyle=$ $\displaystyle 0.652=0.993\,F(2.49),$ (27)

which has an rms relative error of about 1.46%, a bit over 10 times the rms
relative error of $F_{2}(\gamma)$. However, since the numerical error of the
data for $F(\gamma)$ in Eq. (21) is likely to be a few percent, it seems that
both formulas fit the three data points of Eq. (21) quite well. Certainly Eq.
(26) for $F_{p}(\gamma)$ looks simpler than Eq. (24) for $F_{2}(\gamma)$, and
appears even simpler than Eq. (22) for $F_{\mathrm{SBCP}}(\gamma)$ that does
not fit quite so well, though since the complication arises mostly from the
complexity of Eq. (17) for $f_{0}(\gamma)$, the fraction of energy radiated in
a head-on collision, which is theoretically determined by the Zero Frequency
Limit of [14] with only the one free parameter $E_{\infty}$ from the frequency
cutoff, there is actually only one free parameter that is fit to the data in
Eq. (22), namely the fraction $0.14/0.34=7/17\approx 0.412$.

The main point is that Eq. (15) has the theoretical justification given in
Section 3, as an approximate asymptotic formula for large
$\gamma=E/M=(\mathrm{total\ energy})/(\mathrm{rest\ mass})$, whereas the
factors of $1-1/\gamma$ and $0.389-0.406/\gamma$ in Eqs. (22) and (24) appear
to be rather ad hoc. And even if Eq. (15) is not quite the correct asymptotic
form for the fraction of energy not radiated when $\gamma\gg 1$, the fact that
during most of the evolution the black holes are expected to be much smaller
than their separations very strongly suggests that they cannot absorb a large
fraction of the initial energy.

A third possible reason for the disputed conclusion of Sperhake, Berti,
Cardoso, and Pretorius [57], that $F=1-f=(E-\Delta E)/E$ remains bounded below
by a positive number when $\gamma$ is taken to infinity, is that the 2007
results of Pretorius and Khurana [37] suggest that $f=\Delta E/E$, the
fraction of the initial total energy $E$ going into the energy $\Delta E$, is
a very sensitive function of the impact parameter $b$ when it is near the
critical impact parameter $b_{c}$. Therefore, it might be the case that for
the crucial data SBCP calculated at $\gamma=2.49$, which was done with impact
parameter $b=2.749E$ (in my notation; they use $M$ for the total ADM mass that
I call $E$), might not have been near enough to the impact parameter
$b_{m}=\beta_{m}E$ that would maximize the fraction $f=\Delta E/E$ going into
gravitational waves at that value of $\gamma=E/M$ (where my $M$ is the sum of
the initial two rest masses of the black holes).

An observation supporting the hypothesis that with the limited computational
resources available, SBCP were not able to tune the impact parameter close
enough to the value $b_{m}$ that would give the maximum energy radiated to
correctly evaluate what that maximum is for each $\gamma$, is the fact that
Figure 1 of [57] shows that the example they calculated for their maximum
$\gamma$, $\gamma=2.49$, does not have the black hole separation decreasing
monotonically before they reach their minimum separation and then move apart.
I would expect that at $b=b_{m}$, and also at $b=b_{c}$, the critical impact
parameter that maximizes $f=\Delta E/E$ out of all critical impact parameters,
the separation between the two black holes would decrease monotonically until
the black holes merge. Therefore, Figure 1 seems to be evidence that SBCP were
not able to tune the impact parameter sufficiently precisely to obtain the
maximum fraction $f$ of energy radiated. Of course, this is not a criticism of
their heroic effort to learn what they could at as large a value of $\gamma$
that was feasible with their calculational resources, but it does suggest that
even more resources will be needed to find numerical evidence convincingly
suggesting whether or not the final black hole (if the impact parameter $b$ is
infinitesimally smaller than the critical impact parameter $b_{c}=\beta_{c}E$
for fixed total energy $E$ that maximizes $f=\Delta E/E$ at that $E$) can have
a mass $M_{f}$ such that $M_{f}/E\rightarrow 0$ as $M/E\rightarrow 0$.

Actually, the ‘zoom-whirl’ behavior of the two black holes SBCP calculated for
$\gamma\equiv E/M=2.49$ (giving $v\approx 0.9158$) and impact parameter
$b=2.755E\approx 1.009E(2.5/v)$ (incidentally fitting very closely to the
estimate of Shibata, Okawa, and Yamamoto [44] that $b_{c}\approx 2.5E/v$)
suggests to me that perhaps that value of $b$ SBCP used is closer to a
critical dimensionless impact parameter that is different from the one that
maximizes $f=\Delta E/E$ and which I would expect would lead to something
close to an equiangular spiral in which the separation of the two black holes
decreases monotonically until they merge (if $b$ is infinitesimally below the
$b_{c}$ that maximizes $f$). So far the data gives me only a weak hint of the
possibility that there may be more than one critical impact parameter, and
that the fraction $f=\Delta E/E$ of energy radiated into gravitational waves
might not have a single local maximum as a function of $\gamma=E/M$. However,
it would be very interesting to explore this intriguing possibility.

I might initially guess that the number $N$ of local maxima of $f$ might grow
approximately linearly with $\gamma$, and that the number of critical impact
parameters might be $2N-1$ (say if there is one local maximum of $f$ for each
odd-numbered critical impact parameter). This would suggest that for fixed
total energy $E\gg M$ and fixed initial total rest mass $M=E/\gamma$, although
the global maximum for $f$ would occur for $b=\beta_{m}E\sim E$, $f(b)$ might
have some oscillatory structure with a period of the order of $\delta b\sim
M=E/\gamma$. On the other hand, this would give a range for all $2N-1$
critical impact parameters of $\Delta b\sim N\delta b\sim E$, whereas I would
expect the ratio of the range of the oscillatory behavior to the value of
$b_{m}$ (with $b_{m}$ giving the global maximum for $f$ for the fixed values
of $E$ and $M$) to decrease as $\gamma$ is increased, so perhaps the typical
period of the oscillations, say $\delta b$, goes as $b_{m}\sim E$ multiplied
by a power of $1/\gamma=M/E$ that is larger than one, so that $\delta b$
becomes far smaller than $M$ for large $\gamma$. Or, perhaps the number $N$ of
local maxima of $f$ might increase slower than linearly with $\gamma$, so that
the typical period of the oscillations, $\delta b$, could remain of the order
of $M$ but the range, $\Delta b\sim N\delta b$, could become much smaller than
$b_{m}$ for large $\gamma$. In any case, it would be very interesting to see
what the behavior of $f(\gamma)$ is as a function of $\gamma=E/M$ when it
becomes large, and to check whether it exhibits any oscillations, giving more
than one local maximum.

Perhaps when the computational resources become available for making many more
calculations at slightly different impact parameters for the same $\gamma$,
and ideally also for larger $\gamma$, it might become more apparent whether or
not the fraction $F$ of the total energy that does not get radiated does
approach zero as $\gamma$ is increased indefinitely with $\beta\equiv b/E$
tuned to the $\gamma$-dependent value $\beta_{m}(\gamma)$ that maximizes the
fraction $f=1-F$ of the total energy that is radiated for each $\gamma=E/M$.

Another argument for $F(\gamma)\rightarrow 0$ as $\gamma\rightarrow\infty$ is
that in the limit $1/\gamma=0$ at fixed total energy $E$, the rest mass of
each black hole, $M/2=E/(2\gamma)$, goes to zero, so initially one has two
incoming Aichelburg-Sexl metrics [13], corresponding to the gravitational
fields of two classical massless point particles. In this case there are, at
least initially, no black holes to absorb any of the energy, so when SBCP
argue [57] that “absorption sets an upper bound on the maximum energy that can
be radiated,” it would seem that no absorption would occur in the $M=0$ limit,
since there are no black holes to absorb the energy (until the two massless
particles form a single black hole).

One weakness of this last argument is that it might occur that in the
Aichelburg-Sexl limit, as massless particle A collides with the gravitational
shock wave of massless particle B, the gravitational shock wave field of A
might focus the gravitational shock wave field of B to form a black hole
separate from a black hole that might form from the field of B focusing the
shock wave of A. Thus perhaps two black holes might form that conceivably each
could have its rest mass be a fraction of the total energy that is bounded
below by a positive number. In this case that positive number would give a
lower bound on the ratio $M_{f}/E$ for the final black hole that forms with
impact parameter $b$ slightly below the critical impact parameter $b_{c}$.

This conceptual possibility raises the question of whether these possible two
distinct holes each engulf the corresponding massless particle, which might
limit the number of black holes that form to that pair, or whether the two
holes that might form from the focusing of the gravitational shock wave of one
massless particle by the shock wave of the other might form behind the
massless particles rather than engulfing them. This latter possibility raises
the question of whether the gravitational fields of the two persisting
massless particles might focus to produce even more pairs of black holes.
Although I am rather sceptical that the gravitational fields of the two
massless particles can form even one pair of black holes that persist before
merging to form one final black hole (if $b<b_{c}$) or eventually flying apart
(if $b>b_{c}$), if two black holes can form without engulfing the massless
particles, it would seem conceivable that an arbitrarily large number of pairs
of black holes could form from suitable tuning of $\beta=b/E$, and even that
an arbitrarily large number could survive the encounter rather than merging if
$b<b_{c}$.

However, the more plausible possibility for $1/\gamma=M/E=0$ seems to be that
the two massless particles that are the source of the original pair of
Aichelburg-Sexl metrics cannot form separate black holes but only a single
black hole if they merge. In this case, there are never any separate black
holes to absorb any of the radiation, and it seems most plausible that by fine
tuning the ratio $\beta=b/E$ of the impact parameter $b$ to the total energy
$E$ to the critical value $\beta_{0}$, all of the energy would be radiated
away before the two particles merge and disappear at a naked singularity
corresponding to a final single black hole in the limit that its mass goes to
zero.

##  5 Conclusions

In this paper I have given arguments that the 2007 conjecture of Pretorius and
Khurana [37] is essentially correct, that all but an arbitrarily small
fraction of the total energy $E$ of ultrarelativistic objects of original
total rest mass $M$ can be converted to gravitational waves if $M/E$ can be
taken small enough. (I do not argue that all of the kinetic energy can be
converted, since the final total rest mass $M_{f}$ could be larger than $M$,
but only that as $M/E$ is taken to zero, $M_{f}/E$ also approaches zero with
the proper fine tuning of the impact parameter.) The argument for two original
black holes with $M/E\ll 1$ is that during most of the gravitational wave
emission, the black holes are so much smaller than the region where most of
the gravitational wave energy is located that they can absorb only a tiny
fraction of it, with this fraction going to zero as $M/E$ is taken to zero.

Contrary arguments in 2012 by Sperhake, Berti, Cardoso, and Pretorius do not
seem sufficient to refute this argument, since (1) they could only calculate
the emission near the critical impact parameter for $\gamma\leq 2.49$, (2)
they used a rather ad hoc fitting function that does not agree with the
theoretical expectations put forward in this paper, and (3) they do not seem
to have been able to tune the impact parameter close enough to the critical
value to get a convincingly good estimate of the maximum fraction of energy
that could be emitted for their finite values of $E/M$.

## Acknowledgments

I am grateful for email discussions with Emanuele Berti, Vitor Cardoso, Frans
Pretorius, Ulrich Sperhake, Gabriele Veneziano, and Huan Yang. This work was
supported in part by the Natural Sciences and Engineering Research Council of
Canada.

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