By carefully studying the classical limit of amplitudes for finite spin, and
computing the momentum and angular impulses, we were able to precisely
reproduce the leading spin, 1PM scattering data for Kerr black holes. The
complete, all-spin classical expressions, (5.5), can be calculated from
amplitudes by applying our methods to massive spinor helicity representations
in the large spin limit [128]. These results are one artefact of a beautiful
relationship between black holes and amplitudes: minimally coupled graviton
amplitudes are the on-shell avatar of the no-hair theorem [221, 227]. This
provocative idea can provide non-geometric insights into intriguing results in
general relativity, such as the Newman–Janis complex map between the
Schwarzschild and Kerr solutions, which is naturally explained by the
exponentiation of the minimally coupled 3–point amplitude in the large spin
limit [129].

We extended this on-shell insight into black hole physics in chapter 6, where
we showed that the Newman–Janis shift could be interpreted in terms of a
worldsheet effective action. This holds both in gravity, and for the single-
copy $\sqrt{\rm Kerr}$ solution in electrodynamics. Moreover, at the level of
equations of motion we showed that the NJ shift holds also for the leading
interactions of the Kerr black hole. These leading interactions were
conveniently described using chiral classical equations of motion with the
help of the spinor-helicity method familiar from scattering amplitudes. These
spinor equations of motion are extremely powerful tools — they offered
remarkably efficient derivations of the on-shell observables calculated from
amplitudes earlier in the thesis, encapsulating the full on-shell data for
non-aligned black hole scattering in the kick of the holomorphic spinor. It
was also a trivial matter to extend the scope of this technology to the
magnetically charged Kerr–Taub–NUT black hole, facilitating the first
calculation of the angular impulse for this solution. It would be interesting
to see if our methods can be generalised to include the full family of
parameters of the famous Plebanski–Demianski family [278, 279].

The full geometry of our worldsheet remains to be explored, in particular its
applicability when higher-order curvature terms are accounted for. Such
corrections are crucial for the extending the power of the chiral spinor
equations. A major motivation for tackling this problem comes from an obstacle
towards progress in amplitudes computations of importance for gravitational
wave astronomy: a well defined, tree-level expression for the four-point
Compton amplitude, valid for any generic spin $s$, is not known. Application
of BCFW recursion with arbitrary spin representations leads to spurious poles
[223, 230]; these poles can only be removed by hand ambiguously [227]. As we
have seen in chapter 3, triangle diagrams play a crucial role in the
computation of either the NLO impulse or potential. The Compton amplitude is a
key component to the computation of these topologies using generalised
unitarity, and thus progress in computing precision post–Minkowksian
corrections for particles with spin is impeded by a crucial tree-level input
into the calculation.

The all-spin Compton amplitude should have a well defined, unambiguous
classical limit. Physically the amplitude corresponds to absorption and re-
emission of a messenger boson, a problem which is tractable in classical field
theory. Our worldsheet construction shows that the Newman–Janis shift holds
for the leading interactions of Kerr black holes, and thus the results of
chapter 6 should offer a convenient way to approach this problem, provided
that higher order contributions can be included. We hope that, in this manner,
our work will be only the beginning of a programme to exploit the Newman–Janis
structure of Kerr black holes to simplify their dynamics.

###  Appendix A Wavefunction integrals

In this appendix we detail the evaluation of the explicit momentum space
wavefunction integrals that played a key role in our disccussion of the
classical limit in chapters 2 and 3.

We begin with the single particle momentum space wavefunction normalisation in
equation (2.43). To determine this we must compute the following integral:

$m^{-2}\int d\Phi(p)\;\exp[-\frac{2p\cdot u}{m\xi}\biggr{]}\,.$ (A.1)

Let us parametrise the on-shell phase space in a similar manner to equation
(3.30), but appropriate for timelike momenta, writing

$p^{\mu}=E_{p}\bigl{(}\cosh\zeta,\,\sinh\zeta\sin\theta\cos\phi,\,\sinh\zeta\sin\theta\sin\phi,\sinh\zeta\cos\theta\bigr{)}\
$ (A.2)

so that

$\displaystyle d\Phi(p)$ $\displaystyle=(2\pi)^{-3}\hat{d}E_{p}d\zeta
d\Omega_{2}\,\hat{\delta}(E_{p}^{2}-m^{2})\Theta(E_{p})\,E_{p}^{3}\sinh^{2}\zeta$
(A.3) $\displaystyle=(2\pi)^{-3}\hat{d}E_{p}d\zeta d\theta
d\phi\,\hat{\delta}(E_{p}^{2}-m^{2})\Theta(E_{p})\,E_{p}^{3}\sinh^{2}\zeta\sin\theta\,.$

Performing the $E_{p}$ integration, we obtain

$d\Phi(p)\rightarrow\frac{m^{2}}{2(2\pi)^{3}}d\zeta d\theta
d\phi\,\sinh^{2}\\!\zeta\sin\theta\,,$ (A.4)

along with $E_{p}=m$ in the integrand. The integral must be a Lorentz-
invariant function of $u$; as the only available Lorentz invariant is
$u^{2}=1$, we conclude that the result must be a function of $\xi$ alone. We
can compute it in the rest frame of $u$, where our desired integral is

$\frac{1}{2}\\!\int_{0}^{\infty}\\!\hat{d}\zeta\,\sinh^{2}\\!\zeta\\!\int_{0}^{\pi}\\!\hat{d}\theta\,\sin\theta\\!\int_{0}^{2\pi}\\!\hat{d}\phi\,\exp[-\frac{2\cosh\zeta}{\xi}\biggr{]}=\frac{1}{2(2\pi)^{2}}\xi\,K_{1}(2/\xi)\,,$
(A.5)

where $K_{1}$ is a modified Bessel function of the second kind. The
normalisation condition (2.19) then yields

$\frac{2\sqrt{2}\pi}{\xi^{1/2}K_{1}^{1/2}(2/\xi)}$ (A.6)

for the wavefunction’s normalisation.

Next, we compute $\langle p^{\mu}\rangle$. Lorentz invariance implies that the
expectation value must be proportional to $u^{\mu}$; again computing in the
rest frame, we find that

$\langle p^{\mu}\rangle=mu^{\mu}\,\frac{K_{2}(2/\xi)}{K_{1}(2/\xi)}\,.$ (A.7)

The phase-space measure fixes $\langle p^{2}\rangle=m^{2}$, so we conclude
that

$\displaystyle\frac{\sigma^{2}(p)}{\langle p^{2}\rangle}=1-\frac{\langle
p\rangle^{2}}{\langle p^{2}\rangle}$
$\displaystyle=1-\frac{K_{2}^{2}(2/\xi)}{K_{1}^{2}(2/\xi)}$ (A.8)
$\displaystyle=-\frac{3}{2}\xi+\mathcal{O}(\xi^{2})\,,$

where as $\xi\rightarrow 0$ we have used the asymptotic approximations

$\displaystyle K_{1}(2/\xi)$
$\displaystyle=\sqrt{\frac{\pi\xi}{4}}\,\exp\left[-\frac{2}{\xi}\right]\left(1+\frac{3\xi}{16}+\mathcal{O}(\xi^{2})\right),$
(A.9) $\displaystyle K_{2}(2/\xi)$
$\displaystyle=\sqrt{\frac{\pi\xi}{4}}\,\exp\left[-\frac{2}{\xi}\right]\left(1+\frac{15\xi}{16}+\mathcal{O}(\xi^{2})\right).$

For the single particle case it remains to compute the double expectation
value in equation (2.49). We can again apply Lorentz invariance, which
dictates that

$\langle p^{\mu}p^{\nu}\rangle=Au^{\mu}u^{\nu}+B\eta^{\mu\nu}\,.$ (A.10)

Contracting with the velocity leads to

$A+B=\frac{\mathcal{N}^{2}}{m^{2}}\\!\int\\!d\Phi(p)\,(u\cdot
p)^{2}\,\exp\left[-\frac{2u\cdot
p}{m\xi}\right]=m^{2}+\frac{3}{2}m^{2}\xi\frac{K_{2}(2/\xi)}{K_{1}(2/\xi)}\,,$
(A.11)

where we used the result in equation (A.5). Combining this constraint with the
trace of (A.10) is then enough to determine the coefficients as

$A=m^{2}+2m^{2}\xi\frac{K_{2}(2/\xi)}{K_{1}(2/\xi)}\,,\qquad
B=-\frac{m^{2}}{2}\xi\frac{K_{2}(2/\xi)}{K_{1}(2/\xi)}\,,$ (A.12)

yielding the result in equation (2.49).

Our discussion of point-particle scattering in section 3.2 hinged upon the
integral

$T(\bar{q})=\frac{\mathcal{N}^{2}}{\hbar m^{3}}\exp[-\frac{\hbar\bar{q}\cdot
u}{m\xi}\biggr{]}\int
d\Phi(p)\;\hat{\delta}(2p\cdot\bar{q}/m+\hbar\bar{q}^{2}/m)\,\exp[-\frac{2p\cdot
u}{m\xi}\biggr{]}\,.$ (A.13)

This integral is dimensionless, and can depend only on two Lorentz invariants,
$\bar{q}\cdot u$ and $\bar{q}^{2}$, along with $\xi$. It is convenient to
write it as a function of two dimensionless variables built out of these
invariants,

$\omega\equiv\frac{\bar{q}\cdot
u}{\sqrt{-\bar{q}^{2}}}\,,\qquad\tau\equiv\frac{\hbar\sqrt{-\bar{q}^{2}}}{2m}\,.$
(A.14)

We again work in the rest frame of $u^{\mu}$, and without loss of generality,
choose the $z$-axis of the $p$ integration to lie along the direction of
$\boldsymbol{{\bar{q}}}$. The only components that appear in the integral are
then $\bar{q}^{0}$ and $\bar{q}^{z}$; after integration, we can obtain the
dependence on $\tau$ and $\omega$ via the replacements

$\bar{q}^{0}\rightarrow\frac{2m\omega\tau}{\hbar}\,,\qquad\bar{q}^{z}\rightarrow\frac{2m\sqrt{1+\omega^{2}}\tau}{\hbar}\,;$
(A.15)

hence,

$-\\!\bar{q}^{2}\rightarrow\frac{4m^{2}\tau^{2}}{\hbar^{2}}\,.$ (A.16)

Using the measure (A.4), we find that

$\displaystyle T_{1}(\bar{q})$ $\displaystyle=\frac{1}{2\pi\,\hbar m\,\xi
K_{1}(2/\xi)}\exp[-\frac{\hbar\bar{q}^{0}}{m\xi}\biggr{]}$ (A.17)
$\displaystyle\qquad\times\int_{0}^{\infty}\\!d\zeta\,\sinh^{2}\\!\zeta\\!\int_{0}^{\pi}\\!d\theta\,\sin\theta\\!\int_{0}^{2\pi}\\!d\phi\,\exp[-\frac{2\cosh\zeta}{\xi}\biggr{]}$
$\displaystyle\qquad\qquad\qquad\times\hat{\delta}(2\bar{q}^{0}\cosh\zeta-2\bar{q}^{z}\sinh\zeta\cos\theta+\hbar\bar{q}^{2}/m)\,.$

The $\phi$ integral is trivial, and we can use the delta function to do the
$\theta$ integral:

$\displaystyle T_{1}(\bar{q})$ $\displaystyle=\frac{1}{2\hbar
m\,\bar{q}^{z}\,\xi
K_{1}(2/\xi)}\exp[-\frac{\hbar\bar{q}^{0}}{m\xi}\biggr{]}\int_{0}^{\infty}d\zeta\;\sinh\zeta\;\exp[-2\frac{\cosh\zeta}{\xi}\biggr{]}$
(A.18)
$\displaystyle\qquad\qquad\qquad\times\Theta\bigl{(}1+\bar{q}^{0}\coth\zeta/\bar{q}^{z}+\hbar\bar{q}^{2}\textrm{cosech}\,\zeta/(2m\bar{q}^{z})\bigr{)}\,$
$\displaystyle\qquad\qquad\qquad\times\Theta\bigl{(}1-\bar{q}^{0}\coth\zeta/\bar{q}^{z}-\hbar\bar{q}^{2}\textrm{cosech}\,\zeta/(2m\bar{q}^{z})\bigr{)}\,.$

In the $\hbar\rightarrow 0$ limit, the first theta function will have no
effect, even with $\bar{q}^{2}<0$. Changing variables to $w=\cosh\zeta$, the
second theta function will impose the constraint

$w\geq\frac{\bar{q}^{z}\sqrt{1-\hbar^{2}\bar{q}^{2}/(4m^{2})}}{\sqrt{-\bar{q}^{2}}}-\frac{\hbar\bar{q}^{0}}{2m}\,.$
(A.19)

In terms of $\omega$ and $\tau$, this constraint is

$w\geq\sqrt{1+\omega^{2}}\sqrt{1+\tau^{2}}-\omega\tau\,.$ (A.20)

Up to corrections of $\mathcal{O}(\hbar)$, the right-hand side is greater than
1, and so becomes the lower limit of integration. The result for the integral
is then

$\displaystyle T(\bar{q})$
$\displaystyle=\frac{1}{8m^{2}\sqrt{1+\omega^{2}}\tau\,K_{1}(2/\xi)}\exp[-\frac{2\omega\tau}{\xi}\biggr{]}$
(A.21)
$\displaystyle\qquad\qquad\qquad\times\exp[-\frac{2}{\xi}\Bigl{(}\sqrt{1+\omega^{2}}\sqrt{1+\tau^{2}}-\omega\tau\Bigr{)}\biggr{]}$
$\displaystyle=\frac{1}{4\hbar m\sqrt{(\bar{q}\cdot
u)^{2}-\bar{q}^{2}}\,K_{1}(2/\xi)}$
$\displaystyle\qquad\qquad\qquad\times\exp[-\frac{2}{\xi}\frac{\sqrt{(\bar{q}\cdot
u)^{2}-\bar{q}^{2}}}{\sqrt{-\bar{q}^{2}}}\sqrt{1-\hbar^{2}\bar{q}^{2}/(4m^{2})}\biggr{]}\,,$

which is the result listed in equation (3.25).

###  Appendix B Worldline perturbation theory

Throughout part I we computed on-shell classical observables from scattering
amplitudes. We justified our results by their agreement with iterative
solutions to the appropriate classical equations of motion, as shown in detail
in refs. [1, 2]. To provide a flavour of these methods, let us calculate the
LO current for gravitational radiation due to the electromagnetic scattering
of Reissner–Nordström black holes, which we studied using the double copy in
section 4.4.1.

Classically, the emitted gravitational radiation will satisfy the linearised
Einstein–Maxwell field equation, which for the trace-reversed perturbation in
de–Donder gauge is

$\partial^{2}\bar{h}^{\mu\nu}(x)=T^{\mu\nu}(x)=\frac{\kappa}{2}\left(T_{\rm
pp}^{\mu\nu}(x)+T_{\rm EM}^{\mu\nu}(x)\right).$ (B.1)

The electromagnetic field strength tensor $T^{\rm EM}_{\mu\nu}$ is that in
equation (4.30), while in gravity

$T_{\rm
pp}^{\mu\nu}(x)=\sum_{\alpha=1}^{n}m_{\alpha}\int\\!\frac{d\tau_{\alpha}}{\sqrt{g}}\,u^{\mu}_{\alpha}(\tau)u^{\nu}_{\alpha}(\tau)\,\delta^{(D)}(x-x_{\alpha}(\tau))$
(B.2)

for $n$ particles, where $g=-\det(g_{\mu\nu})$. We are seeking perturbative
solutions, and therefore expand worldline quantities in the coupling:

$\displaystyle x_{\alpha}^{\mu}(\tau_{\alpha})$
$\displaystyle=b_{\alpha}^{\mu}+u_{\alpha}\tau_{\alpha}+\Delta^{(1)}x_{\alpha}^{\mu}(\tau_{\alpha})+\Delta^{(2)}x_{\alpha}^{\mu}(\tau_{\alpha})+\cdots\,,$
(B.3) $\displaystyle v_{\alpha}^{\mu}(\tau_{\alpha})$
$\displaystyle=u_{\alpha}+\Delta^{(1)}v_{\alpha}^{\mu}(\tau_{\alpha})+\Delta^{(2)}v_{\alpha}^{\mu}(\tau_{\alpha})+\cdots\,.$

Here $\Delta^{(i)}x_{\alpha}^{\mu}$ will indicate quantities entering at
$\mathcal{O}(\tilde{g}^{2i})$, for couplings $\tilde{g}=e$ or $\kappa/2$. We
also work with boundary conditions
$x^{\mu}_{\alpha}(\tau_{\alpha}\rightarrow-\infty)=b^{\mu}_{\alpha}+u^{\mu}_{\alpha}\tau_{\alpha}$,
so $v_{\alpha}^{\mu}(\tau_{\alpha}\rightarrow-\infty)=u^{\mu}_{\alpha}$. At
leading order we can treat the contributions to the field equation separately,
splitting the problem into gravitational radiation sourced by either the
particle worldlines or the electromagnetic field.

Of course we are interested in electromagnetic interactions of two particles,
and thus require the leading order solution to the Maxwell equation in Lorenz
gauge, $\partial^{2}A^{\mu}(x)=eJ^{\mu}_{\textrm{pp}}(x)$, where the (colour-
dressed) current appears in (2.21c) This is a very simple calculation, and
once quickly finds that the field sourced by particle 2, say, is

$F^{\mu\nu}_{2}(x)=ie\\!\int\\!\hat{d}^{4}\bar{q}\,\hat{\delta}(u_{2}\cdot\bar{q})e^{i\bar{q}\cdot
b_{2}}e^{-i\bar{q}\cdot
x}\frac{\bar{q}^{\mu}u_{2}^{\nu}-u_{2}^{\mu}\bar{q}^{\nu}}{\bar{q}^{2}}\,;$
(B.4)

see [1] for details. The key point is that we can now obtain the leading
deflections simply by solving the classical equation of motion, the Lorentz
force, which yields

$\frac{d\Delta^{(1)}v^{\mu}_{1}}{d\tau}=\frac{ie^{2}}{m_{1}}\\!\int\\!\hat{d}^{4}\bar{q}\,\hat{\delta}(u_{2}\cdot\bar{q})e^{-i\bar{q}\cdot(b_{1}-b_{2})}\frac{e^{-i\bar{q}\cdot
u_{1}\tau}}{\bar{q}^{2}}\Big{(}\bar{q}^{\mu}\,u_{1}\cdot
u_{2}-u_{2}^{\mu}\,u_{1}\cdot\bar{q}\Big{)}\,.$ (B.5)

We now have all the data required to determine the full LO gravitational
current. Focusing on the radiation sourced from the point-particle worldlines,
we can expand (B.2) to find

$T^{\mu\nu}_{\textrm{pp},(1)}(\bar{k})=-2e^{2}\\!\int\\!\hat{d}^{4}\bar{q}\,\hat{\delta}((\bar{k}-\bar{q})\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i(\bar{k}-\bar{q})\cdot
b_{1}}e^{i\bar{q}\cdot b_{2}}}{\bar{q}^{2}(\bar{q}\cdot
u_{1})}\bigg{[}u_{1}\cdot u_{2}\,\bar{q}^{(\mu}u_{1}^{\nu)}\\\
-\frac{1}{2}u_{1}\cdot
u_{2}\,\frac{\bar{k}\cdot\bar{q}\,u_{1}^{\mu}u_{1}^{\nu}}{\bar{q}\cdot
u_{1}}-\bar{q}\cdot u_{1}\,u_{1}^{(\mu}u_{2}^{\nu)}+\frac{1}{2}\bar{k}\cdot
u_{2}\,u_{1}^{\mu}u_{1}^{\nu}\bigg{]}+(1\leftrightarrow 2)\,.$ (B.6)

It is then convenient to relabel $\bar{q}=\bar{w}_{2}$, and introduce a new
momentum $\bar{w}_{1}=\bar{k}-\bar{w}_{2}$. In these variables, the analogous
contribution from the EM stress tensor in (B.4) to the gravitational radiation
is

$T^{\mu\nu}_{\textrm{pp},(1)}(x)=-e^{2}\\!\int\\!\hat{d}^{4}\bar{w}_{1}\hat{d}^{4}\bar{w}_{2}\,\hat{\delta}(\bar{w}_{1}\cdot
u_{1})\hat{\delta}(\bar{w}_{2}\cdot u_{2})\frac{e^{i\bar{w}_{1}\cdot
b_{1}}e^{i\bar{w}_{2}\cdot
b_{2}}}{\bar{w}_{1}^{2}\bar{w}_{2}^{2}}e^{-i(\bar{w}_{1}+\bar{w}_{2})\cdot
x}\\\ \times\bigg{[}2\bar{w}_{1}\cdot
u_{2}\,u_{1}^{(\mu}\bar{w}_{2}^{\nu)}-\bar{w}_{1}\cdot\bar{w}_{2}\,u_{1}^{\mu}u_{2}^{\nu}-u_{1}\cdot
u_{2}\,\bar{w}_{1}^{\mu}\bar{w}_{2}^{\nu}+(1\leftrightarrow 2)\bigg{]}\,.$
(B.7)

Summing the two pieces and restoring classical momenta
$p_{\alpha}=m_{\alpha}u_{\alpha}$ yields

$T^{\mu\nu}_{(1)}(\bar{k})=-\frac{e^{2}\kappa}{4}\\!\int\\!\hat{d}^{4}\bar{w}_{1}\hat{d}^{4}\bar{w}_{2}\,\hat{\delta}(\bar{w}_{1}\cdot
p_{1})\hat{\delta}(\bar{w}_{2}\cdot
p_{2})\hat{\delta}^{(4)}(\bar{k}-\bar{w}_{1}-\bar{w}_{2})\,e^{i\bar{w}_{1}\cdot
b_{1}}e^{i\bar{w}_{2}\cdot b_{2}}\\\
\times\bigg{[}\frac{Q_{12}^{\mu}P_{12}^{\nu}+Q_{12}^{\nu}P_{12}^{\mu}}{\bar{w}_{1}^{2}\bar{w}_{2}^{2}}+(p_{1}\cdot
p_{2})\left(\frac{Q_{12}^{\mu}Q_{12}^{\nu}}{\bar{w}_{1}^{2}\bar{w}_{2}^{2}}-\frac{P_{12}^{\mu}P_{12}^{\nu}}{(\bar{k}\cdot
p_{1})^{2}(\bar{k}\cdot p_{2})^{2}}\right)\bigg{]}\,,$ (B.8)

where we have adopted the gauge invariant functions defined in (4.58). Upon
contraction with a graviton field strength tensor this current then agrees
(for $b_{2}=0$, and up to an overall sign) with equation (4.60), the LO
gravitational radiation kernel for electromagnetic scattering of
Reissner–Nordström black holes.

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