Quantum gravity on the black hole horizon

Inelastic collision of high energy bodies is expected to result in production of black holes. Moreover, at large impact parameters (compared to the Schwarzschild radius), the collapse is expected to be described by classical physics, dominated by eikonalised single graviton exchange [1, 2]. Of course, Hawking famously claimed that semi-classical physics results in information loss in the final state [3]. This may be seen as a thermalisation of the initial states among the large number of degrees of freedom associated to the intermediate macroscopic black hole. Smaller impact parameters and presumably complicated quantum corrections are often accused of hiding the mysteries of information loss in all aspects of scattering after collapse, in the evaporation process [4].

Interestingly enough, the cross-section of the inelastic collisions producing black holes resembles scattering of the initial states on an existing, already formed Schwarzschild background [5]. This is owed to the fact that, during the collapse of a spherical shell for instance, the (apparent) horizon forms long before the shell has fallen past the Schwarzschild radius ($R_{S}$) of the eventual black hole. Therefore, one expects to capture a part of the physics associated with scattering in flat space with impact parameter of the order of the Schwarzschild radius by studying scattering on the intermediate black hole state (already at a perturbative level). The conventional eikonal approximation [6] has been used to study physics (at impact parameters larger than $R_{S}$) in flat space gravitational scattering [7, 8, 9]. In that case, the scattering energies are necessarily Planckian $\sqrt{s}\gg M_{Pl}$ and the impact parameter larger than the Schwarzschild radius $b\gg R_{S}$.

In the present article, we consider scattering near the Schwarzschild horizon in a partial wave basis. Therefore, the impact parameter is necessarily of the order of the Schwarzschild radius or less: $b\leq R_{S}$. We work in an approximation where transverse momentum transfer on the sphere is small, implying $b\gg L_{Pl}$. There are three scales left in the game: the centre of mass energy of the scattering (say $\sqrt{s}$), the mass of the intermediate black hole state $M_{BH}$, and the Planck mass $M_{Pl}$. From these, two dimensionless parameters arise, say $\gamma\coloneqq\sqrt{8\pi G_{N}}/R_{S}\sim M_{Pl}/M_{BH}$ and $s/(\gamma^{2}M^{2}_{Pl})$. Remarkably we find a result that is non-perturbative in $\gamma$, by summing over an infinite number of graviton-exchanging ladder diagrams. This implies that the result is also non-perturbatively resummed in $\hbar$ because the ladder diagrams include infinitely many loop contributions. At every order in $\gamma$ in the ladder diagrams, we work to leading order in $s\gg\gamma^{2}M^{2}_{Pl}$. However, unlike the conventional eikonal approximation, this is far easier to satisfy for black holes much bigger than Planck size. For a solar mass black hole, for instance, the center of mass energy required for this approximation is $\sqrt{s}\gg 10^{-38}M_{Pl}$; this is already satisfied for all the massive particles in the standard model with their rest mass alone. Moreover, the energy of particles diverges near the horizon, validating this approximation further. Therefore, this is indeed the strong gravity regime where the mysteries of black hole information loss must necessarily be resolved.

We compute a four-point correlation function of matter fields on the Schwarzschild horizon. This includes multi-particle states owing to the fact that we work with fields in a partial wave basis. The correlator captures scattering processes in two different channels (much like in familiar nucleon-nucleon scattering), both exchanging virtual gravitons. The result is surprisingly simple and lends support to the S-matrix approach for quantum black holes and generalises earlier results in [10, 11, 12, 13]. Furthermore, this work naturally provides a completely second-quantised description of the earlier quantum mechanical results. Since the black hole geometry is locally weakly curved, one may not expect this calculation to be very instructive. However, as the scattering matrix maps asymptotic states, very rich physics emerges that is characteristically different to that of flat space. For instance, we find that the amplitude is entirely dominated by low angular momentum modes; in contrast, the flat space amplitude is dominated by very large angular momentum modes. While certain numerical factors require detailed calculations which we leave to a companion article [14], the conceptual idea and the results (up to explicit numerical factors) can be obtained from symmetry principles as we show in this paper. The theory under consideration is in Sec. 2, the ladder diagrams to be computed in Sec. 3, and a discussion on the results in Sec. 4.

We consider the Einstein-Hilbert action coupled minimally to matter, say a massless scalar field for simplicity:

$$S~{}=~{}\dfrac{1}{2}\int\sqrt{-g}\,\mathrm{d}^{4}x\left(\dfrac{R}{8\pi G_{N}}-~{}\nabla_{\mu}\phi\nabla^{\mu}\phi\right)\,.$$

We focus on the fluctuations about a spherically symmetric background $\left(g_{\mu\nu}\rightarrow g^{0}_{\mu\nu}+\kappa h_{\mu\nu}\right)$ and a vanishing scalar profile, where

$$g^{0}_{\mu\nu}~{}=~{}-2A\left(x,y\right)\mathrm{d}x\mathrm{d}y+r\left(x,y\right)\mathrm{d}\Omega^{2}_{2}\,,$$

where $\mathrm{d}\Omega^{2}_{2}$ is the round metric on the two-sphere, and $\kappa=\sqrt{8\pi G_{N}}$. For the Schwarzschild background, we have that $A=\exp\left(1-r/R_{S}\right)R_{S}/r$ and $r$ is defined via

$$xy~{}=~{}2R^{2}_{S}\left(1-r/R_{S}\right)\exp\left(r/R_{S}-1\right)$$

(2.1)

in the familiar Kruskal-Szekeres coordinates. While we will later focus on the vacuum Schwarzschild solution in this note, we present a method that applies to generic spherically symmetric solutions supported by matter. Of course, the fluctuations $h_{\mu\nu}$ are not all physical and the field needs to be gauge-fixed, an issue we now turn to.

The tree-level transfer channel ladder diagram i {fmffile}feyn_tree

{fmfgraph*}

(80,80) \fmfstraight\fmflefti1,i2 \fmfrighto1,o2 \fmflabel$p_{2}$i1 \fmflabel$p_{2}$o1 \fmflabel$p_{1}$i2 \fmflabel$p_{1}$o2 \fmfdashes_arrowi1,v1 \fmffermionv1,o1 \fmffermioni2,v2 \fmfdashes_arrowv2,o2 \fmfgluon,label=$k=0$v1,v2

The amplitude is straight forward to calculate and is given by $i\mathcal{M}=4is^{2}\gamma^{2}R^{2}_{S}/\left(\lambda+1\right)=4is^{2}\kappa^{2}/\left(\lambda+1\right)$, where we defined the Mandelstam variable that gives the center of mass energy of the collision: $s=-\frac{1}{2}\left(p_{1}+p_{2}\right)^{2}=p^{1}_{x}p^{2}_{y}$; this is owed to the fact that $p^{1}_{y}=0=p^{2}_{x}$ as discussed in Sec. 2.2. Moving on to loop diagrams, a two loop diagram in the transfer channel is of the following form: {fmffile}feyn_transfertwoloop

{fmfgraph*}

(200,90) \fmfstraight\fmflefti1,i2 \fmfrighto1,o2 \fmfdashes_arrowi1,v1 \fmffermionv5,o1 \fmffermioni2,v2 \fmffermionv1,v3 \fmfdashesv2,v4 \fmffermionv4,v6 \fmfdashesv3,v5 \fmfdashes_arrowv6,o2 \fmfgluon, tension=0v1,v2 \fmfgluon, tension=0v3,v4 \fmfgluon, tension=0v5,v6

There are six such diagrams with all possible attachments for the virtual gravitons. It is also evident that a one-loop diagram contributes to the conserved channel and not to the transfer channel. The solid line is not transferred across to the bottom; it stays on top instead. This is true of all odd-loop diagrams. Therefore, the transfer channel contributes only to the even loop diagrams and the conserved channel only to the odd ones. In a manner similar to that of [6], the general order-$n$ amplitude can be written as

$\displaystyle i\mathcal{M}_{n}$

$\displaystyle=\left(is\gamma\right)^{2n}\int\prod_{j=1}^{n}\left(\frac{\mathrm{d}^{2}k_{j}}{\left(2\pi\right)^{2}}if_{\ell}\left(k_{j}\right)\right)\times I\times\left(2\pi\right)^{2}\delta^{(2)}\left(\sum_{j=1}^{n}k_{j}\right)\,,$

where $I$ contains all possible matter propagators; these can be derived following the techniques of [6]. Noticeably, we have inserted a factor of $is\gamma$ for each vertex following the assumption of Sec. 2.2 that internal vertices contributing to additional virtual graviton momenta are sub-leading at each order in perturbation theory. After some algebra [6, 14], the amplitude simplifies remarkably

$\displaystyle i\mathcal{M}_{n}~{}=~{}i\dfrac{\gamma^{2}s^{2}\left(i\chi\right)^{n-1}}{n!}f_{\ell}\left(0\right)~{}=~{}4s\dfrac{\left(i\chi\right)^{n}}{n!}\,,$

(3.1)

where we have defined $\chi=-\frac{1}{4}\gamma^{2}sf_{\ell}\left(0\right)$. These expressions are, at a technical level, identical 2d analogs of those in [6] and hold for arbitrary $f_{\ell}(0)$. The nonperturbative transfer channel amplitude is now straightforward to write down:

$$i\mathcal{M}_{\text{transfer}}~{}=~{}4s\sum_{n~{}\text{odd}}^{\infty}\dfrac{\left(i\chi\right)^{n}}{n!}~{}=~{}4is\sin\left(\chi\right)\,.$$

(3.2)

A similar calculation returns an analogous result for the conserved channel:

$$i\mathcal{M}_{\text{conserved}}~{}=~{}4s\left(\cos\left(\chi\right)-1\right)\,.$$

(3.3)

The complete amplitude is therefore a sum of these two channels:

$$i\mathcal{M}~{}=~{}4s\left(e^{i\chi}-1\right)\,.$$

(3.4)

This result may also be seen as the four-point amplitude $\langle\phi_{\ell m}\left(p_{2}\right)\phi_{0}\phi_{\ell m}\left(p_{1}\right)\phi_{0}\rangle$. Note that the only non-vanishing component of $p_{1}$ is along the $x$ direction, which is in-going into the black hole; similarly $p_{2}$ is only non-vanishing along $y$, which is out-going from the black hole. Therefore, the amplitude can be written as

$$\langle\phi_{\ell m}\left(p_{\text{out}}\right)\phi_{0}\phi_{\ell m}\left(p_{\text{in}}\right)\phi_{0}\rangle\sim\left(e^{i\frac{\gamma^{2}R^{2}_{S}}{\lambda+1}p_{\text{in}}p_{\text{out}}}-1\right).$$

Fourier transforming the out-going field, we find that the amplitude is only non-vanishing when

$$y_{\text{out}}~{}=~{}\tilde{\lambda}\,p_{\text{in}}\quad\text{with}\quad\tilde{\lambda}=\dfrac{8\pi G_{N}}{\ell^{2}+\ell+2}\,,$$

(3.5)

implying that $\langle\phi_{\ell m}\left(y_{\text{out}}\right)\phi_{0}\phi_{\ell m}\left(p_{\text{in}}\right)\phi_{0}\rangle$ is given by

$$\langle\phi_{\ell m}(\tilde{\lambda}p_{\text{in}})\phi_{0}\phi_{\ell m}\left(p_{\text{in}}\right)\phi_{0}\rangle\,.$$

This result is remarkably similar to those obtained from a first quantised formalism in [10, 11, 12], with an important difference in the pre-factor. At first sight, this may appear to be an inconsistency. However, it can be checked that both results are indeed correct. Moreover, a detailed analysis shows [14] that restricting the graviton fluctuations to be of the form $\mathfrak{h}_{xx}\left(x,y\right)=\mathfrak{h}_{xx}\left(y\right)\delta\left(x\right)$ on the past horizon and $\mathfrak{h}_{yy}\left(x,y\right)=\mathfrak{h}_{yy}\left(x\right)\delta\left(y\right)$ on the future horizon results in the factor ($\ell^{2}+\ell+1$) found in the previous literature [10, 11, 12], for the Dray-’t Hooft shockwave metric. The result of the present article is therefore a generalisation that includes arbitrary off-shell graviton fluctuations.

In this article, we studied scattering processes with an impact parameter less than $R_{S}$ but greater than Planck scale. This is the regime where quantum gravity effects are important. Remarkably, the amplitude can be computed non-perturbatively in $\hbar$ and $\gamma\sim M_{Pl}/M_{BH}$, provided that the centre of mass energy of the collision satisfies $s\gg\gamma^{2}M^{2}_{Pl}$; this relation is easily satisfied for large black holes. The non-perturbative amplitude captures many-particle scattering states as it is obtained for second quantised fields in a partial wave basis. It results in a remarkable relation between the in-going and out-going fields (3.5) that is a generalisation of the analogous result in a first quantised formalism [10, 11, 12]. This lends further support to the scattering matrix approach to quantum black holes as a resolution to the information paradox for the external observer. Despite the expected large collision energies, information transfer is carried out by soft gravitons at all orders in perturbation theory. This implies that no momentum is transferred to or from the infalling observer. In itself, the $2-2$ amplitude is expected to be suppressed. But the black hole eikonal phase we develop allows for $2-N$ amplitudes with particle production to be calculated; such high point functions are expected to resolve the information problem.

We are thankful to Panos Betzios, Anne Franzen, Steve Giddings, Olga Papadoulaki, and Tomislav Prokopec. This work is supported by the Delta-Institute for Theoretical Physics (D-ITP) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).