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aainstitutetext: Department of Physics, Arizona State University, AZ 85287,
United Statesbbinstitutetext: Beyond: Center for Fundamental Concepts in
Science, Arizona State University, Tempe, Arizona 85287, USAccinstitutetext:
Indian Institute of Technology, Gandhinagar, Gujarat 382355, India

# Quantum-Gravitational Null Raychaudhuri Equation

Sang-Eon Bak a, b Maulik Parikh c Sudipta Sarkar a Francesco Setti
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider a congruence of null geodesics in the presence of a quantized
spacetime metric. The coupling to a quantum metric induces fluctuations in the
congruence; we calculate the change in the area of a pencil of geodesics
induced by such fluctuations. For the gravitational field in its vacuum state,
we find that quantum gravity contributes a correction to the null Raychaudhuri
equation which is of the same sign as the classical terms. We thus derive a
quantum-gravitational focusing theorem valid for linearized quantum gravity.

##  1 Introduction

The trajectories of freely-falling test particles — geodesics — are of special
importance in general relativity; among the fundamental equations in the
subject are the geodesic equation for a single particle, the geodesic
deviation equation for a pair of particles, and the Raychaudhuri equation for
a congruence of particles. These equations pre-suppose that spacetime has a
definite geometry: the spacetime metric is taken to be a classical field. A
question of great interest then is: what happens to these equations when the
metric is treated as a quantum field? Precisely this question was addressed in
Parikh:2020nrd ; Parikh:2020fhy ; Parikh:2020kfh , in which the effective
dynamics of a pair of non-relativistic particles coupled to a quantized metric
was obtained. After integrating out the quantized metric, one finds that the
geodesic separation no longer obeys the geodesic deviation equation. Instead,
even classical test particles behave stochastically, obeying a Langevin-like
equation. Intuitively, the stochastic behavior of the particles arises from
the probabilistic nature of the quantum field they are coupled to. This result
can be obtained in different ways, either by using the Feynman-Vernon
influence functional approach Feynman:1963fq  to integrate out gravity
Parikh:2020nrd ; Parikh:2020fhy ; Parikh:2020kfh ; Cho_2022 , or by treating
the geodesic deviation equation itself as an operator equation Kanno:2020usf .
Subsequently, these results were extended to the case of a single non-
relativistic particle falling in a weakly curved spacetime — yielding a
stochastic quantum-gravity correction to the geodesic equation in its
Newtonian limit Chawla:2021lop  — as well as to a congruence of massive
particles, yielding stochastic quantum-gravitational corrections to the
timelike Raychaudhuri Bak:2022oyn  and tidal evolution equation Cho:2023dmh .

Here we consider the especially important case of a congruence of massless
particles. When such test particles propagate in a classical spacetime, the
area of the congruence evolves via the null Raychaudhuri equation, an equation
of great significance in general relativity. Our goal in this paper is to
determine how perturbative quantum fluctuations of the spacetime geometry
modify that equation in linearized quantum gravity. Previous approaches to
quantize the (time-like) Raychaudhuri equation have considered the congruence
as evolving non-classically because its constituent matter is quantized
Ahmadi:2006nq ; Das:2013oda ; Vagenas:2017fwa ; Choudhury:2021huy . By
contrast, the thing that is quantum in this paper is the spacetime metric; our
quantum null Raychaudhuri equation therefore describes the spread of a beam of
light in quantum spacetime.

Our results are as follows. We first obtain a quantum-gravity correction to
the geodesic deviation equation for a pair of null particles, (22). As in the
massive case, the coupling to a quantized metric induces random fluctuations –
noise – to the geodesic deviation, turning the deterministic classical
geodesic equation into a Langevin-like stochastic equation. The statistical
properties of the noise depend on the quantum state of the gravitational
field. In particular, for the Poincaré-invariant vacuum state, the noise
contributes a term to the null Raychaudhuri equation, (50), that has the same
sign as the usual classical terms. This results in a quantum-gravitational
focusing theorem, (54), in the regime of linearized perturbative quantum
gravity.

##  2 A Pair of Null Particles in a Quantized Gravitational Field

Consider a null geodesic congruence in an arbitrary spacetime. We would like
to determine the evolution of the area of the congruence when the spacetime
metric is treated as a quantum field in some quantum field state
$|\Psi\rangle$. Our strategy is to first determine how the geodesic deviation
between a pair of null geodesics is affected by the quantum nature of the
metric, and then to use that result to determine the behavior of a congruence
of null geodesics.

###  2.1 Classical equations of motion

The dynamics of the geodesic deviation of a pair of non-relativistic massive
particles was calculated in Parikh:2020nrd ; Parikh:2020fhy ; Parikh:2020kfh ;
Cho_2022 ; Kanno:2020usf  and used to calculate the quantum-corrected
evolution of a congruence of timelike geodesics Bak:2022oyn ; Cho:2023dmh .
However, the null case is sufficiently different that it is worth starting
anew from first principles. To that end, we begin by writing down the action
for Einstein gravity coupled to a pair of null particles:

$\displaystyle S=\frac{1}{\kappa^{2}}\int R\sqrt{-g}d^{4}x+\int
d\lambda\left[\frac{1}{2e}g_{\mu\nu}\frac{dX^{\mu}}{d\lambda}\frac{dX^{\nu}}{d\lambda}\right]+\int
d\lambda^{\prime}\left[\frac{1}{2e^{\prime}}g_{\mu\nu}\frac{dY^{\mu}}{d\lambda^{\prime}}\frac{dY^{\nu}}{d\lambda^{\prime}}\right]$
(1)

where $\kappa^{2}=16\pi G_{N}$, and $e$ and $e^{\prime}$ are einbeins. Since
we are interested only in the relative motion (geodesic deviation), we can
regard $X$ as a reference geodesic and put $X$ and $e$ on shell. The einbein
equation gives

$\displaystyle-\frac{1}{2e^{2}}g_{\mu\nu}\frac{dX^{\mu}}{d\lambda}\frac{dX^{\nu}}{d\lambda}=0\,.$
(2)

This is a constraint equation. Setting $\frac{dX^{\nu}}{d\lambda}$ to be null,
the einbein $e$ can be arbitrary. We will consider an affine parametrization
so that $\frac{de}{d\lambda}=0$. The equation of motion of $X$ is then just
the geodesic equation with affine parameterization:

$\displaystyle\frac{d^{2}X^{\rho}}{d\lambda^{2}}+\Gamma_{\mu\nu}^{\rho}\frac{dX^{\mu}}{d\lambda}\frac{dX^{\nu}}{d\lambda}=0$
(3)

Since we have in mind that our two null geodesics are part of a congruence of
geodesics, we can use the same parameterization for the second geodesic:
$\lambda^{\prime}\equiv\lambda$.

For a sufficiently small pencil of geodesics, we can always – by virtue of
local flatness – write the metric locally as a perturbation of the Minkowski
metric:

$\displaystyle g_{\mu\nu}(x)=\eta_{\mu\nu}+h_{\mu\nu}(x),$ (4)

More precisely, we can set up null Fermi normal coordinates anchored to our
reference geodesic $X$. Taylor expansion of the metric in directions
transverse to the null geodesic gives Blau:2006ar

$\displaystyle\begin{aligned} &g_{uv}=g_{vu}=1+\mathcal{O}(x^{3})\\\
&g_{uu}=-R_{uaub}(u)x^{a}x^{b}+\mathcal{O}(x^{3})\\\
&g_{ua}=-\frac{2}{3}R_{ubac}(u)x^{b}x^{c}+\mathcal{O}(x^{3})\\\
&g_{ab}=\delta_{ab}-\frac{1}{3}R_{abcd}(u)x^{c}x^{d}+\mathcal{O}(x^{3})\end{aligned}$
(5)

where we consider the light cone coordinates $u=t+z,\,v=t-z$ and the
transverse coordinates $x^{a}$ for $a=1,2$ with the parameter of the reference
geodesic $\lambda=u$. Define

$\displaystyle Y^{\mu}=X^{\mu}+\xi^{\mu}$ (6)

Concretely, we have $X^{\mu}=(u,0,0,0)$ and $Y^{\mu}=(u,\xi^{v},\xi^{a})$.
Then, to quadratic order in $\xi$, we have

$\displaystyle
g_{\mu\nu}\frac{dY^{\mu}}{d\lambda}\frac{dY^{\nu}}{d\lambda}=-R_{uaub}\xi^{a}\xi^{b}+\frac{d\xi^{v}}{du}+\delta_{ab}\frac{d\xi^{a}}{du}\frac{d\xi^{b}}{du}$
(7)

Since $\lambda=u$ and $e^{\prime}(\lambda)$ is a constant (by virtue of
$\lambda$ being affine), the linear term is a total derivative, and the action
for the second particle thus reads

$\displaystyle S[\xi]=\int
d\lambda\left[\frac{1}{2e^{\prime}}\left(\delta_{ab}\frac{d\xi^{a}}{d\lambda}\frac{d\xi^{b}}{d\lambda}-R_{uaub}\xi^{a}\xi^{b}\right)\right]$
(8)

The equation of motion of $\xi^{a}$ now gives

$\displaystyle\frac{d^{2}\xi^{a}}{d\lambda^{2}}=-\tensor{R}{{}^{a}_{ubu}}\xi$

This is precisely the geodesic deviation equation, consistent with
Witten:2019qhl , which we have here derived directly from the action as an
Euler-Lagrange equation. Note that $a=1,2$ in accordance with the fact that
the geodesic deviation for null vectors is a spacelike vector that lives in a
codimension-2 hypersurface.

In order to quantize the metric, it will be helpful to express everything in
Cartesian coordinates and transverse-traceless (TT) gauge. To leading order,
the Einstein-Hilbert action in TT gauge is

$\displaystyle
S_{\text{E-H}}\left[h_{\mu\nu}\right]=-\frac{1}{4\kappa^{2}}\int
d^{4}x\partial_{\mu}h_{ij}\partial^{\mu}h^{ij}$ (9)

where $\kappa^{2}=16\pi G_{N}$. We can decompose $h_{ij}$ into the Fourier
modes:

$\displaystyle
h_{ij}\left(x^{i},t\right)=\frac{\kappa}{\sqrt{V}}\sum_{\mathbf{k},s}h_{\mathbf{k}}^{s}(t)e^{i\mathbf{k}\cdot\mathbf{x}}e_{ij}^{s}(\mathbf{k})$
(10)

where we are working in a finite box $V=L_{x}L_{y}L_{z}$ so that the discrete
modes are $k_{i}=\frac{2\pi n_{i}}{L_{i}}$ for $i=1,2,3$. Here, we defined the
polarization tensor $e_{ij}^{s}(\mathbf{k})$ with its normalization
$e_{ij}^{*s}(\mathbf{k})e_{ij}^{s^{\prime}}(\mathbf{k})=\delta^{ss^{\prime}}$,
and we used the linear polarization for index $s=\times,+$.

We then would like to express our null geodesic deviation equation in
Cartesian coordinates. By using the coordinate transformation $u=t+z,v=t-z$,
the left hand side in (2.1) becomes

$\displaystyle\frac{d^{2}\xi^{a}}{du^{2}}=\frac{d^{2}\xi^{a}}{dt^{2}}$ (11)

where we used $\frac{d\xi^{a}}{dt}=\frac{d\xi^{a}}{dz}$ since
$\xi(u)=\xi(t+z)$. Due to the coordinate transformation, the components of the
Riemann tensor in (2.1) becomes

$\displaystyle\tensor{R}{{}^{a}_{ubu}}=\frac{\partial x^{\mu}}{\partial
u}\frac{\partial x^{\nu}}{\partial u}\tensor{R}{{}^{a}_{\mu b\nu}}$ (12)

One can compute the components of the Riemann tensor in the linearized
gravity. Since $h_{\mu\nu}=h_{\mu\nu}(u)$ along the particle world-line, we
obtain

$\displaystyle\tensor{R}{{}^{a}_{tbt}}=\tensor{R}{{}^{a}_{tbz}}=\tensor{R}{{}^{a}_{zbt}}=\tensor{R}{{}^{a}_{zbz}}=-\frac{1}{2}\tensor{\ddot{h}}{{}^{a}_{b}}$
(13)

where $\dot{}$ denotes a derivative with respect to the coordinate time $t$.
Combining these, the geodesic deviation equation (2.1) becomes

$\displaystyle\ddot{\xi}^{a}=\frac{1}{2}\tensor{\ddot{h}}{{}^{a}_{b}}\,\xi^{b},$
(14)

where $\tensor{h}{{}^{a}_{b}}$ is in the transverse-traceless (TT) gauge.

Similarly, from (9) and (8), one can find the equation of motion of the
Fourier components of the metric perturbation:

$\displaystyle\ddot{\hat{h}}^{s}(\mathbf{k},t)+k^{2}\hat{h}^{s}(\mathbf{k},t)=\frac{\kappa}{2e^{\prime}\sqrt{V}}e_{ab}^{*s}(\mathbf{k})\frac{d^{2}}{dt^{2}}\left(\hat{\xi}^{a}(t)\hat{\xi}^{b}(t)\right)$
(15)

where $e^{\prime}$ is the einbein, and $e_{ab}^{*s}$ is a complex conjugate of
the polarization tensor. The right-hand side is a source term which gives the
backreaction of the fluctuating separation of the null particles on the metric
perturbation. In turn, its effect on the geodesic deviation equation will be
to introduce a fifth derivative radiation reaction term Parikh:2020fhy . We
will ignore this term hereafter, as it is unimportant except for rapidly
varying changes in the deviation.

###  2.2 Canonical quantization and Langevin equation

Now we would like to quantize this, by promoting the metric to an operator:

$\displaystyle\hat{g}_{\mu\nu}(x)=\eta_{\mu\nu}+\hat{h}_{\mu\nu}(x)$ (16)

where

$\displaystyle
h^{\text{cl}}_{\mu\nu}(x)\equiv\langle\Psi|\hat{h}_{\mu\nu}(x)|\Psi\rangle$
(17)

Following Kanno:2020usf , let us promote $h^{s}(\mathbf{k},t)$ to a quantum
operator so that

$\displaystyle\hat{h}^{s}(\mathbf{k},t)=\hat{a}_{s}(\mathbf{k})u_{k}(t)+\hat{a}^{\dagger}_{s}(\mathbf{-k})u^{*}_{k}(t),$
(18)

where the creation and annihilation operators follow the usual commutation
relations,

$\displaystyle[\hat{a}_{s}(\mathbf{k}),\hat{a}^{\dagger}_{s^{\prime}}(\mathbf{k^{\prime}})]=\delta_{\mathbf{k}\mathbf{k^{\prime}}}\delta_{ss^{\prime}},\,\,\,\,\,\,\,\,\,\,[\hat{a}_{s}(\mathbf{k}),\hat{a}_{s^{\prime}}(\mathbf{k^{\prime}})]=[\hat{a}^{\dagger}_{s}(\mathbf{k}),\hat{a}^{\dagger}_{s^{\prime}}(\mathbf{k^{\prime}})]=0$
(19)

Since $\xi^{a}$ depends on $h_{ab}$, as shown by equation (14), then $\xi^{a}$
must also be promoted to a quantum operator. Formally, the equations of motion
of $\hat{\xi}^{a}$ are in the Heisenberg picture, while equation (18) is in
the interaction picture, but since we are ignoring the radiation reaction
terms in equation (14), the Heisenberg and interaction pictures are
equivalent.

Let the quantum state of the gravitational field be $|\Psi\rangle$. We can
express the quantum operator $\hat{h}$ as the sum of a classical part and a
quantum part (where we omit the identity operator multiplying the classical
part):

$\displaystyle\hat{h}\equiv h_{\text{cl}}^{\Psi}+\delta\hat{h}^{\Psi}$ (20)

With this decomposition, we have taken the quantum fluctuations to be zero on
average: $\langle\delta\hat{h}^{\Psi}\rangle=0$. Similarly, $\hat{\xi}^{a}(t)$
can be decomposed as

$\displaystyle\hat{\xi}^{a}(t)=\xi^{a}_{0}(t)+\delta\hat{\xi}^{a}(t),$ (21)

We will assume that the classical part $\xi^{a}_{0}(t)$ is much larger than
the quantum fluctuations $\delta\hat{\xi}^{a}(t)$. Inserting these in the
equation of motion (14) and using (15), we obtain

$\displaystyle\ddot{\xi}_{0}^{a}(t)+\delta\ddot{\hat{\xi}}^{a}(t)=\delta^{ab}\left[\frac{1}{2}\ddot{h}^{\text{cl}}_{bc}(t)+2\hat{\mathcal{N}}_{bc}(t)\right]\left(\xi^{c}_{0}(t)+\delta\hat{\xi}^{c}(t)\right),$
(22)

where

$\displaystyle h_{bc}^{\text{cl}}\left(t\right)$
$\displaystyle=\frac{\kappa}{\sqrt{V}}\sum_{\mathbf{k},s}e_{bc}^{s}(\mathbf{k})h_{\text{cl}}^{s}(\mathbf{k},t)$
(23) $\displaystyle\hat{\mathcal{N}}_{bc}(t)$
$\displaystyle=-\frac{\kappa}{4\sqrt{V}}\sum_{s}\sum_{\mathbf{k}\leq\Omega_{\mathrm{m}}}k^{2}e_{bc}^{s}(\mathbf{k})\delta\hat{h}_{s}^{\Psi}(\mathbf{k},t)$
(24)

where $k=|\mathbf{k}|$. Since we are interested in the quantum-gravitational
contribution to the geodesic deviation, we will assume for simplicity that the
background does not depend on time, so there is no classical contribution.
Then (22) becomes

$\displaystyle\delta\ddot{\hat{\xi}}^{a}(t)=2\delta^{ab}\hat{\mathcal{N}}_{bc}(t)\left(\xi^{c}_{0}(t)+\delta\hat{\xi}^{c}(t)\right)$
(25)

Notice that $\ddot{\xi}^{a}_{0}$ is zero since we imposed that the classical
part of the metric is constant in time. Because of this, any change in the
motion of $\hat{\xi}^{a}(t)$ is only due to the quantum fluctuations described
by the above equation.

Since $\delta\hat{h}$ are the quantum fluctuations of the gravitational field,
we can interpret $\hat{\mathcal{N}}_{ab}$ heuristically as the noise in the
geodesic separation due to gravitons. In addition, since the quantum
fluctuations in the gravitational field have an expectation value of zero, it
follows that

$\displaystyle\left\langle\Psi\left|\hat{\mathcal{N}}_{ab}(t)\right|\Psi\right\rangle=0,$
(26)

where, in the operator formalism we are using here, $\langle\cdot\rangle$ now
indicates the expectation value of a quantum mechanical operator, not the
statistical average of a stochastic function Bak:2022oyn . The auto-
correlation function, or noise kernel, is given by the two-point correlation
function of the noise tensors Kanno:2020usf .

$\displaystyle
K_{abcd}^{\Psi}\left(t,t^{\prime}\right)\equiv\left\langle\Psi\left|\left\\{\hat{\mathcal{N}}_{ab}(t),\hat{\mathcal{N}}_{cd}\left(t^{\prime}\right)\right\\}\right|\Psi\right\rangle=\frac{\kappa^{2}}{80\pi^{2}}\left(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc}-\frac{2}{3}\delta_{ab}\delta_{cd}\right)F^{\Psi}(t,t^{\prime})$
(27)

where we used the
anticommutator111$\\{\hat{A},\hat{B}\\}=\frac{1}{2}(\hat{A}\hat{B}+\hat{A}\hat{B})$
since two noise operators do not commute. Since $F^{\Psi}(t,t^{\prime})$ is
calculated by acting the creation and annihilation operators contained in
$\hat{h}$ on the state $|\Psi\rangle$, its form depends is state-dependent (as
indicated by the superscript $\Psi$). In addition, its time dependence is
$(t-t^{\prime})$ if we assume time-translation symmetry. The pre-factor is a
result of the angular integration of $\mathbf{k}$ with isotropy, and is the
same for all rotationally symmetric states.

In general, $F(t,t^{\prime})$ is a divergent function which needs to be
regulated. One way to regularize the function is to introduce an ultraviolet
cut-off frequency $\Lambda$. The cut-off is determined by the length scale of
the deviation between two geodesics. This relation comes from the assumption
that we take when we use the Fermi normal coordinates. One can understand this
relation in the following way. The Fermi normal coordinates in (5) imply the
condition $|\ddot{h}\,\xi_{0}^{2}|\ll\mathcal{O}(1)$ where $\xi_{0}$ is the
length of the initial geodesic deviation vector. If we only keep track of the
$\omega$ dependence in (10), we have

$\displaystyle|h(0)|\sim\left|\int^{\Lambda}_{0}d\omega\omega^{2}A(\omega)\right|$
(28)
$\displaystyle|\ddot{h}(0)|\sim\left|\int^{\Lambda}_{0}d\omega\omega^{4}A(\omega)\right|$
(29)

where $A(\omega)$ is an amplitude for each modes $\omega$. For the Poincaré-
invariant (“Minkowski”) vacuum state, $|0\rangle$, we have
$A(\omega)\sim\omega^{-1/2}$ so that $|\ddot{h}(0)|\sim|h(0)\Lambda^{2}|$.
Since we have $h(0)\ll 1$ for the linearized perturbation, the assumptions in
the Fermi normal coordinate $|\ddot{h}\,\xi_{0}^{2}|\ll\mathcal{O}(1)$ implies
$\Lambda\sim\xi_{0}^{-1}$. In this sense, the UV cut-off is determined by the
length scale of the geodesic deviation.

In the Minkowski vacuum, $F(t,t^{\prime})$ is obtained as Kanno:2020usf

$\displaystyle
F^{|0\rangle}[\Lambda(t-t^{\prime})]=\int_{0}^{\Lambda}\,dkk^{5}\cos[k(t-t^{\prime})]$
(30)

where we used the fact that the mode function in equation (18) in Mikowski
vacuum is given by

$\displaystyle u_{k}(t)=\frac{1}{\sqrt{2k}}e^{ikt}\,.$ (31)

###  2.3 Quantum fluctuations in the equations of motion

We will solve the equations of motion (25) perturbatively. Let

$\displaystyle\delta\hat{\xi}^{a}(t)=\hat{\xi}_{1}^{a}(t)+\hat{\xi}_{2}^{a}(t)+\dots,$
(32)

where the subscript indicates the order of each term. For example, the
classical part $\xi^{a}_{0}$ is the zeroth order term, while
$\hat{\xi}_{1}^{a}(t)$ is the first order quantum correction to
$\hat{\xi}^{a}(t)$. This approach leads to ambiguities in the higher-order
terms due to the fact that our operators do not commute. To avoid these
issues, we will introduce an anticommutator in (25), so that the quantum
fluctuations are now given by

$\displaystyle\delta\ddot{\hat{\xi}}^{a}(t)=2\delta^{ab}\left\\{\hat{\mathcal{N}}_{bc}(t),\hat{\xi}^{c}(t)\right\\}$
(33)

Up to second order, this gives us

$\displaystyle\ddot{\xi}^{a}_{0}(t)$ $\displaystyle=0$ (34)
$\displaystyle\ddot{\hat{\xi}}^{a}_{1}(t)$
$\displaystyle=2\delta^{ab}\left\\{\hat{\mathcal{N}}_{bc}(t),\xi^{c}_{0}(t)\right\\}$
(35) $\displaystyle\ddot{\hat{\xi}}^{a}_{2}(t)$
$\displaystyle=2\delta^{ab}\left\\{\hat{\mathcal{N}}_{bc}(t),\hat{\xi}^{c}_{1}(t)\right\\}$
(36)

To avoid cumbersome notation, we will avoid writing the anticommutators
explicitly in the following calculations; however, whenever we have a product
of two operators, it should be understood as the anticommutator of the two
operators.

Putting everything together, we have

$\displaystyle\begin{aligned}
\xi^{a}(t)=\xi_{0}^{a}(t)&+2\delta^{ab}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\hat{\mathcal{N}}_{bc}\left(\tau_{2}\right)\xi_{0}^{c}(\tau_{2})\\\
&+4\delta^{ab}\delta^{cd}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}\int_{0}^{\tau_{3}}d\tau_{4}\hat{\mathcal{N}}_{bc}\left(\tau_{2}\right)\hat{\mathcal{N}}_{de}\left(\tau_{4}\right)\xi_{0}^{e}(\tau_{4})+\cdots\end{aligned}$
(37)

One point function of the first-order term vanishes since the average of the
noise operator $\hat{\mathcal{N}}_{ab}$ is zero (26).

$\displaystyle\langle\Psi|\hat{\xi}_{1}^{a}(t)|\Psi\rangle=0.$ (38)

However, the one-point function of the second-order term includes the auto-
correlation function, so it does not vanish.

$\displaystyle\langle\Psi|\hat{\xi}_{2}^{a}(t)|\Psi\rangle=4\delta^{ab}\delta^{cd}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}\int_{0}^{\tau_{3}}d\tau_{4}K_{bcde}^{\Psi}\left(\tau_{2},\tau_{4}\right)\xi_{0}^{e}(\tau_{4})$
(39)

There is also a non-zero two-point function of the first-order terms.

$\displaystyle\langle\Psi|\hat{\xi}_{1}^{a}(t)\,\hat{\xi}_{1}^{b}(t^{\prime})|\Psi\rangle=4\delta^{ac}\delta^{bd}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{t^{\prime}}d\tau_{3}\int_{0}^{\tau_{3}}d\tau_{4}K_{cedf}^{\Psi}\left(\tau_{2},\tau_{4}\right)\xi_{0}^{e}\left(\tau_{2}\right)\xi_{0}^{f}(\tau_{4})$
(40)

Figure 1: The diagram shows the evolution of a null congruence in three-
dimensional space. The red arrows indicate the classical trajectories of null
particles moving in the $z$-direction. The black arrows indicate the geodesic
deviation vectors influenced by the quantum fluctuations of spacetime. Quantum
fluctuations cause the geodesic separation to have a stochastic motion, with
one realization schematically portrayed by the blue lines. The time evolution
of the area of the congruence is represented by the blue sheets. Because of
correlations in the fluctuations in different directions, the area change can
have a definite sign.

##  3 Null Congruences in a Quantized Gravitational Field

Having determined how quantum-gravitational fluctuations affect the geodesic
deviation equation, we turn now to the problem of interest, which is to
calculate the quantum-gravitational evolution of the area of a future-directed
null geodesic congruence. Consider a pencil of light-like geodesics with
rectangular transverse area; this could be the beam of a rectangular
headlight. This area can be expressed in terms of a product of geodesic
deviations as follows. Choose the geodesic at one of the corners of the
rectangle as a reference geodesic. Consider two other geodesics at the two
adjacent corners of the rectangle. Then, the length and width of the rectangle
correspond to the geodesic deviations of these geodesics from the reference
geodesic; see Figure 1.

The area of the rectangle is therefore just the product of the geodesic
distances, so we can define the area operator:

$\displaystyle\mathcal{\hat{A}}(t)=\hat{\xi}(t)\hat{\eta}(t)$ (41)

where again the product is to be understood as an anticommutator:
$\mathcal{\hat{A}}(t)=\\{\hat{\xi}(t),\hat{\eta}(t)\\}$. For simplicity,
choose the basis vectors in the codimension$-2$ subspace to lie along $\xi$
and $\eta$, so that they have only nonzero spatial component. If the geodesics
have no shear or initial expansion, then the classical Raychaudhuri equation
indicates that this transverse area will stay constant:
$\mathcal{A}_{c}=\xi_{0}\eta_{0}$. However, as we have seen, quantum-
gravitational fluctuations make the geodesic deviation operators evolve via
the quantum geodesic deviation, (25). This can be solved perturbatively.

Expanding the geodesic deviation vectors in order of the noise terms as

$\displaystyle\begin{aligned}
&\hat{\xi}(t)=\xi_{0}+\hat{\xi}_{1}(t)+\hat{\xi}_{2}(t)+\cdots\\\
&\hat{\eta}(t)=\eta_{0}+\hat{\eta}_{1}(t)+\hat{\eta}_{2}(t)+\cdots\end{aligned}$
(42)

Then, taking the expectation value of the area operator, we find that

$\displaystyle\begin{aligned}
\langle\Psi|\mathcal{\hat{A}}(t)|\Psi\rangle&=\xi_{0}\eta_{0}+\langle\Psi|\hat{\xi}_{1}\hat{\eta}_{1}|\Psi\rangle+\eta_{0}\langle\Psi|\hat{\xi}_{2}|\Psi\rangle+\xi_{0}\langle\Psi|\hat{\eta}_{2}|\Psi\rangle+\cdots\\\
&\equiv\mathcal{A}_{c}+\mathcal{A}_{q}(t)\end{aligned}$ (43)

Even though the noise averages to zero, (26), we see that the expectation
value of the quantum-gravitational contribution to the area does not vanish
because it picks up the two-point function of geodesic deviations, as well as
the one-point function of the second-order deviations.

###  3.1 Quantum-gravitational Raychaudhuri equation

In order to determine the quantum-corrected time evolution of the area, we see
from (43) that we need to first compute the two-point function and one-point
function. These depend on $|\Psi\rangle$, the quantum state of the
gravitational field. To be concrete, we will hereafter take the state to be
the Poincaré-invariant vacuum state, $|0\rangle$.

Then, using (27), (40) and (30), we find that

$\displaystyle\langle
0|\hat{\xi}_{1}(t)\,\hat{\eta}_{1}(t)|0\rangle=-\frac{\kappa^{2}\xi_{0}\eta_{0}}{30\pi^{2}}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{t}d\tau_{3}\int_{0}^{\tau_{3}}d\tau_{4}F^{|0\rangle}\left(\tau_{2},\tau_{4}\right)\simeq-\frac{\kappa^{2}\xi_{0}\eta_{0}}{120\pi^{2}}\Lambda^{4}t^{2}$
(44)

The minus sign of the two-point function implies that there is a negative
correlation between two different geodesic deviations. If one of the lengths
is shrinking, then the other one should be expanding. Similarly, we obtain the
one-point function of the second-order terms:

$\displaystyle\begin{aligned} &\langle
0|\hat{\xi}_{2}(t)|0\rangle=\frac{\kappa^{2}\xi_{0}}{6\pi^{2}}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}\int_{0}^{\tau_{3}}d\tau_{4}F^{|0\rangle}\left(\tau_{2},\tau_{4}\right)\simeq-\frac{\kappa^{2}\xi_{0}}{48\pi^{2}}\Lambda^{4}t^{2}\\\
&\langle
0|\hat{\eta}_{2}(t)|0\rangle=\frac{\kappa^{2}\eta_{0}}{6\pi^{2}}\int_{0}^{t}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}\int_{0}^{\tau_{3}}d\tau_{4}F^{|0\rangle}\left(\tau_{2},\tau_{4}\right)\simeq-\frac{\kappa^{2}\eta_{0}}{48\pi^{2}}\Lambda^{4}t^{2}\end{aligned}$
(45)

Unlike the two-point function, the angular pre-factor in (27) results here in
an overall plus sign.

Thus, by (43), the quantum correction to the area is

$\displaystyle\mathcal{A}^{|0\rangle}_{q}(t)\simeq-\frac{\kappa^{2}}{20\pi^{2}}\mathcal{A}_{c}\,\Lambda^{4}t^{2}$
(46)

We define the quantum expansion of the null congruence via

$\displaystyle\Theta(t)\equiv\frac{1}{\langle\hat{\mathcal{A}}(t)\rangle}\frac{d\langle{\hat{\mathcal{A}}}(t)\rangle}{dt}$
(47)

By taking the time derivative and keeping track of the leading order
corrections in $A_{q}/A_{c}$, the expansion is written as

$\displaystyle\Theta(t)=\frac{\dot{\mathcal{A}}_{c}(t)}{\mathcal{A}_{c}(t)}+\frac{d}{dt}\left(\frac{\mathcal{A}_{q}(t)}{\mathcal{A}_{c}(t)}\right)\equiv\theta_{c}(t)+\theta_{q}(t)$
(48)

where, for the simple case of a static classical area, the quantum correction
to the expansion is

$\displaystyle\theta^{|0\rangle}_{q}(t)\simeq-\frac{\kappa^{2}}{10\pi^{2}}\Lambda^{4}t$
(49)

Then, differentiating again, we find

$\displaystyle\dot{\Theta}^{|0\rangle}(t)=-\frac{1}{2}\theta_{c}^{2}-\sigma^{2}-R_{\mu\nu}k^{\mu}k^{\nu}-\frac{\kappa^{2}}{10\pi^{2}}\Lambda^{4}$
(50)

This is the quantum-gravity corrected Raychaudhuri equation when the
gravitational field is in the Poincaré-invariant vacuum state. Here we have
allowed the classical area to be time-dependent, so that $\theta_{c}$ obeys
the classical Raychaudhuri equation, and we have set the vorticity term,
$\omega^{2}$, to zero by hypersurface-orthogonality. We see that the effect of
quantum fluctuations of spacetime is to introduce an additional term on the
right-hand side of Raychaudhuri’s equation. The term depends on $\Lambda$, the
frequency cut-off scale which, as noted, is inversely proportional to the
characteristic size of the congruence. Notably, the new term has – at least
for the vacuum state – the same negative sign as the usual classical terms. It
would be very interesting to determine the sign of this term for other classes
of states, to see whether a general statement can be made.

##  4 A Quantum-Gravitational Focusing Theorem

#### Classical Focusing Theorem

Consider the classical Raychaudhuri equation for a null congruence:

$\displaystyle\dot{\theta}_{c}=-\frac{\theta_{c}^{2}}{2}-\sigma^{2}-R_{\mu\nu}k^{\mu}k^{\nu}$
(51)

where again we have discarded the vorticity term by hypersurface-
orthogonality. By Einstein’s equations, we have
$R_{\mu\nu}k^{\mu}k^{\nu}=\frac{\kappa^{2}}{2}T_{\mu\nu}k^{\mu}k^{\nu}$. If we
now impose the null energy condition $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$, the
null Raychaudhuri equation implies that the classical expansion of the
congruence always decreases, i.e.,

$\displaystyle\dot{\theta}\leq 0$ (52)

This is known as the classical focusing theorem: it implies that the expansion
must decrease during the congruence’s evolution. Thus, an initially diverging
congruence will diverge less rapidly in the future, while an initially
converging congruence will converge more rapidly in the future. This theorem
is quite generic in general relativity for which, of course, there is a
definite classical geometry. However, to the best of our knowledge, it has not
yet been investigated how the classical focusing theorem is modified in
situations where quantum fluctuations of the metric are taken into account.

#### A Quantum Focusing Theorem

In this work, we have derived a quantum-gravitational modification of the
Raychaudhuri equation for a pencil of null geodesics (50), simply

$\displaystyle\dot{\theta}_{c}+\dot{\theta}_{q}=-\frac{\theta_{c}^{2}}{2}-\sigma^{2}-R_{\mu\nu}k^{\mu}k^{\nu}-\frac{4\hbar
G_{N}}{\mathcal{A}_{0}}\left(\frac{32\pi^{3}}{5\mathcal{A}_{0}}\right)$ (53)

where we used $\mathcal{A}_{0}\sim\left(\frac{2\pi}{\Lambda}\right)^{2}$ and
have explicitly restored $\hbar$. The first three terms on the right-hand side
of the equation (53) originate from the classical contribution, while the last
term involves a quantum-gravitational contribution. It is important to note
that this quantum correction comes from the quantum nature of spacetime
itself, not from quantum matter fields. If we take into account only classical
matter, we can use the Einstein equation
$R_{\mu\nu}k^{\mu}k^{\nu}=\frac{\kappa^{2}}{2}T_{\mu\nu}k^{\mu}k^{\nu}$ and
the null energy condition $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$. Remarkably, the
quantum-gravitational correction has a negative sign, so that the right-hand
side of (53) is always negative. Therefore, the quantum-gravitational null
Raychaudhuri equation also implies a focusing theorem:

$\displaystyle\dot{\Theta}^{|0\rangle}\leq 0$ (54)

where $\Theta=\theta_{c}+\theta_{q}$ with the definition (47). It appears
that, at least for the Minkowski vacuum state, perturbative quantum
corrections do not spoil the classical focusing theorem. This also suggests
the formation of a caustic in the future, as in the case of timelike geodesics
Bak:2022oyn . It would be interesting to study how this affects the black hole
singularity theorems.

If we take into account the effect of quantized matter fields and examine the
contribution from the renormalized energy-momentum tensor within a semi-
classical framework, we can obtain a deviation from the classical null energy
condition Birrell:1982ix . This results in a potential alteration of the sign
in the Raychaudhuri equation. What we have found is that, perhaps
surprisingly, the effect of linearized perturbative quantum gravity is
qualitatively different. The quantum fluctuation of the metric simply
contributes a term that aligns with the sign of the classical contributions.

There is also a quantum focusing conjecture motivated by the holographic
principle and generalized second law Bousso:2015mna , which is related to
Bousso:2015wca ; Bousso:2022tdb . In this conjecture, a quantum expansion is
introduced, incorporating a contribution from quantum matter fields in the
generalized entropy associated with a holographic screen. The aim is to extend
the focusing theorem to scenarios where quantum matter might violate the null
energy condition and the validity of the generalized second law. A quantum
null energy condition Bousso:2015wca  was proposed as a sufficient condition
for the quantum focusing conjecture (similarly, the focusing theorem can also
be motivated on thermodynamic grounds Parikh:2015ret ). Our conjecture differs
from their framework in several aspects. First, in our approach, the quantum
correction arises from perturbative quantum gravity, and we consider the
expansion of a pencil of the congruence rather than a holographic screen,
representing the boundary of certain subregions. Second, we assume that the
matter sector continues to obey the usual null energy condition. However,
despite these differences, the ultimate outcome in both situations appears to
be linked by the non-violation of classical focusing of null geodesics in the
presence of quantum effects. Therefore, it might be an interesting future work
to investigate a setup that has an overlap between the two works.

##  5 Summary

In this paper, we have considered the effect of spacetime fluctuations on the
deviation of null geodesics in the regime of linearized perturbative quantum
gravity, and used that to obtain a quantum-gravitational null Raychaudhuri
equation. Our equation has an additional term on the right which originates in
stochastic fluctuations of the geodesic separation induced by the quantum
fluctuations of the metric. The details of the new term depend on the quantum
state of the gravitational field. Remarkably, when the gravitational field is
in the usual vacuum state, the sign of the correction is the same as that of
the classical terms and therefore does not significantly alter the conclusion
of the classical focusing theorem. It would be very interesting to be able to
prove that this true in general for all quantum states of the gravitational
field. But if that were to be true, then presumably the singularity theorems –
which rely crucially on the Raychaudhuri equation – would also continue to
hold, and linearized quantum gravity at least would then be unable to save
general relativity from the menace of singularities.

Acknowledgments We thank George Zahariade for helpful discussions. MP is
supported in part by Heising-Simons Foundation grant 2021-2818, Department of
Energy grant DE-SC0019470 and Government of India DST VAJRA Faculty Scheme
VJR/2017/000117. The research of S.S. is supported by the Department of
Science and Technology, Government of India under the SERB CRG Grant
CRG/2020/004562.

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