# Minimal length scale correction in the noise of gravitons

Soham Sen<EMAIL_ADDRESS>Department of Astrophysics and High Energy
Physics, S. N. Bose National Centre for Basic Sciences, JD Block, Sector-III,
Salt Lake City, Kolkata-700 106, India  Sunandan Gangopadhyay
<EMAIL_ADDRESS>Department of Astrophysics and High Energy
Physics, S. N. Bose National Centre for Basic Sciences, JD Block, Sector-III,
Salt Lake City, Kolkata-700 106, India

###### Abstract

In this paper we have considered a quantized and linearly polarized
gravitational wave interacting with a gravitational wave detector
(interferometer detector) in the generalized uncertainty principle (GUP)
framework. Following the analysis in Phys. Rev. Lett. 127 (2021) 081602, we
consider a quantized gravitational wave interacting with a gravitational wave
detector (LIGO/VIRGO etc.) using a path integral approach. Although the
incoming gravitational wave was quantized, no Planck-scale quantization
effects were considered for the detector in earlier literatures. In our work,
we consider a modified Heisenberg uncertainty relation with a quadratic order
correction in the momentum variable between the two phase space coordinates of
the detector. Using a path integral approach, we have obtained a stochastic
equation involving the separation between two point-like objects. It is
observed that random fluctuations (noises) and the correction terms due to the
generalized uncertainty relation plays a crucial role in dictating such
trajectories. Finally, we observe that the solution to the stochastic equation
leads to time dependent standard deviation due to the GUP insertion, and for a
primordial gravitational wave (where the initial state is a squeezed state)
both the noise effect and the GUP effects exponentially enhance which may be
possible to detect in future generation of gravitational wave detectors. We
have also given a plot of the dimensionless standard deviation with time
depicting that the GUP effect will carry a distinct signature which may be
detectable in the future space based gravitational wave observatories.

Introduction: It is a well known fact that a single particle freely falling
under the effect of gravity, follows the geodesic equation, and in the case of
a pair of point particles, they follow the geodesic deviation equation. These
trajectories of freely falling objects under the effect of gravity is
deterministic in nature and same follows for the geodesic deviation equation.
These facts have been verified experimentally at a classical level, however
their status at the quantum level has been the subject of intense research
theoretically. Although the general theory of relativity gives a perfect
description of gravity at a macroscopic level, the quantum nature of gravity
is still unknown to the physics community. It is expected that gravity must
have a quantum nature and therefore the search for a quantum theory of gravity
is pursued. Recently there has been a very important work QGravNoise ;
QGravLett ; QGravD  which demonstrated the effect on falling bodies due to the
quantization of the gravitational field. It is observed that the dynamics of
separation of a pair of falling bodies have a probabilistic nature which is
different from the deterministic nature observed in the case of classical
Einstein’s gravity. It is also observed that the separation of two particles
follows a Langevin-like stochastic equation with a random fluctuation term
which is claimed as the quantum generalization of the classical geodesic
deviation equation. The investigation reveals that linearized quantum theory
of gravity can indeed have a startling effect on the motion of falling bodies.
In this regard, we would like to stress that low energy quantum gravity
corrections have been predicted by all the existing candidates for a quantum
theory of gravity, such as string theory String1 ; String2 , loop quantum
gravity Loop1 ; Loop2 , and noncommutative geometry Snyder ; Snyder2 ; AC ;
NCG1 . These theories indicate the presence of an observer-independent
fundamental length scale (the Planck length $\approx 10^{-33}$ m) in nature.
However, for some string theories with dynamical string tensions, the
existence of a fundamental length scale may be avoided. For example, in
Guendelman , a minimum time in the Euclidean space has been avoided, which
would in principle be applicable to any spatial minimum length as well. By
modifying the Heisenberg uncertainty principle (HUP), one can incorporate this
minimal length. This modified HUP is known as the generalized uncertainty
principle (GUP). The relation between gravity and this minimal length scale
was first shown in BRON1 ; BRON2  and later in MEAD . In the gedanken
experiments, a very strong signature of the existence of this minimal length
scale was obtained. There have also been several investigations in different
areas of theoretical physics to exploit this GUP framework, such as black hole
physics MAGGIORE ; SCARDIGLI ; ADLERSAN ; ADLERCHENSAN ; RABIN ; SG1 ;
SCARDIGLI2 ; SG2 ; Ong ; EPJC ; BMajumder , harmonic oscillators DAS1 ; DAS2 ,
optomechanical systems IVASP ; IVASP2 ; KSP ; OTM0 , and gravitational wave
bar detectors SG3 ; SG4 ; OTM . Recently there have been a few efforts to
construct a laboratory-based test to investigate the effects of GUP in
optomechanical systemsPetruzzeillo ; DasModak .

The general structure of the modified uncertainty relation is given by the
following relation Farag ; Farag2 ; SGSB

$\Delta\tilde{\xi}_{i}\Delta\tilde{\pi}_{i}\geq\frac{\hbar}{2}\left[1+\gamma\left(\Delta\tilde{\pi}^{2}+\langle\tilde{\pi}\rangle^{2}\right)+2\gamma\left(\Delta\tilde{\pi}_{i}^{2}+\langle\tilde{\pi}_{i}\rangle^{2}\right)\right]$
(1)

where the index $i$ runs from $1$ to $3$, and $\tilde{\xi}_{i}$ and
$\tilde{\pi}_{i}$ are the phase space position and the conjugate momenta. The
GUP parameter $\gamma$ in terms of the dimensionless parameter $\gamma_{0}$
can be recast as $\gamma=\frac{\gamma_{0}}{m_{p}^{2}c^{2}}$, where $m_{p}$
denotes the Planck mass and $c$ denotes the speed of light. The GUP modified
variables $\\{\tilde{\xi}_{i},\tilde{\pi}_{i}\\}$ satisfies the following
commutation relation consistent with the uncertainty product given in eq.(1)

$[\tilde{\xi}_{i},\tilde{\pi}_{j}]=i\hbar\left[\delta_{ij}+\gamma\left(\delta_{ij}\tilde{\pi}^{2}+2\tilde{\pi}_{i}\tilde{\pi}_{j}\right)\right]~{}.$
(2)

The above commutation relation reduces to the following form in one spatial
dimension Farag ; Farag2 ; SGSB ; Kempf

$[\tilde{\xi},\tilde{\pi}]=i\hbar[1+3\gamma\tilde{\pi}^{2}]~{}.$ (3)

Following from the above commutation relation, the GUP modified variables
$(\tilde{\xi},\tilde{\pi})$ in terms of the variables $(\xi,\pi)$ obeying the
Heisenberg’s uncertainty principle can be expressed as

$\tilde{\xi}=\xi~{},~{}\tilde{\pi}=\pi(1+\gamma\pi^{2})~{}.$ (4)

It is important to note that the coordinates $\\{\tilde{\xi},\tilde{\pi}\\}$
obeying the modified commutation relation (eq.(3)) are being represented in
terms of the coordinates $\\{\xi,\pi\\}$ which obey the usual commutation
relation $[\xi,\pi]=i\hbar$. Owning to the above discussion, it is natural to
carry out the analysis in QGravNoise ; QGravLett ; QGravD  in the GUP
framework. This would then incorporate effects from both the quantization of
linearized gravity and modification of the Heisenberg’s uncertainty relation.
The model considered in QGravNoise ; QGravLett ; QGravD  is as follows. The
mirrors in the arm of the falling particles are considered as two freely
falling particles. One of the particles has a heavier mass and the other
particle has a comparatively smaller mass. The incoming gravitational wave is
treated quantum mechanically, and a perturbative approach is taken to include
the quantum effects. Our analysis is significantly different from the previous
analysis QGravNoise ; QGravLett ; QGravD  in the sense that we have now
considered quantum gravity effects in the detector arm also by considering the
existence of a minimal length scale. In order to truly incorporate the effects
of the minimal length scale (having a direct connection to gravity), we have
replaced the phase space coordinates of the detector part in the particle-
graviton interaction Hamiltonian by GUP modified phase space coordinates. In
order to analyze the system, we have considered a path integral approach and
calculated the Feynman-Vernon influence functional FeynmanVernon  for the
modified Hamiltonian. The incoming states have now been considered as minimum
uncertainty states or coherent states which most closely resemble the
classical gravitational wave. In our analysis, we observe as in QGravNoise ;
QGravLett ; QGravD  that the separation between the two particles obeys a
Langevin-like stochastic equation. However, we find terms containing both the
effects of the modified uncertainty relation and the quantization of the
gravitational waves. It is also observed that in the modified geodesic
deviation equation, instead of second order time derivatives of the stochastic
term, one can also observe first order time derivatives of the stochastic term
solely due to the consideration of the modified uncertainty relation. We at
first considered the graviton to exist in a coherent state and then we have
extended our analysis for gravitons in squeezed states. In principle, one can
find that the noise in the case of coherent states is very small but if one
considers squeezed vacuum states, by tuning the squeezing parameter one may
really detect such quantum gravitational effects in the next generation of
gravitational wave detectors which could in principle give us hints about both
the generalized uncertainty relation and the gravitons. Although having a
squeezed vacuum state also have the limitation that there must be a source
from which such gravitons in squeezed states are being generated. It is though
assumed that primordial gravitational waves can be considered to exist in such
squeezed states which can be a future candidate for the detection of such
enhanced noise spectrum. At this point we would like to mention that inclusion
of a minimal length scale with unmodified uncertainty principle and dispersion
relation has also been reported in the literature Douglas . However, the
standard way of incorporating a minimal length scale is through the
introduction of a GUP Kempf  or through a noncommutative algebra AC ; Snyder ;
Snyder2 . In this work, we have followed this path. Background: The main goal
of this paper is to quantize the arm length of a gravitational wave detector
(with arm length $\tilde{\xi}$) where the phase space coordinates of the
detector follow the modified Heisenberg uncertainty priciple. The complete
action of the system can be obtained by combining the Einstein-Hilbert action
with the action of the detector. In terms of the GUP modified variables one
can obtain the action of the system (considering only one polarization of the
gravitational wave) as

$S_{\mathcal{A}}=\int
dt\left(\frac{m}{2}\left(\dot{q}^{2}-\omega^{2}q^{2}\right)+\frac{m_{0}}{2}\dot{\tilde{\xi}}^{2}-\mathcal{G}\dot{q}\dot{\tilde{\xi}}\tilde{\xi}\right)$
(5)

with $q$ denoting the configuration space variable of the gravitational wave,
$\hbar\omega$ being the energy of the gravitational wave mode, and $m_{0}$
denoting the smaller mass between the two masses representing the
interferometer. Here, $\mathcal{G}=\frac{m_{0}}{2l_{p}}$ denotes the graviton-
detector coupling constant with $l_{p}=\sqrt{\frac{\hbar G}{c^{3}}}$ being the
Planck length and $m=\frac{L^{3}c^{5}}{16\hbar G^{2}}$. The Hamiltonian from
the action in terms of the phase space variables of the gravitational wave
($\\{q,p\\}$) and the GUP modified phase space variables of the gravitational
wave detector ($\\{\tilde{\xi},\tilde{\pi}\\}$) is given by

$H=\left(\frac{p^{2}}{2m}+\frac{\tilde{\pi}^{2}}{2m_{0}}+\frac{\mathcal{G}p\tilde{\pi}\tilde{\xi}}{mm_{0}}\right)\left[1-\frac{\mathcal{G}^{2}\tilde{\xi}^{2}}{mm_{0}}\right]^{-1}+\frac{1}{2}m\omega^{2}q^{2}~{}.$

The above Hamiltonian in terms of the unmodified detector variables
($\\{\xi,\pi\\}$) upto $\mathcal{O}(\gamma)$ can be recast as

$H=\frac{\frac{p^{2}}{2m}+\frac{\pi^{2}}{2m_{0}}+\frac{\mathcal{G}p\pi\xi}{mm_{0}}+\gamma\left(\frac{\pi^{4}}{m_{0}}+\frac{\mathcal{G}p\pi^{3}\xi}{mm_{0}}\right)}{1-\frac{\mathcal{G}^{2}\xi^{2}}{mm_{0}}}+\frac{1}{2}m\omega^{2}q^{2}$
(6)

where we have used $\tilde{\pi}=\pi(1+\gamma\pi^{2})$. Here, $\xi$ denotes the
geodesic separation of the lighter mass $m_{0}$ from the heavier mass. One can
now elevate this Hamiltonian from classical description to quantum description
by introducing appropriate commutation relations between the coordinate
variables and their conjugate momenta. In this analysis we consider the
initial state of the gravitational wave as $|\psi_{\omega}\rangle$ and that of
the final state as $|\mathcal{F}\rangle$. As we do not know the final state of
the gravitational wave, we need to sum over all $\lvert\mathcal{F}\rangle$
states. As initially there was no coupling between the initial gravitational
wave state and the detector state, we will consider them as a tensor product
state. Now via spontaneous and stimulated emission procedures the detector
masses will both emit and absorb gravitons. Here, our aim is to compute the
probability and the form of the probability is given as follows

$P_{\psi_{\omega}}^{[\phi_{i}\rightarrow\phi_{f}]}=\sum\limits_{\lvert\mathcal{F}\rangle}\left|\langle\mathcal{F},\phi_{f}|\hat{U}(T+\Delta,-\Delta)|\psi_{\omega},\phi_{i}\rangle\right|^{2}$
(7)

where $|\phi_{f}\rangle$ and $|\phi_{i}\rangle$ are the final state and the
initial states of the particle at times $t=T+\Delta$ and $t=-\Delta$
respectively. $\hat{U}$ is the unitary time evolution operator in eq.(7) and
associated with the quantum mechanical analogue to the Hamiltonian in eq.(6).
It is very important to note that the interaction is turned on in the interval
$t=0$ to $t=T$. Inserting complete set of joint position eigenstates and
summing over all the final gravitational wave states $|\mathcal{F}\rangle$ in
eq.(7), one can rewrite the transition probability as

$\begin{split}&P_{\psi_{\omega}}^{[\phi_{i}\rightarrow\phi_{f}]}=\int
dq_{i}dq_{i}^{\prime}dq_{f}d\xi_{i}d\xi_{i}^{\prime}d\xi_{f}d\xi_{f}^{\prime}\psi_{\omega}(q_{i})\psi^{*}_{\omega}(q_{i}^{\prime})\phi_{i}(\xi_{i})\\\
&\times\phi_{i}^{*}(\xi^{\prime}_{i})\phi_{f}^{*}(\xi_{f})\phi_{f}(\xi_{f}^{\prime})~{}\langle
q_{i}^{\prime},\xi_{i}^{\prime}|\hat{U}^{\dagger}(T+\Delta,-\Delta)|q_{f},\xi_{f}^{\prime}\rangle\\\
&\times\langle
q_{f},\xi_{f}|\hat{U}(T+\Delta,-\Delta)|q_{i},\xi_{i}\rangle~{}.\end{split}$
(8)

Rewriting each of the amplitudes present in the canonical path integral form
and executing the path integral over $\pi$, one can now recast the amplitude
in a much compact form given by

$\begin{split}&\langle
q_{f},\xi_{f}|\hat{U}(T+\Delta,-\Delta)|q_{i},\xi_{i}\rangle=\int\tilde{\mathcal{D}}\xi\exp\Bigr{[}\frac{im_{0}}{2\hbar}\int_{-\Delta}^{T+\Delta}dt\\\
&\times[\dot{\xi}^{2}-2\gamma
m_{0}^{2}\dot{\xi}^{4}]\Bigr{]}\int\mathcal{D}q\mathcal{D}pe^{\frac{i}{\hbar}\int_{-\Delta}^{T+\Delta}dt[p\dot{q}-H^{\gamma}_{\xi}[q,p]]}~{}.\end{split}$

In the above expression, the form of the reduced Hamiltonian is given by

$H^{\gamma}_{\xi}[q,p]=\frac{(p+\mathcal{G}\xi\dot{\xi}(1-3\gamma
m_{0}^{2}\dot{\xi}^{2}))^{2}}{2m}+\frac{1}{2}m\omega^{2}q^{2}~{}.$

We can finally obtain the form of the probability in eq.(8) as

$\begin{split}&P_{\psi_{\omega}}^{[\phi_{i}\rightarrow\phi_{f}]}=\int
d\xi_{i}d\xi_{i}^{\prime}d\xi_{f}d\xi_{f}^{\prime}\phi_{i}(\xi_{i})\phi_{i}^{*}(\xi^{\prime}_{i})\phi_{f}^{*}(\xi_{f})\phi_{f}(\xi_{f}^{\prime})\\\
&\int\tilde{\mathcal{D}}\xi\tilde{\mathcal{D}}\xi^{\prime}e^{\frac{i}{\hbar}\int_{-\Delta}^{T+\Delta}dt\frac{m_{0}}{2}\left[(\dot{\xi}^{2}-\dot{\xi}^{\prime
2})-2\gamma m_{0}^{2}(\dot{\xi}^{4}-\dot{\xi}^{\prime
4})\right]}F^{\gamma}_{\psi_{\omega}}[\xi,\xi^{\prime}]\end{split}$ (9)

where we have defined the following two quantities,

$\displaystyle\langle
q_{f}|\hat{U}^{\gamma}_{\xi}(T+\Delta,-\Delta)|q_{i}\rangle\equiv\int\mathcal{D}q\mathcal{D}pe^{\frac{i}{\hbar}\int_{-\Delta}^{T+\Delta}dt[p\dot{q}-H^{\gamma}_{\xi}[q,p]]},$
$\displaystyle
F^{\gamma}_{\psi_{\omega}}[\xi,\xi^{\prime}]\equiv\langle\psi_{\omega}|\hat{U}^{\gamma\dagger}_{\xi^{\prime}}[T+\Delta,-\Delta]\hat{U}^{\gamma}_{\xi}[T+\Delta,-\Delta]|\psi_{\omega}\rangle~{}.$

Here, $F^{\gamma}_{\psi_{\omega}}[\xi,\xi^{\prime}]$ gives the Feynman-Vernon
influence functional. It is very important to observe that influence
functional is the only term in the probability containing the harmonic
oscillator state ($|\psi_{\omega}\rangle$) and providing a coupling between
the harmonic oscillator and the detector variable. With the generic structure
of the probability in hand, we shall now proceed to explicitly compute the
form of the influence functional. It is important to observe that we are
currently running our analysis for a single mode of the gravitational wave and
we shall extend it for multiple modes also. The influence functional: From the
form of the Hamiltonian $\hat{H}^{\gamma}_{\xi}(q,p)$, it can be seen that the
instantaneous eigenstates are general harmonic oscillator eigenstates
generated by the shift of the momentum in the momentum space via a parameter
$\mathcal{G}\xi\dot{\xi}(1-3\gamma m_{0}^{2}\dot{\xi}^{2})$. In order the
convert the the integrals from $-\Delta$ to $T+\Delta$ to integrals from $0$
to $T$, we need to redefine the Heisenberg eigenstates from
$|\psi_{\omega}\rangle$ to
$e^{-\frac{i}{\hbar}\hat{H}_{0}\Delta}|\psi_{\omega}\rangle$ and we also need
to introduce the modified wave functions given by the following two relations

$\displaystyle\tilde{\phi}_{i}(\tilde{\xi}_{i})$ $\displaystyle=\int
d\xi_{i}\phi_{i}(\xi_{i})\int\tilde{\mathcal{D}}\xi
e^{\frac{i}{\hbar}\int_{-\Delta}^{0}dt\frac{m_{0}}{2}(\dot{\xi}^{2}-2\gamma
m_{0}^{2}\dot{\xi}^{4})}~{},$ $\displaystyle\tilde{\phi}_{f}(\tilde{\xi}_{f})$
$\displaystyle=\int d\xi_{f}\phi_{f}(\xi_{f})\int\tilde{\mathcal{D}}\xi
e^{-\frac{i}{\hbar}\int_{T}^{T+\Delta}dt\frac{m_{0}}{2}(\dot{\xi}^{2}-2\gamma
m_{0}^{2}\dot{\xi}^{4})}~{}.$

In the above two relations, we have defined $\xi(-\Delta)=\xi_{i}$,
$\xi(T+\Delta)=\xi_{f}$, $\xi(0)=\tilde{\xi}_{i}$, and
$\xi(T)=\tilde{\xi}_{f}$. The modified probability formula takes the form as
follows

$\begin{split}&P_{\psi_{\omega}}^{[\phi_{i}\rightarrow\phi_{f}]}\equiv\int
d\xi_{i}d\xi_{i}^{\prime}d\xi_{f}d\xi_{f}^{\prime}\phi_{i}(\xi_{i})\phi_{i}^{*}(\xi^{\prime}_{i})\phi_{f}^{*}(\xi_{f})\phi_{f}(\xi_{f}^{\prime})\\\
&\int[\tilde{\mathcal{D}}\xi]_{\xi_{i},0}^{\xi_{f},T}[\tilde{\mathcal{D}}\xi^{\prime}]_{\xi^{\prime}_{i},0}^{\xi^{\prime}_{f},T}e^{\frac{i}{\hbar}\int_{0}^{T}dt\left[L_{0}^{\gamma}(\xi)-L_{0}^{\gamma}(\xi^{\prime})\right]}F^{\gamma}_{\psi_{\omega}}[\xi,\xi^{\prime}]\end{split}$
(10)

where $L_{0}^{\gamma}(\xi)=\frac{m_{0}}{2}(\dot{\xi}^{2}-2\gamma
m_{0}^{2}\dot{\xi}^{4})$ and we have redefined the detector variable from
$\tilde{\xi}$ to $\xi$ (with $\xi(0)=\xi_{i}$, $\xi(T)=\xi_{f}$ and
$\xi^{\prime}(0)=\xi^{\prime}_{i}$, $\xi^{\prime}(T)=\xi^{\prime}_{f}$). Due
to the change in the integration limit, the Feynman-Vernon influence
functional also gets modified. The modified Feynman-Vernon influence
functional has the form

$\begin{split}&F^{\gamma}_{\psi_{\omega}}[\xi,\xi^{\prime}]=\langle\psi_{\omega}|e^{\frac{i}{\hbar}\hat{q}\mathcal{G}\xi_{i}^{\prime}\dot{\xi}^{\prime}_{i}[1-3\gamma
m_{0}^{2}\dot{\xi}^{\prime
2}_{i}]}\hat{U}^{I\dagger}_{\gamma\xi^{\prime}}(T,0)\\\ &\times
e^{-\frac{i}{\hbar}\hat{q}_{I}(T)\mathcal{G}\xi_{f}^{\prime}\dot{\xi}^{\prime}_{f}[1-3\gamma
m_{0}^{2}\dot{\xi}^{\prime
2}_{f}]}e^{\frac{i}{\hbar}\hat{q}_{I}(T)\mathcal{G}\xi_{f}\dot{\xi}_{f}[1-3\gamma
m_{0}^{2}\dot{\xi}^{2}_{f}]}\\\
&\times\hat{U}^{I}_{\gamma\xi}(T,0)e^{-\frac{i}{\hbar}\hat{q}\mathcal{G}\xi_{i}\dot{\xi}_{i}[1-3\gamma
m_{0}^{2}\dot{\xi}^{2}_{i}]}|\psi_{\omega}\rangle\end{split}$ (11)

where $\hat{U}^{I}_{\gamma\xi}(T,0)$ gives the unitary time evolution operator
in the interaction picture and
$\hat{q}_{I}(T)=e^{\frac{i}{\hbar}\hat{H}_{0}T}\hat{q}e^{\frac{i}{\hbar}\hat{H}_{0}T}$.
In order to proceed further, we need to decompose the unitary time evolution
operators in terms of the time ordered exponential functions in the
interaction picture and finally write the form of the unitary time evolution
operator in the interaction picture as follows

$\begin{split}&\hat{U}^{I}_{\gamma\xi}(T,0)\equiv
e^{-\frac{i\mathcal{G}}{\hbar}\hat{q}_{I}(T)\xi_{f}\dot{\xi}_{f}[1-3\gamma
m_{0}^{2}\dot{\xi}_{f}^{2}]}e^{\frac{i\mathcal{G}}{2\hbar}\int_{0}^{T}dt~{}\hat{q}_{I}(t)Z(t)}\\\
&e^{\frac{i\mathcal{G}}{\hbar}\hat{q}_{I}(0)\xi_{i}\dot{\xi}_{i}(1-3\gamma
m_{0}^{2}\dot{\xi}_{i}^{2})}e^{-\frac{\mathcal{G}^{2}}{8\hbar^{2}}\int_{0}^{T}dt\int_{0}^{t}dt^{\prime}Z(t)Z(t^{\prime})[\hat{q}_{I}(t),\hat{q}_{I}(t^{\prime})]}\end{split}$
(12)

where $X(t)=\frac{d^{2}}{dt^{2}}(\xi^{2})$ and
$Y(t)=\frac{d}{dt}(\dot{\xi}^{2}\frac{d}{dt}(\xi^{2}))$, and
$Z(t)=X(t)-3\gamma m_{0}^{2}Y(t)$. By the repeated use of the Baker-Campbell-
Hausdorff formula and substituting the form of the unitary time evolution
operator from eq.(12) into the Feynman-Vernon influence functional, we obtain
the modified form of the influence functional as follows

$\begin{split}F^{\gamma}_{\psi_{\omega}}[\xi,\xi^{\prime}]=\langle\psi_{\omega}|e^{-V^{*}\hat{a}^{\dagger}}e^{V\hat{a}}|\psi_{\omega}\rangle
e^{i\Phi^{\gamma}_{0_{\omega}}[\xi,\xi^{\prime}]}\end{split}$ (13)

where

$V=\frac{i\mathcal{G}}{\sqrt{8\hbar m\omega}}\int_{0}^{T}dte^{-i\omega
t}(Z(t)-Z^{\prime}(t))$ (14)

and the phase factor in the exponential term of eq.(13) is given by the
following relation

$\begin{split}i\Phi^{\gamma}_{0_{\omega}}[\xi,\xi^{\prime}]\equiv&-\frac{\mathcal{G}^{2}}{8\hbar
m\omega}\int_{0}^{T}dt\int_{0}^{t}dt^{\prime}(Z(t)-Z^{\prime}(t))\\\
&\times(Z(t^{\prime})e^{-i\omega(t-t^{\prime})}-Z^{\prime}(t^{\prime})e^{i\omega(t-t^{\prime})})~{}.\end{split}$
(15)

With the form of the Feynman-Vernon influence functional in hand, we are now
in a position to consider different cases for the state
$|\psi_{\omega}\rangle$ of the gravitational wave. We now consider a
gravitational wave-mode in a coherent state
$|\psi_{\omega}\rangle=|\alpha_{\omega}\rangle$ with eigenvalue
$\alpha_{\omega}=\sqrt{\frac{m\omega}{2\hbar}}\zeta_{\omega}e^{-i\phi_{\omega}}$
where the form of the classical gravitational wave mode is given by
$q_{cl}(t)=\zeta_{\omega}\cos(\omega t+\phi_{\omega})$. One can now easily
compute the form of the influence functional in the single-mode analysis and
proceed to compute the influence functional for a continuum of such modes (or
a gravitational field). The influence functional of the field can be
considered as a product of the influence functional for the individual modes
($F^{\gamma}_{\Psi}[\xi,\xi^{\prime}]=\prod\limits_{\vec{k}}F^{\gamma}_{\psi_{\omega(\vec{k})}}[\xi,\xi^{\prime}]$)
where the gravitational field is given as
$|\Psi\rangle=\mathop{\otimes}\limits_{\vec{k}}|\psi_{\omega(\vec{k})}\rangle$.
One can finally compute the transition probability for a gravitation field
with its field modes in coherent states as follows

$\begin{split}&P_{\Psi}\equiv\int
d\xi_{i}d\xi^{\prime}_{i}d\xi_{f}d\xi^{\prime}_{f}\phi_{i}(\xi_{i})\phi_{i}^{*}(\xi_{i}^{\prime})\phi_{f}^{*}(\xi_{f})\phi_{f}(\xi_{f}^{\prime})\int\tilde{\mathcal{D}}\xi\tilde{\mathcal{D}}\xi^{\prime}\\\
&\times\int\tilde{\mathcal{D}}\mathcal{N}_{0}~{}\exp\left[-\frac{1}{2}\int_{0}^{T}dt\int_{0}^{T}dt^{\prime}\mathcal{A}_{0}^{-1}(t,t^{\prime})\mathcal{N}_{0}(t)\mathcal{N}_{0}(t^{\prime})\right]\\\
&\times\exp\biggr{[}\frac{im_{0}}{2\hbar}\int_{0}^{T}dt\biggr{[}(\dot{\xi}^{2}-\dot{\xi^{\prime}}^{2})-2\gamma
m_{0}^{2}(\dot{\xi}^{4}-\dot{\xi^{\prime}}^{4})\\\
&+\left[\frac{(\bar{h}(t)+\mathcal{N}_{0}(t))}{2}-\frac{m_{0}G}{4}[\dot{Z}(t)+\dot{Z}^{\prime}(t)]\right][Z(t)-Z^{\prime}(t)]\biggr{]}\end{split}$
(16)

where $\bar{h}(t)=\frac{1}{l_{p}}\sum\limits_{\omega}\zeta_{\omega}\cos(\omega
t+\phi_{\omega})$. To obtain the final form of the probability in eq.(16), we
have made use of the Feynman-Vernon trick. The function $\mathcal{N}_{0}(t)$
has the interpretation of a noise term, that is a stochastic random function
with a Gaussian probability density. Indeed, one can define the average of
$\mathcal{N}_{0}(t)$ which vanishes QGravD

$\displaystyle\langle\mathcal{N}_{0}(t)\rangle$
$\displaystyle\equiv\int\mathcal{D}\mathcal{N}_{0}\exp\Bigr{[}-\frac{1}{2}\int_{0}^{T}\int_{0}^{T}dt^{\prime}dt^{\prime\prime}\mathcal{A}_{0}^{-1}(t^{\prime},t^{\prime\prime})$
$\displaystyle\times\mathcal{N}_{0}(t^{\prime})\mathcal{N}_{0}(t^{\prime\prime})\Bigr{]}\mathcal{N}_{0}(t)=0~{}.$
(17)

The function $\mathcal{A}_{0}(t,t^{\prime})$ is the autocorrelation function
of $\mathcal{N}_{0}(t)$ as QGravD

$\displaystyle\langle\mathcal{N}_{0}(t)\mathcal{N}_{0}(t^{\prime})\rangle\equiv\int\mathcal{D}\mathcal{N}_{0}\exp\Bigr{[}-\frac{1}{2}\int_{0}^{T}\int_{0}^{T}dt^{\prime\prime}dt^{\prime\prime\prime}$
$\displaystyle\times\mathcal{A}_{0}^{-1}(t^{\prime\prime},t^{\prime\prime\prime})\mathcal{N}_{0}(t^{\prime\prime})\mathcal{N}_{0}(t^{\prime\prime\prime})\Bigr{]}\mathcal{N}_{0}(t)\mathcal{N}_{0}(t^{\prime})=\mathcal{A}_{0}(t,t^{\prime})~{}.$
(18)

This autocorrelation function will be of immense importance as we shall see
below. In case of the coherent state analysis, the auto correlation function
has the form QGravD

$\mathcal{A}_{0}(t,t^{\prime})=\frac{4\hbar
G}{\pi}\int_{0}^{\infty}d\omega\omega\cos(\omega(t-t^{\prime}))$ (19)

which is in general divergent in nature. But for a gravitational wave
detector, it is sensitive to a certain range of the gravitational wave
frequency and therefore one can regularize the integral by applying a maximum
cut-off frequency $\omega_{max}$. Due to the usage of a dipole-like
approximation in our current analysis, we can use the maximum value of the
frequency as $\frac{2\pi c}{\xi_{0}}$ with $\xi_{0}$ being the resting arm
length of the detector. Dynamics of the arm length: With the form of the
probability in hand, we can now compute the quantum-dynamics of the detector
arm length $\xi$. In order to obtain the effective stochastic equation, we
consider the saddle point approximation. The saddle point gives the maximum
contribution to the path integral in eq.(16). For the gravitational wave in
the coherent state, following the procedure in QGravD , one can obtain the
following differential equation for the detector variable $\xi(t)$

$\begin{split}&\ddot{\xi}-\frac{1}{2}\biggr{[}\left(\ddot{\bar{h}}(t)+\ddot{\mathcal{N}}_{0}(t)-\frac{m_{0}G}{c^{5}}\frac{d^{5}}{dt^{5}}(\xi^{2})\right)(1+3\gamma
m_{0}^{2}\dot{\xi}^{2})+\\\ &\frac{3\gamma
m_{0}^{3}G}{c^{5}}\frac{d^{4}}{dt^{4}}\left(\frac{d}{dt}(\xi^{2})\dot{\xi}^{2}\right)\biggr{]}\biggr{]}\xi+3\gamma
m_{0}^{2}(\dot{\xi}^{3}+3\xi\dot{\xi}\ddot{\xi})\biggr{[}\dot{\bar{h}}(t)+\\\
&\dot{\mathcal{N}}_{0}(t)-\frac{m_{0}G}{c^{5}}\frac{d^{4}}{dt^{4}}(\xi^{2})\biggr{]}=0~{}.\end{split}$
(20)

Eq.(20) is one of the main results in our paper. It is important to observe
that the geodesic deviation equation is replaced now by a quantum stochastic-
equation. The most important observation is that now the quantum geodesic
equation is governed by the terms coupling the effects of the GUP parameter
and the noise term. Unlike QGravNoise ; QGravLett ; QGravD , there are also
terms involving the first order derivative of the noise term with respect to
time. The $\ddot{\bar{h}}(t)\xi$ term in eq.(20) is a tidal acceleration term
due to the passing of a classical gravitational wave and the fifth order time
derivative term is the dissipative gravitational radiation reaction term
KSThorne ; WLBurke ; Chandrasekhar ; Sasaki . This term $\ddot{\bar{h}}(t)\xi$
will be important as will be clear in the subsequent discussion. The other
higher derivative terms are corrections to the gravitational radiation
reaction due to the modification in the uncertainty relation of the detector
variables.

For a gravitational wave in the coherent state, the noise spectrum is of the
order of the Planck length making it a near impossible task to detect the
signatures of the gravitons, let alone the generalized uncertainty relation.
Therefore, we shall consider the gravitational wave in a squeezed state
$\hat{S}_{z_{\omega}}|0_{\omega}\rangle$, where
$\hat{S}_{z_{\omega}}=e^{\frac{1}{2}(z_{\omega}^{*}\hat{a}^{2}-z_{\omega}\hat{a}^{\dagger
2})}$ gives the squeezing operator with
$z_{\omega}=r_{\omega}e^{i\phi_{\omega}}$ being the squeezing parameter. The
quantum geodesic equation for a gravitational wave in a squeezed state gives
the following equation

$\begin{split}&\ddot{\xi}-\frac{1}{2}\biggr{[}\left(\ddot{\mathcal{N}}^{n.s.}(t)+\sqrt{\cosh
2r}\ddot{\mathcal{N}}_{0}(t)-\frac{m_{0}G}{c^{5}}\frac{d^{5}}{dt^{5}}(\xi^{2})\right)(1+\\\
&3\gamma m_{0}^{2}\dot{\xi}^{2})+\frac{3\gamma
m_{0}^{3}G}{c^{5}}\frac{d^{4}}{dt^{4}}\left(\frac{d}{dt}(\xi^{2})\dot{\xi}^{2}\right)\biggr{]}\biggr{]}\xi+3\gamma
m_{0}^{2}\biggr{[}\dot{\mathcal{N}}^{n.s.}(t)\\\ &+\sqrt{\cosh
2r}\dot{\mathcal{N}}_{0}(t)-\frac{m_{0}G}{c^{5}}\frac{d^{4}}{dt^{4}}(\xi^{2})\biggr{]}(\dot{\xi}^{3}+3\xi\dot{\xi}\ddot{\xi})=0\end{split}$
(21)

where $r$ denotes the real part of the squeezing parameter for the quantum
field and $\mathcal{N}^{n.s.}(t)$ denotes the non-stationary noise generated
due to the time modulation of the noise in squeezed states. It is very
important to observe in eq.(21) that the associated noise term now can be
exponentially increased along with the terms generated due to the generalized
uncertainty principle for values of $r$ greater than unity. This induces an
increased chance in the detection of such minuscule corrections present in
such gravitational wave detection scenarios. To proceed further, it is
important to note that we can obtain a solution of the given Langevin-like
equations by means of perturbative calculations. We can get rid of the
dissipative gravitational radiation reaction terms in eq.(s)(20,21) QGravD .
In order to do so one needs to consider that $\xi$ is measured in a coarse
grained manner which in turns result in the higher derivatives to be
negligible. We shall try to obtain an approximate solution of the time
dependent geodesic separation for the gravitational wave to be initially in a
coherent state. In order to find a solution to eq.(s)(20,21), we use an
iterative approach. For the base equation $\ddot{\xi}(t)=0$ without the higher
order terms, we can obtain a zeroth order solution of the form
$\xi^{(0)}(t)=\xi_{0}+\lambda t$, where the constant $\lambda$ has the
dimension of velocity and can have a maximum value $\lambda=c$. The maximum
interaction time between the graviton being absorbed and released by the
detector is $t_{\text{max}}\sim\frac{\xi_{0}}{c}$ and therefore the linear
time dependent term in $\xi(t)$ can go up to $\lambda t_{\text{max}}$.
Following the same iterative procedure, we can obtain a most general solution
of eq.(20) up to
$\mathcal{O}(\gamma,\mathcal{N}_{0},\bar{h},\gamma\mathcal{N}_{0},\gamma\bar{h})$
as follows

$\displaystyle\xi(t)\cong$ $\displaystyle(\xi_{0}+\lambda
t)\left[1+\frac{1}{2}\left[1+3\gamma
m_{0}^{2}\lambda^{2}\right])(\bar{h}(t)+\mathcal{N}_{0}(t))\right]$
$\displaystyle-\lambda(1+6\gamma
m_{0}^{2}\lambda^{2})\int_{0}^{t}dt^{\prime}(\bar{h}(t^{\prime})+\mathcal{N}_{0}(t^{\prime}))~{}.$
(22)

Now, the higher limit of the integral in eq.(22) has a cut-off at
$t=t_{\text{max}}$. Our aim now is to calculate the standard deviation
$\sigma=\sqrt{\left<(\xi(t)-\left<\xi(t)\right>)^{2}\right>}$. We can separate
the standard deviation term into two parts as
$\sigma(t)\cong\sigma_{0}(t)+\sigma_{\gamma}(t)$. The $\sigma_{0}(t)$ part has
been calculated in QGravLett ; QGravD  and reads

$\sigma_{0}(t)\sim\sqrt{2\pi}l_{p}\sim 10^{-35}\text{ m}~{}.$ (23)

In case of the GUP contribution of the standard deviation, we need to consider
the time dependent parts and ignore the linear time-dependent contribution. We
then obtain the form of the dimensionless
parameter$\frac{\sigma_{\gamma}(t)}{\sqrt{2\pi}l_{p}}$ to be (with $\lambda$
set to its maximum value)

$\begin{split}&\frac{\sigma_{\gamma}(t)}{\sqrt{2\pi}l_{p}}\cong 3\gamma
m_{0}^{2}c^{2}\biggr{[}1-\frac{2}{\pi(1+\frac{ct}{\xi_{0}})}\frac{\xi_{0}\sin^{2}\left[\frac{\pi
ct}{\xi_{0}}\right]}{\pi ct}\\\
&+\frac{4}{\pi^{2}\left(1+\frac{ct}{\xi_{0}}\right)^{2}}\left(\gamma_{\varepsilon}-\text{Ci}\left[\frac{2\pi
ct}{\xi_{0}}\right]+\ln\left[\frac{2\pi
ct}{\xi_{0}}\right]\right)\biggr{]}\end{split}$ (24)

where $\gamma_{\varepsilon}$ gives the Euler constant and Ci denotes the
cosine integral function Gradshteyn . For a gravitational wave in the squeezed
coherent state, we obtain the form of the standard deviation to be
(considering the static part only)

$\sigma_{\text{Squeezed}}(t)=\sqrt{\cosh 2r}\sigma(t)$ (25)

where $\sigma(t)=\sigma_{0}(t)+\sigma_{\gamma}(t)$. Phenomenological aspects
of the model:  It is important to note that the gravitational wave
observatories LIGO (/VIRGO) has an $L$ shaped structure with the arm length at
rest to be $\xi_{0}=4\text{ km}$ ($\xi_{0}=3\text{ km}$ for VIRGO). For the
mirror suspended at the both ends of the Fabry-Perot cavity, the mirror
coating is made up of fused Silica (mass of a single $\text{SiO}_{2}$ molecule
is $m_{\text{SiO}_{2}}\sim 10^{-25}\text{ kg}$) which serves as the low-index
layer and Tantalum pentoxide (mass of a single $\text{Ta}_{2}\text{O}_{5}$
molecule is $m_{\text{Ta}_{2}\text{O}_{5}}\sim 10^{-24}\text{ kg}$) which
serves as the high-index layer Accadia . We can indeed obtain a bound on the
GUP parameter using these parameters from the existing gravitational wave
observatories. Note that, $\gamma m_{0}^{2}c^{2}\sim\gamma_{0}\times 10^{-33}$
(for $m_{0}\sim 10^{-24}\text{ kg}$) and from the requirement $\zeta\gamma
m_{0}^{2}c^{2}<1$ (where the dimensionless constant $\zeta$ is a number of
order 10), we can impose a bound on the dimensionless quadratic GUP parameter
to be $\gamma_{0}<10^{31}$ which is weaker than the bound obtained earlier for
a resonant bar detector interacting with a gravitational wave in
SukantaDaSunandan  but tighter than the bound obtained in GWaveDas  using
gravitational wave observation data. We shall now try to give a basic estimate
on the detectability of the GUP effect from the standard deviation
$\sigma(t)$. For the gravitational wave to be in coherent state, it is
important to understand that $\sigma\cong\sqrt{2\pi}l_{p}\sim
10^{-35}\text{m}$ and current detectability lies around $10^{-18}$ m. In case
of the initial graviton state being in a squeezed state, we can indeed observe
that $\sigma^{\text{Squeezed}}(t)=\sqrt{\cosh 2r}\sigma(t)$ which indicates
that for a sufficiently high squeezing parameter the standard deviation due to
the induced noise and the GUP effect may be detectable.

Figure 1: $\frac{\sigma(t)}{\sqrt{2\pi}l_{p}}$ vs $t$ plot for gravitational
wave with maximum frequency $\omega_{\text{max}}\sim 1\text{ Hz}$. The inset
image depicts the long term behaviour of the time dependent standar deviation
and it signifies that the standard deviation asymptotes towards the value of
the standard deviation obtained in QGravD ; QGravLett  (dotted line) with
increase in time $t$.

In general such states can only generate in post-inflationary scenarios PI1 ;
PI2 ; PI3  leading to a very rare chance of detection of such primordial
gravitational waves. For a “grand unified theory” inflation, the frequency is
at $\omega\sim 0.1Hz$ which is beyond the frequency range of both LIGO and
VIRGO. This frequency range can although be detectable by the future space
based gravitational wave observatories DECIGO or LISA111DECIGO: Decihertz
Interferometer Gravitational Wave Observatory and LISA: Laser Interferometer
Space Antenna.. For such a primordial gravitational wave $e^{r}\sim 10^{18}$
Soda  which results in the enhancing parameter to be of the order of
$\sqrt{\cosh 2r}\cong\frac{1}{\sqrt{2}}e^{r}\sim 10^{18}$. We will primarily
consider the LISA interferometer with $\xi_{0}\sim 10^{6}\text{ km}$ and
maximum frequency $\omega_{\text{max}}\sim 1\text{ Hz}$ QGravD . LISA has a
projected sensitivity at $10^{-18}$ m QGravD . With the inherent squeezing in
the primordial gravitational wave and a maximum interaction time of
$t_{\text{max}}\sim 3.33\text{ sec}$ one can estimate that the standard
deviation is around (while the detector has just stopped interacting with the
gravitational wave)
$\sigma_{0}^{\text{Squeezed}}(t_{\text{max}})\sim\sqrt{\cosh 2r}\times
10^{-35}\text{ m}\sim 10^{-17}\text{ m}$ which is in the detectable range of
the LISA observatory. A very important result can be observed in terms of the
GUP part of the standard deviation. The contribution to the standard deviation
due to the GUP effect can be observed (from eq.(24)) to have a value
$\sigma_{\gamma}^{\text{Squeezed}}(t_{\text{max}})\sim\sqrt{\cosh
2r}~{}10^{-37}\text{ m}\sim 10^{-19}\text{ m}$. Hence, we find that for
primordial gravitational waves, the standard deviation carrying the signature
of the graviton has a value $\sigma_{0}^{\text{Squeezed}}\sim 10^{-17}\text{
m}$ and the GUP effect lies in the range
$\sigma^{\text{Squeezed}}_{\gamma}\sim(10^{-19}-10^{-20})\text{ m}$ which is
just one order of magnitude beyond the projected sensitivity of the LISA
observatory. The ratio of $\sigma_{\gamma}^{\text{Squeezed}}(t_{\text{max}})$
with $\sigma_{0}^{\text{Squeezed}}(t_{\text{max}})$ is
$\sigma_{\gamma}^{\text{Squeezed}}/\sigma_{0}^{\text{Squeezed}}\sim 10^{-2}$.
The analysis has been done for a primordial gravitational wave with squeezing
parameter $r\sim 42$. If it is possible to detect a primordial gravitational
wave with a squeezing parameter $r\sim 44$, we then find that the GUP
contribution to have a value $\sigma^{\text{Squeezed}}_{\gamma}(t)\sim
10^{-18}\text{ m}$ which indeed will be in the projected sensitivity range of
the LISA observatory. It will then be possible to detect the stochastic noise
effect due to the existence of gravitons as well as the existence of GUP.
Plotting the standard deviation due to the GUP contribution divided by
$\sqrt{2\pi}l_{p}$ with respect to time, we can observe a very unique
behaviour based on the time of sampling of the $\xi(t)$ data in Fig.(1). We
observe from Fig.(1) that the value of the dimensionless number
$\frac{\sigma(t)}{\sqrt{2\pi}l_{p}}$ decreases and then increases with time.
Here, we have compared the result of the standard deviation obtained in our
analysis with the result obtained in QGravD ; QGravLett . Here,
$\sigma(t)=\sigma_{0}(t)+\sigma^{\prime}_{\gamma}(t)$ with the definition
$\sigma^{\prime}_{\gamma}(t)\equiv\sigma_{\gamma}(t)-\sigma_{\gamma}(0)$.
Fig.(1) (and the small inset Figure depicting long time behaviour) suggests
that the sudden increase in the value of the standard deviation decreases
gradually with time and returns back to the value of the standard deviation
$\sigma_{0}(t)=\sqrt{2\pi}l_{p}$, given in QGravD ; QGravLett , in the long
time limit. If the standard deviation can be calculated at a fixed time
interval during when the interaction happens and such a dip in the standard
deviation value is observed, we can claim the existence of the generalized
uncertainty principle. Hence, for an advanced gravitational wave observatory,
it may be possible to detect both the existence of gravitons as well as the
existence of a fundamental minimal length scale correction in the Heisenberg
uncertainty principle. This would indeed point towards a quantum nature of
gravity. Summary: In this paper, we have considered a linearly polarized
gravitational wave interacting with a gravitational wave detector. In the
current model a quantized gravitational wave interacts with a detector where
the detector variables obey the modified Heisenberg uncertainty relation (also
known as the generalized uncertainty principle). In our analysis, we have
considered that initially the gravitational wave was not interacting with the
detector making it possible to write the initial state of the system as a
tensor product state of the state corresponding to the graviton and the
initial state corresponding to the detector. Following the approach in
QGravNoise ; QGravLett ; QGravD , we have summed over all the final states of
the graviton. We then integrate out the coordinates and momenta corresponding
to the graviton to obtain the influence functional involved in this entire
process using a path integral approach. Following this one graviton analysis,
we have extended our study to a gravitational field where the field modes are
in coherent states and then squeezed states respectively. We finally obtain a
stochastic Langevin-like equation including a noise term which can be
considered as a quantum gravitational correction to the classical geodesic
deviation equation. Our work focusses in the inclusion of a minimal length
scale correction in the Heisenberg uncertainty principle, corresponding to the
detector variables, leading to a GUP modified stochastic Langevin equation. We
then obtain an approximate solution of the time dependent detector arm length
and calculate the corresponding standard deviation in it. Although in case of
the field modes being in a coherent state leads to a minuscule correction to
the classical geodesic deviation equation, for a squeezed state analysis we
find that along with the noise term the GUP effect also gets an exponential
boost due to the existence of a tunable squeezing parameter. It is surprising
in a sense that the GUP effect is coming from the detector variables spanning
the modified phase space only and therefore the squeezing embedded in the
field states can amplify hidden signatures of the minimum length scale
corrections considered in the detector. We then obtain a bound on the
dimensionless GUP parameter which is tighter than the bounds obtained earlier
using gravitational wave data. We finally plot the dimensionless standard
deviation term due to the GUP effect with respect to time and observe that it
may be possible to detect hidden GUP signatures while detecting primordial
gravitational waves (with a squeezing parameter $r\simeq 44$) in future
generation of gravitational wave detectors (LISA) along with the detection of
gravitons.

## Acknowledgement

We thank the referees for constructive and valuable comments which have helped
us to improve the manuscript substantially.

## References

  * (1) M. Parikh, F. Wilczek, and G. Zahariade, Int. J. Mod. Phys. D 29 (2020) 2042001.
  * (2) M. Parikh, F. Wilczek, and G. Zahariade, Phys. Rev. Lett. 127 (2021) 081602.
  * (3) M. Parikh, F. Wilczek, and G. Zahariade, Phys. Rev. D 104 (2021) 046021.
  * (4) D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 216 (1989) 41.
  * (5) K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234 (1990) 276.
  * (6) C. Rovelli, Living Rev. Relativ. 1 (1998) 1.
  * (7) S. Carlip, Rep. Prog. Phys. 64 (2001) 885.
  * (8) H. S. Snyder, Phys. Rev. 71 (1947) 38.
  * (9) H. S. Snyder, Phys. Rev. 72 (1947) 68.
  * (10) A. Connes. Noncommutative Geometry (1994), Academic Press.
  * (11) F. Girelli, E. R. Livine, and D. Oriti, Nucl. Phys. B 708 (2005) 411.
  * (12) E. I. Guendelman, Int. J. Mod. Phys. D 30 (2021) 2142028.
  * (13) M. P. Bronstein, Zh. Eksp. Teor. Fiz. 6 (1936) 195.
  * (14) M. P. Bronstein, Phys. Z. Sowjetunion 9 (1936) 140.
  * (15) C. A. Mead, Phys. Rev. B 135 (1964) 849.
  * (16) M. Maggiore, Phys. Lett. B 319 (1993) 83.
  * (17) F. Scardigli, Phys. Lett. B 452 (1999) 39.
  * (18) R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14 (1999) 20.
  * (19) R. J. Adler, P. Chen, and D. I. Santiago, Gen. Relativ. Gravit. 33 (2001) 2101.
  * (20) R. Banerjee and S. Ghosh, Phys. Lett. B 688 (2010) 224.
  * (21) S. Gangopadhyay, A. Dutta, and A. Saha, Gen. Relativ. Gravit. 46 (2014) 1661.
  * (22) F. Scardigli and R. Casadio, Eur. Phys. J. C 75 (2015) 425.
  * (23) R. Mandal, S. Bhattacharyya, and S. Gangopadhyay, Gen. Relativ. Gravit. 50 (2018) 143.
  * (24) Y. C. Ong, JCAP 09 (2018) 015.
  * (25) L. Buoninfante, G. G. Luciano, and L. Petruzzeillo, Eur. Phys. J. C 79 (2019) 663.
  * (26) B. Majumder, Phys. Lett. B 701 (2011) 384.
  * (27) S. Das and E. C. Vagenas, Phys. Rev. Lett. 101 (2008) 221301.
  * (28) S. Das and E. C. Vagenas, Can. J. Phys. 87 (2009) 233.
  * (29) I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, Nat. Phys. 8 (2012) 393.
  * (30) P. Bosso, S. Das, I. Pikovski, and M. R. Vanner, Phys. Rev. A 96 (2017) 023849.
  * (31) S. P. Kumar and M. B. Plenio, Phys. Rev. A 97 (2018) 063855.
  * (32) S. Sen, S. Bhattacharyya, and S. Gangopadhyay, Class. Quant. Grav. 39 (2022) 075020.
  * (33) S. Gangopadhyay and S. Bhattacharyya, Phys. Rev. D 99 (2019) 104010.
  * (34) S. Bhattacharyya, S. Gangopadhyay, and A. Saha, Class. Quant. Grav. 37 (2020) 195006.
  * (35) S. Sen, S. Bhattacharyya, and S. Gangopadhyay, Universe 8 (2022) 450.
  * (36) L. Petruzzeillo and F. Illuminati, Nat. Commun. 12 (2021) 4449.
  * (37) S. Das and S. K. Modak, Class. Quant. Grav. 39 (2022) 015005.
  * (38) A. F. Ali, S. Das, E. C. Vagenas, Phys. Lett. B 678 (2009) 497.
  * (39) A. F. Ali, S. Das, E. C. Vagenas, Phys. Rev. D 84 (2011) 044013.
  * (40) S. Gangopadhyay and S. Bhattacharyya, Phys. Rev. D 104 (2021) 026003.
  * (41) A. Kempf, G. Mangano, and R. Mann, Phys. Rev. D 52 (1995) 1108.
  * (42) R. P. Feynman and F. L. Vernon, Jr., Ann. Phys. 24 (1963) 118.
  * (43) M. Bishop, J. Contreras, and D. Singleton, Int. J. Mod. Phys. D 31 (2022) 2241002.
  * (44) K. S. Thorne, Astrophys. J. 158 (1969) 997.
  * (45) W. L. Burke, J. Math. Phys. (N.Y.) 12 (1971) 401.
  * (46) S. Chandrasekhar and F. Paul Esposito, Astrophys. J. 160 (1970) 153.
  * (47) Y. Mino, M. Sasaki, and T. Tanaki, Phys. Rev. D 55 (1997) 3457.
  * (48) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, Amsterdam, 2007), 6th ed.
  * (49) T. Accadia et. al., J. Inst. 7 (2012) P03012.
  * (50) S. Bhattacharyya, S. Gangopadhyay, and A. Saha, Class. Quant. Grav. 37 (2020) 195006.
  * (51) A. Das, S. Das, N. R. Mansour, and E. C. Vagenas, Phys. Lett. B 819 (2021) 136429.
  * (52) A. Albrecht, P. Ferreira, and M. Joyce, and T. Prokopec, Phys. Rev. D 50 (1994) 4807.
  * (53) L. P. Grishchuk and Y. V. Sidorov, Phys. Rev. D 42 (1990) 3413.
  * (54) B. L. Hu, G. Kang, and A. Matacz, Int. J. Mod. Phys. A 09 (1994) 991.
  * (55) S. Kanno, J. Soda, and J. Tokuda, Phys. Rev. D 103 (2021) 044017.


# Schwarzschild black-hole immersed in uniform electric or magnetic
backgrounds in Entangled Relativity

Olivier Minazzoli<EMAIL_ADDRESS>Artemis, Université Côte d’Azur, CNRS,
Observatoire Côte d’Azur, BP4229, 06304, Nice Cedex 4, France,
Bureau des Affaires Spatiales, 2 rue du Gabian, 98000 Monaco.  Maxime
Wavasseur  Departament de Física Quántica i Astrofísica (FQA), Universitat de
Barcelona (UB), Carrer de Martí i Franquès, 1, Barcelona, 08028, Spain,
Artemis, Université Côte d’Azur, CNRS, Observatoire Côte d’Azur, BP4229,
06304, Nice Cedex 4, France.

###### Abstract

In this paper, we present the solution for a Schwarzschild black-hole immersed
in an electric or magnetic background field within the framework of Entangled
Relativity. Previous solutions in Entangled Relativity required black-holes to
be charged for the matter field to be defined everywhere. This is because the
theory precludes the existence of vacuum solutions, thereby satisfying
Einstein’s definition of Mach’s Principle. The current black-hole solutions
represent the first exact and neutral black-hole solutions of Entangled
Relativity discovered to date. The Schwarzschild black-hole of General
Relativity emerges as a limit of these solutions when the background field
approaches zero, whereas the Melvin solution of General Relativity does not
emerge as a limit when the black hole’s size approaches zero. This finding
suggests that astrophysical black-holes in Entangled Relativity are
indistinguishable from those in General Relativity, given the generally weak
interstellar density of matter fields.

##  I Introduction

Entangled Relativity is a reformulation of Einstein’s General Theory of
Relativity that posits a very specific non-linear coupling between the Ricci
scalar curvature and matter fields, instead of the usual linear coupling found
in Einstein’s theory Ludwig  _et al._ (2015). This non-linear coupling
prevents the theory from being defined without matter fields from the onset,
ensuring that Entangled Relativity satisfies Einstein’s original requirement
that the metric field is wholly determined by matter, such that no vacuum
solution should exist in a satisfying theory of relativity Einstein (1918a,
b). This requirement for a relativistic theory is known as (Einstein’s
definition of) Mach’s principle Einstein (1918a, b); Pais (1982); Hoefer
(1995). Because matter and spacetime curvature are intertwined at the level of
the theory’s formulation, it has been named ‘Entangled Relativity’ Arruga  _et
al._ (2021).

A major motivation for studying Entangled Relativity is that it requires fewer
fundamental universal dimensionful constants than standard physics, while
still recovering standard physics in a generic limit.111While standard physics
relies on three fundamental universal dimensionful constants, $G$, $\hbar$,
and $c$, Entangled Relativity, in a relevant quantum field theory framework
depicting the entire scope of physics, requires only two: $\epsilon$ and $c$,
where $\epsilon$ is the reduced Planck energy Minazzoli (2022, 2023)—see Eq.
(1). Thanks to a cancellation in the field equations—originally found in
scalar-tensor theories and named ‘intrinsic decoupling’ in Minazzoli and Hees
(2013)—the phenomenology of the theory is surprisingly similar to that of
General Relativity in many relevant situations, such as in the solar system
Minazzoli and Hees (2013) and with neutron stars Arruga  _et al._ (2021);
Arruga and Minazzoli (2021). The issue of the accelerated expansion of the
universe remains unclear within this framework, although there are already a
few proposals Minazzoli (2018, 2021, 2024). Entangled Relativity naturally
converges toward General Relativity without a cosmological constant during the
universe’s expansion. However, the structure of the theory prevents adding a
cosmological constant by hand if one wants Newton’s constant, $G$, to remain
constant in the dark energy cosmological era Minazzoli (2018, 2021, 2024).
Therefore, in Entangled Relativity, the acceleration of the expansion of the
universe cannot simply be accounted for by a cosmological constant but must
originate from something else. Hence, if the theory provides a better
depiction of the laws of physics than General Relativity, the $\Lambda$-CDM
cosmological model cannot be the most accurate cosmological model possible.

Since the theory prohibits vacuum solutions a priori, the vacuum solutions of
General Relativity are not solutions of Entangled Relativity. However, we now
have increasing evidence, notably through the observation of the shadows of
supermassive black-holes Event Horizon Telescope Collaboration (2019);
Collaboration (2019) and the ringdown of gravitational waves from the merger
of two black-holes Isi  _et al._ (2019); LVK collaboration (2019, 2021), that
General Relativity’s black-holes provide a good description of very massive
compact objects in our universe. Therefore, it is necessary to verify that
black-hole solutions in Entangled Relativity do not deviate significantly from
those in General Relativity under astrophysical conditions, and/or to predict
potential observational deviations that might be expected from Entangled
Relativity.

To investigate this issue, previous studies have focused on charged black-
holes Minazzoli and Santos (2021); Wavasseur  _et al._ . This approach ensures
that the matter field—here, the electromagnetic field—is defined everywhere in
the solution. These studies notably showed that the Schwarzschild black-hole
solution is a limit of the charged black-hole of Entangled Relativity in the
vanishing charge limit, which corresponds to the vanishing matter field limit
in this case. Since astrophysical black-holes are not expected to have
significant charges—either because they are not formed with substantial
charges or because they tend to discharge over time by attracting more
particles with opposite charges—this result suggests that Entangled Relativity
black-holes may be indistinguishable from those of General Relativity under
astrophysical conditions.

In the work presented in this communication, we used an alternative strategy.
Instead of looking for charged black-hole solutions, we sought neutral
(spherical) solutions immersed in either an electric or a magnetic field. The
results of these investigations are presented below.

##  II Field equations

The quantum phase that defines the theory is given by Minazzoli (2022, 2023)

$\Theta=-\frac{1}{2\epsilon^{2}}\int
d^{4}_{g}x\frac{\mathcal{L}^{2}_{m}(f,g)}{R(g)},$ (1)

where $R$ is the usual Ricci scalar that is constructed upon the metric tensor
$g$, $\mathrm{d}^{4}_{g}x:=\sqrt{-|g|}\mathrm{d}^{4}x$ is the spacetime volume
element, with $|g|$ the metric $g$’s determinant, and $\mathcal{L}_{m}$ is the
Lagrangian density of matter fields $f$—such as gauge bosons, fermions and the
Higgs—which could be the current standard model of particle physics Lagrangian
density, but most likely a completion of it. It also depends on the metric
tensor, a priori through to the usual comma-goes-to-semicolon rule Misner  _et
al._ (1973) in order to recover General Relativity in some limit.222Strictly
speaking, this condition is only necessary in some limit of the theory, but
could perhaps be relaxed in general, as long as it then emerges in the
required limit. Let us note that, like General Relativity, Entangled
Relativity does not specify what $\mathcal{L}_{m}$ should be.

Neither Newton’s constant $G$, nor Planck’s constant $\hbar$ appear in the
formulation of the quantum phase of the theory Eq. (1). Therefore, none of
them are fundamental constant in Entangled Relativity Ludwig  _et al._ (2015);
Minazzoli (2022, 2023). From now on, we will refer to them as Newton’s
parameter or Planck’s parameter.

Classical physics correponds to the paths with stationary phases
$\delta\Theta=0$. Those paths correspond to the following field equations
(Ludwig  _et al._ , 2015)

$G_{\mu\nu}=\kappa
T_{\mu\nu}+f_{R}^{-1}\left[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box\right]f_{R},$
(2)

with

$\displaystyle\kappa=-\frac{R}{\mathcal{L}_{m}},\qquad
f_{R}=\frac{1}{2\epsilon^{2}}\frac{\mathcal{L}_{m}^{2}}{R^{2}}=\frac{1}{2\epsilon^{2}\kappa^{2}},$
(3)

with the following stress-energy tensor

$T_{\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}_{m}\right)}{\delta
g^{\mu\nu}},$ (4)

which is not classically conserved

$\nabla_{\sigma}\left(\frac{\mathcal{L}_{m}}{R}T^{\alpha\sigma}\right)=\mathcal{L}_{m}\nabla^{\alpha}\left(\frac{\mathcal{L}_{m}}{R}\right).$
(5)

In the particular case of eletromagnetism, using units such that the magnetic
permeability of vacuum is $\mu_{0}=1$, one has

$\mathcal{L}_{m}=-\frac{1}{2}F_{\mu\nu}F^{\mu\nu},$ (6)

with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and $A^{\alpha}$
the four-vector electromagnetic potential. The stress-energy tensor is

$T_{\mu\nu}=2\left(F_{\mu\sigma}F_{\nu}^{~{}\sigma}+g_{\mu\nu}\frac{\mathcal{L}_{m}}{2}\right).$
(7)

The electromagnetic field equation on the other hand reads

$\nabla_{\sigma}\left(\frac{\mathcal{L}_{m}}{R}F^{\mu\sigma}\right)=0.$ (8)

##  III Black-hole in a uniform magnetic field

We seek for a solution akin to the Schwarzschild-Melvin solution of General
Relativity Griffiths and Podolský (2009). Using units such that

$\lim_{A^{\alpha}\rightarrow 0}\kappa=1,$ (9)

where $A^{\alpha}$ is the electromagnetic 4-vector, the solution for the
metric is

$\displaystyle
ds^{2}=\Lambda^{-\frac{28}{13}}r^{2}sin^{2}(\theta)d\varphi^{2}$
$\displaystyle+\Lambda^{\frac{20}{13}}\left[\left(1-\frac{\mathit{r}_{S}}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}-\left(1-\frac{\mathit{r}_{S}}{r}\right)c^{2}dt^{2}\right],$

where $\mathit{r}_{S}$ is the Schwarzschild radius and with

$\Lambda=1+\frac{13}{48}B^{2}r^{2}sin^{2}\theta.$ (11)

The solution for the electromagnetic 4-vector is

$A=\frac{B}{2\Lambda}r^{2}\sin^{2}\theta~{}d\varphi.$ (12)

Regarding the Petrov classification of the spacetime described by the metric
(III), we find that the external part is algebraically general. Let us note
that the solution is such that

$\mathcal{L}_{m}=-B^{2}\Lambda^{-44/13}\left(1-\frac{\mathit{r}_{S}~{}\sin^{2}\theta}{r}\right),$
(13)

and

$R=B^{2}\Lambda^{-46/13}\left(1-\frac{\mathit{r}_{S}~{}\sin^{2}\theta}{r}\right),$
(14) 333Whereas $R=0$ and
$\mathcal{L}_{m}=-B^{2}\Lambda^{-4}\left(1-\mathit{r}_{S}\sin^{2}\theta/r\right)$
for the Schwarzschild-Melvin solution of General Relativity.

such that one has

$\frac{R}{\mathcal{L}_{m}}=-\Lambda^{-2/13}.$ (15)

Therefore, it is clear from the onset that the magnetic solution is well
defined in the $\mathcal{L}_{m}\rightarrow 0$ limit. It is also worthwhile to
note that the solution is such that $\mathcal{L}_{m}<0$ and $R>0$, such that
$R/\mathcal{L}_{m}<0$. In particular, for $\theta=0$, one has
$\mathcal{L}_{m}=-B^{2}$ and $R=B^{2}$, such that $\kappa=1$.444The solution
and its properties have been checked using SageManifolds Gourgoulhon  _et al._
(2015). The notebook is freely accessible at the following url :
https://github.com/ominazzoli/ER-Schwarzschild-Melvin.

### Limiting cases

####  III.0.1 $B\rightarrow 0$

For $B\rightarrow 0$, one has $\Lambda\rightarrow 1$, such that the
Schwarzschild black-hole is a limit of the solution Eq. (III) when the
magnetic field disappears. In that limit, both $R$ and $\mathcal{L}_{m}$ tend
to zero, but their ratio remains well defined in that limit, as one can see in
Eq. (15). Given the existence of cosmological magnetic fields with extremely
weak strengths in our universe, the solution Eq. (III) with $B\rightarrow 0$
is a good approximation of astrophysical (spherical) black holes. This is
because astrophysical black holes have negligible charges due to their
formation processes and their tendency to attract oppositely charged particles
more than similarly charged ones. At first, given that the field equations of
Entangled Relativity are different from those of General Relativity, it may be
surprising that one obtains a metric of General Relativity as a limiting case
when the magnetic field disappears. However, this can be easily understood by
taking the trace of the metric field equation. It reads

$3\square\frac{\mathcal{L}_{m}^{2}}{R^{2}}=-\frac{\mathcal{L}_{m}}{R}\left(T-\mathcal{L}_{m}\right).$
(16)

This equation shows that the ratio between $\mathcal{L}_{m}$ and $R$ is a new
scalar degree of freedom in Entangled Relativity Ludwig  _et al._ (2015). With
a magnetic field, such a scalar degree of freedom is sourced because
$\mathcal{L}_{m}\neq 0$ whereas $T=0$. However, when
$(\mathcal{L}_{m},T)\rightarrow 0$, the scalar degree of freedom is no longer
sourced and remains constant. When this happens, one can check that the whole
set of field equations becomes equivalent to those obtained in the framework
of General Relativity minimally coupled to matter fields.

That being said, it would not have been possible to start finding a solution
from the vacuum assumption—that is $(\mathcal{L}_{m},T)=0$—as the ratio
between $R$ and $\mathcal{L}_{m}$ would not have been well-defined in Eq. (2)
in this case. Hence, we still argue that Entangled Relativity prevents the
existence of vacuum solutions,555Moreover, as soon as $\mathcal{L}_{m}$ is
defined in the path integral formulation of the theory using Eq. (1) Minazzoli
(2022, 2023), there will always be a non-null effective background field
composed of the quantum vacuum. The issue of the effective contribution of the
quantum vacuum to the field equations in this theory is yet to be
investigated. but it does have the vacuum solutions of General Relativity as
asymptotic solutions in the $(\mathcal{L}_{m},T)\rightarrow 0$ limit.

####  III.0.2 $\mathit{r}_{S}\rightarrow 0$

When the size of the black-hole is set to zero, the metric becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Lambda^{\frac{20}{13}}\left[dr^{2}+r^{2}d\theta^{2}-dt^{2}\right]$
(17) $\displaystyle+\Lambda^{-\frac{28}{13}}r^{2}sin^{2}(\theta)d\varphi^{2}.$

It deviates from the Melvin solution Griffiths and Podolský (2009) because the
extra degree-of-freedom is sourced—see Eq. (16)—such that the field equations
deviate from the ones of General Relativity. Nevertheless, the structure is
quite similar to the Melvin solution of General Relativity.

####  III.0.3 $(B,\mathit{r}_{S})\rightarrow 0$

Quite interestingly, since $\lim_{B\to 0}\Lambda=1$, we also find that
Minkowski is a limit when both $B$ and $\mathit{r}_{S}$ tend to zero, for
which the ratios $R/\mathcal{L}_{m}$ and $\mathcal{L}_{m}/R$ remain finite all
along.

##  IV Black-hole in a uniform electric field

The electric version of the solution can be obtained through the following
transformation

$\displaystyle F_{\mu\nu}\longrightarrow
F^{e}_{\mu\nu}=-\frac{1}{2}\frac{\mathcal{L}_{m}}{R}~{}\epsilon_{\mu\nu\kappa\lambda}F^{\kappa\lambda},$
(18a) $\displaystyle g_{\mu\nu}\longrightarrow
g^{e}_{\mu\nu}=\left(\frac{\mathcal{L}_{m}}{R}\right)^{4}g_{\mu\nu}.$ (18b)

The resulting solution for the metric is

$\displaystyle
ds^{2}=\Lambda^{-\frac{20}{13}}r^{2}sin^{2}(\theta)d\varphi^{2}$
$\displaystyle+\Lambda^{\frac{28}{13}}\left[\left(1-\frac{\mathit{r}_{S}}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}-\left(1-\frac{\mathit{r}_{S}}{r}\right)c^{2}dt^{2}\right],$

with

$\Lambda=1+\frac{13}{48}E^{2}r^{2}sin^{2}\theta.$ (20)

Whereas, the resulting solution for the electromagnetic 4-vector is

$A=E\left(1-\frac{\mathit{r}_{S}}{r}\right)r\cos\theta~{}dt.$ (21)

We find that the external part ($r>\mathit{r}_{S}$) of the spacetime in Eq.
(III) is algebraically general. This is expected, given that the magnetic and
electric metric solutions differ by a conformal factor in Eq. (18), while the
Petrov classification is invariant under conformal transformations. Let us
note that the solution is such that

$\mathcal{L}_{m}=E^{2}\Lambda^{-56/13}\left(1-\frac{\mathit{r}_{S}~{}\sin^{2}\theta}{r}\right),$
(22)

and

$R=-E^{2}\Lambda^{-54/13}\left(1-\frac{\mathit{r}_{S}~{}\sin^{2}\theta}{r}\right),$
(23)

such that one has

$\frac{R}{\mathcal{L}_{m}}=-\Lambda^{2/13}.$ (24)

Therefore, it is clear from the onset that the electric solution is also well
defined in the $\mathcal{L}_{m}\rightarrow 0$ limit. It is also worthwhile to
notice that the solution is such that $\mathcal{L}_{m}>0$ and $R<0$—that is,
the opposite situation with respect to the magnetic case—such that one still
has $R/\mathcal{L}_{m}<0$ nonetheless. In particular, for $\theta=0$, one has
$\mathcal{L}_{m}=E^{2}$ and $R=-E^{2}$, such that $\kappa=1$.666The notebook
that checks this solution and its properties can be found at the following
address : https://github.com/ominazzoli/ER-Schwarzschild-Melvin.

### Limiting cases

As in the magnetic case, one obtains the Swcharzschild black-hole in the
$E\rightarrow 0$ limit, and Minkowski in the $(E,\mathit{r}_{S})\rightarrow 0$
limit. However, the solution without a black-hole is not the same as in Eq.
(17), since it reads as follows

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\Lambda^{\frac{28}{13}}\left[dr^{2}+r^{2}d\theta^{2}-dt^{2}\right]$
(25) $\displaystyle+\Lambda^{-\frac{20}{13}}r^{2}sin^{2}(\theta)d\varphi^{2}.$

##  V Discussion

Before the present result, it was not known whether or not Entangled
Relativity could have neutral black-hole solutions. Now we know this is the
case for spherical black-holes, and we can expect that it should also be the
case for rotating black-holes—although rotating neutral solutions remain to be
found. As with the case of charged black-hole solutions studied in Minazzoli
and Santos (2021); Wavasseur  _et al._ , we found that the vacuum black-hole
solutions of General Relativity are limits of black-hole solutions of
Entangled Relativity when the value of the matter field amplitude goes to
zero. Since such a value is expected to be extremely small in the
universe—whether one is talking about the charge of a black-hole, or the
background density of magnetic or electric fields surrounding black-holes—the
present results are yet another indication that vacuum black-hole solutions of
General Relativity are good approximations of black-holes in Entangled
Relativity. Therefore, it presently seems unlikely to us that one may be able
to discriminate between the two theories through the observations of black-
holes and associated phenomena. Neutron stars seem to be better targets in
that respect Arruga  _et al._ (2021); Arruga and Minazzoli (2021).

One interesting point with the solutions presented in this paper is that they
are such that the matter Lagrangian is either positive
($\mathcal{L}_{m}\propto E^{2}$) or negative ($\mathcal{L}_{m}\propto-B^{2}$)
depending on whether one is dealing with an electric or magnetic field. This
is interesting because one can see in Eq. (3) that the sign of the
gravitational coupling parameter $\kappa$ is proportional to the ratio between
$\mathcal{L}_{m}$ and $R$. Indeed, Entangled Relativity does not prevent, a
priori, a negative value the gravitational parameter $\kappa$. This would,
however, be quite inconsistent with any observations so far, as a negative
parameter would imply repulsive gravity. Furthermore, one might naïvely expect
the sign of $\kappa$ to change with a change in the sign of $\mathcal{L}_{m}$,
and the definition of $\kappa$ to be singular when $\mathcal{L}_{m}\rightarrow
0$.777These arguments have notably been used by a referee to prevent the
publication of Arruga and Minazzoli (2021) in Phys. Rev. D. But for the
solutions found in the present paper—as well as for the charged black-hole
solutions Wavasseur  _et al._ —this is not the case. This is due to the fact
that when $\mathcal{L}_{m}$ changes sign or goes to zero, so does $R$, such
that the ratio is always well-defined and the sign of $\kappa$ remains the
same in all cases.

Given our present understanding of Entangled Relativity, we do not expect
$\kappa$ to vary much in the observable universe. Indeed, it has been found
that it only varies by a few percent within the densest objects of the
observable universe Arruga  _et al._ (2021); Arruga and Minazzoli (2021).
However, it cannot be excluded a priori that the value of $\kappa$ varies much
more widely in the collapsing dense compact object inside a black-hole, that
is, in the collapsing object after the formation of the event horizon.
Furthermore, we currently do not see any reason to exclude a priori that
$\kappa$ might even have its sign changed, potentially leading to repulsive
gravity that could prevent the formation of a singularity. One could use the
same line of argument for the primordial universe. These prospects are
fascinating future avenues of research in Entangled Relativity.

## References

  * Ludwig  _et al._ (2015) Hendrik Ludwig, Olivier Minazzoli, and Salvatore Capozziello, “Merging matter and geometry in the same Lagrangian,” Physics Letters B 751, 576–578 (2015), arXiv:1506.03278 [gr-qc] .
  * Einstein (1918a) A. Einstein, “Prinzipielles zur allgemeinen Relativitätstheorie,” Annalen der Physik 360, 241–244 (1918a), translation available at https://einsteinpapers.press.princeton.edu.
  * Einstein (1918b) Albert Einstein, “Kritisches zu einer von Hrn. de Sitter gegebenen Lösung der Gravitationsgleichungen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin , 270–272 (1918b), translation available at https://einsteinpapers.press.princeton.edu.
  * Pais (1982) Abraham Pais,  _Subtle is the Lord. The science and the life of Albert Einstein_ (Oxford University Press, Oxford, 1982).
  * Hoefer (1995) C. Hoefer, “Einstein’s Formulations of Mach’s Principle,” in  _Mach’s Principle: From Newton’s Bucket to Quantum Gravity_ , edited by Julian B. Barbour and Herbert Pfister (Birkhäser, Boston University, 1995) p. 67.
  * Arruga  _et al._ (2021) Denis Arruga, Olivier Rousselle, and Olivier Minazzoli, “Compact objects in entangled relativity,” Phys. Rev. D 103, 024034 (2021), arXiv:2011.14629 [gr-qc] .
  * Minazzoli (2022) Olivier Minazzoli, “Quantum of action in entangled relativity,” arXiv e-prints , arXiv:2206.03824 (2022), arXiv:2206.03824 [gr-qc] .
  * Minazzoli (2023) Olivier Minazzoli, “Standard quantum field theory from entangled relativity,” Contribution to the 2023 Gravitation session of the 57th Rencontres de Moriond , arXiv:2304.09482 (2023), arXiv:2304.09482 [gr-qc] .
  * Minazzoli and Hees (2013) Olivier Minazzoli and Aurélien Hees, “Intrinsic Solar System decoupling of a scalar-tensor theory with a universal coupling between the scalar field and the matter Lagrangian,” Phys. Rev. D 88, 041504 (2013), arXiv:1308.2770 [gr-qc] .
  * Arruga and Minazzoli (2021) Denis Arruga and Olivier Minazzoli, “Analytical external spherical solutions in entangled relativity,” European Physical Journal C 81, 1027 (2021), arXiv:damour2106.03426 [gr-qc] .
  * Minazzoli (2018) Olivier Minazzoli, “Rethinking the link between matter and geometry,” Phys. Rev. D 98, 124020 (2018), arXiv:1811.05845 [gr-qc] .
  * Minazzoli (2021) Olivier Minazzoli, “De sitter space-times in entangled relativity,” Classical and Quantum Gravity 38, 137003 (2021).
  * Minazzoli (2024) Olivier Minazzoli, “Cosmology in Entangled Relativity,” Contribution to the 2024 Cosmology session of the 58th Rencontres de Moriond , arXiv:2405.02051 (2024), arXiv:2405.02051 [gr-qc] .
  * Event Horizon Telescope Collaboration (2019) Event Horizon Telescope Collaboration, “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” ApJ 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA] .
  * Collaboration (2019) The Event Horizon Telescope Collaboration, “First m87 event horizon telescope results. vi. the shadow and mass of the central black hole,” The Astrophysical Journal Letters 875, L6 (2019).
  * Isi  _et al._ (2019) Maximiliano Isi, Matthew Giesler, Will M. Farr, Mark A. Scheel, and Saul A. Teukolsky, “Testing the No-Hair Theorem with GW150914,” Phys. Rev. Lett. 123, 111102 (2019), arXiv:1905.00869 [gr-qc] .
  * LVK collaboration (2019) LVK collaboration, “Tests of general relativity with the binary black hole signals from the LIGO-Virgo catalog GWTC-1,” Phys. Rev. D 100, 104036 (2019), arXiv:1903.04467 [gr-qc] .
  * LVK collaboration (2021) LVK collaboration, “Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog,” Phys. Rev. D 103, 122002 (2021), arXiv:2010.14529 [gr-qc] .
  * Minazzoli and Santos (2021) Olivier Minazzoli and Edison Santos, “Charged black hole and radiating solutions in entangled relativity,” European Physical Journal C 81, 640 (2021), arXiv:2102.10541 [gr-qc] .
  * (20) Maxime Wavasseur, Théo Abrial, and Olivier Minazzoli, “Slowly rotating Black-holes in Entangled Relativity,” in preparation .
  * Misner  _et al._ (1973) Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,  _Gravitation_ (W.H. Freeman and Compagny, 1973).
  * Griffiths and Podolský (2009) Jerry B. Griffiths and Jirí Podolský,  _Exact Space-Times in Einstein’s General Relativity_ (Cambridge University Press, 2009).
  * Gourgoulhon  _et al._ (2015) Eric Gourgoulhon, Michal Bejger, and Marco Mancini, “Tensor calculus with open-source software: the SageManifolds project,” in  _Journal of Physics Conference Series_, Journal of Physics Conference Series, Vol. 600 (IOP, 2015) p. 012002, arXiv:1412.4765 [gr-qc] .