Observational Signatures of Quantum Gravity in Interferometers

Erik P. Verlindea, Kathryn M. Zurekb

aInstitute of Physics, University of Amsterdam, Amsterdam, The NetherlandsbWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA USA

Abstract

We consider the uncertainty in the arm length of an interferometer due to metric fluctuations from the quantum natureof gravity, proposing a concrete microscopic model of energy fluctuations in holographic degrees of freedom on thesurface bounding a causally connected region of spacetime. In our model, fluctuations longitudinal to the beam directionaccumulate in the infrared and feature strong long distance correlation in the transverse direction. This leads to a signalthat could be observed in a gravitational wave interferometer. We connect the positional uncertainty principle arisingfrom our calculations to the ’t Hooft gravitational S-matrix.

1. Introduction

The quantum mechanical description of gravity togetherwith the other forces remains one of the most importantquestions in physics. While general relativity can be quan-tized as an effective field theory valid at low energies, andthere has been significant theoretical progress in under-standing other aspects of quantum gravity, signatures ofthe quantum nature of gravity have so far remained stub-bornly immune to observation.

An important clue towards the ultimate theory of quan-tum gravity is provided by the holographic principle [1, 2].One of its implications is the covariant entropy bound [3],which states that the entropy associated to region boundedby null geodesics does not exceed A/4GN , where A denotesthe area of the surface and Newton’s constant is identifiedwith the square of the Planck length via 8πGN ≡ l2p, withlp ' 10−35m. The holographic principle suggests that thetotal number of microscopic degrees of freedom associatedto a given region of space (defined by the maximal entropy)is given by the area of the surrounding surface in Planckunits. In this form the holographic principle is known tobe realized in spacetimes with negative cosmological con-stant [4], and is firmly incorporated in the framework ofthe AdS/CFT correspondence [5].

Motivated by the holographic principle, one is temptedto postulate that the microscopic spacetime degrees of free-dom, also in flat spacetime, can be identified with Plancksize pixels on the surface bounding a causally connectedpart of space. The spacetime volume would then emergein the infrared from these holographic spacetime quanta.One of the intriguing aspects of the holographic principleis that, to ensure the validity of the entropy bound, thespacetime degrees of freedom are necessarily correlated inthe infrared. This raises the question of whether Planckscale physics could appear at much longer, potentially ob-servable, length scales.

Our goal in this Letter is to investigate whether fluctu-ations due to the graininess of spacetime can potentiallylead to observable signatures. Normal intuition would saythat, since the natural length and time scale associatedwith the quantum nature of spacetime is Planckian, thatno feasible experiment exists that could measure its ef-fects. We will argue, however, that when combined withimportant infrared effects naturally expected from holog-raphy, the accumulative effect of Planck scale fluctuationscan be transmuted to observable time and length scales.Because of their sensitivity to exquisitely short distancescales, gravitational wave interferometers are an ideal test-ing ground for these ideas.

We will identify the required theoretical conditions thatneed to be satisfied to obtain observable effects, and con-struct a concrete holographic model that obeys those con-ditions. After showing that uncorrelated Planckian fluctu-ations are not macroscopically observable, we demonstratethat fluctuations with sufficient transverse correlations inthe infrared do lead to observable effects. The appearanceof transverse correlations is crucial and suggests a holo-graphic description in which the longitudinal and trans-verse behavior of the spacetime degrees of freedom aretreated on a different footing.

We will build an explicit holographic model in termsof Planck-size pixels that saturate the holographic boundand have energy fluctuations that cause the spatial lengthL of a causally connected region of space to fluctuate. Thetransverse correlations are generated through the Newto-nian potential of these energy fluctuations, allowing usto make a concrete prediction for the spectrum of lengthfluctuations in an interferometer. These fluctuations im-ply a spacetime uncertainty relation in the longitudinaldirection, which we connect, albeit in a modified form,to the gravitational S-matrix approach of ’t Hooft (seeRefs. [6, 7]).

While there have been several previous studies seeking

Preprint submitted to Elsevier October 22, 2021

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to heuristically connect holography to interferometry (e.g.[8, 9, 10, 11, 12]), our theoretical description is structurallyunique in its holographic set-up. And although the firststeps we take–employing a Planckian random walk–sharescommonalities with these works, our approach differs inthe sense that we present a concrete theoretical modelleading to length fluctuations along the longitudinal di-rection with a distinctive signature for strong transversecorrelations, which is, as a consequence, macroscopicallyobservable in an interferometer. A phenomenological re-sult is that constraints from the images of distant astro-physical sources derived for uncorrelated fluctuations inRefs. [13, 14] do not apply to our model.

2. Length Fluctuations with Planckian White Noise

In this paper we consider a toy experimental set-up,shown in Fig. 1, in which the arm length L of an inter-ferometer is measured after a single light crossing. In thisidealized scenario the length fluctuations δL due to quan-tum fluctuations in the metric is given by

δL(t) =1

2

∫ L

0

dz h(t+z−L) (1)

where h ≡ hzz is the metric component along the lightbeam propagation (see e.g. [15]). The magnitude of theselength fluctuations is normally expressed in terms of thepower spectral density (PSD)

S(ω, t) =

∫ ∞−∞

dτ

⟨δL(t)

L

δL(t− τ)

L

⟩e−iωτ . (2)

Let us first consider a simple model with a white noisesignal of Planckian amplitude⟨

h(t+z1−L)h(t+z2−L−τ)⟩

= Clpδ(τ+z1−z2), (3)

where lp =√

8πGN . This leads to a PSD of the form

S(ω) =Clp4

sin2 ωL

ω2L2. (4)

In this simple model the length fluctuations 〈δL2〉 obey⟨δL2(t)

L2

⟩=

1

2π

∫ ∞−∞dω S(ω) =

Clp8L

, (5)

and thus grow linearly with L [8, 9, 10, 11, 12]. This signalcould in principle be observable, since the peak sensitiv-ity for gravitational wave interferometers is right aroundthe Planck scale: S(ω, t) . lp. Over the next sections ourgoal will be to show how some of the generic behavior inEqs. 4, 5 can arise from a holographic model, motivatingthe size of the constant C, with crucial observational ef-fects arising from angular correlations. In addition, in ex-periments like LIGO and Virgo a typical photon traversesthe interferometer arm multiple times before being mea-sured. In this paper we continue to focus on our simple setup and defer the detailed discussion of multiple crossingsto future work.

3. Holographic Scenario and Basic Postulates

Our aim in the following is to derive a result similarto Eq. 5 from a holographic scenario, in which the holo-graphic surface is fixed by the light path of a photon, asdepicted in Fig. 1. In order to clearly delineate betweentheoretical input and observational consequences, we willstate here our basic postulates:

1. Holographic principle in flat spacetime. We postu-late that the holographic principle also applies toMinkowski spacetime. It states that the maximalentropy carried by the microscopic degrees of free-dom associated with a finite region of flat spacetimebounded by null geodesics is S = A/4GN . Thisbound is saturated for a region of space whose nullboundary coincides with a horizon.

2. Universality of metric fluctuations at horizons. Wepostulate, as a corollary of the first postulate, thatmetric fluctuations near null surfaces associated withthe boundary of a finite region follow from the en-tropy and temperature using standard thermody-namic considerations. This postulate implies thatmetric fluctuations near a Rindler-type horizon areidentical to those near a black hole horizon with thesame temperature and entropy.

Note that we are treating the metric fluctuations atthe holographic surface separating the inside of the causaldiamond from the outside as if it were a black hole hori-zon (see Ref. [16]), even though we are considering thevacuum of Minkowski space. The basic reason we believethese are reasonable postulates is that a finite causal dia-mond in Minkowski space, when suitably foliated, can berecast in the metric of a so-called topological black hole[17]. Furthermore, a conformal field theory restricted tothe diamond behaves as a thermal field theory [17], andthe quantized Einstein-Hilbert metric in the infrared be-haves as a conformal field theory. In related work [18], weshow that these postulates are justified in the context ofAdS/CFT. That they hold for the Einstein-Hilbert metricin Minkowski space must, at the present time, be ulti-mately verified by experiment. Fortunately, we show thatthe experimental signatures associated with a spacetimeobeying these postulates are within reach with current in-terferometer technology.

4. Towards Macroscopic Effects in Interferometers

The results in Eqs. 3-5, that were derived from thesimple 1D-model, are by themselves not sufficient to showan effect. In order to be observable in a realistic experi-mental set up, the fluctuations must be coherent at macro-scopic spacetime distances. To examine the conditions un-der which such coherent fluctuations occur, we extend ourmodel by including the two spatial directions transverseto the beam direction. Anticipating our holographic de-scription, we consider metric fluctuations that depend on

2

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endmirror

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time

space

laser

Figure 1: The interferometer together with the spacetime diagram fora single crossing of a photon in the signal beam. The interferometerat time t is contained in a causal diamond centered at the beamsplitter and with the photon path on its null boundary.

only three coordinates, one longitudinal null direction andtwo transversal directions, corresponding to the outsideboundary of the causal diamond in Fig. 1,⟨(

δL

L

)1

(δL

L

)2

⟩=

1

16L2

∫ L

−L

∫ L

−Ldu1du2 (6)∫

d3k1(2π)3

d3k2(2π)3

〈h(k1)h(k2)〉eik1·x1eik2·x2 .

We first consider an Ansatz corresponding to uncorrelatedwhite noise in those three dimensions:⟨

h(k1)h(k2)⟩

= (2π)3δ3(k1 + k2)Cl3p. (7)

This power spectrum implements the principle of statisti-cal independence both in the longitudinal as well as thetransversal directions. This can be seen directly by com-puting the PSD and RMS length fluctuations:⟨(

δL

L

)1

(δL

L

)2

⟩=

Clp16πL

1

(∆x2T /l2p + 1)3/2

. (8)

In the limit ∆xT → 0, we recover a signal of an amplitudethat is in principle within the observable range and is con-sistent with Eqs. 4-5. However, for a realistic macroscopicinterferometer, with the beam size centimeters across suchthat ∆xT /lp�1, this signature would be unobservable.

Let us consider an alternative Ansatz for the metricfluctuations in which the transversal directions are treateddifferently:⟨

h(k1)h(k2)⟩

= (2π)3δ3(k1 + k2)Clp

(k2T + k2IR), (9)

where kIR acts as a regulator. Then Eq. (8), in the limitthat kIR∆xT � 1, becomes⟨(

δL

L

)1

(δL

L

)2

⟩∼ Clp

16πLlog [1/∆xT kIR] . (10)

Already this result shows important features that theunderlying theory must give, notably that the longitudi-nal and transverse directions appear on a different footing.The metric fluctuation in the transverse direction must becorrelated, while the metric fluctuations in the longitudi-nal direction accumulate, as in a random walk, and aretransmuted to a low-energy, long-distance signature. Wewill show over the next sections how these features arisenaturally from energy fluctuations on a holographic sur-face.

5. From Minkowski to Schwarzschild-like Metric

The central part of our argument involves utilizinga correspondence between any horizon and a black holehorizon. To show concretely how this applies to the caseat hand, we make two metric transformations, which aredescribed below. First we define light cone coordinatesu = r + t and v = r − t so that metric becomes

ds2 = dudv + dy2 + huudu2 + hvvdv

2 + . . . (11)

where the dots denote the angular components. In thismetric the light paths on the lower and upper half of thecausal diamond shown in Fig. 1 are given by

v = L+ δv(u) and u = L+ δu(v)

The total length fluctuation δL can be expressed as

δL = (δv(L) + δu(L)) /2.

It turns out that only one metric component contributesto the time delay along each light path. As we will showin a companion paper, the values for δv(L) and δu(L) canbe expressed in terms of the metric fluctuations via

δv(L) =

∫ L

−Ldu huu(u, L) (12)

δu(L) =

∫ L

−Ldv hvv(L, v).

As a next step, to employ our postulates, we recast themetric in the Schwarzschild-like form

ds2 = −f(R)dT 2 +dR2

f(R)+ r2(dθ2 + sin2 θdφ2). (13)

in such a way that the light paths of the photon are mappedonto the event horizon located at f(R) = 0. This isachieved by making the coordinate transformation

(u− L)(v − L) = 4L2f(R), logu− Lv − L =

T

L(14)

where the function f(R) is given by

3

f(R) = 1− R

L+ 2Φ. (15)

Here Φ plays the role of the Newtonian potential andparametrizes the deviations in the geometry due to vac-uum fluctuations in the energy conjugate to the time T .

Without any quantum gravity effects, the horizon islocated at R = L. In general, its location is determinedby f(R) = 0. This leads to the following relationshipbetween the product of the lightcone time variations δu(L)and δv(L) and the value of Newton’s potential

δv(L)δu(L)

L2= 2Φ(L). (16)

This equation should be regarded as an operator identity.Since 〈Φ〉 = 0 in vacuum, the right-hand-side of this equa-tion actually represents a fluctuation around the vacuum,whose amplitude is given by squaring the operators andtaking its expectation value:⟨(

δv(L)δu(L)

L2

)2⟩

=⟨

4Φ(L)2⟩. (17)

For a more detailed and formal discussion of this pointin the context of AdS/CFT, we encourage the reader toconsult Sec IV of our companion paper Ref. [18].

The goal of the next section is to determine the root-mean-square value of the fluctuations in Φ in an ensembleaveraged over many interferometer light crossings.

6. Holographic Model for Spacetime Fluctuations

We are now ready to employ all of our postulates to-gether to compute the deviations in the Newtonian poten-tial, Eq. 16. The fluctuations in Φ(L) will be induced byvacuum fluctuations in the energy conjugate to the timecoordinate T . Eq. 16. In the following analysis we followclosely the reasoning of Marolf in Ref. [16] for the quan-tum thickness of black hole horizons. Directly applying theholographic principle to the horizon of the causal diamondgives

Shor =A

4GN=

8π2L2

l2p. (18)

Now the fluctuations in the Newtonian potential on thehorizon obeys

2Φ(L) = − l2p∆M

4πL, (19)

where ∆M represents the energy fluctuations in the holo-graphic degrees of freedom. Heuristically, one expects theRMS value of ∆M to scale as the square root of the num-ber of pixels on the horizon, times the typical energy of thefluctuation, which is given by the Hawking temperature.

One of the standard methods to determine the Hawk-ing temperature is to go to Euclidean time and impose

that the resulting metric is free from conical singularities.In this way one finds

Thor =|f ′(L)|

4π=

1

4πL. (20)

In the present situation the temperature Thor is measuredby an accelerated observer whose event horizon coincideswith the photon trajectory and whose own trajectory passesthrough the origin at T = 0. This observer stays at R = 0and has T as proper time coordinate.

We now calculate the RMS value of the fluctuations,by assuming that the vacuum energy E vanishes. Thisimplies that the free energy F (β) equals

F (β) = −ThorShor = − β

2l2p(21)

where in the last step we eliminated the length L in favorof the inverse temperature β = 1/Thor = 4πL. In thecanonical ensemble the mass fluctuations ∆M are obtainedby taking the second derivative of the free energy. Onethus obtains

〈∆M2〉 = − ∂2

∂β2(βF ) =

1

l2p. (22)

Note that ∆M ∼ Thor√Shor, as expected from the heuris-

tic argument. We now assume that at a coincident point,δv(L) and δu(L) take the same value δL. In this situation,combining Eqs. 16-22, we learn that the amplitude of thelength fluctuation is⟨

δL2

L2

⟩=l2p∆M

4πL=

lp4πL

, (23)

where here ∆M =√〈∆M2〉 is interpreted as the root-

mean-square of the mass fluctuation. Note this has pre-cisely the behavior shown in Eq. 5 needed to be observable,where now we can fix the constant C via the holographicprinciple. We will propose in the next section that angu-lar correlations between the interferometer arms respectthe spherical symmetry of the measuring apparatus, andwould give rise to a distinctive experimental signature.

7. Angular Correlations and ’t Hooft’s S-matrix

We have considered so far the amplitude of the fluctu-ations only as a function of the longitudinal coordinates.Physically it is clear that the fluctuations will also have anangular dependence, which can be straightforwardly deter-mined for an interferometer with two arms of equal lengthL. In this case, a spherical coordinate system, with originat the beamsplitter, is appropriate, with the far mirrorslocated at two positions r̃1, r̃2 on the sphere. In this ex-perimental configuration, the angular information can bedetermined with the help of the Newtonian potential Φ de-composed in terms of spherical harmonics, thus respectingthe spherical symmetry of the measuring apparatus.

4

Accordingly, we propose the following natural Ansatzas a generalization of Eq. 23:⟨

δL(r̃1)δL(r̃2)⟩

=lpL

4πG(r̃1, r̃2). (24)

This equation can be further justified from a generalizationof Eq. 17 to give the four-point correlation at separatedpoints:⟨

δu(r̃1)δv(r̃1)δu(r̃2)δv(r̃2)⟩

=l2pL

2

16π2G2(r̃1, r̃2). (25)

If we again assume as above that at a coincident pointδu(r̃) = δv(r̃) = δL(r̃), the two-point Eq. 24 is seen to bethe factorization of Eq. 25 into the product of two two-point functions.

Our suggestion, based on the symmetries of the sys-tem, is to identify the function G(r̃1, r̃2) with the Greenfunction of a modified Laplacian on the sphere. It obeys(

−∇2r̃1 +

1

L2

)G(r̃1, r̃2) = δ(2)(r̃1, r̃2), (26)

and can be obtained by integrating the 3D Green functionalong the radial direction corresponding to the beam. Atshort distances it behaves as the normal Green functionon the 2D-plane

G(r̃1, r̃2) ∼ 1

2πlog

(L

|r̃1−r̃2|

)for |r̃1−r̃2|<<L. (27)

In terms of spherical harmonics it has the expansion

G(r̃1, r̃2) =∑`,m

Y`,m(r̃1)Y ∗`,m(r̃2)

`2 + `+ 1.. (28)

We can write this result alternatively in terms of the co-efficients, δL`m and δ`′m′ , of the decomposition of δL(r̃)and δL(r̃) in to spherical harmonics as⟨

δL`mδL`′m′⟩

=1

4π

lpL

`2 + `+ 1δ``′δmm′ . (29)

This relation tells us that much of the power in the fluctua-tions is contained in the low ` modes, and thus appears onthe largest scales, contrary to one’s intuition about Planck-ian effects. While we think that our Ansatz of fluctuationsobeying the Green function on the 2-d sphere is naturalgiven the symmetries of the system, we have not rigor-ously derived this result here; we leave more detailed workin this direction for the future.

Our result implies a fundamental uncertainty relationbetween the longitudinal spacetime components. As ’tHooft showed, the in-going and out-going radiation at thehorizon causes a spacetime shift due to gravitational shockwaves. He then went on to postulate that there is an inher-ent uncertainty in the values of the position of the horizon.In fact, ’t Hooft’s uncertainty relations described in e.g.[6, 7], when translated into a correlation function, have anidentical angular dependence as Eq. 29, though a differentnormalization. Due to our assumption of statistical inde-pendence and the resulting accumulation of the spacetimefluctuations, we find an extra factor L/lp compared to ’tHooft.

8. Conclusion and Discussion

In this Letter we have constructed a concrete holo-graphic model of transversally correlated longitudinal dis-tance fluctuations, due to vacuum energy fluctuations ofthe (holographic) degrees of freedom associated with acausally connected volume of spacetime. By assuming thatthe energy of these fluctuations is of the order 1/L, whereL is the length of the interferometer arm, and that num-ber of fluctuating degrees of freedom is L2/l2p, we havederived length fluctuations of size δL2 ∼ lpL. The strongtransverse correlation implies that in interferometer ex-periments the length fluctuations are sufficiently coherentacross the light beam, so that they are in principle observ-able.

If a signal with the characteristic features of our modelis observed it would be a confirmation that our postulatesregarding the theory of quantum gravity in flat spacetimeare realized. Conversely, if no signal is observed in anappropriately sensitive interferometer, it would tell us thatone our proposed postulates does not hold. In either case,we will have obtained concrete experimental informationabout the underlying theory of quantum gravity.

Our results were derived for a simple toy Michelsoninterferometer, so the next step is to concretely connectthe result in Eq. 29 to a power spectral density and torealistic interferometers. The closest experimental set-upto our toy is the “Holometer” [19], but to make a con-crete comparison to experimental results requires at mini-mum a computation extending Eq. 29 to include the (likely`-dependent) frequency information in the PSD. Gravi-tational wave interferometers like LIGO and Virgo havemultiple light-crossings; in our model we expect the sig-nal from each subsequent light-crossing to be statisticallyuncorrelated, effectively reducing the signal in Eq. 29 bya factor of the number of light crossings (as discussed inRef. [12]). To fully determine the observational implica-tions of our results one needs to incorporate these andother experimental aspects in to our model.

The transversal correlation of our model also has otherimportant phenomenological implications. Previous at-tempts to consider phenomenological effects from Planck-ian Brownian noise (see Refs. [20, 13], and as suggestedby Eq. 3-5) were stymied by the blurring of images fromdistant astrophysical sources. If the fluctuations are un-correlated in the transverse direction, this implies largedeviations in the phase of the light [14]. In the model un-der consideration here, however, most of the power in thelength fluctuations is contained in low `-modes, as shownin Eq. 29, that are coherent across the aperture diameterD of an optical device, namely those with ` ≤ 2πL/D.These modes do not affect the quality of the image ofastronomical objects. In our model the image quality istherefore improved compared to previously considered sit-uations. To fully establish that the transversal correlationsare sufficient to evade the astrophysical constraints needsfurther study.

5

On the theoretical side the most important open ques-tion is the precise nature of the holographic degrees of free-dom that are responsible for the spacetime fluctuations. Apossible route to gain more control over the microscopictheory is to consider an interferometer in Anti-de Sitterspace, and to reformulate the problem in terms of observ-ables of the dual conformal field theory. The question isthen to isolate the microscopic degrees of freedom dualto a small causal diamond deep inside the AdS geometry.Another promising direction is to relate our descriptionto the recent works on the BMS-group, soft gravitons andgravitational memory effects [21, 22]. In this context one isdealing with coordinate shifts at spatial infinity, or at eventhorizons of black holes [23]. The theoretical challenge inthis case is to generalize these studies to finite size causaldiamonds, and determine the fluctuation spectrum of thecoordinate shifts. We will leave these theoretical, as wellas the phenomenological, analyses to future work.

Acknowledgments. We thank Rana Adhikari for dis-cussions on the experimental prospects, Cliff Cheung fordiscussions and comments on the draft, as well as CraigHogan and Grant Remmen for conversation at the veryearly stages of this work. KZ thanks the CERN theorygroup for hospitality over much of the duration of thiswork. The research of EV is supported by the grant “Scan-ning New Horizons” and a Spinoza grant of the Dutch Sci-ence Foundation (NWO).

References

[1] G. ’t Hooft, Dimensional reduction in quantum gravity, Conf.Proc. C930308 (1993) 284–296. arXiv:gr-qc/9310026.

[2] L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995)6377–6396. arXiv:hep-th/9409089, doi:10.1063/1.531249.

[3] R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999)004. arXiv:hep-th/9905177, doi:10.1088/1126-6708/1999/

07/004.[4] L. Susskind, E. Witten, The Holographic bound in anti-de Sitter

space (1998). arXiv:hep-th/9805114.[5] J. M. Maldacena, The Large N limit of superconformal field the-

ories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113–1133,[Adv. Theor. Math. Phys.2,231(1998)]. arXiv:hep-th/9711200,doi:10.1023/A:1026654312961,10.4310/ATMP.1998.v2.n2.a1.

[6] G. ’t Hooft, The Scattering matrix approach for the quantumblack hole: An Overview, Int. J. Mod. Phys. A11 (1996) 4623–4688. arXiv:gr-qc/9607022, doi:10.1142/S0217751X96002145.

[7] G. ’t Hooft, Discreteness of Black Hole Microstates (9 2018).arXiv:1809.05367.

[8] Y. J. Ng, H. Van Dam, Limit to space-time measure-ment, Mod. Phys. Lett. A9 (1994) 335–340. doi:10.1142/

S0217732394000356.[9] Y. J. Ng, H. Van Dam, Remarks on gravitational sources,

Mod. Phys. Lett. A10 (1995) 2801–2808. doi:10.1142/

S0217732395002945.[10] G. Amelino-Camelia, An Interferometric gravitational wave de-

tector as a quantum gravity apparatus, Nature 398 (1999) 216–218. arXiv:gr-qc/9808029, doi:10.1038/18377.

[11] C. J. Hogan, Measurement of Quantum Fluctuations in Ge-ometry, Phys. Rev. D77 (2008) 104031. arXiv:0712.3419,doi:10.1103/PhysRevD.77.104031.

[12] O. Kwon, C. J. Hogan, Interferometric Tests of PlanckianQuantum Geometry Models, Class. Quant. Grav. 33 (10)(2016) 105004. arXiv:1410.8197, doi:10.1088/0264-9381/33/10/105004.

[13] E. S. Perlman, Y. J. Ng, D. J. E. Floyd, W. A. Christiansen,Using Observations of Distant Quasars to Constrain QuantumGravity, Astron. Astrophys. 535 (2011) L9. arXiv:1110.4986,doi:10.1051/0004-6361/201118319.

[14] E. S. Perlman, S. A. Rappaport, W. A. Christiansen, Y. J.Ng, J. DeVore, D. Pooley, New Constraints on Quantum Grav-ity from X-ray and Gamma-Ray Observations, Astrophys. J.805 (1) (2015) 10. arXiv:1411.7262, doi:10.1088/0004-637X/805/1/10.

[15] LIGO, Response of ligo to gravitational waves at high frequen-cies and in the vicinity of the fsr (37.5 khz), Technical NoteLIGO-T0060237-00 (2005).

[16] D. Marolf, On the quantum width of a black hole horizon,Springer Proc. Phys. 98 (2005) 99–112. arXiv:hep-th/0312059,doi:10.1007/3-540-26798-0_9.

[17] H. Casini, M. Huerta, R. C. Myers, Towards a derivation ofholographic entanglement entropy, JHEP 05 (2011) 036. arXiv:1102.0440, doi:10.1007/JHEP05(2011)036.

[18] E. Verlinde, K. M. Zurek, Spacetime Fluctuations in AdS/CFT,JHEP 04 (2020) 209. arXiv:1911.02018, doi:10.1007/

JHEP04(2020)209.[19] A. Chou, et al., Interferometric Constraints on Quantum Geo-

metrical Shear Noise Correlations, Class. Quant. Grav. 34 (16)(2017) 165005. arXiv:1703.08503, doi:10.1088/1361-6382/

aa7bd3.[20] W. A. Christiansen, Y. J. Ng, D. J. E. Floyd, E. S. Perlman,

Limits on Spacetime Foam, Phys. Rev. D83 (2011) 084003.arXiv:0912.0535, doi:10.1103/PhysRevD.83.084003.

[21] T. He, V. Lysov, P. Mitra, A. Strominger, BMS supertransla-tions and Weinberg?s soft graviton theorem, JHEP 05 (2015)151. arXiv:1401.7026, doi:10.1007/JHEP05(2015)151.

[22] A. Strominger, A. Zhiboedov, Gravitational Memory, BMSSupertranslations and Soft Theorems, JHEP 01 (2016) 086.arXiv:1411.5745, doi:10.1007/JHEP01(2016)086.

[23] S. W. Hawking, M. J. Perry, A. Strominger, Soft Hair on BlackHoles, Phys. Rev. Lett. 116 (23) (2016) 231301. arXiv:1601.

00921, doi:10.1103/PhysRevLett.116.231301.

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