Short-range forces due to Lorentz-symmetry violation Quentin G. Bailey1 Jennifer L. James2 Janessa R. Slone1 and

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Short-range forces due to Lorentz-symmetry

violation

Quentin G. Bailey1, Jennifer L. James2, Janessa R. Slone1, and

Kellie O’Neal-Ault1

E-mail: baileyq@erau.edu

1Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, AZ,

86301, USA

2Vanderbilt University, 2201 West End Avenue, Nashville, TN, 37235 USA

Abstract. Complementing previous theoretical and experimental work, we explore

new types of short-range modiﬁcations to Newtonian gravity arising from spacetime-

symmetry breaking. The ﬁrst non-perturbative, i.e., to all orders in coeﬃcients for

Lorentz-symmetry breaking, are constructed in the Newtonian limit. We make use of

the generic symmetry-breaking terms modifying the gravity sector and examine the

isotropic coeﬃcient limit. The results show new kinds of force law corrections, going

beyond the standard Yukawa parameterization. Further, there are ranges of the values

of the coeﬃcients that could make the resulting forces large compared to the Newtonian

prediction at short distances. Experimental signals are discussed for typical test mass

arrangements.

1. Introduction

Presently, the nature of gravity is still largely unknown on length scales less than

micrometers. In fact, new types of forces many times stronger than the Newtonian

gravitational force could exist on short length scales and still be consistent with current

experimental limits [1]. Suggestions for hypothetical new forces that could modify

gravity at short ranges abound in the literature [2, 3, 4, 5, 6, 7, 8]. In particular,

miniscule but potentially detectable violations of fundamental symmetries underlying

General Relativity (GR) can arise in a plethora of ways [9, 10, 11, 12, 13, 14, 15]. The

breaking of local Lorentz symmetry, for instance, can modify gravity on short ranges

while being consistent with longer range measurements [16, 17].

To categorize the phenomenology of spacetime symmetry breaking one needs a

comprehensive test framework. Eﬀective ﬁeld theory (EFT) is a widely used tool

for describing potentially detectable new physics [18]. EFT descriptions of spacetime-

symmetry breaking, including local Lorentz symmetry breaking, are based on including

the action of GR and a standard matter sector action [19]. To these basic pieces,

are added a series of symmetry breaking terms that can be organized by number

of derivatives, curvature, mass dimensions, and so on [20, 21, 22]. This approach

arXiv:2210.00605v1 [gr-qc] 2 Oct 2022

has the advantage that one can in principle calculate the eﬀect on some observable

due to some symmetry breaking terms, which can then be compared with entirely

diﬀerent observables in diﬀerent scenarios, for measurements of the same coeﬃcients

controlling the size of the eﬀects. Other formalisms for testing symmetries in gravity

are parametrized directly from the form of a GR observable [23, 24, 25], or are based

on speciﬁc models of alternatives to GR [26, 27, 28, 29].

We will consider in this work modiﬁcations to the gravity sector that, contrary to

standard GR, break local Lorentz symmetry and diﬀeomorphism symmetry explicitly or

spontaneously. These spacetime symmetries can be thought of as gauge symmetries for

gravity, and thus GR is a gauge theory of gravity with local Lorentz and diﬀeomorphism

symmetries as the gauge symmetries, analogous to Standard Model physics based on

gauge groups [30]. The subtle issue of the role of broken spacetime symmetries in the

context of curved spacetime, particularly when assuming asymptotically ﬂat scenarios

or not, has been discussed at length elsewhere [22, 31, 32]. While we do not fully

discuss these concepts and subtleties here, we shall refer to conventions and categories

of transformations in these references as needed.

In the EFT approach taken here, we highlight comparison of short-range (SR)

gravity tests with gravitational wave (GW) observations, thus comparing two tests

“across the universe” for measuring the same quantities describing spacetime-symmetry

breaking for gravity. In fact, we show certain rotational scalar coeﬃcients that can

be measured in GW tests can also be probed in SR tests. Further, there are some

coeﬃcients that cannot be completely disentangled with GW tests alone, but using also

SR gravity tests could accomplish this.

In references [16] and [17] solutions for short-range gravity tests were found, but

these used an approximation of leading order in the coeﬃcients. We show here that

exact, non-perturbative, solutions can reveal where other combinations of coeﬃcients,

not yet disentangled, can show up in experiment. As we are concerned in this paper

with modiﬁcations to gravity that do not break the Weak-Equivalence Principle, we do

not discuss WEP violations here. The connection between Lorentz violation and WEP

has been discussed at length elsewhere [33, 34, 35, 36].

Since we examine non-perturbative solutions, the results in this work also touch

on the nature of higher than second order derivatives in the action and how that might

aﬀect gravity. For this latter topic, we do not attempt a comprehensive investigation

of these issues but simply note where results exhibit behavior expected of such models

[37, 38, 39], and how they might be consistent with perturbative approaches.

The paper is organized as follows. In section 2, we review two commonly used EFT

schemes for the description of spacetime symmetry breaking in gravity and we discuss

prior results in short-range gravity signals for Lorentz violation. In section 3, we explore

non-perturbative solutions with a special case model to identify key features. Following

this, we go on to solve the general EFT framework in the static, isotropic coeﬃcient

limit. Features of the solutions are discussed and explained with several plots. We

discuss attempting exact solutions with anistropic coeﬃcients in section 4, and compare

to perturbative methods. For Section 5, we apply the theoretical results to simulate

the signal of the gravitational ﬁeld above a ﬂat plate of mass, and comment on the

experimental signatures. A summary and outlook is provided in section 6. Finally, in the

appendix we include a review of relevant diﬀerential equations, the details of the tensor

analysis for isotropic coeﬃcients used, and special cases of the SR gravity solutions. In

this work, we assume 4 dimensional spacetime with metric signature −+ ++ and units

where ¯h=c= 1. Latin letters are used for 3 dimensional space, and Greek letters for

spacetime indices.

2. Background theory

2.1. Action and ﬁeld equations

One can work with an observer covariant EFT expansion or an action designed for

weak-ﬁeld applications, the latter formulated in a quadratic action expansion. The

two approaches are overlapping descriptions of physics beyond GR and the SM when

spacetime symmetries are broken. We display both approaches here, to emphasize recent

points of view in the literature, and because we use them in this work.

It is a basic premise that in the EFT context, a breaking of spacetime symmetries

is indicated by the presence of a background tensor ﬁeld of some kind that couples

to matter or gravity or both [9, 19, 20]. The details and subtleties of this premise

have been discussed at length elsewhere [22, 31, 32]. Suﬃce it to say here that the

EFT maintains coordinate invariance of physics (observer invariance) while the action

may not be invariant under symmetry transformations of localized ﬁeld conﬁgurations

(particle transformations). The latter violation is due to the presence of the background

tensor ﬁelds, which remain ﬁxed under such transformations.

The observer covariant expansion has a Lagrange density that takes the form of a

series of terms:

L=√−g

2κ(R+k(4)

αβγδRαβγδ +k(5)

αβγδκ∇κRαβγδ

+k(6)

κλµναβγδRκλµν Rαβγδ +...) + L0.(1)

In this expression, the determinant of the metric is √−g,Rαβγδ is the Riemann curvature

tensor, Ris the Ricci scalar, and k(4)

αβγδ,k(5)

αβγδκ, and k(6)

κλµναβγδ are the coeﬃcients

controlling the degree of symmetry breaking [22, 16]. The coupling is κ= 8πGN, where

GNis the gravitational constant. The ﬁrst term is the Einstein-Hilbert lagrange density,

while the remaining terms are the symmetry-breaking terms. Note that additional terms

for the coeﬃcients can be included in L0. For instance, a general expansion for such

terms exists, for the case of a two-tensor sµν ∝k(4)α

µαν , and takes the form

L0=√−g

2κha31

2(∇µsνλ)(∇µsνλ) + a41

2(∇µsµλ)(∇λsβ

β)

+... +a7sµν sκλRµκνλ +a8sµν sµ

λRνλ +...i,(2)

which can be viewed as terms of second order in the coeﬃcients or as dynamical terms

[40, 32]. Alternatives to (2) can adopt the explicit symmetry breaking scenario, where

the coeﬃcients in (1) are given a priori, this latter possibility given emphasis more

recently [41, 42, 43, 44].

An alternative overlapping approach, the quadratic action approach, assumes an

expansion around ﬂat spacetime ηµν , of the standard form

gµν =ηµν +hµν .(3)

We examine the quadratic action [45, 46] in the limit that maintains the usual linearized

gauge invariance of GR: hµν →hµν −∂µξν−∂νξµ. The Lagrange density for this approach

takes the form

L=−1

4κhαβGαβ +1

8κhµν (ˆsµρνσ + ˆqµρνσ +ˆ

kµρνσ )hρσ,(4)

where Gαβ is the linearized Einstein tensor. The “hat” operators are built from

background coeﬃcients for spacetime-symmetry breaking and partial derivatives. The

three types appearing in (4) are given by,

ˆsµρνσ =s(d)µρ1νσ2...d−2∂1...∂d−2,

ˆqµρνσ =q(d)µρ1ν2σ3...d−2∂1...∂d−2,

kµνρσ =k(d)µ1ν2ρ3σ4...d−2∂1...∂d−2.(5)

While the expansions in (5) appear similar for the three types of coeﬃcients, the s,q,

and kin fact diﬀer by symmetry and tensor properties. The detailed tensor properties

of these terms are described in the Young Tableau of Table 1 of Ref. [45], (some samples

are included in appendix (59)). In particular, ˆsµρνσ is anti-symmetric in the pairs of

indices µρ and νσ, while ˆqµρνσ is anti-symmetric in µρ and symmetric in νσ, and ﬁnally

kµνρσ is symmetric in the pairs of indices µρ and νσ. In terms of discrete spacetime

symmetries, The ˆsoperators have even CPT symmetry and mass dimension d≥4; ˆq

operators have odd CPT and mass dimension d≥5; ˆ

koperators have even CPT and

mass dimension d≥6.

The phenomenology of the terms in (1) and (4) has been studied in a number

of works. Observable eﬀects in weak-ﬁeld gravity tests have been established for a

subset of the possible terms [47, 48, 16] and some work has been done on strong-ﬁeld

gravity regimes like cosmology [49, 42, 50, 51]. Eﬀects on gravitational waves have

been studied, showing that dispersion and birefringence occur generically as a result of

CPT and Lorentz violation [45]. Analysis has been performed in tests such as lunar

laser ranging [52], gravimetry [53], pulsars [54], and using the catalog of GW events

[55, 56, 57, 58, 59]. An exhaustive list of up to date experimental limits and papers on

gravity sector coeﬃcients can be found in [60].

On the theory side, explicit local Lorentz and diﬀeomorphism symmetry cases have

been explored various contexts. A “3+1” formulation of the EFT framework has been

explored in Refs. [42, 43, 61]. Extensive work has been completed mapping out the

approach to explicit symmetry breaking with Finsler geometry [62, 63, 64, 65]. Other

work includes much attention to vector and tensor models of spontaneous symmetry

breaking [26, 66, 67, 68, 27, 69, 70] and how these models can be matched to the EFT

expansion above [71, 72, 42, 44]. More recently, black hole solutions have been studied

[73, 74, 75]. Also, the systematic construction of dynamical terms for the spontaneous

symmetry breaking scenario, like in (2), has been undertaken in the gravity sector

[40]. Finally we note some recent theoretical work has identiﬁed general properties of

backgrounds in eﬀective ﬁeld theory [32], and new types of tests are possible that search

for non-Riemann geometry [76].

Of the two approaches identiﬁed above, the latter, equation (4), is appropriate

for short-range gravity tests. Such tests involve weak gravitational ﬁelds in the Earth

laboratory setting, thus the typical size of components of hµν are much less than unity,

in cartesian coordinates. Furthermore, to keep a reasonable scope we will truncate the

series (5) to mass dimensions 4, 5, and 6.

Any study of actions with higher than second order derivatives is subject to well-

known results, such as Ostragradsky instabilities [39]. In the present paper, while the

test framework (4) is viewed perturbatively, with the higher derivative terms as small

corrections [77], our discussion of solutions beyond leading order in coeﬃcients will

overlap with features in higher derivative models. Some features are discussed in our

results in Section 3 and Section 4.

2.2. Prior short-range gravity results

In references [16] and [17], Lorentz-symmetry breaking solutions for short-range gravity

tests were found using an approximation of ﬁrst order in the coeﬃcients. We summarize

these results brieﬂy here for comparison. Assuming a static matter source and using

the framework of (4), one solves the ﬁeld equations perturbatively assuming any

modiﬁcations to the ﬁeld equations from symmetry-breaking terms are small [47, 16, 17].

The leading order modiﬁed Newtonian potential from a point mass mat the origin can

be written in terms of Newton spherical coeﬃcients kN(d)lab

jm as a series

U=GNm

r+X

djm

GNm

rd−3Yjm(θ, φ)kN(d)lab

jm ,(6)

where the angular dependence θ, φ in the spherical harmomics Yjm(θ, φ) pertains to

the vector from the origin to the ﬁeld point ~r =r(sin θcos φ, sin θsin φ, cos θ) and

r=|~r|. The spherical coeﬃcients kN(d)lab

jm are related to the coeﬃcients in Eq.

(5) as linear combinations, but the expressions are lengthy and omitted here, and

relations between the dimension label (d) and the allowed values of jcan be found

in [17]. The superscript “lab” means that the coeﬃcients are written in the laboratory

coordinate system. Typically, the lab frame coeﬃcients are re-expressed in terms of

the Sun-centered Celestial Equatorial Frame coeﬃcients using an observer Lorentz

transformation, revealing harmonic time dependence [78, 79, 80].

The result in equation (6) has already been used for analysis in experiments

[81, 82, 83, 84]. In fact, new experiments can be designed to maximize the type of

anisotropic signal in (6) [83, 85, 86]. Recent result place limits on 14 kN(6)

jm coeﬃcients

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Short-rangeforcesduetoLorentz-symmetryviolationQuentinG.Bailey1,JenniferL.James2,JanessaR.Slone1,andKellieO'Neal-Ault1E-mail:baileyq@erau.edu1Embry-RiddleAeronauticalUniversity,3700WillowCreekRoad,Prescott,AZ,86301,USA2VanderbiltUniversity,2201WestEndAvenue,Nashville,TN,37235USAAbstract.Complementin...

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Short-range forces due to Lorentz-symmetry violation Quentin G. Bailey1 Jennifer L. James2 Janessa R. Slone1 and

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