Relativistic Theory of Elastic Bodies in the Presence of Gravitational Waves Mario Hudelist1 Thomas B. Mieling2 and Stefan Palenta1

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Relativistic Theory of Elastic Bodies in the Presence of

Gravitational Waves

Mario Hudelist1, Thomas B. Mieling2, and Stefan Palenta1

1University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria

2University of Vienna, Faculty of Physics, Vienna Doctoral School in Physics (VDSP), Boltzmanngasse 5,

1090 Vienna, Austria

April 17, 2023

Abstract

The equations of motion governing small elastic oscillations of materials, induced by grav-

itational waves, are derived from the general framework of Carter and Quintana. In transverse-

traceless gauge, no bulk forces are present, and the gravitational wave is found to act as an eﬀective

surface traction. For thin rods, an equivalent description is given, in which there is no surface trac-

tion, but a bulk acceleration, which is related to the Riemann curvature of the gravitational wave.

The resulting equations are compared to those of the Synge–Bennoun elasticity theory.

1 Introduction

For the theoretical description of elastic materials in curved space-time, there are multiple approaches.

Here, we distinguish what we refer to as the Synge–Bennoun (SB) theory from the Carter–Quintana

(CQ) theory.

The SB theory, essentially laid out in Refs. [1;2], dispenses with a notion of strain (which, in the

classical theory, quantiﬁes the deviation of the spatial metric from a reference conﬁguration), but uses

a notion of rate of strain. Hooke’s law is then reinterpreted as not to provide a linear relationship

between stress and strain, but rather as a linear relationship between rates of stress and strain.

The CQ theory [3] on the other hand, introduces the notion of a three-dimensional material

manifold. By endowing this material manifold with a Riemannian metric, one is able to deﬁne a

notion of strain, and hence can use direct generalisations of non-relativistic stress-strain relationships

to obtain a stress tensor. Even in the absence of a preferred Riemannian reference metric, this theory

allows deﬁning a notion of stress via what might be called a stress-deformation-relationship. Moreover,

unlike the SB theory, the CQ theory is not limited to the regime of linear elasticity, but also allows

for strong deformations.

While most current texts on mathematical aspects of relativistic elasticity are based on the CQ

theory (and are seemingly unaware of the SB theory), see e.g. Refs. [4–7] and references therein, articles

focusing on experimental applications with regards to gravitational wave detection mostly rely on the

SB theory instead [8;9].

The aim of this text is to describe the eﬀect of weak gravitational waves on elastic objects using

the CQ theory, and to compare the resulting description with the SB theory, and also with related

equations put forward by various authors on heuristic grounds (in fact, Weber’s original papers were

not based on any fully developed relativistic elasticity theory).

We start in Section 2by laying out the mathematical description of elastic materials in a general

curved space-time. In this setting, we naturally expect the material to be subject to deformations.

These deformations determine the stresses, with details depending on the material properties. For the

considered case of hyperelastic materials, this stress-deformation relationship is fully determined by

an energy density function. Section 3specialises the general theory to linearized elastic perturbations

induced by weak, but otherwise arbitrary perturbations in the space-time metric.

arXiv:2210.04618v2 [gr-qc] 14 Apr 2023

In Section 4we apply the general formalism developed in the previous sections to the case of

plane gravitational waves, where we recover the classical Navier–Cauchy equations, supplemented by

boundary conditions where the gravitational wave acts as an eﬀective external traction force.

To illustrate these equations, in Section 5we explicitly solve the simple example of an elastic

thin rod, responding to gravitational radiation. Finally, we compare with Papapetrou’s equations for

gravitational-wave-induced elastic oscillations in Section 6.

Compared to previous treatments of the problem within the CQ theory, this work emphasises the

importance of boundary conditions at the material’s surface, without which no unique solution to the

elasticity equations can be obtained.

This text uses geometric units in which the speed of light is set to unity, and the metric signature

is taken to be “mostly positive.”

2 Review of Relativistic Elasticity

This section reviews the theory development of general relativistic elasticity provided in Ref. [10],

while maintaining the deﬁnition of deformation and the notation of various deformation tensors from

Ref. [7], which are closely related to the elasticity theory notions from classical continuum mechanics.

Other related works such as Refs. [4;6;11] follow essentially the same theory development, but use

slightly altered nomenclatures and conventions.

The main object in the CQ theory of relativistic elasticity is the deformation map f:M→B,

where Mis the four-dimensional space-time manifold (with local coordinates xa) and Bis a three-

dimensional manifold, the “body” (with local coordinates XA). The conﬁguration is required to

be such that there is a unique unit timelike vector ﬁeld va(the material’s four-velocity) satisfying

∂afAva= 0. Conceptionally, this means that fassigns to each point in space-time the material point

present there, and the world-lines of material points are given by lines of constant f.

The velocity vector ﬁeld gives rise to the spatial projection `a

band the spatial metric `ab via

b=δa

b+vavb, `ab =gab +vavb.(1)

where gab is the space-time metric. The tensor ﬁeld `a

bprojects vectors onto v⊥, the subspace or-

thogonal to va, and `ab is the restriction of the space-time metric to v⊥and thus encodes the spatial

geometry of the medium. One then deﬁnes the spatial deformation gradient HAaand the material

deformation gradient Fa

Athrough the equations

HAa=∂afA, HAcFc

B=δA

B, F a

CHCb=`a

b.(2)

These quantities can be viewed as mappings which translate tensor ﬁelds between the conﬁguration

manifold Band the space-time M. Starting from the deformation gradient, one can introduce various

types of deformation tensors as in classical continuum mechanics.

For the purposes of this paper, we will restrict the discussion to the Green deformation CAB and

the Piola deformation BAB, deﬁned as

CAB =Fa

AFb

B`ab , BAB =HAaHBb`ab ,(3)

which also encode the spatial geometry and are essential to obtain a notion of strain. Since `ab and

`ab are inverse to each other, Eq. (2) implies that CAB and BAB are mutually inverse.

To further quantify compression of elastic bodies, and to formulate a continuity equation, a volume-

form on Bis necessary. If Ωis such a form, then its pull-back along the conﬁguration, f∗Ω, is

necessarily proportional to the spatial volume form φ=vyΦ, where Φis the space-time volume form

and ydenotes the interior product. The Jacobian Jis then deﬁned as the proportionality factor in

φ=Jf ∗Ω.(4)

Here, we take Ωto have units of volume, so that Jis dimensionless. The Jacobian Jthus compares the

“actual volume” φwith the “reference volume” Ωand thus measures expansion (J > 1) or contraction

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RelativisticTheoryofElasticBodiesinthePresenceofGravitationalWavesMarioHudelist1,ThomasB.Mieling2,andStefanPalenta11UniversityofVienna,FacultyofPhysics,Boltzmanngasse5,1090Vienna,Austria2UniversityofVienna,FacultyofPhysics,ViennaDoctoralSchoolinPhysics(VDSP),Boltzmanngasse5,1090Vienna,AustriaApril17...

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Relativistic Theory of Elastic Bodies in the Presence of Gravitational Waves Mario Hudelist1 Thomas B. Mieling2 and Stefan Palenta1

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