---
abstract: 'Using the observational properties of Einstein’s gravitational field it is shown that a minimum of four non-coplanar mass probes are necessary for the Michelson and Morley interferometer to detect gravitational waves within the context of General Relativity. With fewer probes, some alternative theories of gravitation can also explain the observations. The conversion of the existing gravitational wave detectors to four probes is also suggested.'
author:
- |
    Ivan S. Ferreira\
    University of Brasilia, Institute of Physics, Brasilia, DF Brasilia, DF 70910-900, ivan@fis.unb.br,\
    C. Frajuca\
    National Institute for Space Research, Sao Jose dos Campos, 12227-010, frajuca@gmail.com,\
    Nadja S. Magalhaes\
    Physics Department, Sao Paulo Federal University, SP09913-030, Brazil, nadjasm@gmail.com,\
    M. D. Maia\
    University of Brasilia, Institute of Physics, Brasilia, DF70910-900, maia@unb.br,\
    Claudio M. G. Sousa\
    Federal University of Para, Santarem, PA 68040-070, claudiogomes@ufpa.br.
title: The Laser Gravitational Compass
---

The Observable Gravitational Wave
=================================

Einstein’s prediction of gravitational waves (gw) was originally derived from arbitrarily small perturbations of the Minkowski metric $g_{\mu\nu}= \eta_{\mu\nu} + h_{\mu\nu}$, such that Einstein’s equations reduce to a linear wave equation, written in a special (de Donder) coordinate gauge, as (Except when explicitly stated, Greek indices run from 0 to 3 and small case Latin indices run from 1 to 3). $$\Box^2 \Psi_{\mu\nu} = 0, \;\;  \; \; \Psi_{\mu\nu}= h_{\mu\nu}-\frac{1}{2}h  \eta_{\mu\nu},\;\; \;\: h=\eta^{\mu\nu}h_{\mu\nu}. \label{eq:waves}$$ The currently operating laser gw observatories are inspired by the Michelson & Morley (M&M) interferometers[@Giazotto; @Pitkin; @Eardley], where the data acquired by the Fabri-Perót interferometer (an etalon) is used to generate a numerical simulation, thus producing a template from which the most probable source is estimated[@Abbot2016]. The purpose of this note is to show that in order to detect gw described by General Relativity (GR) with an M&M interferometer requires a minimum of four non-coplanar mass probes.

The observables of Einstein’s gravitational field are given by the eigenvalues of the Riemann curvature[@Zakharov; @Pirani1956; @Pirani1957], defined by $R(U,V)W = [\nabla_U, \nabla_V]W - \nabla_{[U,V]}W$, whose components in any given basis $ \{e_\mu \} $ are $R(e_\mu,e_\nu)e_\rho = R_{\mu\nu\rho\sigma}e^\sigma$. Then we find that there are at most six independent eigenvectors $X_{\mu\nu}$ and six eigenvalues $\lambda$, solutions of the eigenvalue equation $ R_{\mu\nu\rho\sigma}X^{\rho\sigma}=\lambda X_{\mu\nu}$, including the zero eigenvalue, corresponding to the absence of gravitation[@Pirani1962; @Sachs1962a; @Pirani1967]. Thus, using the language of field theory, Einstein’s gravitation is said to have five non-trivial observables or degrees of freedom $(dof)$. The spin-statistics theorem relates the $dof$ to the helicity or the orbital spin of the field as $s=(dof-1)/2$, so that Einstein’s gravitation is also said to be a spin-2 field.

Alternative theories of gravitation may have distinct definitions of observables (not necessarily related to curvature) and their gravitational waves, if they exist, may require different methods of observations. Well known examples include: the spin-1 gauge theories of gravitation (there are several of them), characterized by dof=3; topological gravitation in three dimensions; projective theories of gravitation; F(R) theories; among many others. Therefore, in order to understand the observation of a gw it is essential to specify the observables of the theory on which the experiment is based.

The most general massless spin-2 field $h_{\mu\nu}$ was defined in the Minkowski space-time by Fierz and Pauli[@FierzPauli], as a trace-free field $h=h_\mu{}^\mu=0$, satisfying the field equations $\Box^2 h_{\mu\nu}=0$. This is a linear field not to be confused with Einstein’s gravitation. Since in the case of the present observations of gw, the supporting theory is Einstein’s gravitation, then the observational signature to be sought is that of a $dof=5$ or of a spin-2 field, characterized by the observable curvature.

The use of the Fierz-Pauli field $h_{\mu\nu}$ as the perturbation of the Minkowski metric makes it possible to free Eq. (\[eq:waves\]) of coordinate gauges so that its solutions can be written as as a superposition of plane polarized gravitational waves, characterized by the Traceless-Transverse-Plane-Polarized (TTPP) gauge conditions[@Giazotto]: $$h=0,\;\;  h_{i0}=0, \;\; \Box^2 h_{\mu\nu}=0,\;\;  h_{\mu\nu;\rho}=0. \label{eq:TTPP}$$ Then, these conditions are used to simulate a template, from which the source source of the gw observations is estimated.

The Equivalence Principle and the M&M gw Detector
=================================================

The use of an M&M detectors for gw is based on the principle of equivalence of GR: Given 2 masses A and B with attached mirrors, under the exclusive action of a known gravitational field, they propagate (or “free fall”) along time-like geodesics, with unit tangent vectors $T_A$ and $T_B$ respectively, satisfying geodesic equations $\nabla_{T_{A}} T_A=0$ and $\nabla_{T_{B}} T_B=0$ . Eventually, probe A sends a light signal with velocity $P$ to particle B along the light geodesic with equation $\nabla_P P=0$. After a while, probe A receives back the reflected signal, so that these geodesics describe a closed parallelogram, with the closing condition $\nabla_T P=\nabla_P T$.

The curvature tensor calculated in that parallelogram is $ R(T,P)T = [\nabla_T ,\nabla_P ]T =  \nabla_T (\nabla_T P)  = \nabla_T a$, where $a = \nabla_T P$ is the acceleration of the signal along the fall. Defining a basis by $\{e_0 =T , e_i= P \} $, where $e_i$ denotes any space-like direction, we obtain the geodesic deviation equation in components $\frac{d a_i}{cdt}=R_{0i0i}$, where $t$ denotes the time parameter of the time-like geodesic. As we can see, the motion of the probes generates a 2-dimensional world-sheet, whose curvature $R_{0i0i}$, is translated directly into the the variation of the fall acceleration.

In the currently operating and some future planned M&M gw detectors, three mass-probes are used, defining a space-like plane in space-time, whose motion under the action of a gw generates a 3-dimensional world-volume, whose curvature is measured by a $2 \times 2$ acceleration array $a_{ij}$, obeying the geodesic deviation equation $$\frac{d a_{ij}}{cdt}= \!\!R_{0i0j}, \; i,j =1,2.   \label{eq:GDE2}$$ Therefore, such detectors are capable to measure only 3 curvature components $R_{0101},R_{0102},R_{0202}$, so that at most three degrees of freedom of Einstein’s gravitational field can be obtained. Since the observed gw are very weak, it has been assumed that the missing degrees of freedom in one detector may be complemented by the data collected on another detector located somewhere else. Such understanding is supported by the numerical simulation, from which an estimate of the wave source capable of reproducing the same observation in two separate detectors is obtained. Although it is possible to parallelly transport the curvature tensor from one detector to another, the left hand side content of Eq. (\[eq:GDE2\]) represents a locally measured quantity which cannot be transported to another detector, under the penalty of breaching the principle of equivalence.

The definitive solution for the missing dof problem can be obtained by a direct measure of the curvature tensor using the geodesic deviation equations Eq. (\[eq:GDE2\]), extended to $i,j =1,2,3$. One such detector, the “gravitational compass” was conceived by P. Szekeres. It consists of four non-coplanar mass probes connected by six dynamometers, to measure the six curvature eigenvectors. By comparing three of these eigenvectors with the three principal directions of curvature, the compass can points directly to the source[@Szekeres].

Szekeres’ gravitational compass was improved by F. Pirani[@Pirani1967], where instead the dynamometers, the four mass-probes had attached mirrors and light sources. Then, applying Eq. (\[eq:GDE2\]) again for $i,j =1,2,3$, all components of the curvature can be read locally, defining the 4-dimensional gravitational field at the center of mass of the four probes. The interesting fact is that four non-coplanar masses define a virtual spheroidal surface in space-time, then the natural oscillation modes of the spheroid give a direct measure of the curvature eigenvalues.

The Szekeres-Pirani gravitational compass can be implemented in the presently operating gw observatories, with the addition of a fourth mass probe not belonging to the plane of the existing 2-dimensional M&M detectors, located either in a tower or well, not necessarily having the same length as the existing arms, thus defining a spheroidal gw detector.\
Summarizing, the presently operating gw detectors are hindered by the fact that they cannot detect directly the all degrees of freedom of Einstein’s gravitational field at each point, thus leaving open the possibility of alternative explanations for the gw. For the specific case of gravitation in the sense of Einstein’s theory, we suggest the addition of a fourth mass to the existing detectors, so that all degrees of freedom of gravitation are directly measured from the five fundamental oscillation modes of a spheroid.

**Acknowledgments**: NSM and CF acknowledge FAPESP for its support to their research through the thematic project 2013/26258-4.

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