Gravitational waves physics using Fermi coordinates: a new teaching perspective
aa r X i v : . [ g r- q c ] J a n Gravitational waves physics using Fermi coordinates: a newteaching perspective
Matteo Luca Ruggiero ∗ Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (Dated: January 19, 2021)
Abstract
The detection of gravitational waves is possible thanks to a multidisciplinary approach, involvingdifferent disciplines such as astrophysics, physics, engineering and quantum optics. Consequently, itis important today for teachers to introduce the basic features of gravitational waves science in theundergraduate curriculum. The usual approach to gravitational wave physics is based on the use oftraceless and transverse coordinates, which do not have a direct physical meaning and, in a teachingperspective, may cause misconceptions. In this paper, using Fermi coordinates, which are simplyrelated to observable quantities, we show that it is possible to introduce a gravitoelectromagneticanalogy that describes the action of gravitational waves on test masses in terms of electric-like andmagnetic-like forces. We suggest that this approach could be more suitable when introducing thebasic principles of gravitational waves physics to students. . INTRODUCTION In the framework of General Relativity (GR) Einstein calculated the emission of grav-itational waves (GWs) on the basis of the quadrupole formula. According to this formula,ideal candidates for the emission of GWs are huge masses moving at highly relativistic ve-locities. These sources are far away from the Earth, and their effects are detected as smallperturbations or ripples in the fabric of space-time. The first indirect evidence of the ex-istence of GWs was obtained studying the binary pulsar B1913+16: the orbits of binarysystems are modified by the emission of gravitational waves, and these modifications can beobserved by accurate timing measurements . It took more or less 100 years after Einstein’sfirst calculation to obtain, in 2015, the first direct detection of GWs , which constitutedthe birth of gravitational waves astronomy . We are currently in the era of multi-messengerastronomy: a given astrophysical source can be detected by means of different messengers. Therefore, it is important today for physics and astronomy teachers to explain the foun-dations of GWs science, starting from the emission process up to their detection, whichrequires a multidisciplinary approach involving also engineering and quantum optics.There are concrete proposals to integrate GWs science into physics and astronomycurricula. To this end, several introductory textbooks can be used, such as Refs. 8 and 9; acomprehensive collection of useful materials can be found in the resource letter in Ref. 10.There is also literature focusing on specific issues: for instance, Ref. 11 suggests that thebasic properties of GWs can be obtained by combining Newtonian gravity with the retarda-tion effects due to the finite size of the speed of light. The detection process is made clearby considering the basic principles of interferometric detectors, as explained by Refs. 12 and13, while Ref. 14 discusses the key features of LIGO on the basis of Newtonian mechanics,dimensional considerations, and analogies between gravitational and electromagnetic waves.The mechanism of emission can be studied on the basis of the post-Newtonian theory, asdiscussed in Ref. 15, while data analysis can be used as a tool to design a GWs laboratory,as suggested in Ref. 16.There are subtle issues connected with the process of detection of GWs, due to thedistinct role of coordinates and observable quantities in GR. In Einstein’s theory, physicalmeasurements are meaningful only when the observer and the object of the observationsare unambiguously identified. Roughly speaking, there are three steps in the measurement2rocess: (i) observers possess their own space-time, in the vicinity of their world-lines;(ii) covariant physics laws are projected onto local space and time; (iii) predictions for theoutcome of measurements in the local space-time of the observers are obtained. Gravitationalwaves are usually described in terms of a transverse and traceless (TT) tensor, which allowsto introduce the so-called TT coordinates (see e.g. Ref. 19 and references therein for athorough discussion on the various coordinates used to describe the interaction with GWs).TT coordinates are used because they do not contain gauge-depending information; however,from a teaching perspective, TT coordinates are difficult to handle, since they are not strictlyrelated to measurable quantities and lack a direct physical meaning. When he was 21 (some months before obtaining his undergraduate degree in physics),Enrico Fermi published an influential paper. In this paper he introduced a quasi-Cartesiancoordinates system in the observer’s neighbourhood to describe the effects of gravitation.Such a set of coordinates, called
Fermi coordinates , adapted to the world-line of an observer,defines a
Fermi frame . Using this approach it is simple to emphasise that what an observermeasures depends both on the background field where he is moving and, also, on his motion.This is quite similar to what happens when we study classical mechanics in non inertialframes: inertial forces appear, depending on the peculiar motion of the frame with respectto an inertial one. Fermi coordinates have a concrete meaning, since they are the coordinatesan observer would naturally use to make space and time measurements in the vicinity ofhis world-line. Fermi coordinates are defined, by construction, as scalar invariants. Theyare of the utmost importance to understand the measurement process in GR, which isrelevant in experimental tests of gravity; moreover, they provide a simple interpretation ofthe equivalence principle.It is possible to show that using Fermi coordinates the effects of a plane gravitationalwave can be described by gravitoelectromagnetic fields: in other words, the wave field isequivalent to the combined action of a gravitoelectric and a gravitomagnetic fields that aretransverse to the propagation direction and orthogonal to each other. The analogy betweenelectromagnetic fields and gravitational fields was already envisaged by Heaviside, on thebasis of the similarity between Newton’s law of gravitation and Coulomb’s law of electrostaticforce (see e.g. Ref. 24 and references therein). General Relativity naturally predicts theexistence of a gravitomagnetic field, produced by mass currents, in analogy to what happensfor the magnetic field, produced by charge currents. Generally, it is possible to describe3ravitational effects on the basis of a gravitoelectrogmagnetic analogy as discussed in Refs.25 and 26.In this paper we introduce a gravitoelectrogmagnetic description of the field of a planegravitational wave using Fermi coordinates, and we show that, in doing so, it is possible tounderstand the basic features of GWs. In particular, this approach describes the interactionwith detectors in terms of a Lorentz-like force equation. Moreover, we show that, while exist-ing detectors, such LIGO and VIRGO, or future ones, such as LISA, reveal the interaction oftest masses with the gravitoelectric components of the wave, there are also gravitomagneticinteractions that could be used to detect the effect of GWs on moving masses and spinningparticles. We believe that this approach can be useful for teaching gravitational waves physicsbecause it directly leads to measurable quantities, avoiding possible misunderstanding de-riving from the use of other types of coordinates. Making an analogy with the more familiarconcept of electromagnetic waves can also help students understand the new concept ofgravitational waves.
The paper is organized as follows: in section II we briefly review the classical approachto gravitational waves, while we discuss Fermi coordinates and the gravitoelectrogmagneticapproach in section III. On the basis of the gravitoelectrogmagnetic analogy, we describein section IV the interaction of GWs with detectors. Conclusions are eventually drawn insection V.We use the convention in which Greek indices refer to space-time coordinates and assumethe values 0 , , ,
3, while Latin indices refer to spatial coordinates and assume the values1 , ,
3, usually corresponding to the Cartesian coordinates x, y, z ; the spacetime signature is( − , , , II. GRAVITATIONAL WAVES IN TRANSVERSE, TRACELESS GAUGE
In this section we briefly recall the standard approach to the description of gravitationalwaves. To this end, we start from Einstein’s equations G µν = 8 πGc T µν , (1)4nd we suppose that the space-time metric g µν is in the form g µν = η µν + h µν , where | h µν | ≪ η µν of flat space-time. Setting¯ h µν = h µν − η µν h , with h = h µµ , Einstein’s field equations (1) in the Lorentz gauge ∂ µ ¯ h µν = 0(where ∂ µ = ∂∂x µ ) turn out to be (cid:3) ¯ h µν = − πGc T µν , (2)where (cid:3) = ∂ µ ∂ µ = ∇ − c ∂∂t is the d’Alambert operator. Gravitational waves propagatethrough empty space and are solutions of equations (2) in vacuum: (cid:3) ¯ h µν = 0 . (3)Typically, these equations are solved using the so-called transverse - traceless coordinates(see e.g. Ref. 19). In particular, we look for plane wave solutions propagating along the x axis. Accordingly, a solution of (3) can be written in the form¯ h µν = − (cid:0) h + e + µν + h × e × µν (cid:1) , (4)with h + = A + cos (cid:0) ωt − kx + φ + (cid:1) , h × = A × cos (cid:0) ωt − kx + φ × (cid:1) , (5)where φ + , φ × are constants, and e + µν = − , e × µν = (6)are the linear polarization tensors of the wave. In the above definitions, A + , A × are theamplitude of the wave in the two polarization states, φ + , φ × the corresponding phases, while ω is the frequency and k the wave number, so that the wave four-vector is k µ = (cid:16) ωc , k, , (cid:17) ,with k µ k µ = 0. Notice that any radiation field of spin s has two states of linear polarizationinclined to each other at an angle of π/ (2 s ): for the photon s = 1 and the linear polarizationstates of electromagnetic waves are orthogonal, while for gravitational waves the linearpolarization states (6) are at π/
4, since s = 2 (see e.g. Ref. 31). Furthermore, by analogywith electromagnetic waves, the two linear polarizations states can be added with phasedifference of ± π/ h + = A + sin ( ωt − kx ) , h × = A × cos ( ωt − kx ) , (7)5hus fixing the phase difference: accordingly, circular polarization corresponds to the condi-tion A + = ± A × .In TT coordinates the gravitational field of the wave is described by the line element ds = − c dt + dx + (1 − h + ) dy + (1 + h + ) dz − h × dydz . (8)Now, in order to understand the effect of a GW on test masses, we focus on the geodesicequation: starting from d x d τ µ + Γ µαβ d x α d τ d x β d τ = 0 , (9)and using the chain ruled x i d τ = d x i d t d t d τ d x i d τ = d x i d t (cid:18) d t d τ (cid:19) + d x i d t d t d τ , (10)the space components of the equation (9) can be expressed in terms of the form:d x i d t (cid:18) d t d τ (cid:19) + d x i d t d t d τ + Γ iαβ d x α d t d x β d t (cid:18) d t d τ (cid:19) = 0 , (11)where a i = d x i d t and v i = d x i d t are, respectively, the coordinate acceleration and velocity. Ifwe also use the time component, we can write1 c d x i d t = − Γ ijk v j c v k c − i j v j c − Γ i + v i c (cid:18) Γ + Γ jk v j c v k c + 2Γ j v j c (cid:19) . (12)We suppose that test masses are moving at non-relativistic velocities, which is reasonable ifwe are dealing with detectors: hence, since | v i | c ≪
1, we can neglect all velocity-dependentterms in Eq. (12) and obtain 1 c d x i d t = − Γ i . (13)Using the metric (8), we get Γ i = 0 and, then, d x i d t = 0. It is interesting to point out thatthis result is true for any gravitational field in the form ds = − c dt + g ij ( x µ ) dx i dx j , (14)i.e. with g = − g i = 0: all test particles that are spatially at rest in such aspacetime follow geodesics. In TT gauge the metric (8) is in the form (14), and this simplymeans that the TT coordinates of a test mass acted upon by a gravitational wave do notchange: but remember that coordinates in general relativity do not have a direct physical6eaning. In order to see the effect of GWs on test masses, we need to evaluate the variationof the physical distance between them, which is defined by the proper length and not by thecoordinate distance. For instance, let us suppose that two test masses are located along the y axis, at P = (0 , ,
0) and P = (0 , L, d y between them is obtainedfrom the line element (8): d y = Z L √ g yy dy = Z L √ − h + dy ≃ (cid:18) − h + (cid:19) L, (15)where we have taken into account the smallness of the perturbation; hence, according to (7)the proper distance changes with time, due to the passage of the GWs. Interferometers likeLIGO and VIRGO are designed to measure this change. As we are going to show below,Fermi coordinates allow a direct description of the effect of GWs on test masses, in termsof measurable quantities.TT gauge is used because of its convenience: as we said, in linearized theory it fixesall gauge freedom so that the metric perturbations are physical and do not contain gauge-depending information; moreover, in this gauge it is manifest that the GWs have two po-larization components and that they are transverse to the propagation direction.For future convenience, we remember that, in linear approximation in the perturbation h µν , the Riemann curvature tensor is written in the form: R αµβν = 12 ( h αν,µβ + h µβ,να − h µν,αβ − h αβ,µν ) . (16)In particular, since in the TT metric h i = 0 and h = −
1, we obtain the following expressionof the Riemann tensor R µβν = 12 ( h µβ,ν − h µν, β ) (17)which will be used below. III. FERMI COORDINATES
The space-time metric in Fermi coordinates, in the vicinity of a given observer’s world-line, depends both on where and how the observer is moving. In other words, the backgroundspace-time and the type of motion within it determine the local metric, whose generalexpression can be found in Ref. 23. Here, since we are concerned with GWs effects, for thesake of simplicity we consider an observer freely falling in the field of a plane gravitational7ave: hence, Fermi coordinates are a geodesic coordinate system based on non rotating framealong the observer’s world-line (the reference world-line).
If we set Fermi coordinates X α = ( cT, X, Y, Z ) = ( cT, X ), the metric can be expressed in a power series in X from thereference world-line, in the form: ds = − (cid:0) R i j X i X j (cid:1) c dT − R jik X j X k cdT dX i + (cid:18) δ ij − R ikjl X k X l (cid:19) dX i dX j . (18)The above expression is valid up to quadratic displacements | X i | from the reference world-line. Notice that R αβγδ = R αβγδ ( T ) is the Riemann curvature tensor evaluated along thereference geodesic, where X = 0, and it depends on T only, which is the observer’s propertime.As discussed in Ref. 26, neglecting the terms g ij related to the spatial curvature, thespace-time element (18) can be recast in terms of the gravito-electromagntic potentials(Φ , A ) ds = − (cid:18) − c (cid:19) c dT − c ( A · d X ) dt + δ ij dX i dX j , (19)where the gravitoelectric potential Φ = Φ( T, X ) isΦ( T, X ) = − c R i j ( T ) X i X j , (20)and the components of the gravitomagnetic potential A = A ( T, X ) turn out to be A i ( T, X ) = c R jik ( T ) X j X k . (21)In close analogy to electromagnetism, the gravitoelectric and gravitomagnetic fields E and B are defined in terms of the potentials by E = −∇ Φ − c ∂∂T (cid:18) A (cid:19) , B = ∇ × A . (22)Using the definitions (20) and (21) we obtain (up to linear order in | X i | ) the followingcomponents E i ( T, X ) = c R i j ( T ) X j , (23)and B i ( T, R ) = − c ǫ ijk R jk l ( T ) X l . (24)8he electromagnetic analogy is useful to describe the motion of free test particles: namely,the geodesic equation of the space-time metric (19) can be written in the form of a Lorentz-like force equation m d X d T = q E E + q B V c × B , (25)up to linear order in the particle velocity V = d X d T , which is actually a relative velocity .In the Lorentz-like force equation, q E = − m is the gravitoelectric charge, and q B = − m is the gravitomagnetic one (the minus sign takes into account the fact that the gravitationalforce is always attractive). We notice that the ratio q B q E = 2, since linearized gravity is aspin-2 field.As a consequence, the Lorentz-like force equation becomes m d X d T = − m E − m V c × B . (26)This equation is the key point of this paper; let us briefly comment on its meaningand implication on the study of GWs. According to our approach, we may say that, inhis reference frame, the observer studies the evolution of a test mass using Eq. (26); inother words, the latter equation describes how the mass coordinates X, Y, Z (which, byconstruction, measure proper distances away from the reference world-line) change due tothe action of the gravitational field.It is important to emphasise that the effects expressed by the gravitoelectrogmagneticfields E and B have a tidal character, since both fields (23)-(24) depend on the location ofthe mass, relative to the observer which is at the origin of the frame. Accordingly, the actionof the GWs is simply described in the Fermi frame in terms of Newtonian gravitoelectrog-magnetic forces; of course, if other forces are present (such as mechanical or electromagneticones) they should be added to the equation of motion.The gravitoelectrogmagnetic fields E and B , according to their expressions (23)-(24),vanish along the reference world-line, where X = 0; then, the Lorentz-like force equation(26) suggests that test masses are freely moving. This is nothing but a rephrasing of theequivalence principle: in local freely falling frames the physics of special relativity holdstrue.In the following section we are going to show how, using the Lorentz-like equation (26)it is possible to describe the interaction of the wave with a detector. To be more specific,in the Riemann curvature tensor needed to the define the gravitoelectrogmagnetic fields, we9ill neglect the contributions due to local gravitational fields (such as the one of the Earth)and we will consider only the contribution of the wave. IV. GRAVITOELECTROGMAGNETIC EFFECTS IN THE FERMI FRAME
According to what we have seen before, in order to study the interaction of GWs withtest masses, we can use the Lorentz-like force equation (26): to this end, we need the explicitexpressions of the gravitoelectrogmagnetic fields appearing therein. We remember that thesefields are defined in terms of the Riemann tensor: E Ci ( T, X ) = c R i j ( T ) X j , B Ci ( T, R ) = − c ǫ ijk R jk l ( T ) X l . (27)To calculate these fields, in principle we need the expression of the Riemann tensor inFermi coordinates. However, in the weak field approximation - that is to say up to linearorder in h µν - the Riemann tensor is invariant with respect to coordinate transformations,hence it has the same expression in terms of the new coordinates. As a consequence, wecan use the TT values for the perturbations h µν given by Eq. (8) and express them inFermi coordinates. Also, dealing with GWs, in what follows we suppose that the extensionof the reference frame is much smaller than the wavelength, so that we may neglect thespatial variation of the wave field: consequently the components of the Riemann tensor areevaluated at the origin of our frame, where X = 0. If this condition is not fulfilled, it isnecessary to use the expression of the Fermi coordinates valid at higher order in the distancefrom the reference world-line (see e.g. Ref. 33) and, as a consequence, additional terms willbe present.Using Eqs. (8) and (17), the components of the gravitoelectric field (23) are E X = 0 , E Y = − ω (cid:2) A + sin ( ωT ) Y + A × cos ( ωT ) Z (cid:3) , E Z = − ω (cid:2) A × cos ( ωT ) Y − A + sin ( ωT ) Z (cid:3) , (28) while those of the gravitomagnetic field (24) turn out to be B X = 0 , B Y = − ω (cid:2) − A × cos ( ωT ) Y + A + sin ( ωT ) Z (cid:3) , B Z = − ω (cid:2) A + sin ( ωT ) Y + A × cos ( ωT ) Z (cid:3) . (29) Notice that both fields are perpendicular to the propagation direction: gravitationalwaves, like electromagnetic ones, are transverse. Taking into account the expressions (28)10nd (29) it is easy to check that E · B = 0: in other words the two fields are orthogonaleverywhere; moreover, we obtain also that | E | − | B | = 0: they have the same magnitude.Notice also that E ( A × ) = B ( A + ) and E ( A + ) = − B ( A × ).In Figures 1 and 2 the components of the gravitoelectric and gravitomagnetic fields areplotted at fixed T : it is manifest that, for both fields, the A × components are obtained fromthe A + with a rotation of π/
4. In Figure 3 we see that, at fixed time, the A + componentsof the fields are orthogonal at any spatial location; the same is true for the A × components.Now, using the gravitoelectrogmagnetic approach, we want to study the interaction ofthe wave with a detector. To begin with, it is important to understand, in this context,the meaning of detector . One of the first extensive analyses of gravitational waves detectorscan be found in the paper by Press and Thorne; here a GW is described as “field of(relative) gravitational forces propagating with the speed of light”. This definition nicelyfits our approach, where it gets a true operational meaning: in fact using Fermi coordinatesrelative to a given observer, physical quantities, such as displacements, are relative to thereference world-line. In particular the gravitoelectric and gravitomagnetic fields are positiondependent and act differently on test masses located at different positions, thus producingtidal effects. In summary, the passage of GWs provokes a space-time deformation which canbe described in terms of tidal forces due to the gravitoelectrogmagnetic fields.That being said, a detector or gravitational antenna is a physical system made of testmasses, on which the wave acts producing displacements and motion relative to the referenceworld-line. Starting from the explicit expressions of the gravitoelectrogmagnetic fields (28)and (29), we are now in a position to use the Lorentz-like force equation (26) to describethe effect of GWs on gravitational antennas. Before doing that, it is important to point outthe limits of our approximation: we work at first order in the wave amplitude, so we haveto deal with equations in a self-consistent way. If we suppose that V is the velocity of atest mass before the passage of the wave, the wave produces a change V ( T ) = V + δ V ( T ),where the variation δ V ( T ) is of the order of the wave amplitude A : δ V ( T ) = O ( A ). As aconsequence, in the linear approximation in the equation of motion (26) we can neglect thecontribution of the gravitomagnetic field if the test masses are at rest before the passageof the wave; things are different if we consider masses in motion before the passage of thewave.The simplest GW-antenna is made of two free masses that are at rest before the passage11f the wave; in particular, we suppose that one of them is at the origin, so we are interested inthe motion of the other mass. Accordingly, the test mass is acted upon by the gravitoelectricfield only, and its equation of motion is d X d T = − E (30)Let us suppose that the polarization of the wave is such that A × = 0 (as we have seenbefore, the effect of the A × polarization is qualitatively the same); according to Eq. (28)the gravitoelectric field is given by E X = 0 , E Y = − ω (cid:2) A + sin ( ωT ) Y (cid:3) , E Z = ω (cid:2) A + sin ( ωT ) Z (cid:3) . (31)Let the location of the mass before the passage of the wave be X = (0 , L, L . Then, the the solution of Eq. (30) up tolinear order in the wave amplitude, is X ( T ) = 0 , Y ( T ) = L (cid:20) − A + ωT ) (cid:21) , Z ( T ) = 0 . (32)The distance between the two masses changes with time. This result is in agreement withEq. (15), obtained using TT coordinates and considering, then, the physical distance: thecoordinate distance in Fermi coordinates is an observable quantity.Another kind of detector, called a heterodyne antenna, was proposed during the 70’s byBraginskij and collaborators; its functioning is based on the resonance principle. It isinteresting to see how our approach allows to simply understand the interaction of GWswith this device, using the equations of basic mechanics in the Fermi frame.The antenna is made of two dumbbells crossed at an angle of π/
2, with length R . Beforethe passage of the wave, they independently rotate in the plane orthogonal to the propagationdirection with the same frequency ω . We suppose that at T = 0 the configuration of thedumbbells is that of Figure 4-(a), i.e. the four masses m = m = m = m = m arealong the axes Y and Z . The coordinates of the mass m , whose position at T = 0 is X = (0 , , R ), are X = 0 , Y = R sin ω T, Z = R cos ω T. (33)We suppose that the wave is circularly polarised, so that A + = A × = A ; using the gravito-electric field (28), the force acting on the mass is F E = − m E F E ,X = 0 , F E ,Y = mω AR ω − ω ) T, F E ,Z = − mω AR ω − ω ) T. (34)12f ω = ω/
2, the above expression becomes F E ,X = 0 , F E ,Y = mω AR ω T, F E ,Z = − mω AR ω T (35)Hence, if the resonance condition ω = ω/ m experiences a force ofconstant magnitude | F E | = mω AR , orthogonal to the dumbbell. A similar approach suggeststhat the other mass m undergoes an equal force in opposite direction. In summary, due tothe action of the wave, a constant torque τ = − mω AR u X (where u X is the unit vectorof the X axis) acts on the dumbbell, with the effect of accelerating its rotation.Applying the same approach to the other dumbbell, we see that it is acted upon by aconstant torque τ = mω AR u X , with the effect of decelerating its rotation.In summary, with this choice of the rotation frequency, one dumbbell is accelerated andthe other is decelerated, so that the masses come closer: the angular separation θ betweenthe two dumbbells evolves with time with the law ∆( θ )( T ) = π − δθ ( T ) = π − ω AT ,which is independent of the length R .However, our approach based on the Lorentz force equation (26) suggests that, sincebefore the passage of the wave the masses are in motion, there is an additional effect, due tothe action of the gravitomagnetic field on the rotating masses. The gravitomagnetic forceacting on a mass moving with speed V , is F B = − m V c × B . Since we are working at linearorder in the wave amplitude, we use in this expression the velocity of the system before thepassage of the wave. Let the rotation frequency be ω/ m : B X = 0 , B Y = − ω AR ω T, B Z = − ω AR ω T. (36)This field has constant magnitude and it is always directed toward the center; hence, themass m undergoes the force F B = mω AR c u X . The other mass m undergoes to the sameforce, so that the total force acting on the first dumbbell is F B = mω AR c u X . If we considerthe other dumbbell, using the same approach we see that it experiences a total force F B = − mω AR c u X . The first dumbbell moves in the direction of propagation of the wave, whilethe other one moves in the opposite direction: accordingly, their distance d changes withtime according to d ( T ) = ω AR c T .This effect can be understood taking into account the expression of the Poynting vector P = c πG E × B . (37)13hich defines the energy per unit time and unit of surface transported by the wave alongits propagation direction (see e.g. Ref. 26). In fact, if we consider the fields acting onthe test masses, as described in Figure 4-(b), it is easy to check that the Poynting vectoracting on the first dumbbell is directed along the direction of propagation of the wave, whileit acts in the opposite direction on the second dumbbell. Since Fermi coordinates have aconcrete meaning and are strictly related to measurable quantities, this result is a simpledemonstration of the wave transmitting linear momentum. The effect of the wave on arotating detector suggests another simple argument that proves the reality of gravitationalwaves in addition to famous sticky bead argument , developed by Feynman and Bondi to showthat gravitational waves can have physical effects: in particular they suggested that if beadssliding on sticky rock move under the effect of the passing wave, they must transfer heat tothe road by friction, which proves that gravitational waves carry energy (see e.g. Ref. 38).The above examples suggest that our approach provides a simple interpretation of theinteraction between GWs and test masses. Indeed, it is important to emphasise that cur-rent detectors are essentially looking for gravitoelectric effects. A proposal to exploit thegravitomagnetic effect is discussed in Ref. 28. V. CONCLUSIONS
Gravitational waves are today a key ingredient of multi-messenger astronomy, whichallows us to study astrophysical sources using different channels and contributes to increasingour understanding of the Universe. As a consequence, we believe that it is important toteach the main features of gravitational waves science in introductory physics and astronomycourses, which is useful also to give students the possibility of understanding the continuousbreakthroughs in this field. The standard approach in teaching gravitational waves physicsis usually based on TT coordinates. On one hand, these coordinates are useful becausethey explain some basic characteristics of GWs, such as their polarizations and the fact thatthey are transverse to the propagation direction. On the other hand, TT coordinates lack aphysical meaning and, in order to understand the interaction of the waves with detectors, it isnecessary to obtain observable quantities, using the standard GR approach. From a teachingperspective, we believe that a different approach, based on the use of Fermi coordinates,would be more suitable: in fact, Fermi coordinates are defined, by construction, as scalar14nvariants and have a concrete meaning, since they are the coordinates an observer wouldnaturally use to make space and time measurements in the vicinity of her/his world-line.Moreover, Fermi coordinates enable simple understanding of the meaning of the principle ofequivalence, on which GR is based.We have shown that, thanks to Fermi coordinates, it is possible to describe the effectsof a plane gravitational wave using an electromagnetic analogy: in fact, the wave field isequivalent to the action of a gravitoelectric and a gravitomagnetic field, that are transverseto the propagation direction and orthogonal to each other. Moreover, the action of the waveon test masses is described in terms of tidal forces, determined by a Lorentz-like equation.Then, it is easy to describe how the physical distance between two test masses changes, dueto the passage of the wave: this can be understood as the action of a gravitoelectric fieldwhich, because of its tidal character, provokes different effects on masses located at differentpositions. Furthermore, on the basis of this approach, it is possible to see that there are alsogravitomagnetic effects, caused by the passage of the wave on moving test masses. Indeed,even if existing detectors, such as LIGO and VIRGO, and future ones, such as LISA, areaimed at detecting gravitoelectric effects, it is possible that new types of detectors could bedesigned to measure also gravitomagnetic effects.In summary, we believe that our approach, which rests upon the analogy with well knownfacts from electromagnetic theory, could help students to better understand and exploregravitational waves physics.
ACKNOWLEDGMENTS
The author thanks Dr. Antonello Ortolan for the stimulating and useful discussions;moreover, the author expresses appreciation for the referees and the editors, whose sugges-tions greatly improved the paper. ∗ [email protected] A. Einstein. Approximative Integration of the Field Equations of Gravitation.
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APS Physics , 9:17, 2016. IG. 1. The gravitoelectric field E : on the left, we suppose that the GW has just the A + polar-ization; on the right, we suppose that the GW has just the A × polarization. Notice that the twopolarizations differ by a rotation of π/ B : on the left, we suppose that the GW has just the A + polarization; on the right, we suppose that the GW has just the A × polarization. Notice that thetwo polarizations differ by a rotation of π/ IG. 3. The gravitoelectric E and gravitomagnetic field B for a wave with A + polarization: noticethat the two fields orthogonal everywhere.FIG. 4. (a): two identical dumbbells are made by test masses m = m = m = m = m at fixeddistance R , and rotated by π/ ω , before the passage of the wave. (b): the gravitoelectromagnetic fields acting on the test masses,due the passage of the wave., before the passage of the wave. (b): the gravitoelectromagnetic fields acting on the test masses,due the passage of the wave.