Kundt geometries and memory effects in the Brans-Dicke theory of gravity
aa r X i v : . [ g r- q c ] N ov Kundt geometries in Brans-Dicke theory and thememory effect
Siddhant Siddhant , Indranil Chakraborty and Sayan Kar , ∗ Department of PhysicsIndian Institute of Technology Kharagpur,721 302, India and Centre for Theoretical StudiesIndian Institute of Technology Kharagpur,721 302, India
Abstract
We study Kundt geometries (
Kundt waves as well as a generalised Kundt metric ) in Brans-Dicke(BD) theory with emphasis on specific solutions and possible memory effects. Three solutions withdifferent values of the BD scalar ( ω = − , +1 , − /
2) are presented for Kundt waves. In particular,for ω = − / ω = −
2. For ω = +1 , − / X , X , X ) is found – a feature resulting from the presence of nonzero gyraton-like terms in the metric. In summary, we confirm that distinct memory effects (different from their GRcounterparts) are indeed present in diverse Kundt geometries in BD theory. ∗ Electronic address: [email protected], [email protected], [email protected] . INTRODUCTION The detection of gravitational waves in binary mergers has opened up new prospects fortesting theories of gravity in the strong field regime [1, 2]. Gravitational memory is onesuch unobserved strong field effect that can be used to test diverse theories of gravity. Thegravitational wave memory effect is the residual permanent DC shift in the position (orvelocity) caused due to the passage of a gravitational wave pulse [3].The study of memory effects began in the work of Zel’dovich and Polnarev [4] who studiedgravitational radiation emitted due to the motion of flybys/collapse of stars in a globularcluster. A few years later, Braginsky and Grishchuk [5] looked at motion of test particles inweak field, linearized gravity and coined the term memory effect to denote the change in themetric perturbation at early and late times. Geodesic deviation of test particles due to lowfrequency gravitational radiation at null infinity was investigated further by Ludvigsen [6].Christodoulou, using full nonlinear GR showed the presence of memory due to the transportof energy and momentum of gravitational waves to null infinity [7]. This effect relatedto non-linearity was ascribed to gravitons produced by the radiation itself [8]. Memoryeffects are also possible in electrodynamics [9] and Yang-Mills theories [10, 11]. Interestingtheoretical links to memory effects have been conjectured, of late, in the context of softtheorems and BMS symmetries [12]. It has been noted that the nonlinear memory effect canalso be understood as a BMS transformation relating two inequivalent Minkowski vacua atfuture null infinity caused by the passage of gravitaional waves (see the review [13] and thereferences cited therein).Memory effects in non-flat backgrounds in GR have been studied in both dS [14, 15] (mo-tivations from cosmology) and AdS spacetimes [16]. In [16], the authors have showed howto isolate the gravitational wave contribution from the background spacetime by resortingto Fermi normal coordinates and solving the geodesic deviation equation. They treated thewave as a perturbation over AdS spacetime caused due to scattering of massive/masslessparticles. In our work, we adopt the same method for studying memory effects in Kundtspacetimes. However, in our case, the setting is non-perturbative, since we deal with exactspacetimes representing gravitational waves.Kundt spacetimes are exact radiative geometries consisting of nonexpanding, nonshearingand nontwisting null geodesic congruences (NGC) [17–19]. The spacetime admits various2ave solutions ( pp waves, Siklos waves [20, 21]) owing to the presence of NGC whose tangentvector is generally not covariantly constant. Hence, the wave surfaces are not Cartesianplanes. This non-planarity can be ascribed to the presence of matter or a cosmologicalconstant [22]. Gyratons (spinning relativistic sources) are solutions obtained as a subclassof Kundt geometries [23–25]. Presence of gyratonic matter in a Kundt geoemtry imparts anangular momentum due to its intrinsic spin. Till date, most of the research around Kundtgeometries have largely been focused on Einstein gravity [26–31]. There exists some recentwork in Gauss-Bonnet [32] and quadratic gravity [33]. However, as far as we know, theredoes not exist any literature on such geometries in scalar tensor theories. Our article is onesuch attempt towards understanding Kundt solutions in the most basic scalar-tensor theory,BD gravity, using the memory effect as a tool.There does exist previous work on memory effects in BD theory. As is well known, the BDscalar field produces a breathing mode along with the two additional polarizations (+ , × )found in GR [34]. Lang computed GW waveforms for scalar and tensor modes separately inthe PN approximation [35, 36]. Du and Nishizawa proposed a test of gravity for scalar tensortheories [37]. They found two distinct sets of memory contributions: T-memory (tensor) andS-memory (scalar). This scalar memory is an unique effect in such theories, unlike GR. Suchscalar memory effect was used as a tool to understand the Vainshtein screening mechanismin BD gravity [38]. Asymptotically flat spacetimes in BD theory have been recently studiedin [39, 40]. The BMS group [41] is retained for the tensorial case. There are degeneratevacua for the scalar sector related via Lorentz transformations. The BMS charge algebrahas also been computed in [42].Studying memory effects for such Kundt wave spacetimes in GR was inititiated by two ofus in [43], by analysing geodesics. Similar to exact plane wave spacetimes one can constructsandwich waves here by choosing appropriate limiting profiles [19, 44, 45]. This servesas a toy model of a gravitational wave burst, qualitatively. Interesting distinctions occurbetween negative and positive constant curvature solutions, particularly for the latter, wherewe found a new frequency memory effect . In this article, our motivation is two fold: a) tryto understand differences in memory effects between GR and BD theory based on Kundtwave geometries b) differentiate between the nature of Kundt wave solutions and generalized3undt metric for the same value of ω . In the Kundt wave scenario, we solve explicitlythree cases with different values of ω (-2,+1,-3/2). The first case ( ω = −
2) resembles aconstant negative scalar curvature. ω = +1 consist of variable positive curvature. In thesetwo cases we perform geodesic analysis to draw parallels between GR and BD theories. Wealso solve for ω = − / ω is arrived at, in the full metric scenario, directly from thefield equations .The cases having variable scalar curvature have singular solutions. Such solutions werealso reported in [31] where they analysed Kundt spacetimes in GR having scalar field asmatter. The Kundt metric obtained here is indeed singular. We discuss the nature ofthese singularities in detail. The coordinate ranges are chosen such that the geodesics arenot inside the singular region. From our geodesic analysis it is shown that there is indeedfocusing towards a singularity.As for the methods employed, in all cases we first find out the metric by solving the relevantfield equations. The scalar field is only dependent on the spatial coordinates in the casesof interest. Then we solve for the geodesic equations and try to analyse memory effectsusing them. Since we are working in non-flat backgrounds, another way to approach theproblem is by solving the deviation equation. This entire calculation is done in Ferminormal coordinates [16]. Here the coordinate system is Cartesian and hence the notionof displacement and velocity memory effect is qualitatively similar to exact plane wavespacetimes [45–47]. In such Fermi coordinates, we construct tetrads along a given timelikegeodesic. For the Kundt wave metric we can construct parallely propagating tetrads. We findthat only for the full metric it is not possible to obtain a set of parallely propagating tetrads . Hence, they are Fermi-Walker transported. Thus, the deviation equation contains extraterms. After obtaining all the tetrads we find the relevant Riemann curvature in the tetradframe. We split the background and gravitational wave terms (this is done by looking intoeach term and separating those terms which have factors proportional to the gravitationalwave part from the metric). The relevant contribution coming from the background deviationis first calculated. Next, we solve for the deviation solely due to gravitational wave. This In this paper, we refer to generalized Kundt metric with ω = − / full metric . Different value of ω gives two different theories. This is due to the presence of the gyraton-like terms in the full metric ω = − , +1 , − / II. BASIC FRAMEWORKA. Brans-Dicke gravity
Brans and Dicke seeking motivation from Mach’s principle proposed this theory [48] wherethe Newtonian gravitational constant (G) is considered as the reciprocal of a scalar field.This is based on the idea of variability of inertial mass at different points in spacetime. Theaction for the BD theory in the Jordan frame is given below. S = Z √− g (cid:20) φR − ωφ ∇ α φ ∇ α φ + 16 π L m (cid:21) d x (1)Here, φ denotes the scalar field, ω is the BD parameter and the L m denotes the matterLagrangian. The value of ω is highly constrained from Solar System observations [49].Different values of ω correspond to different theories. We initially work in the generalscenario where it can take any general value. However we do analyse solutions for specificvalues ω = − , +1 , − /
2. The motivations for choosing such values are discussed in therelevant sections. In all the cases considered here we solve for vacuum solutions ( L m = 0).An interesting point to note is that the case of ω = − / g µν and φ . After performing a little algebra,we can write them in the standard form as shown below.5 µν = ωφ [ φ, µ φ, ν − g µν φ, α φ, α ] + 1 φ ( φ, µ ; ν ) (2) (cid:3) φ = 0 (3)The box operator is constructed using the Kundt spacetime metric. B. Kundt Solutions
We try to solve for Kundt geometries [17–19] in Brans-Dicke theory. The generalized space-time metric is given below. ds = − Hdu − dudv − W dudx − W dudy + 1 P ( dx + dy ) (4) P ≡ P ( u, x, y ) , H ≡ H ( u, v, x, y ) , W i ≡ W i ( u, v, x, y ) , ∀ i ǫ { x, y } The vector field k = ∂ v gives the NGC. The tangent to the spatial surfaces ( P ∂ x , P ∂ y )and k are orthogonal to each other. W , W are gyraton-like terms [23, 25]. These termsintroduce angular momentum in the spacetime and correspond to spinning null sources.We also work with Kundt wave metrics where the cross terms ( W , W ) are set to zero. Theline element is given below. ds = − H ( u, x, y ) du − dudv + dx + dy P ( u, x, y ) (5)The waves (denoted via the term H ( u, x, y )) are viewed as propagating in the backgroundspacetime [22, 27, 43].We work with both these classes of spacetimes given in Eqs. (4) and (5). First, we considerKundt wave spacetimes. In such cases one can set the BD parameter ( ω ) by hand. Hence, using this freedom we construct positive and negative curvature solutions. Analysesof memory effects using geodesics in such spacetimes helps us in qualitatively understandingthe differences w.r.t. GR [43]. The choices ω = − , +1 correspond to constant negative andvariable positive scalar curvature respectively. For gyratons W , W , H have no dependence on coordinate v . They are not fixed from the field equations. ω = − / ω for the latter. C. Memory effects and geodesic deviation equation
Apart from a geodesic analysis, one can understand memory from geodesic deviation. Thememory effect for spacetimes having non-flat backgrounds have been analysed in [16] usinggeodesic deviation. In [16], the authors have given the motivation to study memory effectssolely due to gravitational wave by going to Fermi normal coordinates and separating thebackground and gravitational wave part. Briefly, we state their methodology and go on tocalculate memory effects for the Kundt metric. The geodesic deviation equation in Fermicoordinates ( t = X , X i ) is given as, d X i dt = − R i j X j (6)The spatial indices associated with the frame are denoted by i, j ( R i j = R µ νρσ e i µ e ν e ρ j e σ ). The tetrads { e α a } are parallelly propagated along the given geodesicwith tangent vector given as e α . The tetrads and metric are related via η ij = e α i e β j g αβ .We assume that the total deviation vector is of the form X i = X iB + X iG , where the suffixes B, G are for background and wave respectively. The same splitting is carried out for theRiemann tensor in the tetrad frame. The splitting of the Riemann tensor is done by notingthe terms which are proportional to H ( u, x, y ) or its derivatives. Such terms denote thegravitational wave contribution while the other terms are due to background curvature orgyraton-like sources. Then, the equation (6) separates into these two equations which areshown below. d X iB dt = − ( R i j ) B X jB (7) d X iG dt = − [( R i j ) B + ( R i j ) G ] X jG − ( R i j ) G X jB (8) t denotes the proper time along the geodesic and X i give the three spatial coordinates. η ab denotes the Minkowski metric with components (-1,1,1,1). In the full metric scenario, the wave like term is H ( u, v, x, y ) given in Eq.(4). t -constant planes correspond to 3d Euclideanspace. Hence, the notion of velocity and displacement memory effects are similar to exactplane wave spacetimes which has been extensively worked out in the literature [44–46]. Animportant feature to note is that Eqs.(6), (7) and (8) take this form only when the con-stucted tetrads are parallely transported. For non-parallel transport all these equationsare modified. In the generalized Kundt metric, we have constructed a set of Fermi-Walkertransported tetrads. The reader may refer to Sec IV D where we have explicitly computedthe modified deviation equation. III. KUNDT WAVE METRIC
We solve for three particular cases ω = − , +1 , − /
2. At first, we rewrite the metric inEq.(5) below. ds = − H ( u, x, y ) du − dudv + dx + dy P ( u, x, y ) We consider the scalar field to be independent of v and hence, φ ≡ φ ( u, x, y ). The compo-nents of Eq.(2) which are relevant for solving the field equations are listed below. G xx = ω φ ( φ, x − φ, y ) + 1 φ ( φ, xx + P, x P φ, x − P, y P φ, y ) (9) G yy = ω φ ( φ, y − φ, x ) + 1 φ ( φ, yy + P, y P φ, y − P, x P φ, x ) (10) G uu = ωφ (cid:18) φ, u + H P ( φ, x + φ, y ) (cid:19) + 1 φ (cid:18) φ, uu − P H, x φ, x − P H, y φ, y (cid:19) (11) G uv = ω φ P ( φ, x + φ, y ) (12) G xu = ωφ (cid:0) φ, x φ, u ) + 1 φ ( φ, xu + P, u P φ, x (cid:1) (13) G yu = ωφ (cid:0) φ, y φ, u ) + 1 φ ( φ, yu + P, u P φ, y (cid:1) (14) The four other equations are the redundancies of the Einsteins field equations and hence are not requiredfor obtaining the solutions.
8e decompose the scalar field and the metric functions as: φ ( u, x, y ) = α ( u ) ψ ( x, y ) , P ( u, x, y ) = ˜ P ( x, y ) U ( u ) , H ( u, x, y ) = H ′ ( u ) h ( x, y ) . (15)Adding equations (9) and (10) and using separation of variables from equation (15) resultsin ψ, xx + ψ, yy = 0 ( we know that G xx = G yy = 0 from the metric). The solution is, ψ ( x, y ) = log( x + y ) (16)From the metric, G uv = P ∆ log P (where, ∆ = ( ∂ xx + ∂ yy )). Using this in equation (12)gives, ˜ P = p x + y [log( x + y )] ω/ (17)The equations for the ‘ xu ’ and ‘ yu ’ components, as in (13),(14) and given the metric (5) weend up with ωφ (cid:0) φ, x φ, u ) + 1 φ ( φ, xu + P, u P φ, x (cid:1) = (cid:18) P, u P (cid:19) , x (18) ωφ (cid:0) φ, y φ, u ) + 1 φ ( φ, yu + P, u P φ, y (cid:1) = (cid:18) P, u P (cid:19) , y (19)Both the above equations reduce to the same equation after using the separation of variables.We have, ( ω + 1) α, u α = U, u U (20)The Ricci scalar curvature is R = 2 P ∆ log P = 4 ωU [log( x + y )] ω +2 (21)The component of G uu from the metric is given below. G uu = P H, xx + H, yy ) + 2 P, uu P − (cid:18) P, u P (cid:19) + H ( − P, x − P, y + P ( P, xx + P, yy ))Using equations (11) and (15) we get˜ P U H ′ ( u )( h, xx + h, yy ) − U, uu U + H ′ ( u ) h (cid:18) ˜ P ( ˜ P , xx + ˜ P , yy ) − ˜ P , x − ˜ P , y U (cid:19) = ω (cid:20)(cid:18) α u α (cid:19) + H ′ h ˜ P U (cid:18)(cid:18) ψ, x ψ (cid:19) + (cid:18) ψ, y ψ (cid:19) (cid:19)(cid:21) + α, uu α − ˜ P U H ′ ( u ) (cid:18) h, x ψ, x ψ + h, y ψ, y ψ (cid:19) (22)We set U = 1. From equations (20), (22) and (16) we find that H ′ ( u ) is unconstrained. The xy dependent part of H ( u, x, y ) becomes h ( x, y ) = log[log( x + y )] (23)9his polarization term h ( x, y ) is different from GR. We will point out the consequences ofthis difference on the nature of the memory effect, contrasting it with GR. Thus from ourgeneric analysis (without specifying the value of ω ) we find that only H ′ ( u ) is unconstrained.We now discuss the various ω solutions. A. Kundt waves for ω = −
1. Metric and geodesic analysis
We consider a scenario where the scalar curvature is constant. Hence, we set ω = − R = − P = p x + y [log( x + y )].We perform a coordinate transformation x = e X cos Y, y = e X sin Y . The metric intransformed coordinates (u,v,X,Y) becomes as follows. ds = − H ′ ( u ) log(2 X ) du − dudv + dX + dY X (24)We choose H ′ ( u ) = sech u as it qualitatively resembles a gravitational wave pulse. Thegeodesic equations for transverse coordinates become ( u is an affine parameter): d Xdu − X (cid:18) dXdu (cid:19) + 1 X (cid:18) dYdu (cid:19) + X sech u = 0 (25) d Ydu − X (cid:18) dXdu (cid:19)(cid:18) dYdu (cid:19) = 0 (26)The above equations are solved numerically for a pair of geodesics in Mathematica 10 . Weset the initial coordinate velocities to zero for making them parallel. The plot for
X, Y directions are given below. We use this coordinate transformation for other values of ω This same initial condition is used in other cases of ω too. - u X (a) Initial positions of x are 2(orange)and5(blue) respectively. - - u Y (b) Initial positions of y are 2(orange),5(blue) respectively. FIG. 1: Displacement memory effect in Kundt waves with spatial 2-surfaces having constantnegative curvature for Brans-Dicke theory.We observe permanent displacement along only X-direction (see Fig.(1a)). There is nochange in position in Y-direction as shown in the Fig.(1b). This is akin to GR where wealso observed constant separation after the passage of the gravitational wave pulse.
2. Geodesic deviation analysis
The construction of a parallel propagating tetrad for Kundt spacetimes has been worked outin [52]. We rewrite the metric in (24) by substituting ξ = X + iY, ¯ ξ = X − iY . For ease ofidentification we name these coordintes ( u, v, ξ, ¯ ξ ) as the Bicak-Podolsky coordinates. ds = − H ′ ( u ) log[ ξ + ¯ ξ ] du − dudv + dξd ¯ ξ ( ξ + ¯ ξ ) (27)This coordinate system exactly mathches with the one presented in [52]. The tetrad con-structed by satisfying the orthonormality condition is as follows. e µ = [ ˙ v, ˙ ξ, ˙¯ ξ, e µ = [ − ˙ X / X, − X, − X, e µ = [ − ˙ Y / X, − iX, iX, e µ = [1 − ˙ v, − ˙ ξ, − ˙¯ ξ, −
1] (28) e µ gives the tangent to the geodesic. It is thus always parallely transported (obeys thegeodesic equations). The tetrad e µ is parallelly propagated while e µ , e µ are not. Hence,this two tetrads are rotated by an angle ˙ θ p = − ˙ Y /X . In the geodesic shown in Fig. 1, we11ave ˙ Y = 0 and so the rotation parameter is a constant. We take θ p = 0 so that the twotetrads e µ , e µ are also parallelly transported.The non-zero Riemann tensor components in the tetrad basis, for both the background andthe wave are shown below.( R ) B = − (cid:18) ˙ YX (cid:19) ( R ) B = ˙ Y ˙ XX ( R ) B = ˙ Y ˙ XX ( R ) B = − (cid:18) ˙ XX (cid:19) ( R ) G = − H ′ ( u ) (29)Substituting these expressions from Eq.(29) of the Riemann tensor components (as given inthe tetrad basis) in Eqs.(7) and (8), we solve for the gravitational wave contribution to thegeodesic deviation. We find that the nontrivial contribution due to the wave comes onlyalong X direction. Thus, both deviation and geodesic equation analysis claim the presenceof memory. - - t X FIG. 2: Deviation due to the gravitational wave only along X direction.Note the presence of a memory effect only along X direction as evident from the plot in Fig.(2). The separation monotonically increases along this coordinate while it remains constantthroughout for the other two directions. 12 . Kundt waves for ω = +1
1. Metric and Geodesic analysis
The curvature scalar and the metric function P ( x, y ) for such a value of ω becomes R = 4[log( x + y )] ˜ P ( x, y ) = p x + y [log( x + y )] / (30)We observe that the Ricci scalar diverges when x + y = 1. The metric in the transformedcoordinates ( X, Y ) is, ds = − H ′ ( u ) log[2 X ] du − dudv + 2 X ( dX + dY ) (31)The geodesic equations for the spatial coordinates X, Y are provided [ H ′ ( u ) = sech ( u )]. d Xdu + 12 (cid:20) X (cid:18) dXdu (cid:19) − X (cid:18) dYdu (cid:19) (cid:21) + sech u X = 0 (32) d Ydu + (cid:18) X (cid:19) dXdu dYdu = 0 (33)Eqs.(32) and (33) are solved numerically in Mathematica 10 . The solutions for the coordi-nates
X, Y are given below. -
10 0 10 20 30 40 502345 u X (a) Initial positions of x are 5(orange)and3(blue) respectively. -
10 0 10 20 30 40 502.02.22.42.62.83.0 u Y (b) Initial positions of y are 3(orange),2(blue) respectively. FIG. 3: Displacement memory effect in Kundt waves with spatial 2-surfaces having variablepositive curvature for Brans-Dicke theory.We find no change in separation along Y direction as shown in Fig.(3b). Along the X direc-tion, we find increasing separation between the geodesics after the departure of the pulse.13his is in sharp contrast to the profiles obtained in GR. In the latter case we found, froma geodesic analysis [43] that for positive curvature scenarios there is presence of frequencymemory effect . This is related to the difference in the nature of the metric for the two the-ories. In BD theory, h ( X, Y ) = log(2 X ) (obtained by solving the field equations) whereasin GR we took it as h ( X, Y ) = ( X − Y ) (usual expression found in + polarization).On extrapolating the geodesic trajectories to higher values of u , we find that the geodesicsare inextendible beyond X = 0, signifying the singular nature of the metric solution at thatpoint.
2. Geodesic Deviation analysis
We use the same technique as for ω = −
2. The metric in Bicak-Podolosky coordinates ( ξ, ¯ ξ )becomes ds = − H ′ ( u ) log[ ξ + ¯ ξ ] du − dudv + ( ξ + ¯ ξ ) dξd ¯ ξ (34)The orthonormal tetrads for this metric turn out to be, e µ = [ ˙ v, ˙ ξ, ˙¯ ξ, e µ = (cid:20) − (2 X ) / ˙ X, − X ) / , − X ) / , (cid:21) e µ = (cid:20) − (2 X ) / ˙ Y , − i (2 X ) / , i (2 X ) / , (cid:21) e µ = [1 − ˙ v, − ˙ ξ, − ˙¯ ξ, −
1] (35) e µ gives the tangent to the timelike central geodesic. Here, too, we need to rotate the dyads e µ , e µ by an angle ˙ θ p = ˙ Y X to make them parallel. ˙ Y is zero as follows from the plot inFig.(3b). The constant θ p is taken to be zero.The nonzero Riemann tensor components in the tetrad basis are given as: Background ( R ) B = 12 (cid:18) ˙ YX (cid:19) ( R ) B = − ˙ Y ˙ X X ( R ) B = − ˙ Y ˙ X X ( R ) B = 12 (cid:18) ˙ XX (cid:19) (36) Gravitational wave ( R ) G = − H ′ ( u ) X ( R ) G = 18 H ′ ( u ) X (37)14ncorporating these expressions into Eqs.(7) and(8) we find the deviation due to the wave.The plots for deviation solely due to gravitational wave are given below. - - t X (a) - - t X (b) FIG. 4: The behaviour of the deviation vectors X , X are shown in the plots.We distinctly observe displacement memory from these plots given in Fig.(4). There is alsono frequency memory here—a fact evident from plotting the geodesics. C. Kundt waves for ω = −
1. Metric and Geodesic analysis
Finally we analyse the scenario where ω = − /
2. Our motivation for choosing such a valueof ω is related to the ’generalised Kundt metric’ (to be discussed later) for which the samevalue of ω is fixed by the field equations. Thus, in order to compare the effects of thepresence and absence of gyratonic terms in the same BD theory (i.e. with the same ω ), weperform the present analysis.The curvature scalar and the metric function P ( x, y ) in this case becomes R = − x + y )] / ˜ P ( x, y ) = p x + y [log( x + y )] / (38)We now have a variable negative curvature solution. The metric in this case is given below(same coordinate transformation from { x, y } → { X, Y } ). ds = − H ′ ( u ) log[2 X ] du − dudv + dX + dY (2 X ) / (39)15ere, too, we find solutions which are singular at X = 0. The geodesic equations for the X, Y coordinates are, d Xdu − (cid:20) X (cid:18) dXdu (cid:19) − X (cid:18) dYdu (cid:19) (cid:21) + sech u √ X / = 0 (40) d Ydu − (cid:18) X (cid:19) dXdu dYdu = 0 (41)As before, solutions are obtained numerically using Mathematica 10 . These are shown below. - - u X (a) Initial positions of x are 8(orange)and10(blue) respectively. - - u Y (b) Initial positions of y are 6(orange),5(blue) respectively. FIG. 5: Displacement memory effect in Kundt waves with spatial 2-surfaces having variablenegative curvature for Brans-Dicke theory.The plots in Fig.(5) highlight the nature of displacement memory. Along the X coordinatewe see a permanent shift while there is no change along Y . There is no trace of velocitymemory from the geodesic analysis. This is analogous to the GR case where we found thatfor both constant and variable negative curvature there is a permanent separation after thedeparture of the pulse. We also find that the geodesics do not cross beyond X = 0. This isdue to the singular nature of the metric as discussed previously.
2. Geodesic Deviation analysis
We use the same methods as prescribed in the previous two subsections for calculatingmemory effects using geodesic deviation. The metric in the Bicak-Podolsky coordinates is16iven below. ds = − H ′ ( u ) log[ ξ + ¯ ξ ] du − dudv + dξd ¯ ξF ( ξ, ¯ ξ ) ; F ( ξ, ¯ ξ ) = ( ξ + ¯ ξ ) / (42)The tetrads constructed along one of the timelike geodesics are shown below. e µ = [ ˙ v, ˙ ξ, ˙¯ ξ, e µ = [ − ˙ X/ √ F , −√ F , −√ F , e µ = [ − ˙ Y / √ F , − i √ F , i √ F , e µ = [1 − ˙ v, − ˙ ξ, − ˙¯ ξ, −
1] (43)The tetrads are such that the tangent to the geodesic is given by e µ . The spatial vectors e µ , e µ are given a rotation of angle ˙ θ p = − Y X to make them parallelly transported. Since,here too, ˙ Y is zero (following the plot of Fig.(5b)), we set the constant θ p to be zero.The nonzero Riemann tensor components in this tetrad basis are given as: Background ( R ) B = − (cid:18) ˙ YX (cid:19) ( R ) B = 3 ˙ Y ˙ X X ( R ) B = 3 ˙ Y ˙ X X ( R ) B = − (cid:18) ˙ XX (cid:19) (44) Gravitational wave ( R ) G = − H ′ ( u ) √ X ( R ) G = − H ′ ( u ) √ X (45)The plots for deviation solely due to gravitational wave are given below. - - t X (a) - - t X (b) FIG. 6: The behaviour of the deviation vectors X , X are shown in the plots.17hus, we clearly see a displacement memory effect in this case as given in the plots of Fig.(6).There is no memory along X direction. This is different from the ω = − IV. GENERALIZED KUNDT METRIC
The generalized Kundt metric is written below along with the functional dependencies ofthe various metric components on the coordinates. ds = − Hdu − dudv − W dudx − W dudy + 1 P ( dx + dy )where P ≡ P ( u, x, y ) , H ≡ H ( u, v, x, y ) , W i ≡ W i ( u, v, x, y ) , ∀ i ǫ { x, y } Such metrics have been previously analyzed in [31] with a minimally coupled scalar field, inGR. Here, we study vacuum solutions in B-D theory in the Jordan frame. We will initiallytry to solve for the metric and the BD scalar field. Later, we will try to analyse memoryeffects using geodesic and deviation equations.
A. Metric functions and the scalar field
From G uv component in equation (2) we find φ, v = 0 . Field equations for G ux and G uy give W , vv = W , vv = 0Hence the functional forms of the W i become: W = vV ( u, x, y ) W = vV ( u, x, y ) (46)This shows that the cross terms W , W have linear dependence on v . Hence, they are notexactly gyratonic solutions [23–25]. We refer them as gyraton-like terms henceforth. The xx and yy components of the field equation (2) yields P V + 3 V ) + P V P, x − P ( P V ) , y + H, vv P (cid:20) ω φ ( φ, x − φ, y ) + 1 φ ( φ, xx − Γ xxx φ, x − Γ yxx φ, y ) (cid:21) (47) We exclude any other analytic functional dependence of φ on v V + 3 V ) + P V P, y − P ( P V ) , x + H, vv P (cid:20) ω φ ( φ, y − φ, x ) + 1 φ ( φ, yy − Γ yyy φ, y − Γ xyy φ, x ) (cid:21) (48)Adding equations (47) and (48) we find P [ V + V − V , x − V , y ] + H, vv = P φ ( φ, xx + φ, yy ) (49)Since P, V , V , φ are all independent of v , we must have H, vv = 0. We consider the form for H as shown below. H ( u, v, x, y ) = vM ( u, x, y ) (50)For Kundt wave spacetimes the metric function H was independent of coordinate v . Wefind that for the full metric there is a linear dependence on v . Hence, for this metric, weconsider H ( u, v, x, y ) as giving the notion of gravitational radiation. The field equation for G uu gives, − P ∆ log( P ) + P V + V − V , x + V , y )) = − P ω φ ( φ, x + φ, y ) − P φ ( V φ, x + V φ, y ) (51)The operator ∆ has the same form as defined in the previous section. Eq.(3) after expandingin terms of metric functions leads to the following φ, xx + φ, yy = V φ, x + V φ, y (52)Using equations (52) and (49) we rewrite equation (51) as − P ∆ log( P ) = P (cid:18) V + V − φ ( V φ, x + V φ, y ) − ωφ ( φ, x + φ, y ) (cid:19) (53)From the G xu component we get P (cid:20) vP (cid:18) V V , x − V V , y + V , yy − V , xy (cid:19) + vP P, y ( V , y − V , x ) + (cid:18) P, u P (cid:19) , x + M, x V P, u P − V , u (cid:21) = P (cid:20) ωφ φ, x φ, u + φ, xu φ + V φ, u φ + P, u φ, x P φ − P v φ φ, y ( V , y − V , x ) (cid:21) (54)Similar to the analysis in the previous section, we decompose the scalar field and metricfunctions here as, φ ( u, x, y ) = α ( u ) ψ ( x, y ) , P ( u, x, y ) = P ′ ( x, y ) U ( u ) , M ( u, x, y ) = h ( u ) N ( x, y ) + µ ( u ) . (55)19e substitute (55) in (54) and consider only v ( v -independent) terms to get, hN, x − V (cid:18) U, u U + α, u α (cid:19) − ψ, x ψ (cid:20)(cid:18) ω + 12 (cid:19) α, u α − U, u U (cid:21) = 0 (56)The solution of (56) can be obtained either by setting the u -dependence or ( x, y ) dependenceto be the same. We take the latter path. Hence, equation (56) breaks up into twoequations which are as follows: N, x − V = − ψ, x ψ (cid:20) N, y − V = − ψ, y ψ (cid:21) (57) h + U, u U + (cid:18) ω + 22 (cid:19) α, u α = 0 (58)Using the equations (57) and (52) we get an equation for ψ ( x, y ). We solve it to obtain ananalytic expression given as, ψ ( x, y ) = 1 a + log( x + y ) (59)This form is quite similar to the one given earlier in equation (16). We get solutions for N, V , V using equations (57) and (59). N ( x, y ) = 4 log[ a + log( x + y )] (60) V = − x ( x + y )[ a + log( x + y )] (61) V = − y ( x + y )[ a + log( x + y )] (62)Using this known functional forms and inserting them in equation (53) we get P ′ ( x, y ). P ′ ( x, y ) = 1[ a + log( x + y )] ω +22 (63)We solve the field equation for G vu by disintegrating the equation in three parts which aredependent on the powers of v . From the v -independent part we get: − U U, u ( µ + hN ) − U U, uu = 2 U ω α, u α + 2 U α, uu α + α, u α U ( µ + hN ) (64)Separating the x, y and u -dependent parts yield two equations which are, − U U, u = α, u α U (65) − U U, uu = 2 U ω α, u α + 2 U α, uu α (66) We initially set V to be independent of u and find the solution to be self consistent. α ( u ) in terms of U ( u ) from (65) in (66) gives ω = − /
2. Sucha value of BD scalar admits traceless matter solution. It is a also a solution of ConformalRelativity [50] and is equivalent to Palatini f(R) gravity [51]. We also find from Eq.(58) that h ( u ) = 0. From the v linear part of the field equation we again obtain Eq.(65). We set U = 1(independent of u ) and thus, α is also a constant (taken equal to unity). The scalar fieldbecomes dependent only on x, y . From the order v part of the field equation, we cannotconstrain µ ( u ) as it reduces to an identity and hence it behaves as a free parameter. Thisparameter µ ( u ) denotes the gravitational wave contribution in the memory effect analysisusing geodesics and geodesic deviations. B. Singular solutions
The Ricci scalar in the metric becomes R = − a + log( x + y )] / (67)We comment on some features of our solutions (possibilities of singularities) obtained below.Such singular solutions have already been analysed in [31]. • There is a singularity in the solution at r (= r ) = e − a/ (where r = p x + y ). • This is a line singularity along the null direction. The circumference of the physicalcylinder in the x − y plane vanishes due to the form of the spatial metric. The singularitycould be attributed to a source moving on a null path (like gyratons). • The same nature of singularity is being observed for other solutions of Kundt waves having ω = +1 , − / r = 0. • In all our analysis we have chosen ranges of x, y to be away from this singularity. Strictly speaking, the solutions obtained for the generalized Kundt metric does not stand on the same footingas compared to Kundt waves in terms of resembling a gravitational wave burst scenario. This is primarilydue to the v -linear dependence in certain terms of the metric. Also, Eq.(70) shows that u does not act asan affine parameter, unlike the case for Kundt wave metrics. Hence, the notion of memory effects discussedfor this solution is still an open issue and yet to be understood fully. . Geodesic analysis We analyse the timelike geodesic equations for such a metric. Proper time is denoted by t and it acts as an affine parameter. The governing equations are as follows:¨ x − ˙ x P, x P − x ˙ y P, y P + ˙ y P, x P − v µP V ˙ u − ˙ v ˙ uP V − vP V ˙ x ˙ u − ˙ u ˙ yvP ( V V + V , y − V , x ) = 0 (68)¨ y − ˙ y P, y P − x ˙ y P, x P + ˙ x P, y P − v µP V ˙ u − ˙ v ˙ uP V − vP V ˙ y ˙ u − ˙ u ˙ xvP ( V V − V , y + V , x ) = 0 (69)¨ u − µ u − V ˙ x ˙ u − V ˙ y ˙ u = 0 (70) − − vµ ˙ u − v ˙ u − vV ˙ u ˙ x − vV ˙ u ˙ y + ( ˙ x + ˙ y ) P (71)We see that u is not an affine parameter from Eq.(70). Substituting equation (71) in (68)and (69) we get ¨ x − ˙ x P, x P − x ˙ y P, y P + ˙ y P, x P − P V − V x + ˙ y ) = 0 (72)¨ y − ˙ y P, y P − x ˙ y P, x P + ˙ x P, y P − P V − V x + ˙ y ) = 0 (73)We solve these equations numerically in Mathematica 10 to analyse their behaviour. Thebehaviour of all the coordinates of the geodesics are shown below. Apart from the transversespatial coordinates x, y , we also show the plots of the coordinate u, v . We also plot theseparation along x, y coordinate between the pair of geodesics for clarity.22
50 100 1502.02.53.03.54.04.55.0 t x (a) t y (b) t δ x (c) t δ y (d) t u (e) t v (f) FIG. 7: The variation of the coordinates are shown for two different geodesics with slightlyvarying initial conditions along x, y coordinates. We also show the change in separationbetween the two geodesics. The initial values are taken as 3,2,5,10 for the orange curvewhile for the blue curve these are 5,3,5,10 for the coordinates x, y, u, v respectively at affineparameter t = −
10. The initial velocities are taken to be zero along x, y coordinates and0 . u coordinate. We consider a = 5 in our calculation.In Fig.(7) we plot all the coordinates u, v, x, y of the generalized Kundt metric. We findthat from (7c) and (7d) the separation along transverse spatial coordinates is increasing.23he other coordinates u, v does not show much difference initially for the two geodesics.Geodesic analysis along x, y directions shows that the curves tend to approach the region ofsingularity. D. Geodesic deviation analysis
We start by rewriting the metric in Eq.(4) below in a modified fashion using the Bicak-Podolsky coordinates. ds = − vµ ( u ) du − dudv + 2 v ¯ Qdudξ + 2 vQdud ¯ ξ + dξd ¯ ξP ( ξ, ¯ ξ ) (74)Like in the previous case we have ξ = x + iy, ¯ ξ = x − iy . The new metric functions Q, ¯ Q arerelated to V , V from (61) and (62) in this way. Q = − V − i V ξξ ¯ ξ [ a + log( ξ ¯ ξ )] (75)¯ Q = − V i V ξξ ¯ ξ [ a + log( ξ ¯ ξ )] (76)The functional form of P ( ξ, ¯ ξ ) is given below from Eq.(63). P ( ξ, ¯ ξ ) = 1( a + log[ ξ ¯ ξ ]) / (77)We write P ′ = P since we have found that U is a constant (taken to be unity) from solvingthe field equations.We construct the tetrads that satisfies the orthonormality relations. e µ = [ ˙ v, ˙ ξ, ˙¯ ξ, ˙ u ] e µ = (cid:20) − u (cid:18) ˙ x P − v ˙ uP V (cid:19) , − P, − P, (cid:21) e µ = (cid:20) − u (cid:18) ˙ y P − v ˙ uP V (cid:19) , − iP, iP, (cid:21) e µ = (cid:20) − ˙ v ˙ u ˙ u , − ˙ ξ, − ˙¯ ξ, − ˙ u (cid:21) (78)We can parallely transport e µ , e µ along the geodesic by rotating the tetrads by an anglegiven below. ˙ θ p = ( x ˙ y − y ˙ x )2[ a + log( x + y )] (79)We cannot transport e µ parallely along e µ . It is only possible if we set Q = ¯ Q = 0(vanishing of the gyartonic-like cross terms). Thus, this tetrad undergoes Fermi-Walker24ransport. Under such a transport, the RHS of the deviation equation gets modified fromthe usual expression (6) in the following way. d X a dt = − R a tbt X b − (cid:18) Ddt e µ b (cid:19) e a µ dX b dt − (cid:18) D dt e µ b (cid:19) e a µ X b (80)We can split up e µ = A µ − e µ . The non-parallel piece is due to A µ = ( u , , , d X a dt = − R a tbt X b − (cid:18) Ddt A µ (cid:19) e a µ dX dt − (cid:18) D dt A µ (cid:19) e a µ X (81)Separating the background and gravitational wave part (as shown in the case for Kundt wavemetric) give us gravitational wave memory. The details of the calculation can be found inAppendix-I.We show below the plots of the deviation vector that give rise to memory effect. -
10 0 10 20 30 40 50020406080100120 t X (a) -
10 0 10 20 30 40 50050100150200250 t X (b) -
10 0 10 20 30 40 50246810 t X (c) FIG. 8: The behaviour of the deviation vectors X , X , X are shown in the plots.From the plots in Fig.(8), it is quite evident there is displacement memory. Unlike the case25f Kundt waves, where there was no change along X direction in all the cases consideredin the paper, here all the spatial coordinates X , X , X show memory effects. V. CONCLUSIONS
In our work we have tried to analyze Kundt geometries in BD theory of gravity. We studyand compare both cases of Kundt waves and the generalized Kundt metric (full metric). Inthe former case, we solve for three particular values of BD parameter ( ω = − , +1 , − / ω = − /
2. The field equations fixes the value of ω asexplicitly shown in Sec IIIA for the full metric. Moreover, we also try to differentiate thenature of memory effect observed here with GR [43] for the Kundt wave spacetimes. Usingdifferent but related methods (i.e. geodesic evolution and geodesic deviation) we have shownhow memory effects can be useful while delineating the differences between the various casesanalysed.All the results in this paper are summarised in the two tables provided below. In the firstone, we tabulate the various solutions of the metric functions obtained from solving thefield equations corresponding to different values of ω . In the second table we provide thenature of memory effects obtained from solving geodesic equations and the geodesic deviationequations in all the cases. KUNDT WAVE METRIC
B. D. Para- Metric functionsmeter ( ω ) g uu g uv g ux g uy g xx g yy − − log[log( x + y )]2 cosh ( u ) -1 0 0 [log( x + y )] − x + y [log( x + y )] − x + y +1 − log[log( x + y )]2 cosh ( u ) -1 0 0 [log( x + y )] x + y [log( x + y )] x + y − − log[log( x + y )]2 cosh ( u ) -1 0 0 [log( x + y )] − x + y [log( x + y )] − x + y GENERALIZED KUNDT METRIC − − v sech ( u )2 -1 vx ( x + y ) − ( a +log( x + y )) 4 vy ( x + y ) − ( a +log( x + y )) [ a + log( x + y )] [ a + log( x + y )] TABLE I: Metric functions obtained as solutions for different ω .26 UNDT WAVE METRIC
B. D. pa-rameter( ω ) Scalarfield( φ ) Memory using geodesics Memory using geodesic deviation Ricci Scalar Along X Along Y Along X Along X Along X − x + y ) Displacementmemory No evolution No deviation Separationincreases No deviation -8+1 log( x + y ) Displacementand velocitymemory No evolution Separationincreases Separationdecreases No deviation x + y )] − log( x + y ) Displacementmemory No evolution Separationincreases Separationincreases No deviation − x + y )] GENERALIZED KUNDT METRICAlong x Along y Along X Along X Along X −
32 1 a +log( x + y ) Displacementand velocitymemory Displacementand velocitymemory Separationincreases Separationincreases Separationincreases − a +log( x + y )] TABLE II: The nature of memory effect obtained for different ω using both geodesicequations and geodesic deviation equations.At first, the field equations for vacuum BD theory in the Jordan frame were written andsolved for the given class of metrics. The scalar field turns to be dependent only on thespatial coordinates ( x, y or X, Y ) as can be seen from Table II. We choose sandwich pulseprofiles (sech ( u )) in all of the cases for understanding memory as it resembles a toy modelfor gravitational wave burst events. We find in the case of Kundt waves that the valueof ω is not fixed from the field equations. After obtaining the metric, we look at therelevant geodesic equations for each of the spatial coordinates, numerically. For the negativecurvature solutions ( ω = −
2: constant curvature, ω = − /
2: varying curvature) we find27onstant separation after the passage of the gravitational wave pulse and eventual focusingalong one direction. In the other direction, the separation is completely unchanged (look atFigs.(1) and (5)). This is in qualitative agreement with the results in GR. For the positivecurvature solution we find monotonically increasing separation (non-zero memory) along onedirection while there is no change in the other as seen from the plots in Fig.(3). There is nopresence of frequency memory effect in BD as was noted earlier, in the GR context. Thisis due to a different analytic form of the h ( X, Y ) = log[2 X ] term in the metric in Eq.(31).In the GR case, this polarization term was h ( X, Y ) = ( X − Y ). It is obvious that thedifferent expressions for this term in different theories are exclusively due to their respectivefield equations.For the generalized Kundt metric, we perform a similar analysis as for the Kundt waves.The BD parameter ( ω = − /
2) is constant. This gives rise to a variable curvature scalarsolution. We point out the singular nature of these solutions. It corresponds to a linesingularity corresponding to a zero circumference cylinder. Such solutions in GR for Kundtspacetimes have been analysed in [31]. In our entire analysis on geodesics and deviation,the range of coordinates are chosen such that they are away from this singular region. Thegeodesic equations for all the coordinates are solved numerically. The geodesic analysis forthe spatial components ( x, y ) show that the trajectories tend to approach the singular region(also true for Kundt waves). Thus, in principle, we study test particle motion in a singularspacetime containing gravitational waves. We plot the evolution for all coordinates sincein this class of metrics, u is not an affine parameter (unlike Kundt waves). We observemonotonic change in separation along both the x, y coordinates (see Fig.(7)). Thus fromthe geodesic analysis we find that in ω = − / timelike Penrose limit of the central timelike geodesic in Kundt spacetimes [47].We construct orthonormal tetrads along the central geodesic and observe they are parallely The varying curvature scenarios for Kundt wave spacetimes are also described. gyraton-like terms present inEq.(4). Hence, these tetrads perform Fermi-Walker transport. This leads to a modificationin the geodesic deviation equation in the tetrad frame with additional corrections arisingdue to non-parallel transport (see Eq.(80)) along the central geodesic. The Riemann tensorin the tetrad frame (along with additional corrections present in the case of the full metric)is decomposed into a gravitational wave part and a background part. This is done by club-bing the terms which are proportional to the H ′ ( u ) part of H ( u, x, y ) in Eq.(5) as givingrise to a net gravitational wave contribution to the curvature. . The rest of the termsare considered coming due to the background. Solving the respective deviation equationfor the background (initially) and the wave (finally) gives us a quantitative measure of thegravitational wave memory effect.The geodesic deviation analysis for the Kundt wave metric shows presence of displacementmemory in all the three cases with certain characteristic differences. In the ω = − ω = +1 , − / ω = − /
2, the separations increase along both directions, in theother case we find both increase and decrease along the two different spatial directions. Inthe full metric scenario we find the separation is non-zero along all three spatial directions.There the deviation monotonically increases along all three coordinates. Thus, we explicitlyfind that displacement memory, though present in all the cases, is quantitatively differentin all these scenarios. A Tissot ring would be a better visual aid for understanding thisdifference. In ω = − ω = − / ω = +1,there may be an increase or decrease in area depending on initial configurations. In the fullmetric scenario, the deformation would take place on a 3-plane.This entire geodesic deviation analysis is non-perturbative and can be readily applied tocalculate memory effects for any spacetime containing gravitational radiation. Moreover,this analysis could also be recast using the B -matrix formalism as introduced in [53] andthe behaviour of the kinematic variables for timelike geodesic congruence can be computed. The same holds true for terms proportional to µ ( u ) (or its derivatives) in H ( u, v, x, y ) given in Eq.(4) ω = − / ACKNOWLEDGEMENTS
S. S. thanks the Department of Physics, IIT Kharagpur, India for providing him with theopportunity to work on this project during his tenure as an integrated M.Sc. student. I.C. acknowledges Matthias Blau for discussions related to Fermi coordinates and associatedtetrads. I. C. also thanks the University Grants Commission (UGC) of the Governmentof India for providing financial assistance through senior research fellowship (SRF) withreference ID: 523711. [1] B. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. , 061102 (2016).[2] E. Berti, K. Yagi, and N. Yunes, Gen. Rel. Grav. , 46 (2018).[3] M. Favata, Class. Qtm. Grav. , 084036 (2010).[4] Y. B. Zel’dovich and A. G. Polnarev, Sov. Astron , 17 (1974).[5] V. B. Braginsky and L. P. Grishchuk, Sov. Phys. JETP , 427 (1985).[6] M. Ludvigsen, Gen. Rel. Grav. , 1205 (1989).[7] D. Christodoulou, Phys. Rev. Lett. , 1486 (1991).[8] K. S. Thorne, Phys. Rev. D , 520 (1992).[9] L. Bieri and D. Garfinkle, Class. Qtm. Grav. , 195009 (2013).[10] M. Pate, A.-M. Raclariu, and A. Strominger, Phys. Rev. Lett. , 261602 (2017).
11] N. Jokela, K. Kajantie, and M. Sarkkinen, Phys. Rev. D , 116003 (2019).[12] A. Strominger and A. Zhiboedov, J. High Energy Phys. (2016), 86.[13] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (2017)arXiv:1703.05448 [hep-th] .[14] Y. Hamada, M.-S. Seo, and G. Shiu, Phys. Rev. D , 023509 (2017).[15] L. Bieri, D. Garfinkle, and N. Yunes, Class. Qtm. Grav. , 215002 (2017).[16] C.-S. Chu and Y. Koyama, Phys. Rev. D , 104034 (2019).[17] W. Kundt, Z. Phys. , 77 (1961).[18] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein’s field equations (Cambridge Univ. Press, Cambridge, England,2003).[19] J. B. Griffiths and J. Podolsky,
Exact Space-Times in Einstein’s General Relativity (Cam-bridge Univ. Press, Cambridge, England, 2009).[20] H. W. Brinkmann, Math. Ann. , 119 (1925).[21] N. Rosen, Phys. Z. Sowjetunion , 366 (1937).[22] J. Podolsk´y and M. Ortaggio, Classical and Quantum Gravity , 1685 (2003).[23] V. P. Frolov and D. V. Fursaev, Phys. Rev. D , 104034 (2005).[24] V. P. Frolov, W. Israel, and A. Zelnikov, Phys. Rev. D , 084031 (2005).[25] H. Kadlecov´a, A. Zelnikov, P. Krtouˇs, and J. Podolsk´y, Phys. Rev. D , 024004 (2009).[26] J. Podolsk´y and M. Ortaggio, Classical and Quantum Gravity , 2689 (2001).[27] M. Ortaggio and J. Podolsk´y, Classical and Quantum Gravity , 5221 (2002).[28] J. B. Griffiths, P. Docherty, and J. Podolsk´y, Class. Qtm. Grav. , 207 (2003).[29] A. Coley, S. Hervik, G. Papadopoulos, and N. Pelavas, Class. Qtm. Grav. , 105016 (2009).[30] J. Podolsk´y and R. ˇSvarc, Class. Qtm. Grav. , 205016 (2013).[31] T. Tahamtan and O. Sv´ıtek, Eur. Phys. J. C , 384 (2017).[32] R. ˇSvarc, J. Podolsk´y, and O. Hruˇska, Phys. Rev. D , 084012 (2020).[33] V. Pravda, A. Pravdov´a, J. Podolsk´y, and R. ˇSvarc, Phys. Rev. D , 084025 (2017).[34] C. M. Will, Living Rev. Rel. , 4 (2014).[35] R. N. Lang, Phys. Rev. D , 084014 (2014).[36] R. N. Lang, Phys. Rev. D , 084027 (2015).[37] S. M. Du and A. Nishizawa, Phys. Rev. D , 104063 (2016).
38] K. Koyama, Phys. Rev. D , 021502 (2020).[39] S. Tahura, D. A. Nichols, A. Saffer, L. C. Stein, and K. Yagi, arXiv:2007.13799 [gr-qc] .[40] S. Hou, in (2020)arXiv:2011.02087 [gr-qc] .[41] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner,Proc. Roy. Soc. Lond. A , 21 (1962).[42] S. Hou and Z.-H. Zhu, (2020), arXiv:2008.05154 [gr-qc] .[43] I. Chakraborty and S. Kar, Phys. Lett. B , 135611 (2020).[44] P.-M. Zhang, C. Duval, G. W. Gibbons, and P. A. Horvathy, Phys. Lett. B , 743 (2017).[45] I. Chakraborty and S. Kar, Phys. Rev. D , 064022 (2020).[46] P.-M. Zhang, C. Duval, G. W. Gibbons, and P. A. Horvathy, Phys. Rev. D , 064013 (2017).[47] G. M. Shore, JHEP (12), 133.[48] C. Brans and R. Dicke, Phys. Rev. , 925 (1961).[49] S. Hou and Y. Gong, Eur. Phys. J. C , 247 (2018).[50] M. P. Dabrowski, T. Denkiewicz, and D. Blaschke, Annalen Phys. , 237 (2007).[51] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. , 451 (2010).[52] J. Bicak and J. Podolsky, J. Math. Phys. , 4495 (1999).[53] M. O’Loughlin and H. Demirchian, Phys. Rev. D , 024031 (2019).[54] P.-M. Zhang, C. Duval, and P. A. Horvathy, Class. Qtm. Grav. , 065011 (2018).[55] S. Bhattacharjee, S. Kumar, and A. Bhattacharyya, Phys. Rev. D , 084010 (2019).[56] P. Mao and X. Wu, JHEP (2019), 058. APPENDIX
Equation (81) can be split up into background and wave parts as was done in equations (7)and (8). The deviation due to background and gravitational wave can be obtained as givenbelow. d X aB dt = − ( R a tbt ) B X bB − (cid:20)(cid:18) Ddt e µ b (cid:19) e a µ (cid:21) B dX bB dt − (cid:20)(cid:18) D dt e µ b (cid:19) e a µ (cid:21) B X bB (82)32 X aG dt = − [( R a tbt ) B + ( R a tbt ) G ] X bG − ( R a tbt ) G X bB − (cid:20)(cid:26)(cid:18) Ddt e µ b (cid:19) e a µ (cid:27) B + (cid:26)(cid:18) Ddt e µ b (cid:19) e a µ (cid:27) G (cid:21)(cid:18) dX b dt (cid:19) G − (cid:20)(cid:18) Ddt e µ b (cid:19) e a µ (cid:21) G (cid:18) dX b dt (cid:19) B − (cid:20)(cid:18) D dt e µ b (cid:19) e a µ (cid:21) G X bB − (cid:20)(cid:26)(cid:18) D dt e µ b (cid:19) e a µ (cid:27) B + (cid:26)(cid:18) D dt e µ b (cid:19) e a µ (cid:27) G (cid:21) X bG (83)We give the non-rotated tetrad components used in the calculation e µ a = ˙ v ˙ ξ ˙¯ ξ ˙ uP (cid:0) − Q − ¯ Q (cid:1) v − ˙ xP ˙ u − P − P iP (cid:0) Q − ¯ Q (cid:1) v − ˙ yP ˙ u − iP iP − ˙ u ˙ v ˙ u − ˙ ξ − ˙¯ ξ − ˙ u (84)The Riemann tetrad components are given below.( R ) G = 1( x + y ) (log( x + y ) + 5) / [ ˙ uv ( − x + y ) − x cos(2 θ ) − xy sin(2 θ ) − y cos(2 θ ))( ˙ ux µ ( u )(log( x + y ) + 5)+ ˙ uy µ ( u )(log( x + y ) + 5))] (85)( R ) G = − x + y ) (log( x + y ) + 5) / [ ˙ uv (2 log( x + y ) + 11)( x sin(2 θ )+ 2 xy cos(2 θ ) − y sin(2 θ ))( ˙ ux µ ( u )(log( x + y ) + 5)+ ˙ uy µ ( u )(log( x + y ) + 5))] (86)( R ) G = − x + y ) (log( x + y ) + 5) / [ ˙ uv (2 log( x + y ) + 11)( x sin(2 θ )+ 2 xy cos(2 θ ) − y sin(2 θ ))( ˙ ux µ ( u )(log( x + y ) + 5)+ ˙ uy µ ( u )(log( x + y ) + 5))] (87)( R ) G = 1( x + y ) (log( x + y ) + 5) / [ ˙ uv (2 log( x + y ) + 11)( x cos(2 θ ) − xy sin(2 θ ) − y cos(2 θ ))( ˙ ux µ ( u )(log( x + y ) + 5)+ ˙ uy µ ( u )(log( x + y ) + 5))] (88)All other non-vanishing components of the gravitational wave is zero.( R ) G = ( R ) G = ( R ) G = ( R ) G = ( R ) G = 0 (89)33 R ) B = 1( x + y ) (log( x + y ) + 5) / [ ˙ uv ( − x ˙ x − y ˙ y )( − x + y ) − x cos(2 θ ) − xy sin(2 θ ) − y cos(2 θ )) −
12 ( x + y )(log( x + y )+ 5)( − xy (2 log( x + y ) + 11)(2 ˙ u ˙ v sin(2 θ ) − p log( x + y ) + 5( ˙ x sin(2 θ ) − x ˙ y + ˙ y sin(2 θ ))) + x (4 ˙ u ˙ v cos(2 θ )(2 log( x + y ) + 11) − p log( x + y ) + 5( ˙ x (cos(2 θ )(4 log( x + y ) + 25) + 4 log( x + y ) + 27)+ ˙ y (cos(2 θ )(4 log( x + y ) + 19) − x + y ) − − x ˙ y sin(2 θ )))+ y ( p log( x + y ) + 5( ˙ x (cos(2 θ )(4 log( x + y ) + 19) + 4 log( x + y ) + 17)+ ˙ y (cos(2 θ )(4 log( x + y ) + 25) − x + y ) −
27) + 6 ˙ x ˙ y sin(2 θ )) − u ˙ v cos(2 θ )(2 log( x + y ) + 11)))] (90)( R ) B = 12( x + y ) (log( x + y ) + 5) / [ − uv ( − x ˙ x − y ˙ y )(2 log( x + y )+ 11)( x sin(2 θ ) + 2 xy cos(2 θ ) − y sin(2 θ )) − ( x + y )(log( x + y )+ 5)(4 xy cos(2 θ )(2 log( x + y ) + 11)(2 ˙ u ˙ v − ( ˙ x + ˙ y ) p log( x + y ) + 5)+ x (4 ˙ u ˙ v sin(2 θ )(2 log( x + y ) + 11) − p log( x + y ) + 5( ˙ x sin(2 θ )(4 log( x + y )+ 25) + ˙ y sin(2 θ )(4 log( x + y ) + 19) + 6 ˙ x ˙ y cos(2 θ )))+ y ( p log( x + y ) + 5( ˙ x sin(2 θ )(4 log( x + y ) + 19) + ˙ y sin(2 θ )(4 log( x + y )+ 25) − x ˙ y cos(2 θ )) − u ˙ v sin(2 θ )(2 log( x + y ) + 11)))] (91)( R ) B = 1( x + y ) (log( x + y ) + 5) / [ x ( − ( ˙ y sin( θ )(2 log( x + y ) + 7)+ ˙ x cos( θ )(2 log( x + y ) + 15))) + 2 xy (2 log( x + y ) + 11)( ˙ x sin( θ ) − ˙ y cos( θ ))+ y ( ˙ y sin( θ )(2 log( x + y ) + 15) + ˙ x cos( θ )(2 log( x + y ) + 7))] (92)34 R ) B = 12( x + y ) (log( x + y ) + 5) / [ − uv ( − x ˙ x − y ˙ y )(2 log( x + y )+ 11)( x sin(2 θ ) + 2 xy cos(2 θ ) − y sin(2 θ )) − ( x + y )(log( x + y )+ 5)(4 xy cos(2 θ )(2 log( x + y ) + 11)(2 ˙ u ˙ v − ( ˙ x + ˙ y ) p log( x + y ) + 5)+ x (4 ˙ u ˙ v sin(2 θ )(2 log( x + y ) + 11) − p log( x + y ) + 5( ˙ x sin(2 θ )(4 log( x + y )+ 25) + ˙ y sin(2 θ )(4 log( x + y ) + 19) + 6 ˙ x ˙ y cos(2 θ )))+ y ( p log( x + y ) + 5( ˙ x sin(2 θ )(4 log( x + y ) + 19) + ˙ y sin(2 θ )(4 log( x + y )+ 25) − x ˙ y cos(2 θ )) − u ˙ v sin(2 θ )(2 log( x + y ) + 11)))] (93)( R ) B = 1( x + y ) (log( x + y ) + 5) / [ ˙ uv ( − x ˙ x − y ˙ y )(2 log( x + y )+ 11)( x cos(2 θ ) − xy sin(2 θ ) − y cos(2 θ )) + 12 ( x + y )(log( x + y )+ 5)( − xy (2 log( x + y ) + 11)(2 ˙ u ˙ v sin(2 θ ) − p log( x + y ) + 5( ˙ x sin(2 θ )+ 2 ˙ x ˙ y + ˙ y sin(2 θ ))) + x (4 ˙ u ˙ v cos(2 θ )(2 log( x + y ) + 11) − p log( x + y ) + 5( ˙ x (cos(2 θ )(4 log( x + y ) + 25) − x + y ) − y (cos(2 θ )(4 log( x + y ) + 19) + 4 log( x + y ) + 17) − x ˙ y sin(2 θ )))+ y ( p log( x + y ) + 5( ˙ x (cos(2 θ )(4 log( x + y ) + 19) − x + y ) − y (cos(2 θ )(4 log( x + y ) + 25) + 4 log( x + y ) + 27) + 6 ˙ x ˙ y sin(2 θ )) − u ˙ v cos(2 θ )(2 log( x + y ) + 11)))] (94)( R ) B = 1( x + y ) (log( x + y ) + 5) / [ x ( ˙ x sin( θ )( − x + y ) − y cos( θ )(2 log( x + y ) + 7)) − xy (2 log( x + y ) + 11)( ˙ x cos( θ ) + ˙ y sin( θ ))+ y ( ˙ x sin( θ )(2 log( x + y ) + 7) − ˙ y cos( θ )(2 log( x + y ) + 15))] (95)( R ) B = 1( x + y ) (log( x + y ) + 5) / [ x ( − ( ˙ y sin( θ )(2 log( x + y ) + 7)+ ˙ x cos( θ )(2 log( x + y ) + 15))) + 2 xy (2 log( x + y ) + 11)( ˙ x sin( θ ) − ˙ y cos( θ ))+ y ( ˙ y sin( θ )(2 log( x + y ) + 15) + ˙ x cos( θ )(2 log( x + y ) + 7))] (96)35 R ) B = 1( x + y ) (log( x + y ) + 5) / [ x ( ˙ x sin( θ )( − x + y ) − y cos( θ )(2 log( x + y ) + 7)) − xy (2 log( x + y ) + 11)( ˙ x cos( θ ) + ˙ y sin( θ ))+ y ( ˙ x sin( θ )(2 log( x + y ) + 7) − ˙ y cos( θ )(2 log( x + y ) + 15))] (97)( R ) B = 4( x + y ) (log ( x + y ) + 5) / (98) (cid:20)(cid:18) Ddt A µ (cid:19) e µ (cid:21) B = 2 (cid:16) y (cid:16) sin( θ ) ˙ u + 2 ˙ y p log ( x + y ) + 5 (cid:17) + x (cid:16) x p log ( x + y ) + 5 − cos( θ ) ˙ u (cid:17)(cid:17) ˙ u ( x + y ) (log ( x + y ) + 5) / (99) (cid:20)(cid:18) Ddt A µ (cid:19) e µ (cid:21) B = − x sin( θ ) ˙ u − y cos( θ ) ˙ u + 4 y ˙ y p log ( x + y ) + 5 + 4 x ˙ x p log ( x + y ) + 5˙ u ( x + y ) (log ( x + y ) + 5) / (100) (cid:20)(cid:18) Ddt A µ (cid:19) e µ (cid:21) B = − u −
2) ( x ˙ x + y ˙ y )˙ u ( x + y ) (log ( x + y ) + 5) (101) (cid:20)(cid:18) Ddt A µ (cid:19) e µ (cid:21) G = (cid:20)(cid:18) Ddt A µ (cid:19) e µ (cid:21) G = (cid:20)(cid:18) Ddt A µ (cid:19) e µ (cid:21) G = 0 (102) (cid:18) D dt e µ b (cid:19) e a µ X b = (cid:18) dB µ dt + Γ µαγ e α B γ (cid:19) e a µ X (103)For background part, the contribution becomes as given below: B v = 12 (cid:18) vP Q ¯ Q + ¯ Q ˙ ξ + Q ˙¯ ξ ˙ u (cid:19) B ξ = P Q B ¯ ξ = P ¯ Q B uu