Constitutive law of nonlocal gravity
aa r X i v : . [ g r- q c ] M a y Constitutive law of nonlocal gravity
Dirk Puetzfeld ∗ ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany
Yuri N. Obukhov † Theoretical Physics Laboratory, Nuclear Safety Institute,Russian Academy of Sciences, B.Tulskaya 52, 115191 Moscow, Russia
Friedrich W. Hehl ‡ Institute for Theoretical Physics, University of Cologne, 50923 Cologne, Germany (Dated: May 10, 2019)We analyze the structure of a recent nonlocal generalization of Einstein’s theory of gravitationby Mashhoon et al. By means of a covariant technique, we derive an expanded version of thenonlocality tensor which constitutes the theory. At the lowest orders of approximation, this leads toa simplification which sheds light on the fundamental structure of the theory and may prove usefulin the search for exact solutions of nonlocal gravity.
PACS numbers: 04.20.Cv; 04.25.-g; 04.50.-hKeywords: Nonlocal gravity; Constitutive law; Approximation methods
I. INTRODUCTION
In a series of works [1–29], eventually culminating inthe book [30], Mashhoon and collaborators proposed anonlocal extension to Einstein’s gravity termed nonlocalgravity (NLcG).In NLcG gravity is assumed to be history dependent,i.e. the gravitational interaction has an additional featureof nonlocality in the sense of an influence (“memory”)from the past that endures. The theory is built uponan ansatz for the so-called nonlocality tensor N ijk , lead-ing to a set of integro-differential field equations. Thecomplexity of these equations surpasses the complexityof the ones encountered in Einstein’s theory of gravityby a great deal. This makes the search for exact or evenapproximate solutions of NLcG a daunting task, even ifadditional symmetry assumptions are made. It is thiscomplexity at a fundamental level, which makes nonlocalgravity a theory for which no exact solutions are knownbeyond the flat case, in other words, no exact solution en-compassing a gravitational field is known. At the sametime much work was put into the linearized version ofthe theory [23], and in this context some very promisingproperties of NLcG – for example addressing the darkmatter problem [13, 16, 19, 22, 29] – have been workedout.However, the fact that no exact solutions exist shouldof course be remedied, for the theory is supposed to bethe successor of General Relativity (GR), for which sev-eral solutions are known, which in turn play a key rolein the conceptual understanding of the theory. In the ∗ [email protected]; http://puetzfeld.org † [email protected] ‡ [email protected] present work we show, that the initial choice for the non-locality is not the “simplest expression” for N ijk , con-trary to what is stated in [30, (6.107)]. We hope that oursimplification will pave the way towards a more manage-able version of the theory, yet retaining its compellingoverall structure.The structure of the paper is as follows: In section II wesummarize the main features of NLcG. This is followedby a brief review of a covariant expansion technique inIII. This technique is then applied in IV to derive an ex-panded and thereby simplified version of the nonlocalitytensor. We conclude our paper in section V with a dis-cussion and outlook. An overview of our notation can befound in table I in appendix A. II. NONLOCAL GRAVITY AS ATELEPARALLEL GRAVITY THEORY
Originally Mashhoon tried to implement the general-ization of the locality principle directly for the field equa-tions of GR. This did not appear to be feasible, and a suc-cessful starting point turned out to be a rather particulartranslational gauge theory of gravity (TG), namely theso-called teleparallel equivalent GR || of Einstein’s GR,see [31–33].For a TG, the translational gauge field potential is rep-resented by the coframe e iα , the translational gauge fieldstrength by its covariant “curl”, the torsion of spacetime: T ijα := 2 (cid:0) ∂ [ i e j ] α + Γ [ i | βα e | j ] β (cid:1) . (1)Here coordinate indices are denoted by i, j, k, ... =0 , , ,
3, and frame indices by α, β, γ, ... = 0 , , ,
3, andΓ iαβ is the Lorentz connection of spacetime, see [34]. Thecurvature tensor of the spacetime vanishes, R ijαβ (Γ) = 2 ∂ [ i Γ j ] αβ + 2Γ [ i | γβ Γ j ] αγ = 0 , (2)that is, we have a teleparallelism, and we can pick a globalgauge – which is denoted by a star over the equality sign– such that at each point in spacetime the Lorentz con-nection Γ iαβ = − Γ iβα vanishes.The inhomogeneous and the homogeneous gravita-tional field equations of NLcG have a Maxwellian struc-ture, see [30, (5.70) and (6.117)], and are given by ∂ j ˇ H ij α − E αi ∗ = T αi , (3) ∂ [ i T jk ] α ∗ = 0 . (4)The gravitational excitation ˇ H ij α , in a Lagrange-Hamilton picture, is the “momentum” conjugate to the“coordinate” e iα , and the “velocity” T ijα : ˇ H ij α := − ∂ L g /∂T ijα , where L g is the gravitational Lagrangiandensity.The nonlinear correction terms in (3) represent theenergy-momentum tensor density of the gravitationalgauge field, E αi := − e iα ( T jkβ ˇ H jkβ ) + T αkβ ˇ H ikβ . (5)As source, we have on the right-hand side of the inhomo-geneous field equation (3) the energy-momentum tensordensity of matter T αi . It has to be assumed symmetric, T [ αβ ] = 0, cf. [32, page 52, 1st paragraph, eqs. (4.42),(4.43) and (4.36)].In a local and linear TG one assumes, as usual ina gauge theory, that the gravitational Lagrangian isquadratic in the field strength – here in the form of thetorsion. Thus, the constitutive law between excitationand field strength is local and linear:ˇ H ij α = 12 χ ijαklβ T klβ . (6)General relativity is recovered, see [35], via GR || χ ij mkln ( g ) = √− g κ (cid:16) − g k [ i g j ] l g mn − δ [ im g b ][ k δ l ] n + 2 δ [ in g j ][ k δ l ] m (cid:17) , (7)where g ij denotes the metric of spacetime, with signa-ture (+1 , − , − , − κ is Einstein’s gravitationalconstant.A nonlocal generalization of a gravity theory is deter-mined by an ansatz, which no longer necessarily can bederived from a Lagrangian,ˇ H y y υ = 12 " χ y y υ y y υ T y y υ − Z σ y x σ y x σ υ ξ K ( x, y ) X x x ξ x x ξ T x x ξ d x , (8)where the integration is performed over a 4-dimensionalvolume; see [30, (6.114)]. Here we use a condensed no-tation (common to the theory of bitensors) in which the point to which the index of a bitensor belongs can bedirectly read from the index itself; e.g., y n denotes in-dices at the spacetime point y . Moreover, in order todistinguish the local frame indices, we use ξ , ξ , . . . and υ , υ , . . . to designate objects with frame indices at thepoint x or y , in complete analogy to the labels x , x , . . . and y , y , . . . used in the holonomic case.In the early works [13, 14] it was suggested to use χ ijαklβ ≡ X ijαklβ ≡ GR || χ ijαklβ (9)as an ansatz for the nonlocal theory. However, it waslater on [23] generalized to χ ijαklβ ≡ GR || χ ij αklβ , (10) X ijαklβ ≡ GR || χ ij αklβ + odd χ ijαklβ , (11)so that odd χ ijαklβ T klβ ∼ ˇ p (cid:0) ˇ T i e jα − ˇ T j e iα (cid:1) , (12)with a new parity-odd coupling parameter ˇ p , see also[30, (6.109)], that controls the contribution of the axialtorsion which is defined asˇ T i := 13 η ijkl T jkl . (13)Another possibility – which, however, was not followed up– would be the additional term odd χ ijαklβ on the right-hand side of eq. (10).Recently, a thorough analysis of the most general lin-ear local constitutive relations in the teleparallel gravityhas been performed in [36], focusing mainly on its ir-reducible decomposition. One can prove that a generalmetric-dependent parity-odd part of the constitutive ten-sor reads odd χ ijαklβ ( g )= √− g κ h β η ijkl g αβ + β η ij [ k [ α e l ] β ] + β (cid:16) e [ i ( α η j ] klβ ) − e [ k ( α η l ] ij β ) (cid:17)i . (14)Accordingly, Mashhoon’s constitutive tensor (11)-(12)encompasses all 6 irreducible parts (principal, skewonand axion, both even and odd parities), and the corre-sponding coupling constants of this irreducible decompo-sition read: β = − , β = − , β = 2 , β = − ˇ p/ , β = − p/ , β = ˇ p/
3. For the complete notational and com-putational details see [36]; note though that there is aconventional overall factor between our and Mashhoon’scoupling constants, and a difference in the definition ofthe torsion. It is particularly noteworthy that the con-stitutive relation in general contains a nontrivial skewonpart which means that such a constitutive law is not re-versible and therefore cannot be derived from a varia-tional principle. For a general introduction to the under-lying premetric framework of electrodynamics and grav-ity see [37–39].Defining the tensors X ijk := 12 X ijkpqr T pqr , (15) χ ijk := 12 χ ij kpqr T pqr , (16)and by switching to holonomic coordinates, we can recast(8) into ˇ H y y y = χ y y y + N y y y . (17)Here we introduced N y y y := − Z σ y x σ y x σ y x K ( x, y ) X x x x d x (18)for the nonlocal part of (8), see also [30, eq. (6.107)].In the rest of this work we are going to focus on thisnonlocality tensor N y y y . III. COVARIANT EXPANSIONS
In the following we make use of a covariant expansiontechnique based on a generalization of Synge’s “worldfunction” σ ( x, y ) [40–42]. Since NLcG is a theory whichis based on a non-Riemannian spacetime, we first need tointroduce the properties of a world function based on au-toparallels in a Riemann-Cartan background. In contrastto a Riemannian spacetime, a Riemann-Cartan space-time is endowed with an asymmetric connection Γ abc ,and there will be differences when it comes to the basicproperties of a world function σ based on autoparallels.The curvature and the torsion are defined w.r.t. thegeneral connection Γ abc as follows: R abcd := 2 ∂ [ a Γ b ] cd + 2Γ [ a | nd Γ b ] cn , (19) T abc := 2Γ [ ab ] c . (20)The symmetric Levi-Civita connection Γ kj i , as well asall other Riemannian quantities, are denoted by an addi-tional overline. For a general tensor A of rank ( n, l ) thecommutator of the covariant derivative thus takes theform:( ∇ a ∇ b − ∇ b ∇ a ) A c ...c n d ...d l = − T abe ∇ e A c ...c k d ...d l + k X i =1 R abec i A c ...e...c k d ...d l − l X j =1 R abd j e A c ...c k d ...e...d l . (21)In addition to the torsion, we define the contortion K kji with the following properties K kji := Γ kj i − Γ kj i , (22) K kji = −
12 ( T kji + T ikj + T ijk ) , (23) T kji = − K [ kj ] i . (24)For a world function σ based on autoparallels, we havethe following basic relations in the case of spacetimeswith asymmetric connections: σ x σ x = σ y σ y = 2 σ, (25) σ x σ x x = σ x , (26) σ x x − σ x x = T x x x ∂ x σ. (27) We denote higher-order covariant derivatives of the worldfunction by σ yx ...y ... := ∇ x . . . ∇ y . . . ( σ y ).For the covariant expansions we need the limiting be-havior of a bitensor B ... ( x, y ) when x approaches thereference point y . This so-called coincidence limit of abitensor B ... ( x, y ) is a tensor[ B ... ] = lim x → y B ... ( x, y ) , (28)at y and will be denoted by square brackets. In partic-ular, for a bitensor B with arbitrary indices at differentpoints (here just denoted by dots), we have the rule [41][ B ... ] ; y = [ B ... ; y ] + [ B ... ; x ] . (29)We collect the following useful identities for the worldfunction σ : [ σ ] = [ σ x ] = [ σ y ] = 0 , (30)[ σ x x ] = [ σ y y ] = g y y , (31)[ σ x y ] = [ σ y x ] = − g y y , (32)[ σ x x x ] + [ σ x x x ] = 0 . (33)Note that up to the second covariant derivative the coin-cidence limits of the world function match those in space-times with symmetric connections. However, at the next(third) order the presence of the torsion leads to[ σ x x x ] = 12 ( T y y y + T y y y + T y y y ) = K y y y , (34)where in the last equality we made use of the contor-tion K . With the help of (29), we can obtain the othercombinations with three indices:[ σ y x x ] = − [ σ y y x ] = [ σ y y y ] = K y y y . (35)At the fourth order we have K y yy K y yy + K y yy K y yy + K y yy K y yy +[ σ x x x x ] + [ σ x x x x ] + [ σ x x x x ] = 0 , (36)and in particular[ σ x y y y ] = − ∇ y ( K y y y + K y y y )+ 13 ∇ y K y y y + 13 ∇ y K y y y + ∇ y K y y y − π y y y y , (37) π y y y y := 13 h K y y y ( K y y y + K y y y ) − K y y y ( K y y y + K yy y ) − K y y y ( K y y y + K yy y ) − K y y y K y y y + K y y y K yy y + K y y y K yy y + R y y y y + R y y y y i . (38)Explicit results for the other index combinations can befound in [43, eqs. (19)-(23)].Finally, let us collect the basic properties of the so-called parallel propagator g yx := e yα e xα , defined interms of a parallely propagated tetrad e yα , which inturn allows for the transport of objects, i.e. V y = g yx V x , V y y = g y x g y x V x x , etc., along an au-toparallel: g y x g xy = δ y y , g x y g yx = δ x x , (39) σ x ∇ x g x y = σ y ∇ y g x y = 0 ,σ x ∇ x g y x = σ y ∇ y g y x = 0 , (40) σ x = − g yx σ y , σ y = − g xy σ x . (41)Note, in particular, the coincidence limits of its deriva-tives [ g x y ] = δ y y , (42)[ g x y ; x ] = [ g x y ; y ] = 0 , (43)[ g x y ; x x ] = − [ g x y ; x y ] = [ g x y ; x x ]= − [ g x y ; y y ] = 12 R y y y y . (44)In the next section we will derive an expanded approx-imate version of the nonlocality tensor. We make use ofthe covariant expansion technique [41, 44] on the basis ofthe autoparallel world function. For a general bitensor B ... with a given index structure, we have the followinggeneral expansion, up to the third order (in powers of σ y ): B y ...y n = A y ...y n + A y ...y n +1 σ y n +1 + 12 A y ...y n +1 y n +2 σ y n +1 σ y n +2 + O (cid:0) σ (cid:1) , (45) A y ...y n := [ B y ...y n ] , (46) A y ...y n +1 := (cid:2) B y ...y n ; y n +1 (cid:3) − A y ...y n ; y n +1 , (47) A y ...y n +2 := (cid:2) B y ...y n ; y n +1 y n +2 (cid:3) − A y ...y n y (cid:2) σ y y n +1 y n +2 (cid:3) − A y ...y n ; y n +1 y n +2 − A y ...y n ( y n +1 ; y n +2 ) . (48)With the help of (45) we are able to iteratively expandany bitensor to any order, provided the coincidence lim-its entering the expansion coefficients can be calculated.We note in passing, that this expansion technique hasalso been applied extensively in the context of the equa-tions of motion of extended test bodies [45–51] and in thegravitational self-force problem [44, 52]. The expansionfor bitensors with mixed index structure can be obtainedfrom transporting the indices in (45) by means of theparallel propagator. IV. CONSTITUTIVE LAW
As was demonstrated in section II, nonlocal gravityis based on an ansatz for the so-called nonlocality ten-sor N y y y , which involves a scalar kernel K ( x, y ) and atensor X x x x . Albeit the form of N y y y given in (18),was declared the “simplest expression” for the nonlocalitytensor in [30], we observe here that a further simplifica-tion can be achieved by performing a covariant expansionof the derivatives of the world function entering (18). Utilizing the general expansion technique from (45)-(48) we have for the derivative of the world functionaround an arbitrary reference world line Y . σ y x = − g y x + g x y [ σ y xy ] σ y + 12 (cid:18) g x y [ σ y xy y ] − g x y g y y [ g y x ; y y ] − g x y [ σ y x ( y ] ; y ) − g x y [ σ y xy ][ σ y y y ] (cid:19) σ y σ y + O (cid:0) σ (cid:1) . (49)With the results for the coincidence limits worked outin the previous section III, we end up with the followingexplicit expansion of the world function derivative up tothe second order: σ y x = − g y x + g x y K y yy σ y + 12 σ y σ y g x y (cid:20) ∇ ( y K | y | y ) y − ∇ y K ( y y ) y − ∇ ( y K y ) yy + K y yy K ( y y ) y − π y ( y y ) y (cid:21) + O (cid:0) σ (cid:1) , (50)with π y ( y y ) y = 13 (cid:20) K y ( y y ′ K y ) y y ′ − K y ( y y ′ K | y ′ | y ) y − K yy y ′ K ( y y ) y ′ − K ( y | y | y ′ K y ) y y ′ + K ( y | yy ′ K y ′ | y ) y + R y ( y y ) y (cid:21) . (51)Inserting (51) into (50) we end up with σ y x = − g y x + g x y K y yy σ y − σ y σ y g x y " R y ( y y ) y + κ y ( y y ) y + O (cid:0) σ (cid:1) , (52)where we collected all contortion terms in the auxiliaryvariable κ y ( y y ) y := K y ( y y ′ K y ) y y ′ − K y ( y y ′ K | y ′ | y ) y − K yy y ′ K ( y y ) y ′ + K ( y | yy ′ K y ′ | y ) y − K ( y | y | y ′ K y ) y y ′ − K ( y y ) y ′ K y ′ yy + ∇ y K ( y y ) y − ∇ ( y K | y | y ) y +3 ∇ ( y K y ) yy . (53) A. Riemann-Cartan spacetime
Plugging in the expansion from (52) into the ansatz forthe nonlocality (18) we end up with: N y y y = Z (cid:26) g y x g y x g y x − σ y ′ h g y x g y x g x y K y ′ yy + g y x g y x g x y K y ′ yy + g y x g y x g x y K y ′ yy i + 16 σ y σ y (cid:20) g x y g y x g y x (cid:0) R y ( y y ) y + κ y ( y y ) y (cid:1) + g x y g y x g y x (cid:0) R y ( y y ) y + κ y ( y y ) y (cid:1) + g x y g y x g y x (cid:0) R y ( y y ) y + κ y ( y y ) y (cid:1) + 6 g y x g x y ′ g x y ′′ K ( y | y ′ y | K y ) y ′′ y + 6 g y x g x y ′ g x y ′′ K ( y | y ′ y | K y ) y ′′ y + 6 g y x g x y ′ g x y ′′ K ( y | y ′ y | K y ) y ′′ y (cid:21) + O (cid:0) σ (cid:1) (cid:27) K ( x, y ) X x x x d x. (54)Different orders in this version of the nonlocality (18) cor-respond to different orders of the approximation in pow-ers of the world function. The expansion (54) clearly ex-hibits the complicated geometrical structure of the orig-inal ansatz (18). The torsion of spacetime, here in theform of the contortion, already enters the picture at thefirst order. This in turn leads to very complicated fieldequations of NLcG. B. Riemannian spacetime
Albeit the latest version of nonlocal gravity describedin [30] uses a Riemann-Cartan spacetime as the geomet-rical setting, our general method also allows for a directspecialization of (18) to a Riemannian background, i.e. N y y y = Z (cid:20) g y x g y x g y x + 16 (cid:18) g y x g y x g x y R y ( y ′ y ′′ ) y + g y x g y x g x y R y ( y ′ y ′′ ) y + g y x g y x g x y R y ( y ′ y ′′ ) y (cid:19) σ y ′ σ y ′′ + O ( σ ) (cid:21) K ( x, y ) X x x x d x. (55)Here R denotes the Riemannian curvature tensor builtfrom the Levi-Civita connection Γ. In contrast to theRiemann-Cartan case – in which the torsion entered atthe first order – the expansion (55) shows that the spe-cialization to a Riemannian background leads to a mildsimplification, in the sense that the geometric terms (i.e.the Riemannian curvature) now enter the nonlocalityansatz only at the second order. TABLE I. Directory of symbols.Symbol ExplanationGeometrical quantities g ab Metric e iα Coframe, tetrad δ ab Kronecker symbol x a , y a Coordinates η abcd Totally antisymm. Levi-Civita tensorΓ iαβ
Lorentz connectionΓ abc
Riemann-Cartan connectionΓ abc
Levi-Civita connection R abcd Curvature T abc Torsionˇ T a Axial torsion K abc Contortion σ ( x, y ) World function g y x Parallel propagatorMiscellaneous L g Gravitational Lagrangianˇ H ij α Gravitational excitation κ Gravitational coupling constant E αi Gauge field energy-momentum T αi Matter energy-momentum N abc Nonlocality tensor K ( x, y ) Causal kernel χ abcdef Constitutive tensor A y ...y n Expansion coefficientˇ p , β , . . . , β Coupling parameters π y y y y , κ y y y y Auxiliary quantitiesOperators ∂ i , “ , ” Partial derivative ∇ i , “ ; ” Covariant derivative“[ . . . ]” Coincidence limit“ ” Riemannian object V. DISCUSSION AND CONCLUSIONS
We have worked out an approximate version of thenonlocality ansatz of NLcG by means of a covariant ex-pansion technique. Our results in the Riemann-Cartan(54), as well as in the Riemannian context (55), pave theway for a refined version of the theory postulated in [30].A natural improvement can be achieved by using just thelowest order in the expansion (54) as a new basic ansatzfor the nonlocality tensor N y y y . Namely, we proposethat the original ansatz (18) should be replaced by N y y y = Z g y x g y x g y x K ( x, y ) X x x x d x. (56)This choice provides an essential development of theNLcG theory since it avoids some of the overwhelminggeometrical complexity of the original ansatz. At thesame time, it is perfectly consistent with all the previousresults of NLcG, in particular, it is important that thenew ansatz (56) is totally compatible with the linearizedsolutions which have been found so far in the context ofNLcG.Furthermore, it is worthwhile to note that the new non-local constitutive law (56) appears to be much more nat-ural from the viewpoint of relativistic multipolar schemes[49, 51] as compared to the original ansatz (18), since itavoids the emergence of derivatives of the world function,which do not have a straightforward interpretation – incontrast to the appearance of the parallel propagator inthe new ansatz (56).With an account of these advantageous properties, onecan expect that our new constitutive law would eventu- ally lead to an exact solution of NLcG, although evenwith the simplified ansatz for the nonlocality, the solu-tion of the full NLcG field equations still appears to bea daunting task. ACKNOWLEDGMENTS
We are grateful to Bahram Mashhoon (Tehran) forhelpful remarks in the context of nonlocal gravity. Fur-thermore we thank Yakov Itin (Jerusalem) and Jens Boos(Alberta) for their comments. This work was supportedby the Deutsche Forschungsgemeinschaft (DFG) throughthe Grant No. PU 461/1-1 (D.P.). The work of Y.N.O.was partially supported by PIER (“Partnership for In-novation, Education and Research” between DESY andUniversit¨at Hamburg) and by the Russian Foundationfor Basic Research (Grant No. 18-02-40056-mega).
Appendix A: Notations and conventions
Table I contains a brief overview of the symbols usedthroughout the work. [1] B. Mashhoon. Nonlocal theory of accelerated observers.
Phys. Lett. A , 47:4498, 1993.[2] C. Chicone and B. Mashhoon. Acceleration-induced non-locality: kinetic memory versus dynamic memory.
An-nalen Phys. , 11:309, 2002.[3] C. Chicone and B. Mashhoon. Acceleration-induced non-locality: Uniqueness of the kernel.
Phys. Lett. A , 298:229,2002.[4] B. Mashhoon. Vacuum electrodynamics of acceleratedsystems: Nonlocal Maxwell’s equations.
Annalen Phys. ,12:586, 2003.[5] B. Mashhoon. Nonlocal electrodynamics of linearly ac-celerated systems.
Phys. Rev. A , 70:062103, 2004.[6] B. Mashhoon. Nonlocality of accelerated systems.
Int.J. Mod. Phys. D , 14:171, 2005.[7] B. Mashhoon. Nonlocal electrodynamics of rotating sys-tems.
Phys. Rev. A , 72:052105, 2005.[8] B. Mashhoon. Toward a nonlocal theory of gravitation.
Annalen Phys. , 16:57, 2007.[9] B. Mashhoon. Nonlocal Dirac equation for acceleratedobservers.
Phys. Rev. A , 75:042112, 2007.[10] B. Mashhoon. Nonlocal electrodynamics of acceleratedsystems.
Phys. Lett. A , 366:545, 2007.[11] C. Chicone and B. Mashhoon. Nonlocal Lagrangians foraccelerated systems.
Annalen Phys. , 16:811, 2007.[12] B. Mashhoon. Nonlocal Special Relativity.
AnnalenPhys. , 17:705, 2008.[13] F. W. Hehl and B. Mashhoon. Nonlocal gravity simulatesdark matter.
Phys. Lett. B , 673:279, 2009.[14] F. W. Hehl and B. Mashhoon. A formal framework fora nonlocal generalization of Einstein’s theory of gravita-tion.
Phys. Rev. D , 79:064028, 2009. [15] B. Mashhoon. Nonlocal transformations for acceleratedobservers.
Annalen Phys. , 18:640, 2009.[16] H.-J. Blome, C. Chicone, F. W. Hehl, and B. Mashhoon.Nonlocal modification of Newtonian gravity.
Phys. Rev.D , 81:065020, 2010.[17] B. Mashhoon. Necessity of acceleration-induced nonlo-cality.
Annalen Phys. , 523:226, 2011.[18] B. Mashhoon. Nonlocal Gravity.
Cosmology and Grav-itation, edited by M. Novello and S. E. Perez Begliaffa(Cambridge Scientific Publishers, UK, 2011) , page 1,2011.[19] C. Chicone and B. Mashhoon. Nonlocal gravity: Modi-fied Poisson’s equation.
J. Math. Phys. , 53:042501, 2012.[20] B. Mashhoon. Observers in spacetime and nonlocality.
Annalen Phys. , 525:235, 2013.[21] B. Mashhoon. Nonlocal gravity: Damping of linearizedgravitational waves.
Class. Quant. Grav. , 30:155008,2013.[22] S. Rahvar and B. Mashhoon. Observational tests ofnonlocal gravity: Galaxy rotation curves and clusters ofgalaxies.
Phys. Rev. D , 89:104011, 2014.[23] B. Mashhoon. Nonlocal gravity: The general linear ap-proximation.
Phys. Rev. D , 90:124031, 2014.[24] B. Mashhoon. Nonlocal General Relativity.
Galaxies , 3:1, 2015.[25] C. Chicone and B. Mashhoon. Nonlocal gravity in thesolar system.
Class. Quantum Grav. , 33:075005, 2016.[26] C. Chicone and B. Mashhoon. Nonlocal Newtonian cos-mology.
J. Math. Phys. , 57:072501, 2016.[27] B. Mashhoon. Virial theorem in nonlocal Newtoniangravity.
Universe , 2:9, 2016.[28] D. Bini and B. Mashhoon. Nonlocal gravity: Conformallyflat spacetimes.
Int. J. Geom. Methods Mod. Phys. , 13:
Astrophys. J. , 872:6, 2019.[30] B. Mashhoon.
Nonlocal gravity . Oxford University Press,Oxford, 2017.[31] F. W. Hehl, J. Nitsch, and P. v.d. Heyde. Gravitation andPoincar´e gauge field theory with quadratic Lagrangian. ”General Relativity and Gravitation. One hundred yearsafter the birth of Albert Einstein.”, A. Held (ed.), PlenumPress, New York , 1:329, 1980.[32] F. W. Hehl. Four lectures on Poincar´e gauge field theory. , page 5, 1980.[33] R. Aldrovandi and J. G. Pereira.
Teleparallel Gravity:An Introduction . Springer, Dordrecht, 2013.[34] M. Blagojevi´c and F. W. Hehl (eds.).
Gauge theories ofgravitation. A reader with commentaries . Imperial Col-lege Press, London, 2013.[35] Y. M. Cho. Einstein Lagrangian as the translationalYang-Mills Lagrangian.
Phys. Rev. D , 14:2521, 1976.[36] Y. Itin, Yu. N. Obukhov, J. Boos, and F. W. Hehl. Pre-metric teleparallel theory of gravity and its local and lin-ear constitutive law.
Eur. Phys. J. C , 78:907, 2018.[37] F. W. Hehl and Yu. N. Obukhov. How does the elec-tromagnetic field couple to gravity, in particular to met-ric, nonmetricity, torsion, and curvature?
In: Gyros,Clocks, Interferometers . . . : Testing Relativistic Gravityin Space. C. L¨ammerzahl et al. (eds.) Lecture Notes inPhysics, Springer (Berlin) , 562:479, 2001.[38] U. Muench, F. W. Hehl, and B. Mashhoon. Accelerationinduced nonlocal electrodynamics in Minkowski space-time.
Phys. Lett. A , 271:8, 2000.[39] F. W. Hehl and Yu. N. Obukhov.
Foundations of classicalelectrodynamics . Birkh¨auser, Boston, 2003.[40] H. S. Ruse. Some theorems in the tensor calculus.
Proc.Lond. Math. Soc. , 31:225, 1930. [41] J. L. Synge.
Relativity: The general theory . North-Holland, Amsterdam, 1960.[42] B. S. DeWitt and R. W. Brehme. Radiation damping ina gravitational field.
Ann. Phys (NY) , 9:220, 1960.[43] D. Puetzfeld and Yu. N. Obukhov. Deviation equationin Riemann-Cartan spacetime.
Phys. Rev. D , 97:104069,2018.[44] E. Poisson, A. Pound, and I. Vega. The motion of pointparticles in curved spacetime.
Living Reviews in Relativ-ity , 14(7), 2011.[45] W. G. Dixon. A covariant multipole formalism for ex-tended test bodies in General Relativity.
Nuovo Cimento ,34:317, 1964.[46] W. G. Dixon. Dynamics of extended bodies in GeneralRelativity. III. Equations of motion.
Phil. Trans. R. Soc.Lond. A , 277:59, 1974.[47] W. G. Dixon. Extended bodies in General Relativity:Their description and motion.
Proc. Int. School of Phys.Enrico Fermi LXVII, Ed. J. Ehlers, North Holland, Am-sterdam , page 156, 1979.[48] W. G. Dixon. Mathisson’s new mechanics: Its aims andrealisation.
Acta Phys. Pol. B Proc. Suppl. , 1:27, 2008.[49] D. Puetzfeld and Yu. N. Obukhov. Equations of motionin metric-affine gravity: a covariant unified framework.
Phys. Rev. D , 90:084034, 2014.[50] W. G. Dixon. The New Mechanics of Myron Mathissonand its subsequent development. ”Equations of Motionin Relativistic Gravity”, D. Puetzfeld et. al. (eds.), Fun-damental theories of Physics, Springer , 179:1, 2015.[51] Yu. N. Obukhov and D. Puetzfeld. Multipolar testbody equations of motion in generalized gravity theo-ries. ”Equations of Motion in Relativistic Gravity”, D.Puetzfeld et. al. (eds.), Fundamental theories of Physics,Springer , 179:67, 2015.[52] A. C. Ottewill and B. Wardell. Transport equation ap-proach to calculations of Hadamard Green functions andnon-coincident DeWitt coefficients.