More on the Geometric Approach for Gravitation Coupled with Electromagnetism within a Riemann-Cartan Vacuum Spacetime Lalaonirina Rakotomanana

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Lalaonirina Rakotomanana. More on the Geometric Approach for Gravitation Coupled with Electro- magnetism within a Riemann-Cartan Vacuum Spacetime. 2021. ￿hal-03216920￿

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More on the Geometric Approach for Gravitation Coupled with Electromagnetism within a Riemann-Cartan Vacuum Spacetime

Lalaonirina R. Rakotomananaa,1 1Univ Rennes, IRMAR (Institut de Recherche Mathematique´ de Rennes), UMR 6625, F-3500 Rennes, France

Received: / Accepted: date

Abstract The goal of this paper is to link the geometric necessary to go further into the unification of field theo- variables of four-dimensional spacetime with electromag- ries e.g. Hammond [1987]. Paraphrasing Tonnelat, there netic and gravitational fields. For that purpose, assuming remains up to now a need of unified theory joining gravi- a Hilbert-Einstein action with Yang-Mills action, we derive tation and electromagnetism whose basic equations repre- both the gravitation and the electromagnetic equation within sent the background required to design geometrical struc- a Riemann-Cartan spacetime where curvature and torsion ture of the spacetime Tonnelat [1955], and it is prior to are present. We thus show that the gravitational and elec- any other unification attempts. Since the pioneering work tromagnetic fields are respectively identified as geometric of Weyl one hundred years ago, there remains among the objects of such a spacetime, namely the curvature for gravi- physics community a great interest to couple Maxwell elec- tation which is a classical result, and the torsion as electro- trodynamics to Einstein gravitation e.g. Weyl [1929]. The magnetic fields. Moreover, the torsion tensor of the space- geometric approach for continuum physics includes at least time is related to the so-called flux spin angular momentum two steps : development of the geometric base by means of of electromagnetic waves propagating within the spacetime. manifold in the framework of differential geometry, and then definition of geometric objects and their identification of Keywords Curvature, torsion gravitation, electromag- · experimental quantities. As earlier as 1921, such approach netism Riemann-Cartan manifold · was already adopted by Eddington to extend the Weyl’s ap- proach by using affine connection with non zero torsion to 1 Introduction develop gravitation and electromagnetism theory within a unique framework e.g. Eddington [1921]. On the one hand, Deflection of light measured as earlier as 1919 was among the development of the Einstein gravitation theory lies upon the first experimental measurements to point out the interac- the Riemann geometry where the curvature of the space- tion between gravitation and electromagnetic waves Dyson time is identified with the gravitation. More precisely, the & al. [1920]. An important fact usually assumed as evi- geodesic equation shows that inertia and gravitation are uni- dent nowadays but that should not be forgotten is that both fied in a unique framework of General Relativity. On the the gravitation and the electromagnetism either separately at other hand, the Special Relativity unified the electricity and their own or in interaction occur in a spacetime, and then il- magnetism. Early, Weyl clearly thought that gravitation and lustrate the concept of field as action at distance phenomenon electromagnetism could be reduced to the geometry of space- MacMullin [2002]. Unification of gravitation and electro- time. Starting with Riemannian geometry, he slightly mod- magnetism draw back to at least one hundred years, as re- ified the metric compatibility equation to —a gbl = sa gbl by introducing a 1-form field s = 0, expecting that this 1- lated in the review paper by Goenner e.g. Goenner [2004] a 6 where the most active period of unifying the two domains form would be an electromagnetic potential. However, re- was between 1910 and 1933. Since then, it is however recog- cent studies highlight strong physics objection to this as- nized that the unification of the gravitation and the electro- sumption, namely on the definition of the proper time and magnetism within the same geometric framework remains geodesic length in the spacetime resulting from the introduc- tion of this 1-form e.g. Romero et al. [2019]. In general rel- ae-mail: [email protected] ativity, the metric is no longer a Minkowskian since g = g(x) 2 depends on the coordinates of the spacetime. In an attempt no contribution from torsion in Maxwell equations. When a to formulate an unified theory of gravitation and electromag- strong magnetic field coexists with matter distribution, there netism, following the idea of Ferraris and Kijowski, Chru- is however a possibility to induce spin polarization of in- sciel proposed a Lagrangian function having depending only dividual particles composing the matter assumed as contin- F l upon the skew-symmetric tensor µn := ¬µnl and on the uum e.g. Prasanna [1975a]. Some authors have even sug- symmetric part of the Ricci curvature tensor with nonzero gested that torsion play a keyrole in electromagnetism when torsion e.g. Chrusciel [1984]. The use of the curvature of considering electromagnetic field within twisted spacetime the spacetime as the only one variable to sketch electro- e.g. Hammond [1989]. They propose that the electromag- magnetism and gravitation might be questionable. Never- netic potential is represented by the torsion vector Aa := theless, it could be checked that the two-covariant skew- b ¿a = ¿ab. The influence of torsion tensor as cosmic dislo- symmetric tensor Fµn satisfies the first set of Maxwell equa- cation (that is a singularity of the curvature tensor) was in- tions (Bianchi equations), and thus could be suggested as vestigated in e.g. Dias & Moraes [2005], or some material the combined electromagnetic field e.g. Hammond [1989], defects such as screw dislocations Fumeron et al. [2015], or whereas the symmetric tensor obtained from Ricci curva- fluids with spin density e.g. Schutzhold et al. [2002]. A con- ture capture the gravitational fields. Implicitly the model venient method for deriving constitutive laws and conserva- proposed by Chrusciel suggests that the origin of the elec- tion laws from a Lagrangian density lies on the concept of tromagnetism comes from the skew-symmetric part of the variation of an action, namely the Lagrangian variation and spacetime connection, and then of the torsion tensor. Recent the Eulerian variation (Poincare´ invariance) e.g. Utiyama works have shown that gravitation and electromagnetism could [1956]. Covariance requires the use of metric, torsion, and be reduced to geometric object of the spacetime, gravitation curvature as arguments of Lagrangian function for a second to curvature and electromagnetism to contortion tensor e.g. gradient continuum e.g. Antonio & Rakotomanana [2011]. Fernando et al. [2012], or torsion tensor Hammond [2018], For the present paper, we are interested in the search of link Rakotomanana [2018]. In Fernando et al. [2012], the au- between the electromagnetic fields and the gradient contin- thors assumed a particular Riemann-Cartan structure where uum geometric variables developed in e.g. Rakotomanana the contortion tensor, is proportionnal to the product of the [2003] but extended to spacetime e.g. Rakotomanana [2019]. Faraday strength tensor with the electromagnetic potential. Electromagnetic waves, including light wave propagation, It seems however necessary to confirm or not this interesting are described by Maxwell equations within Minkowskian, assumption. Riemannian or Riemann-Cartan spacetime. We consider in Historically, the interaction of relative gravitation and this work some elements of the theory of interaction be- electromagnetism was often considered in a curved Rieman- tween gravitation and electromagnetism respectively based nian spacetime e.g. Fernandez-Nunez & Bulashenko [2016]. on the classical Hilbert-Einstein action Hilbert [1915] and It is usually assumed that the electromagnetic field is of the the Yang-Mills action Yang & Mills [1954]. For the sake order of small perturbation of the spacetime metric. Only the of the clarity, we focus on the gravitation and electromag- influence of the metric on electromagnetic field is mostly ac- netism within spacetime although results are easily extended counted for, not the converse. Second, the influence of the to other continuum media at least from a conceptual point of Riemann-Cartan geometry on the electromagnetic field is view. Indeed, from a philosophical point of view, the analy- not so easy. The hope to relate electromagnetism to affine sis and then the physics of fields remains a most passionat- connection of spacetime draw back to Einstein himself, a ing domain of physics e.g. MacMullin [2002]. The appar- long debate among others between Einstein and Reichen- ent simplicity hides a deep concept surrounding action-at- bach relates the fact that Reichenbach showed that both the distance. The paper is organized as follows: Section 2 is de- metric g and the affine connection — of the spacetime, which voted to basic geometrical concepts of spacetimes. Section 3 is independent on the metric g, could be derived entirely deals with the classical electromagnetism within Minkowski i from tetrads Fa and their partial derivatives with respect spacetime. Il also presents the combined gravitation theory to coordinates Reichenbach [1929] and they are respec- with electromagnetism in a Riemann curved spacetime. In tively related to gravitation and electromagnetism. Implic- section 4, focus is on the geometric development of grav- itly, these results suggested that metric, torsion, and curva- itation and electromagnetism deduced from the additional ture of the spacetime constitute geometric objects of gravi- action including Hilbert-Einstein and Yang-Mills contribu- tation and electromagnetism. Although the relation between tions. Section 5 considers the extension of worldlline (au- gravitation and spacetime curvature was clearly stated by toparallel) deviation in presence of torsion and curvature. Einstein’s theory, the link between electromagnetism and Last section gives some concluding remarks. other geometric variables remains not clear and evident. A free electromagnetic field is suggested to not produce torsion e.g. Hehl et al. [1976], and there is in principle 3

2 Basics on spacetime geometry [1996]. We consider the spacetime (or space) endowed with the metric g which may depend or not of the coordinates x. Space and time may be considered as the most fundamental For gravitation theory, associated to this metric can be de- concepts of physics and even most of natural sciences. New- fined connection —, with zero torsion ¿ 0 but nonzero ⌘ ton (1687) assumed that an absolute space and time (asymp- curvature ¬ = 0. However, to begin with, it is usual to start 6 totic and idealized) exists prior to any other concept. The with the Minkowskian flat spacetime with the metric gˆ with theory of special relativity is based on two assumptions : (1) zero torsion, and zero curvature. Then we consider the ac- the laws of physics are the same for all inertial observers; tual spacetime metric g := gˆ + h, where h(x) is a pertur- and (2) the speed of light is constant and the same in a vac- bation. With the metric and the curvature, Einstein built the uum for all inertial observers independent of their motion theory of general relativistic gravitation. The spacetime may relative to the source of the light. Theory of relative gravita- also have non symmetric connection but compatible with the tion lies on two postulates: (1) the relativity principle saying metric. that the local physics is governed by the theory of special From the differential geometry point of view, it is then relativity; and (2) the equivalence principle assuming that it clear that three tensor variables constitute the underlying is impossible to distinguish gravity and acceleration locally. structure of the spacetime : the metric, the torsion and cur- vature. The metric compatibility condition is an essential property of a spacetime continuum to survive as a spacetime 2.1 Affinely connected spacetime continuum after non holonomic pertubation h e.g. Verc¸yn [1990]. By adopting the spacetime model with affinely con- Mikkowski spacetime allows us to describe both particle nected manifold, we implicitely consider space of a aggre- mechanics and electromagnetism. It constitutes the funda- gate of microcosms Gonseth [1926] where connection links mental geometry of special relativistic mechanics (without each microcosm with its neighbour. Relative motions of mi- gravitation) and electrodynamics. Governing equations are crocosms may induce discontinuity of relative displacements derived in the local coordinates of Minkowski spacetime M and rotations e.g. Rakotomanana [1997], called translational endowed with the metric denotedg ˆµn := +1, 1, 1, 1 . and rotational dislocations. The non compatibility —g = 0 { µ } 6 Coordinates within the spacetime manifold are denoted x := between the metric and the connection leads to another more 0 1 2 3 (x = ct,x ,x ,x ) where c is the light speed in a Minkowski extended version of dislocations, called Somigliana disloca- spacetime, considered as vacuum. More generally, space- tions, which is a continuum mechanics version of the Weyl time is described as a 4-dimensional manifold affinely con- theory Weyl [1929]. The geometric background associated nected where any event can be described by coordinates to Somigliana field of dislocation is the the Weyl manifold 0 1 2 3 4 (x ,x ,x ,x ) of R , where index 0 stands for time e.g. Rako- where additional primal variables are the non-metricity ten- tomanana [2019]. Namely, from Galilean view, space is sor Q := —g. A Somigliana dislocation is idealized in terms a tridimensional Euclidean manifold and time is a one di- of a closed volume V of ”added / substracted matter” located mensional manifold, they are are assumed to exist indepen- within a continuum. After cutting the body and separating dently each other. In the Minkowskian point of view, space- two opposite faces, the operation is represented by a small time is a four-dimensional manifold where time is coupled displacement field b(x), the empty space is then filled with with space. Presence and motion of a structureless particle ”added matter” (or ”substracted matter” if the two faces pen- are assumed to not influence the spacetime geometry. In the etrate each other after the cutting). presence of matter, an initially flat spacetime is bent. This results into a curved spacetime which can be understood as gravity. In a more extended point of view, affinely connected Cartan parallelogram Figure (1) may be better interpreted manifold constitutes the main geometry background for ana- with the concept of Cartan parallelogram. Say a scalar field lyzing the spacetime structure with a more general purpose. q and a vector field w within the spacetime. Consider two On an affinely connected manifold endowed with a met- vectors f1 and f2 at any point M of M , they define two paths ric g and an independent connection —, two tensors are asso- of length e1 and e2 that may be considered as two adja- ciated to this connection (which is not a tensor), the torsion cent sides of an infinitesimal parallelogram. Non zero tor- tensor ¿ and the curvature tensor ¬ defined by the follow- sion and curvature fields induce the following relationships ing relations respectively e.g. Nakahara [1996]: Rakotomanana [1997]:

¿(u,v) := —uv —vu [u,v] (1) lim (q 0 q 00)/e1e2 = ¿(f1,f2)[q] ¬(u,v,w) := —u—vw —v—uw —[u,v]w (e ,e ) 0 ⇢ 1 2 ! g g g 8 lim [f (w0) f (w”)]/e1e2 = ¬(f1,f2,w,f ) (2) where [,] denotes the Lie-Jacobi bracket, and (u,v,w) an > (e1,e2) 0 > ! arbitrary triplet of vectors on the manifold e.g. Nakahara < fg — w ¿(f1,f2) > :> 4

ever the general form is far from tractable. It is worth to in- l troduce the Ricci curvature tensor ¬ab := ¬lab. For com- patible connection, say —g 0, the Ricci curvature tensor ⌘ is symmetric. Curvature of a three dimensional manifold is uniquely determined by the Ricci tensor. The scalar curva- ab ture is defined by the contraction ¬ := g ¬ab. For the metric-affine gravity theory, the Lagrangian density takes the form of L (g,¬), owing that metric and curvature are in- dependent variables. In this paper we do not consider quan- tum physics which describes phenomena physics at the small- Fig. 1: Riemann-Cartan spacetime as a set of microcosms. The space- est scales of energy levels of atoms and subatomic particles. time M is viewed as connected chunks of micro-spacetimes having locally their own metric, torsion and curvature, and where relative dis- We rather consider classical physics in the sense where it placements and rotations are allowed. The unusual feature not encoun- describes phenomena at macroscopic and microscopic scale tered before in Minkowski spacetime, namely a closure failure of par- levels, namely gravitation and electromagnetism. allelograms, implies the non vanishing of torsion and curvature.

3 Gravitation and electromagnetism within curved where q (x), and w(x) are respectively a scalar field and spacetime a vector field on M . We notice q 0 := q (M0), and q”:= q (M”); and w0 := w(M0), and w”:= w(M”). The system We remind in this section the covariant form of the laws of of equations (2) are the continuum extension of the discrete classical electromagnetism in spacetime which are compat- dislocation loop induced by the discontinuity of the displace- ible with gravitation. It should be stressed that originally, ment field within spacetime composed with microcosms. Fun- the Newton absolute space, and therefore Minkowski space- damentally, the possibility of relative motion between two time, enters implicitly into the general theory of gravitation neighboured microcosms induces a non zero torsion and cur- and electromagnetism. We do not enter into the old debate vature. whether vacuum space is full of substance or empty. We merely proceed step by step by considering a gradual com- plexity of the spacetime from Minkowski, Riemann and then 2.2 Modelling continuum physics by Lagrangian Riemann-Cartan geometry.

Action principles are widely used to express the laws of physics, namely those of general relativity and electromag- 3.1 Basic electromagnetism in a Minkowski spacetime netism. Action is a functional basically defined on the space- The Minkowski flat spacetime, say g = uniform, ¿ 0 and time. The Einstein-Cartan theory is the extension of the rel- ⌘ ¬ 0, of special relativistic physics constitutes the right ativistic gravitation theory, allowing the spacetime to have ⌘ nonzero torsion, it was suggested by Cartan in 1922. The spacetime model for this purpose. Electromagnetism theory application of the covariance theorem states that any La- is built upon electric field E, and magnetic field B. Both of grangian function L (g,—,—) should be written as L (g,¿,¬) them depend on the space coordinate and the time in the to be diffeomophism invariant Antonio & Rakotomanana general case. The general form of the Maxwell equations is [2011]. When the connection is that of Levi-Civita1, among intimately linked to the geometry of the spacetime charac- the arguments of the Lagrangian function we should con- terized by the Minkowski metric, they constitute the funda- sider are second order derivatives of the metric as additional mental basis of classical electrodynamics. Before going into variable. Let a spacetime continuum modeled by a metric- the derivation of Maxwell equation, it is worth to remind the affine manifold endowed with a connection compatible with notion of proper time in the framework of special relativ- the metric. (a) L (g,¿ = 0,R = 0) corresponds to an strain ity. Consider a body / or a referential frame moving with a energy function in elastic spacetime; (b) L (g,¿) is associ- uniform velocity v with respect to M . The proper time is given by : dt := 1 (v2/c2)dt, where the proper time t ated to an elastic spacetime with dislocation singularity. (c) along a timelike world line in the spacetime M is the laps of L (g,¿,R) is associated to an elastic spacetime with dis- p location and disclination singularity. The elasticity refers to time measured by a clock following that line. As such, the the metric as argument of the Lagrangian function. How- Maxwell equations are the conservation laws, and then by accounting for constitutive laws, we obtain the partial differ- ential equations where the electric and magnetic intensities 1The Euclidean connection derived from the metric tensor of a spacetime body was mostly the connection used in continuum mechan- are the unknowns. By using the form notation in the four- ics for over two centuries, e.g. Rakotomanana [2003]. dimensional spacetime, the electromagnetic strength or also 5

Faraday tensor is a 2-form: By introducing the definitions (4) and (6) the three dimen- 1 0 2 0 3 0 sional formulation of the Lagrangian density function re- F = E1dx dx + E2dx dx + E3dx dx ^ ^ ^ duces to : L = 1 (D E B H). From relations (4), (6), and + B1dx2 dx3 + B2dx3 dx1 + B3dx1 dx2 (3) 2 · · ^ ^ ^ (7), it is easy to derive the electromagnetic wave equation in from which we can easily check the correspondence of the the Minkowski spacetime e.g. Rakotomanana [2019]: components Fµn with the components of tri-dimensional gnb—ˆ —ˆ Aµ = 0 (8) space vector fields E and B. The covariant formulation of n b electromagnetism theory in the Minkowski spacetime M is by accounting for the Lorenz gauge condition —ˆ An 0. n ⌘ classically obtained by considering the Faraday tensor (3) The equation (8) governs the electromagnetic wave travel- from the four-dimensional electromagnetic potential A = ling at the speed c within Minkowski spacetime. (Aµ ): Remark 2 The covariance of the Maxwell 3D equations with 0 E1 E2 E3 respect to rotations, space reflection, time reversal, and charge 3 2 ˆ ˆ E1 0 B B conjugation (modification of positive charge to negative one) Fµn := —µ An —n Aµ , Fµn = 2 3 1 3 (4) E2 B 0 B may be checked by means of a four-dimensional covariance 6 2 1 7 6 E3 B B 0 7 analysis with respect to Lorentz group of transformations. 4 5 where —ˆ is the (flat) connection of the Minkowski space- In presence of gravitation phenomena, the flat Minkowski time, and where the combined electromagnetic field (E,B) spacetime becomes curved, metric components vary with lo- do not transform as 3-vectors but as the six components of cation in the spacetime. Gravitation is related to the non null the skew-symmetric tensor Fµn. We conform here to the curvature, ¬ = 0, within the spacetime. In the present pa- 6 convention in e.g. Hehl [2008], Obukhov [2008]. In this per, electromagnetic fields equations are formulated within way the skew symmetric tensor Fµn is chosen as primal vari- curved spacetime of general relativity. ables of the theory. Let us now define the dual variable Hµn constructed from the electric displacement D and the mag- netic intensity H. Independently on the constitutive laws, the 3.2 Gravitation within curved spacetime : Einstein fields classical electromagnetism theory considers the electromag- equation netic excitation as a two-form Hµn : Although special relativity is mainly based on the Minkowski H := H dx1 dx0 H dx2 dx0 H dx3 dx0 1 ^ 2 ^ 3 ^ spacetime, which is a flat manifold, relative gravitation lies + D3dx1 dx2 + D1dx2 dx3 + D1dx2 dx3 (5) upon the Riemannian manifold (or more precisely pseudo- ^ ^ ^ in the same way as the definition of the electromagnetic Riemannian). Let us consider a torsionless but curved space- strength (3). The variable we are interested in is in fact the time (M ,g,—). The spacetime structure is summarized by dual variable Hµn in order to be able to link it with the pri- the Riemannian spacetime structure with the metric g, the torsion ¿, and the curvature ¬: mal variable Fab. From (5), we easily obtain the two con- travariant components as: —g = 0, ¿ 0, ¬ = 0 (9) ⌘ 6 0 D1 D2 D3 The Lagrangian function L usually depends on the metric, D1 0 H H the torsion and the curvature in the general case. The rel- Hµn 2 3 2 3 = 2 (6) evant connection for Einstein relativistic gravitation is the D H3 0 H1 6 3 7 Levi-Civita connection, with a non zero curvature describ- 6 D H2 H1 0 7 6 7 ( , ) 4 5 ing the gravitational field, with the Lagrangian L g ¬ . Let Remark 1 For Minkowski spacetime, the relations between consider a Lagrangian corresponding to the Einstein-Hilbert fields are particular D = e0E, and B = µ0H. We notice that action of classical relativistic gravitation Hilbert [1915], that the electric permittivity and the magnetic permeability and with a retrospective analysis in Brading & Ryckman are related by e0µ0 = 1 when we choose coordinate system [2008]:2 with x0 := ct. It is a particular linear constitutive law. Co- variant formulation of constitutive laws may be derived by SG :=(1/2c) R wn (10) means of the electromagnetic Lagrangian, and the covariant Z Maxwell equations recast in 4-dimension accordingly: where R is the scalar curvature and wn a volume-form e.g. 0 3 Nakahara [1996], for example wn := pDetg dx dx 1 ^···^ L := FµnF , —ˆ Fµn = 0 (7) 4 µn µ 2By using a variational procedure, Hilbert preceeded Einstein with µn the publication of the gravitational field equations of relative gravita- where F denotes the electromagnetic tensor including the tion by five days, however both authors arrived at same field equations electric displacement field and the magnetic field. along very different paths. 6 when the spacetime is endowed with the metric g. The con- spacetime metric. The deformation does not stay only near stant factor c := 8pG/c4 is introduced to reproduce the clas- the massive body. The field equations of Einstein suggested sical mechanics of Newton when some matter is moving that the deformation can propagate throughout the entire within this Minkowski spacetime reduced to an Euclidian spacetime. The main difference compared to seismic waves space and a time interval e.g. Ryder [2009]. is that gravitational waves can travel in empty space at the First we introduce the metric variation g g + light speed. This is typical example where the gauge invari- ab ! ab dgab (corresponding to the Eulerian variation of the metric ance is useful for deriving the wave equations of relativistic at a fixed point of the spacetime). A straightforward calculus gravitation. The method is based on linear perturbation of gives the variation of the connection and the Ricci curvature the metric, the Ricci curvature tensor, and the Einstein ten- (we omit bar overline for connection and curvature for sim- sor. plicity): Consider a weak field gravitation where metric is close to Minkowski metric g gˆ +2e , with e << 13. dGg =(1/2) ggl — dg + — dg — dg (11) ab ' ab ab k abk ab b al a lb l ab It is also usual to assume that at large distance from sources, d¬ = — (dGl ) — (dGl ) (12) the spacetime becomes Minkowskian e.g. Dixon [1975]. ab l ba b la Theory of special gravitation allows us to obtain the con- where the covariant derivative is related to the unperturbed servation laws associated to the linearized part of Hilbert- (metric compatible) connection. The equation (12) is known Einstein Lagrangian L (g ,∂ ∂ g ) :=(1/2c) R with: as Palatini identity, showing that for torsionless spacetime ab g l ab the variation of the Ricci tensor may be transferred to the l ¬ = gˆls ∂ ∂ e ∂ ∂ e + ∂ ∂ e ∂ ∂ e boundary condition terms via the divergence theorem. The abµ µ a sb µ b sa s b µa s a µb principle of least action dS 0 for arbitrary variation of (14) G ⌘ the metric gives the Einstein field equation of the general relativity e.g. Lovelock [1969]: The equation (13) governs the dynamics of Riemann space- time in relativistic gravitation. the unknowns are the metric G := ¬ (1/2) R g = 0 (13) components. In the presence of moving bodies, the prob- ab ab ab lem in relativistic gravitation is to solve, at the same time, which is the field equation within a Riemann spacetime, the the gravitation fields induced by the bodies and the motion Euler-Lagrange equations associated to the Einstein-Hilbert of the bodies e.g. Papapetrou [1951]. From the previous µ action. Metric components gab(x ) are the unknown vari- equation, we deduce the linear scalar curvature e.g. Rako- ables in this field equation. To this end, given a curvature tomanana [2019]: field ¬ on a metric manifold M , determination of the 10 a n an metric components g needs integration of system of 20 sec- R = 2[∂ ∂ ean gˆ ∂a ∂n (Tre)] (15) ond order partial differential equations. For this to be possi- The linearized Einstein’s equation of gravity G = 0 is ob- ble, additional integrability must be satisfied for third-order bµ tained accordingly after some rearrangements to highlight derivatives of metric components assuming that they are C3. the D’Alembertian operator: These are the Bianchi identities e.g. Rakotomanana [2003]. an n n Gbµ = gˆ ∂a ∂n ebµ (Tre)gˆbµ +∂ ∂µ ebn + ∂b ∂ enµ Remark 3 First, the derivation of the field equation of gen- eral relativity, is obtained accounting that the variation of D⇥0Alembertian ⇤ g a n ∂b ∂µ (Tre) (∂ ∂ ean)gˆbµ (16) the connection dGab is shifted to the boundary by means | {z } of the divergence operator. Second, the Einstein’s gravita- Despite the fact that we have ten equations for ten unknowns, tional equations are invariant under the group of (passive) it is not yet possible, at this step, to solve them because we diffeomorphims (covariance), and not only for the group of have first to define a coordinate system. Indeed, the decom- Lorentz transformations. Under an arbitrary change of coor- position of the metric into two terms as the flat Minkowskian a a µ dinatex ˜ = x˜ (x ), the same shape of equations is obtained metric h , and a perturbation 2e is not unique. Depend- ˜ bµ bµ to give exactly Gab = 0. ing on the choice of a coordinate system (xl ) the shape of the perturbation may be different. For this purpose, let intro- duce the Lorentz gauge (also called Einstein gauge, Hilbert 3.3 Linearized gravitation equations : Gravitational waves 3Application of the Lagrangian formalism in general relativity may In this subsection, we derive the basic equations due to linear induce some difficulties, because physical quantities in classical or spe- perturbation of the Minkowskian metric as for linear grav- cial relativity framework require fixed geometric background (New- ity phenomenon. Gravity is the consequence of how mas- tonian or Minkowskian spacetime). Indeed, for general relativity the spacetime geometry is itself a dynamical object. Separation of the met- sive object deforms the spacetime. Near any massive body, ric into two parts that may be respectively assigned to inertia and grav- the spacetime becomes curved following the change of the ity is an affair of taste e.g. Shen & Moritz [1996]. 7 gauge, de Donder gauge or Fock gauge). Starting from the where no matter action is considered, and where the Faraday l coefficients of connection Gµn, the Lorentz gauge imposes tensor and the scalar curvature are defined by the relation- that the skew-symmetry part of the connection is equal to ships: µn l zero g Gµn 0, together with its linearized version: µn ⌘ Fµn := —µ An —n Aµ , R := g ¬µn (20) µn rl gˆ gˆ ∂µ ern + ∂n eµr ∂r eµn µn lr µ l where the Faraday tensor F is calculated with the connec- = gˆ ∂ erµ (1/2)∂ Tr(e) 0 (17) ⌘ tion with zero torsion. First, the Lagrangian variation of the The requirement of gauge invariance condition based on the action (19) allows us to obtain the expression: Lorentz gauge, the linearized gravitation takes the form of 1 1 after some straightforward calculus : DS = FµnDF + F gµlFrn + Fµrgln Dg µn µn lr ZM 2 4 an an ˆ ˆ ⇢ ⇣ ⌘ gˆ ∂a ∂n ebµ = 0 = = gˆ —a —n ebµ = 0 (18) 1 R 1 ) + ¬lr glr Dg + Fµn F glrDg 2c 2 lr 8 µn lr This is a wave equation in the Minkowskian spacetime where ✓ ◆ the perturbed metric is the unknown. Obviously, it takes the 1 µn l l + g — DG —µ DG wn form of classical equation of a wave (18) travelling at the 2c l µn ln h ⇣ ⌘ ⇣ ⌘i speed c of the light as electromagnetic wave. For the variation of the Lagrangian (19), it is worth to re- Remark 4 The source-free field equations (13) and accord- mind the independent variations of the metric and the four- ingly the gravitational wave equation (18) were obtained potential vector. The Lagrangian variation of the Faraday along independent paths by Hilbert and Einstein. To account tensor takes the form of : for the sources (“forces”) terms other than gravitation, the DFµn = —µ (DAn ) —n (DAµ ) (21) associated Lagrangian L may be split into the sum of two contributions, a gravitational part given by the Riemann cur- This relation is obtained by directly writing: vature scalar and a source part which they left unspecified r r DF = D ∂ A G A D ∂ A G A except for the postulation that it depend only on the com- µn µ n µn r n µ nµ r ponents of the metric and the components of the electro- ⇣ r ⌘ ⇣r ⌘ = ∂ DA G DA DG A magnetic vector potential and its first derivatives e.g. Sauer µ n µn r µn r [2005]. Both Hilbert and Einstein had left the source term ⇣ r r ⌘ ∂n DAµ G nµDAr DG nµAr undetermined to some extent. Einstein had not specified his ⇣ ⌘ r source term at all. Hilbert proposed that the source term de- accounting for that the connection variation DG µn induces a pend only on the electromagnetic variables and implicitly variation of the field DFµn. Second, the two systems of con- assumed that all matter is of electromagnetic origin. In the servation laws associated to the unknown primal variables following, electromagnetism coupled with gravitation is in- (say the 4-vector potential Aµ , and the Riemannian metric vestigated. gµn) are derived by varying the Lagrangian along the Lie- derivative variations Lx Aµ , and Lx gµn. Shifting the diver- gence terms at the boundary of the spacetime and assuming 3.4 Covariant Maxwell equations in curved spacetime a zero divergence at this boundary allow us to obtain the conservation laws. We can rearrange the Lagrangian varia- Various phenomenae may have effects on electromagnetic tion of the action to give: waves such as the presence of gravitational field. The consti- tutive equations should be worthily changed to account for 1 lr R DS = — Fµn DA w + ¬ glr the modification of the spacetime environment in presence n µ n 2c 2 ZM ZM  ✓ ◆ of gravity for instance (curvature). Here, we thus consider 1 Fµn + FµnF glr + gµlFrn + Fµrgln Dg (22)w the electromagnetic constitutive laws D = eE and B = µH 8 µn 4 lr n where e and µ are the electric and magnetic parameters of ⇣ ⌘ owing that the Faraday tensor is in fine expressed in terms the spacetime replacing e and µ . For the sake of the sim- 0 0 of the potential A by means of equation (20). Due to the plicity, let us consider the simplest example of action for free µ arbitrariness of the metric and potential variations, we obtain electromagnetic field without sources and occurring within the classical (and covariant) Einstein-Maxwell equations: a Riemann curved spacetime where we assume an action in- cluding both Hilbert-Einstein Hilbert [1915] and a Yang- µn —n F = 0 Mills Yang & Mills [1954] contributions, say e.g. Ham- R 8 1 lr lr 1 µn lr mond [1987],: ¬ g + F Fµn g (23) > 2c 2 8 > ✓ ◆ 1 µn 1 < 1 lµ nr ln µr S := L wn with L := F Fµn + R (19) g FµnF + g FnµF = 0 M 4 2c 4 Z > > ⇣ ⌘ :> 8 where the first equation is the covariant Maxwell equations in a three-dimensional formulation. By introducing the elec- in a Riemann spacetime, such as second gradient continuum. tromagnetic tensors (4) and (6) into the expression of the The second equation governs the interaction of electromag- energy-momentum tensor, we have the Minkowski tensor netism with gravitation. The unknowns in first term of the T µn is not symmetric when considering the time index 0. second equation are the spacetime metric. The electromag- It is worth to express the energy momentum as: netic source (including both the second and the third terms) E E H in the second equation constitutes the energy-momentum T µn = ⇥ (27) M D BTM tensor. They influence the gravitation field and vice versa the  ⇥ metric field has also some influence on the electromagnetic where E is the energy, and M is the Maxwell tensor with field via the Levi-Civita covariant derivative —. Maxwell T contraviant components T ij. The (nonsymmetric) energy- equations (23) (first row) are used to analyze the electromag- momentum such defined is called Minkowski energy mo- netic wave within a Riemann spacetime. The first equation mentum. may be re-written: µn µa an µa n nb µ Remark 5 The Poynting vector S := E H (originally dis- —n F = —n g g Fab = —n g —a A g —b A ⇥ covered by JH Poynting in 1884) represents the rate of en- h i = gµa — — An +¬n Ag gnb— — Aµ = 0 ergy in the i-direction. However, it is well-known that the a n nag n b Maxwell equations admit only the fundamental fields E and where we haveh used the Schouten relationsi e.g. Rakotomanana B, without considering any model of material media where [2003] with a zero torsion. Maxwell equations include a the electromagnetic fields occur. In such a case, the Poynt- classical wave part, a divergence term, and the contribution 1 ing vector is rigorously defined as S = E B, where µ0 of the Ricci curvature of the spacetime: µ0 ⇥ is the vacuum permeability (Minkowski spacetime). This is gnb— — Aµ + gµa— — An + gµa¬ Ag = 0 (24) considered as the general expression of the Poynting vec- n b a n ag tor. The moment of the Poynting vector J := x Sdv is The first term expresses a D’Alembertian operator. The sec- ⇥ defined as the angular momentum including the so-called ond term may be dropped if we assume a Lorenz gauge R orbital angular momentum and the spin angular momentum — An = 0. We then obtain the electromagnetic wave propa- n e.g. Barnett [2002]. gation equation within curved spacetime: We relate the electromagnetic fields T lr as source of the gnb— — Aµ + gµa¬ Ag = 0 (25) n b ag bending of the spacetime, we can multiply this equation by in which we notice the direct influence of the gravitation the covariant components of the metric glr to obtain with- (represented by Ricci curvature) on the electromagnetic wave out difficulty the Ricci curvature and then the curvature of propagation. In the following we will consider an extension the spacetime: of the equation (25) in the framework of Riemann-Cartan lr spacetime. R = T := g T = 2cF F lr = c (D E B H) lr M lr · · (28) 3.5 Some remarks on electromagnetic fields which is exactly c times twice of the electromagnetic part of the Lagrangian. The second row of system (23) is the field equation which extends the Einstein equation for Riemann-Cartan space- Remark 6 In this section the goal is not reached because the time, where the term represents the energy momentum anal- electromagnetic variables should be considered as additional ogous of the Maxwell energy-momentum for the space part, variables. They are not directly related to the spacetime ge- and with nonsymmetric property when considering the time- ometry. On should add the electromagnetic potential A (or like part: the electromagnetic Faraday strength F ) as arguments of 1 F the Yang-Mills action. T lr := FµnF glr µn gµlFrn + Fµrgln (26) 4 µn 2 ⇣ ⌘ It is the Minkowski energy-momentum tensor due to electro- magnetic field. It modifies the gravitational field as source 4 Gravitation and electromagnetism within twisted and whereas the spacetime modifies the electromagnetic field curved spacetime according to (25). The temporal component of the energy- Analysis of electromagnetic fields in presence of extremely momentum (26) holds: T 00 = 1 DiE + BiH , which is ex- 2 i i massive gravitation remains a relevant topic in relativistic actly the electromagnetic energy density astrophysics. Propagation of electromagnetic waves governed T 00 =(1/2)(D E + B H) := E by Maxwell equations within a curved spacetime constitutes · · 9 a fundamental basis for studying signals received from neu- spacetime. In this framework, two of the Maxwell equa- tron stars and black holes to name but a few in astrophysics. tions dF = 0 would be expected since the Faraday tensor Other methods consist in measuring the signal due to grav- 2-form F is exact, say F := dA, and hence closed, dF = itational waves. It is now admitted that the influence of the d (dA)=0. In order to investigate electromagnetic waves non-Minkowskian metric of the curved spacetime is much within twisted and curved spacetime matter (which may be a stronger on the electromagnetic field Fµn(x ) than the in- considered as a Riemann-Cartan manifold), it is assumed fluence of this field on the bending of the spacetime M . that the electromagnetic field is described by an electro- Including the torsion of continuum recently gains interest in magnetic 2-form Fµn. It constitutes an extended model of the propagation of light in twisted medium e.g. Zhang & al. electromagnetism within curved spacetime as earlier as in [2019]. In this section we consider the gravitation electro- e.g. Plebanski [1960], and in the framework of differential magnetism interaction within a Riemann-Cartan spacetime forms e.g. Prasanna [1975a]. Prasanna [1975a] has derived µ g µ endowed with metric gab(x ) and connection Gab(x ).We the Maxwell equations in a Riemann-Cartan spacetime. The consider curved spacetime (M ,g,—) with non zero torsion. Faraday tensor (minimally coupled to the gravitation in a As previously, the spacetime structure is caracterized by the Riemann-Cartan spacetime via the torsion) and the scalar Riemann-Cartan spacetime with the metric g, the torsion ¿, curvature are defined by the relationships: and the curvature ¬: F := — A — A = — A — A ¿r A , µn µ n n µ µ n n µ µn r —g = 0, ¿ = 0, ¬ = 0 (29) µn 6 6 R := g ¬µn (31) In the following, we derive the Maxwell equations in a curved and twisted manifold M . By using a formalism based on where the contravariant components of Fµn are calculated exterior calculus, Maxwell equations were established for by means the connection with torsion e.g. Smalley & Krisch various continua (Minkowski, Riemann, and almost post- [1992]. Definition of Faraday tensor (30) in Riemann space- Riemann) Puntigam et al. [1997] where they considered time holds for both Euclidean and (pseudo)-Riemannian and as basic axioms the conservation of electric charge and the also even proposed in some post Riemannian spacetimes e.g. conservation of magnetic flux. This allows them to put aside Puntigam et al. [1997]. As extension the definition (31) is the connection structure of the spacetime. Third, either for valid for both Euclidean, Riemannian and Riemann-Cartan metric-based energy, or metric-torsion based energy, it is spacetime. This again illustrates the fact that the extension worth to define a Lagrangian L (F , ) associated to the of physical variables as F can be done in many ways µn ··· µn electromagnetic fields when we face the question of vari- (as a 2-form in e.g. Puntigam et al. [1997] or as a twice ational formulation. To relate electromagnetism with rela- the skew-symmetric part of the gradient in e.g. Smalley & tivistic gravitation, it is interesting to remind that applica- Krisch [1992]). tion of the gauge invariance principle for the group of trans- lation (corresponding to torsion) of the spacetime M with Yang-Mills type Lagrangian, quadratic in the field strengths Fµn (as for electromagnetism), allows us to deduce the Ein- 4.2 Field equations, wave equations stein’s theory of gravitation, based on the Einstein-Hilbert action e.g. Cho [1976a]. For the sake of the simplicity, consider the action for free gravito-electromagnetic field without sources and occurring within a Riemann-Cartan spacetime (curved and twisted), 4.1 Faraday tensor in twisted spacetime the Yang-Mills and Hilbert-Einstein action e.g. Charap & Duff [1977],: Formulation of Maxwell equations by means of differential forms may be not equivalent to formulation by means con- nection in Riemann-Cartan spacetime e.g Vandyck [1996]. 1 1 S := L w with L := Fµn F + R (32) In a Riemann-Cartan spacetime, the Faraday tensor is cal- n µn ZM 4 2c culated as follows e.g. Prasanna [1975a], Smalley [1986]: F := — A — A = ∂ A ∂ A +¿r A . It is rather µn µ n n µ µ n n µ µn r where no action due to matter is considered. For the vari- different if calculated by means of an exterior derivative of ation of (32), it is worth to remind that the metric and the the 1-form A =(A ) e.g. Prasanna [1975a] : µ torsion are independents primal variables as well as the elec- tromagnetic four-potential. The Lagrangian variation of this F := dA = Fµn = ∂µ An ∂n Aµ (30) ) 2-form and curvature include both the variation of the po- where, in such a case, we have exactly the same form of tential Aµ , the variation of the Riemann metric gab, and also g Faraday tensor in either Minkowski spacetime or Riemann the variation of the connection Gab. First, the Lagrange vari- 10 ation of the action (32) allows us to obtain : of the variation of primal variables, we deduce the system of 1 1 partial differential equations: = Fµn F + F µlFrn + Fµr ln DS D µn µn g g Dglr µn M 2 4 F = Z ⇢ —n 0 ⇣ ⌘ R 1 lr R lr 1 µn lr 1 lr lr 1 µn lr + ¬ g Dg + F Fµn g Dg 8 ¬ g + F Fµn g 2c 2 lr 8 lr > 2c 2 8 ✓ ◆ > ✓ ◆ µn > 1 (36) g r > lµF Fnr lnF Fµr + — DGl — DGl ¿ DGl w = 0 > g µn + g nµ = 0 2c l µn µ ln lµ rn n < 4 ⇣ 1 µ ⌘ h ⇣ ⌘ ⇣ ⌘ i (Fµn Fnµ) A + ¿ grn = 0 where the last line is deduced from extended version of Pala- > l c lr > tini relation. Indeed, we can formulate the variation of pri- > g where> we notice a slightly extension of the fields equations mal independent variables as dg and the connection dG . : ab ab in Charap & Duff [1977] for Riemann-Cartan spacetime. For this purpose, we remind the relations e.g. Rakotomanana The first row of the system (36) expresses the Maxwell equa- [1997]: tions in Riemann-Cartan spacetime, and it should be stressed g g g D¿ = DG DG , that for this Lagrangian (model), the potential A may be ap- ab ab ba µ parently calculated independently on the gravitation (except D¬l = — DGl — DGl ¿n DGl (33) abµ a bµ b aµ ab nµ eventual coupling at the boundary ∂M ). The connection ap- where the covariant⇣ derivatives⌘ ⇣ use the⌘ connection with non proach for Faraday tensor is equivalent to the differential zero torsion. The second equation (33) extends the Palatini form approach when the spacetime is Riemannian without identity when continuum has torsion. The Lagrangian varia- torsion (Vandyck [1996]), or when the non metricity of the tion of the Faraday tensor takes the form of : connection is traceless. The Maxwell equations of (36) show that the connection approach with torsion is ”naturally” de- DF = — (DA ) — (DA )+D¿r A (34) duced from a variation principle, and the same form as the µn µ n n µ µn r form approach is obtained. This relation is obtained by directly writing: Remark 7 It is stressed that for non-symmetric tensors, care DF = D ∂ A G r A D ∂ A G r A µn µ n µn r n µ nµ r should be taken about the placement of the indices. = ∂ DA G r DA DGr A µ n µn r µn r r r Non symmetry of the Ricci tensor First, the difference be- ∂n DAµ GnµDAr DGnµAr tween the spacetime connection connection and the Levi- accounting for that the variation of the geometric structure, Civita connection is the contortion tensor: r say D¿µn, induces a variation of the field DFµn. At a sec- g Tg := G g G (37) ond step, the three systems of conservation laws associated ab ab ab to the unknown primal variables (say the 4-vector potential g where G are the Christoffel symbols associated to the A , the Riemannian metric g , and the torsion ¿r ) are ab µ µn µn metric g . The contortion tensor describes the deviation derived by varying the Lagrangian along the Lie-derivative ab of the matter geometry from the Riemannian geometry one, variations L A , L g , and L ¿r . Now we factorize x µ x µn x µn whose connection reduces to the Christoffel symbols. By the variation with respect to the Lagrangian variations of choosing a metric connection on the manifold, we have: the electromagnetic potential DAµ , the metric Dg , and lr g g g g g l ¬ = ∂a (G + T ) ∂b (G + T ) the connection DGµn respectively. The presence of the term abl bl bl al al µ g DGl means that the torsion and curvature may evolve since (G + Tµ )(G + Tg ) µn al al bµ bµ they are independent primal variables of the theory. By shift- µ g +(G + Tµ )(G + Tg ) ing divergence terms at the boundary of the spacetime M bl bl aµ aµ we can rearrange the Lagrangian variation of the action to We arrive to the expression of the curvature strain which is give: defined as the difference between the matter and the curva- ture calculated with Levi-Civita connection: µn 1 lr R lr DS = — F DA wn + ¬ g n µ 2c 2 g g g g g µ g µ ZM ZM ¬ = ¬ + —a T — T (T T T T )  ✓ ◆ abl abl bl b al bµ al aµ bl 1 Fµn + FµnF glr + gµlFrn + Fµrgln Dg w (38) 8 µn 4 lr n ⇣ ⌘ where another interest appears when we calculate the Ricci µn nµ 1 rn µ l (F F ) Al + g ¿ DG wn (35) curvature which is obviously non symmetric : c lr µn ZM ✓ ◆ ¬ := ¬a = ¬ + — Ta — Ta owing that the Faraday tensor is in fine expressed in terms of bl abl bl a bl b al the potential A by means of (31). Due to the arbitrariness (Ta Tµ Ta Tµ ) (39) µ bµ al aµ bl 11 where we notice that the Levi-Civita covariant derivative is may be dropped if we assume a null divergence as a gauge used for calculating the contortion. In view of the relation condition. For a non twisted and non curved spacetime, the (39), we notice the possibility of the Ricci tensor to be not electromagnetic wave propagation equation reduces to ⇤ Aµ = symmetric in presence of torsion. The gravitational part in 0. The third term introduces a first covariant derivative which the equation (36 2) is then not symmetric too, and this is leads to a diffusion of the wave (spacetime attenuation), and coherent to the fact that the Minkowski energy-momentum the last term points out a breathing mode whenever the bound- may be not symmetric too. ary conditions allow it e.g. Futhazar et al. [2014]. What The curvature (39) allows us to calculate de scalar cur- should be observed too is that the torsion and Ricci curva- bl vature R := g ¬bl. Since the both two connections are ture influence the wave propagation linearly. It should be initially assumed metric compatible, and by shifting the di- stressed that the spacetime geometry and in fine the gravi- vergence terms to the spacetime boundary ∂M , we notice tation is in fact tightly linked to the electromagnetism phe- that the gravitation part of the Hilbert-Einstein action seem nomenon. This may not be perceived at a first sight. In sum, to be not equivalent when using the two connections : Levi- the second row of the system (36) gives the coupling equa- Civita and the connection with non zero torsion. However, tion of the electromagnetic field and the gravitational field S A the symmetric ¬bl and anti-symmetric ¬bl parts of the one. The electromagnetic terms act as a source-term for the Ricci curvature are obtained from (39): gravitation. They act as a kind of electromagnetic energy generating evolution of the spacetime metric. We recognize ¬S = ¬ bl bl the Einstein field equation in the absence of the electromag- ¬A = — Ta — Ta (Ta Tµ Ta Tµ ) ( bl a bl b al bµ al aµ bl netic field. Despite its apparent relative simplicity, the sys- tem of partial differential equations (36) remains complex where we notice that the symmetric part reduces to the Ricci since the connection, and by the way the Ricci and total cur- curvature associated to the Levi-Civita connection whereas vatures, includes both the (gravitational) metric g and the the skew-symmetric part includes all the remaining terms µn contortion tensor Tg . of the equation (39). We easily deduce that the two scalar µn bl bl curvatures are equal R := g ¬bl = g ¬bl = R and the Remark 8 Most importantly, the torsion ¿ of the spacetime same Lagrangian density of the gravitation by using the two is related to the flux spin angular momentum of electro- connections. We nevertheless draw attention that the choice magnetic waves e.g. Allen & al. [1992], Barnett [2002], of the volume-form wn for integrating 4-forms should be un- Milonni & Boyd [2010] as shown in the equation (36-3). dertaken carefully to ensure compatibility e.g. Mosna & Saa This equation is deduced directly from a variational princi- [2005]. ple by assuming an additive action including both the Hilbert- Einstein and Yang-Mill actions. In a recent work, by using Electromagnetic wave propagation The Maxwell equations another meyhod based on the fields equations, it has also (36) (first row) may be used to analyze the electromagnetic been shown that electromagnetic spin creates torsion within wave propagation within a twisted and curved spacetime. spacetime Hammond [2018], and accordingly the presence Let consider a spacetime M endowed with a metric gab g of the torsion as argument of the Lagrangian L is required and a connected with Gab, this later is compatible with the to ensure gauge invariance of the electromagnetism theory. metric. Maxwell equations may be re-written as follows: Electromagnetic waves and particularly, light beams have µn µa an µa n nb µ angular momentum, corresponding to the Poynting vector. —n F = —n g g Fab = —n g —a A g —b A There are two particularly manners in which a electromag- µa n g h n n g i = g —a —n A ¿ —g A + ¬ A netic waves can rotate: if every polarization vector rotates, na nag nb µ the wave has spin; if the phase structure rotates, the wave g ⇥ —n —b A = 0 ⇤ has orbital angular momentum. What would be interesting where we used the Schouten’ relations e.g. Rakotomanana too is that spin angular moment as well as orbital angular [2003]. By arranging the previous relationships, we notice momentum have mechanical effects, this may open to some that the Maxwell equations include, as for elastic wave prop- research in the domain of relativistic optomechanics at least agation, a classical wave part, a divergence term, and the from a basic theoretical point of view. contribution of the twisting and the Ricci curvature of the spacetime: g 4.3 Remarks on the extended electromagnetism wave gnb— — Aµ gµa¿ — An + gµa¬ Ag = 0 (40) n b na g ag equation where the condition — An 0 was used to extend the Lorenz n ⌘ condition in the framework of Riemann-Cartan geometry, The third row of (36) gives the equation to calculate the more specifically in the way of Gauss units system. The first torsion field. It is striking its analogy with the relation as- term expresses a D’Alembertian operator. The second term sumed by Fernando et al. [2012] by considering a partic- 12 ular Riemann-Cartan spacetime and working with contor- dielectric disk placed in a crossed oscillating radial electric tion tensor. It is a link between electromagnetic fields and and longitudinal magnetic fields), and James (which is based the twisting of the spacetime. What is interesting is that the on a similar experimental jig but with radial electric field and electromagnetic field allows us to calculate with an algebraic azimuthal magnetic field) Obukhov & Hehl [2003]. explicit formula the torsion field by means of the third row. Once the torsion is obtained, we can apply covariant deriva- Remark 11 In the previous equations, the torsion does not tive within Riemann-Cartan geometry. By multiplying with propagate. In order to account for the torsion propagation, i.e., a well-known method would be to add a scalar bilinear gns, the explicit formula for calculating the torsion is ob- term of the covariant derivatives of the torsion e.g. Ham- tained accordingly by means of an algebraic relation: µ mond [1987] where the trace of the torsion ¿n := ¿nµ ¿µ = c g (Fµn Fnµ)A ls sn l can be considered as the electromagnetic four-potential, and µa nb µb na the skew-symmetric part of the Ricci curvature tensor as = 2c gsn g g g g Al —a Ab (41) proportional to the electromagnetic Faraday tensor. For that ⇣ ⌘ R owing the expression of the electromagnetic strength in terms purpose, he has considered the Lagrangian: S := M ( c + of potential. It may be noticed that the contribution of the a G G µn) with G := ∂ ¿ ∂ ¿ where arbitrariness µn µn µ n n µ R electromagnetic potential to the torsion field is of second or- of the metric and the torsion variations hold. In his approach der ”Al —a Ab ”. the electromagnetic variables are also deduced from space- time geometry. Remark 9 The investigation of the interaction of electromag- netic masses with Riemann-Cartan continuum was done by To go further let us consider the first and third equations numerous authors for charged and spinning ”static” dust (here of the system of equations relating gravitation and electro- static means here no displacement of the center of mass), magnetism. As for the Maxwell equations within Minkowski for perfect fluids with spin density e.g. Smalley & Krisch spacetime, the above equation may be formulated by means [1992]. By analyzing the solutions of Maxwell equations, of four-potential vector Aµ by introducing properties of cur- the torsion field together with the spin of Einstein-Cartan vature tensor, the metricity of the connection and the Lorenz gravitation theory may be suggested as produced by the elec- gauge (— An 0). The third equation may be re-arranged n ⌘ tromagnetic field e.g. Tiwari & Ray [1997]. Paraphrasing to isolate the torsion. The first and third equations thus give, n these authors, it was concluded that in the absence of elec- by assuming a null divergence for the potential —n A = 0,: tromagnetic fields, the body has a vanishing spin density which itself is associated to the continuum torsion e.g. Hehl gnb— — Aµ gµa¿g — An + gµa¬ Ag = 0 n b na g ag & von der Heyde [1973]. The third algebraic equation of g g g ( 2ec(— Aa —a A )Ab = ¿ab (36) conforms this conclusion concerning the torsion field. (43) Remark 10 From the system of equations (36), we notice that the electromagnetic energy-momentum in a vacuum has where, in the Maxwell equations, the first term represents a the same shape as for as for electromagnetic within a con- wave equation, the second term a diffusion contribution due tinuum matter Obukhov & Hehl [2003]: to the torsion field, and the last term with the Ricci curvature introduces a ”breathing” mode due to the non vanishing of 1 1 curvature tensor. It should be pointed out that the torsion T lr = FµnF glr glµF Fnr + glnF Fµr 4 µn 2 µn nµ field is of second-order with respect to the potential Aa . ⇣ ⌘(42) Remark 12 In the present model, we define the Faraday ten- This is a (non) symmetric Minkowski (canonical) energy- sor as F := — A — A where the connection has tor- ab a b b a momentum tensor e.g. Obukhov [2008] for the free electro- sion. The U(1) gauge invariance of Maxwell equations may magnetic field occurring within spacetime. There is a con- be violated without cautions with this choice. Indeed by mod- troversy between the version of Minkowski and that of Abra- ifying the potential as A A + g —gL where L(xµ ) b ! b bg ham, not deduced from a Lagrangian. We do not enter into is an arbitrary function, we get: Fab = —a Ab —b Aa g this long last debate, which was done in the past. We have ¿ab—gL where the last term vanishes if and only if the tor- just to remind that the Minkowski version is defined in the sion is zero or the function L is covariantly uniform. Some framework of Lagrange-Noether conforming to the invari- previous authors propose to define F := — A — A ab a b b a ance approach we adopt in this work. Obukhov & Hehl sug- as Faraday tensor even in Riemann-Cartan spacetime e.g.de gested the adoption of the Minkowski version (42) which Andrade & Pereira [1999], Smalley [1986]. Further anal- is motivated by the Lagrangian axiomatic approach, and by yses are required in the future. Results in the framework of the experimental evidence conducted in the past by Walker Riemann-Cartan Gravitation (e.g. Sotiriou & Liberati [2007]) & Walker (which is based on experimental measurements of may highlight some problems of gauge invariance since the 13

magnetic fields. It means that the torsion is identically zero Minkowski Special Relativity within the spacetime. Further investigations should be done about the definition of the Faraday tensor, there is yet a lot to Spacetime gab := be done in this domain. This may hurt at first sight, however, metric +, , , g := Detg { } | ab| more investigations should be conducted since the concept q Faraday of magnetic monopole enters into the discussion because the g tensor G 0 F = ∂ A ∂ A Gauss law on magnetic flux should be re-analyzed in such a ab ⌘ ab a b b a case e.g. Fernando et al. [2012]. L := Constitutive 1 laws F abF g F ab = e gaµgbnF 4 ab 0 µn 5 Geodesic and autoparallel deviation for gravitational Conservation and electromagnetic waves ab nb µ laws —b F = 0 g —n —b A = 0 Detection of gravitational waves illustrates a well-known Riemann Einstein Gravitation application of the geodesic deviation equation e.g. Nieto et al. [2007]. The measuring of the separation of two neigh- Spacetime bored geodesic curves in a Riemann spacetime may be eval- metric g := g (xµ ) g := Detg ab ab | ab| uated by means of the separation acceleration. Some pre- q vious studies have extended this deviation equation to in- Faraday g clude the relativistic top moving in a gravitational field e.g. tensor G F = —a A — Aa ab ab b b Nieto et al. [2007], or to reformulate the geodesic devia- Constitutive L := tion in terms of teleparallel gravity Darabi et al. [2015]. 1 ab ab aµ bn laws F Fab g F = e0 g g Fµn Geodesic deviation equation describes the relative motion of 4 two structureless particles determined by the spacetime ge- Conservation ometry with non zero curvature. When particles are charged laws — F ab = 0 gnb— — Aµ gµa¬ Ag = 0 b n b ag electrically, the deviation equation should account for the charge q e.g. Balakin et al. [2000] where geodesic curves Riemann- are called autoparallel curves or worldlines e.g. Balakin et Cartan Einstein-Cartan Gravitation al. [2000]. In the present subsection, we are interested in de- veloping the extension of the deviation equation for geodesics Spacetime to autoparallel curves where the spacetime is curved with metric g := g (xµ ) g := Detg ab ab | ab| torsion. In view of the previous results where the torsion ten- q Faraday sor of the spacetime is directly related, identified, with spin g tensor G F = — A — A angular momentum of the electromagnetic field by means of ab ab a b b a the equation (41), it is necessary to check what about the in- L := Constitutive 1 fluence of the torsion on the deviation of autoparallel curves. laws F abF g F ab = e gaµgbnF 4 ab 0 µn

nb µ µa g n Conservation g —n —b A + g ¿na—g A 5.1 Geodesic deviation equation in Riemannian manifold ab µa g laws —b F = 0 g ¬agA = 0 First, let consider two geodesic curves in the pseudo-Riemann spacetime (M ,g) denoted by g0 and g1 respectively. At the Table 1: Theories of electromagnetism interacting with gravitation in same propertime t, we define the separation four-vector x twisted and curved spacetime : Minkowski(flat), Riemann (curved) and Riemann-Cartan (twisted and curved) of the spacetime M which connects a point (event) x(t) of the geodesic g0 to a point (event) x(t)+x(t) of a nearby geodesic g1. The separation x is small in such a way that any expansion of tensor function of x with respect to x can electromagnetic tensor does not satisfy the Lorenz gauge in- be truncated to only the first-order terms. The relativistic variance (say U(1) gauge invariance) e.g. Puntigam et al. acceleration a of two material points is defined as the sec- [1997]. Choosing the Faraday tensor as F := dA or equiv- ond derivative of the separation vector x as the two material alently F := — A — A = — A — A ¿r A µn µ n n µ µ n n µ µn r points move along their respective geodesics. Let remind the allows to obtain a U(1) invariant model but induces the fol- separation velocity and deduce the separation acceleration lowing field equations: — F µn 2¿µ F rn = 0 and ¿r = n rn µn along the geodesic curve as follows: 0 replacing the first Maxwell equations and the third equa- tion coupling the torsion within spacetime and the electro- v := —ux, a := —uv (44) 14 where the components u := dx/dt define the four-vector of particles in gravitational field, such as the chaotic behav- velocity (timelike vector). Then the separation acceleration ior of particles orbits but they are not well-suited to study takes the general form : spinning particles, either for microscopic with intrinsic spin or macroscopic bodies with intrinsic spin e.g. Leclerc [2005]. D2x := — — x (45) Dt2 u u Remark 13 For Newton mechanics, a geodesic is a curve along which a particle moves as free falling particle. The which remains valid for either Riemann or Riemann-Cartan concept of geodesic deviation is based on the comparison spacetime. For this purpose, let us worthly remind some ba- of two geodesic curves in the spacetime (t,xa) for Newton sic assumptions and relations: spacetime. Say two particles within a potential F(xµ ), their 1. A geodesic on the spacetime M endowed with an affine motion equations hold respectively: connection — is defined as a curve g(t) such that parallel a a a a a transport along the curve preserves the tangent vector to x¨ (t)= ∂ F(P), z¨ (t)= ∂ F(Q)= ∂ F(P)+x¨(t) the curve. Say u a four-vector along a geodesic curve, (51) then we have : Expanding the potential about the point P gives: F(Q)= — u = 0 (46) F(P) ∂ F(P) x b O(x). This implies the expression u b of the acceleration of the geodesic deviation: 2. The orthogonality condition implies the nullity of the 2 a Lie derivative of the separation vector x along the vec- d x a b = ∂ ∂bFx (52) tor tangent to the geodesic curve: L x := — x — u dt2 u u x ¿(x,u)=0. This allows us to deduce two equivalent which represents the distance acceleration between two par- relations : ticles falling in a nonuniform gravitation field F(t,xa). Ex- tension of the special non-relativistic spacetime to include — u = — x ¿(x,u) x u (47) gravitation is first due to Cartan Cartan [1986] and later — x = — u + ¿(x,u) ( u x in e.g. Havas [1964], and for continuum mechanics in e.g. Duval & Kunzle [1978]. In weak field condition (for earth 3. We also remind the basic relation for the curvature oper- 2 ator for any three vectors : gravitation this means that GM/(c R) << 1) (G is the con- stant of gravitation, M earth mass, R earth radius, and c light

—u—vw —v—uw —[u,v]w = ¬(u,v,w) (48) speed) and the low speed motion i.e. v/c << 1 the difference between Newton gravitation and general relativistic gravi- From the equation (48), we have: tation may be neglected e.g. Shen & Moritz [1996]. For relating the Newtonian description with the relativistic ap- — — u — — u — u = ¬(u,x,u) (49) u x x u [u,x] proach, let us remind the time variable as dt = cdt where where the second and the third terms vanish because vec- c is the light speed, and to rewrite the geodesic equation of tor u is tangent to a geodesic, and the Lie derivative of x Newtonian limit case as: along the geodesic curve is zero respectively. We deduce the d2x a = ¬a x b (53) intrinsic Jacobi equation so-called geodesic deviation equa- dt2 b00 tion, expressed by means of the Levi-Civita connection e.g. a = 2 a Levi-Civita [1927] (1927), Synge [1934],: where we introduce ¬b00 c ∂ ∂bF. Such equation was used to analyse the oblatness of the earth e.g. Greenberg 2 2 a D x D x a [1974]. = ¬(u,x,u), = ¬ ub ud x g (50) Dt2 Dt2 dgb which is valid within a Riemannian spacetime without tor- sion. ¬(u,˙,u) is called Jacobi operator along the curve g(t) 5.2 Autoparallel deviation in Riemann-Cartan spacetime with the tangent u. The geodesic deviation equation (50) shows that the curvature produces acceleration of the sep- The idea is now to detect the relativistic acceleration of two aration between two neighboring geodesics g0 and g1. This nearby particles when the spacetime is curved with torsion. provides a geometrical interpretation of the curvature tensor. First of all, it is worth to remind that in a Riemann-Cartan The geodesic deviation equation constitutes a fundamental manifold the deviation from an autoparallel curve is obtained equation for relativistic gravitation since it relates the rela- from the definition of the deviation (component and intrinsic tivistic acceleration of two nearby particles in presence of forms): gravitation field. In a flat spacetime, the separation will be D2x := — (— x) (54) linear. Equation (50) allows us to analyze numerous motions Dt2 u u 15

Theory Potential Geodesic deviation g1

d2x a g0 Newton F = ¬a x b x dt2 00b D2x Einstein g, — = ¬(u,x,u) t Dt2 x + dx 2 u D x 2 = Einstein- Dt t + dt Dx Cartan g , — ¬(u,x,u)+¿ ,u +(— ¿)(x,u) Dt u ✓ ◆ u + du

Table 2: Expression of the geodesic and autoparallel deviation equa- Fig. 2: Autoparallel curves. The gap vector x separating autoparallel tion for Newton (N), Einstein (E), and Einstein-Cartan (EC) theories. timelike curves respectively g0 and g1, defined by —uu = 0. So, the acceleration of separation is completely determined by the cur- vature and the torsion of the spacetime. associated to gravitation and electromagnetic fields respectively. where the connection have torsion and curvature. In the fol- lowing, this intrinsic definition can be also used to obtain in the geometric backgrounds of the spacetime two kinds with a straightforward calculus the result (50) on a Riemann of action-at-distance physics fields : gravitation and electro- manifold. Let now extend to Riemann-Cartan spacetime. magnetism. In view of the system of equations (43), it is obvious that the torsion of the spacetime is due to the spin Theorem 1 Let (M ,g,G ) a Riemann-Cartan spacetime with of angular momentum, and the solving of such equation re- u := dx/dt, four-vector velocity (timelike vector), and x mains a great challenge. Notice that gravitational waves are the separation between two autoparallel curves g , and g . 0 1 described by linearized gravitation field equation deduced We assume the Lie derivative of x along u vanishes (as for from curvature (18) whereas electromagnetic waves may also classical assumption in general relativity) L x 0. Then u ⌘ described by further linearized version of (43) involving tor- the acceleration of the separation between two auto-parallel sion. For the sake of the completeness, elastic waves are curves g , and g takes the form of: 0 1 associated to linearized equations of Euler continuum me- D2x Dx chanics equations. Indeed, by considering the relation be- = ¬(u,x,u)+¿ ,u + — ¿(x,u) (55) Dt2 Dt u tween torsion tensor and electromagnetism (41), it is hope- ✓ ◆ fully guessed that the last two terms of the (55) represent which extends the equation (50) from Riemann to Riemann- forces due to electromagnetism. Symmetries may reduce the Cartan spacetime. number of components. For example, within Einstein-Cartan spacetime with a spherical symmetry where the metric is of Proof: The relation (48) induces —u—x u = ¬(u,x,u). The Schwarzschild type, the simplest case where the spins of in- nullity of the Lie derivative also holds : —ux = —x u+¿(x,u). The separation acceleration writes in such a case: dividual particles, or fluid elements composing the contin- uum are all aligned in the radial direction, only the compo- 2 D x 0 0 = ( )= + ( , ) nent ¿23 = ¿32 := ¿0 is not equal to zero e.g. Prasanna 2 —u —ux —u —x u ¿ x u Dt [1975b]. = ¬(u,x,u)+—u[¿(x,u)] For the sake of clarity, we calculate separately the last term : 5.3 Summary [ ( , )] = ( )( , )+ ( , )+ ( , ) —u ¿ x u —u¿ x u ¿ —ux u ¿ x —uu Geodesic deviation is present for any gravitational theory. where the third last term vanishes since u is tangent to the We can sketch the analogy between Newton, Einstein, and Einstein-Cartan gravitation in the table below. In the follow- geodesic. Owing that —ux := Dx/Dt, we get the result ⇤ ing table we resume the different expressions of the geodesic Remark 14 Each term in the equation (55) has its physical deviation where t is the proper time of relativistic theory. interpretation when identified with the results of Balakin et These three formulae express how the spacetime curvature al. [2000] (page 5011, equation (8)), where the torsion ¿ and torsion influence two nearby geodesic or autoparallel which is proportional to the Faraday tensor F appears as curves, making them converge to or diverge from each other. well as its covariant derivative —¿ related to —F . Assum- The right-hand side terms may be considered as tidal forces. ing a Riemann-Cartan manifold allows us to entirely include The analogies between the tidal forces resulting from the 16 previous three theories is obvious4. For Newtonian gravita- propagating. Considering a very simple shape of the La- tion, the tidal forces do not explicitly depend on the velocity grangian (the same form for all the models), we extend the u conversely to Einstein and to Einstein-Cartan gravitation. geometry structure from the curved spacetime and then to curved spacetime with torsion. In order to analyze the inter- Remark 15 In the deviation equations (53), (50), and (55), action of electromagnetism and gravitation, the development the vector x may be physically interpreted as the vector sep- of the Maxwell equations within curved spacetime shows aration of two moving objects (ideally two mass points) near the electromagnetism and gravitation mutual influence by each other, and vector u represents their initial motions. The means of the geometry characterized by metric, Levi-Civita second term is linear with respect to the separation vec- connection, and associated Ricci curvature. When dealing tor constitutes the influence of the spacetime geometry on with spacetime analogous to continuum with continuous dis- this separation acceleration. For Einstein-Cartan spacetime, tribution of singularity e.g. Rakotomanana [1997], where we again observe and stress that a non curved spacetime abrupt gradients of physical properties may occur, the ex- with torsion may induce a separation acceleration between tension of the Maxwell equations, namely the resulting wave the two moving objects due to gravitation and electromag- propagation, is necessary to account for the non zero torsion netism, but described entirely and solely with spacetime ge- ¿ = 0 and non zero curvature ¬ = 0. Among numerous ap- ometry. 6 6 proaches, the use of Riemann-Cartan manifold as underly- ing geometrical structure seems worth. First, ”geometriza- 6 Discussion tion” of gravity developed by Einstein by considers the de- pendence of the Lagrangian on the curvature tensor as the The unified theory in physics gave rise to long debate in sci- starting point for deriving the field equations (Einstein equa- ence history. The need of unified theory joining gravitation tions). Second, introducing the tensor Faraday including elec- and electromagnetism prior to any other unification attempts tric and magnetic fields - within the Lagrangian, it is rec- remains up-to-date in order to design the geometrical struc- ognized that the light, a particular case of electromagnetic ture of the spacetime. The Einstein-Cartan theory is prob- waves, bends if viewed from a uniformly accelerating frame ably the most natural extension of the relative gravitation and then accordingly that the gravity would therefore bend theory in order to include the non symmetry of the affine the light. The interaction of gravitation and electromagnetic connection, considered as independent variable with respect waves are described Einstein-Maxwell equations. The ”ge- to metric field. Cartan introduced additionally the torsion ometrization” of the electromagnetic fields constitutes the tensor as arguments of the theory in 1922 and it was pro- third step when these fields are present in the spacetime. posed before the discovery of the electron spin. On the one For that purpose, we have considered an extended spacetime hand, numerous works devoted to the extension of relative where curvature and torsion are present, a Riemann-Cartan gravitation did not treat the electromagnetic part and might spacetime. By observing the fields equation, we find that face a problem of lack of gauge invariance. On the other the gravitational and electromagnetic fields are respectively hand, there are many studies on the coupling of gravitation identified as geometric objects of such a spacetime, namely g g with electromagnetism in a Riemann spacetime. With this the curvature ¬abl and the torsion ¿ab. Further studies are approach, it is necessary to have the presence of matter (by required for the invariance aspects. The equation (36) we ob- the way the electron spin) in order to point out the torsion tained, is analogous to the particular contortion tensor found tensor. In the present paper, we choose to avoid introduction in a paper by Fernando et al. [2012]. Indeed, they have de- g g of matter and then provide a spacetime geometry as initial duced that a particular connection defined by Gab := G ab + g background without reference to matter either continuous of Tab in Riemann-Cartan spacetime allowed them to derive discrete. The guideline is to electromagnetism interacting Maxwell equations. It is shown that the torsion is nothing with gravity such as source of change of Riemann space- more than the spin angular momentum of an electromag- time of gravitation theory to Riemann-Cartan spacetime. We netic wave. The first goal of this work is reached in the sense have considered in the present study the Hilbert-Einstein that the electromagnetic variables are defined entirely from action (which is the simplest case among numerous grav- the spacetime geometry. More precisely the geometry of the itation theories) combined to Yang-Mills action (again the spacetime is deduced from the electromagnetic variables. simplest case of electromagnetism theories) to relate elec- The work should not be considered as a ”geometrization” of tromagnetism and gravitation within the extended spacetime gravitation and electromagnetism, but should be considered with torsion. 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Optical angular-momentum flux, Journal of as a part of the trace of the torsion itself e.g. Poplawski Optics B : Quantum and Semiclassical Optics 4, 2002, [2010]. All these aspects will certainly constitute future re- pp 7-16. search topics. We observe that both the torsion and the cur- Brading KA, Ryckman TA. Hilbert’s ”Foundations of vature influence the electromagnetic wave propagation in a Physics” : Gravitation and electromagnetism within the Riemann-Cartan spacetime. Despite the crucial point on the axiomatic method, Studies in History and Philosophy of Lorenz gauge invariance, this model seems to extend and Modern Physics 39, 2008, pp 102-153. thus include all previous models. Capoziello S, Cardone VF, Piedipalumbo E, Sereno M, Last but not least, we investigate the influence of tor- Troisi A. Matching torsion Lambda-term with observa- sion and curvature on the deviation equation. 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