aa r X i v : . [ phy s i c s . g e n - ph ] S e p On generalized forces in higher derivative Lagrangian theory.
M. Beau
1, 2 Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA
In this article, we introduce higher derivative Lagrangians of this form α A µ ( x ) ˙ x µ , α G µ ( x ) ¨ x µ , α B µ ( x )... x µ , α K µ ( x ).... x µ , · · · , that generalize the electromagnetic interaction to higher order derivatives. We show that oddorder Lagrangians describe interactions analog to electromagnetism while even order Lagrangians are similarto gravitational interaction. From this analogy, we formulate the concept of the generalized induction principleassuming the coupling between the higher fields U ( n ) ,µ ( x ) , n ≥ j ( n ) µ = ρ ( x ) d n x µ / ds n ,where ρ ( x ) is the spatial density of mass ( n even) or of electric charge ( n odd). In short, this article is aninvitation to reflect on a generalization of the concept of force and of inertia. We discuss the implications ofthese paradigms more in depth in the last section of the paper. CONTENTS
I. Introduction 2II. Equations of motion of a massive particle 2A. Second and third order Lagrangians 2B. Higher order Lagrangians 3III. General fields hypothesis 4A. Construction of the n = I. INTRODUCTION
In 1850, Mikhaïl Ostrogradsky introduced the idea of higher derivative Lagrangians, see [1]. The concept generalizes New-tonian mechanics to the case of forces that can depends explicitly on velocity, acceleration, and higher order derivatives, suchas jerk, snap, etc... Generalized mechanics with higher derivative Lagrangians has been extensively studied and is ubiquitous.Some examples include, classical mechanics [2–6], quantum mechanics [7–9], relativistic mechanics [10, 11], the well-knownPais-Uhlenbeck oscillator [12, 13] and generalized electrodynamics [14, 15], higher-derivative scalar field model [16, 17], andclassical rigid string [18–20]. Interestingly, the model is also applied to polymer physics [21], formation of microemulsions[22], and membrane biology [23–25]. However, to my knowledge, there is no article dealing with relativistic higher derivativeLagrangians of this form: e L ( ˙ x , ¨ x , · · · , x ( n ) ) = α A µ ( x ) ˙ x µ + α G µ ( x ) ¨ x µ + · · · + α n U ( n ) µ ( x ) x ( n ) µ (1)where x ( n ) ( s ) ≡ d n x ( s ) / ds n are the n -derivatives of the position ( ds = cd τ , where τ is the proper time), and U ( n ) µ ( x ) , n = , , , .. are the generalized vector -fields coupling linearly with the x ( n ) vectors. Here, we denote the field U (1) µ ( x ) = A µ ( x ) to refer to theelectromagnetic potential and U (2) µ ( x ) = G µ ( x ) by analogy with the geodesic equations we obtain in equation (4). In this article,we first derive the equations of motion of the a massive particle experiencing interactions described by the Lagrangians of theform (1) and show that odd and even orders Lagrangians are analogous to the electromagnetic and the gravitational interaction,respectively. Next, we generalize the concept of electromagnetic induction assuming the coupling between the higher fields U ( n ) ,µ ( x ) , n ≥ higher derivative currents j ( n ) µ = ρ ( x ) d n x µ / ds n , where ρ ( x ) is the spatial density of mass ( n even) or ofelectric charge ( n odd). In the last section, we discuss the applications to the model to microscopic physics and general relativity. II. EQUATIONS OF MOTION OF A MASSIVE PARTICLE
In this section, we analyze the equations of motion of a massive particle experiencing the action defined by S = Z dsL ( ˙ x ) + Z ds e L ( ˙ x , ¨ x , · · · , x ( n ) ) , where L ( ˙ x ) ≡ mc ˙ x µ ˙ x µ and where the interaction Lagrangian e L is given by equation (1). A. Second and third order Lagrangians
First, we consider the case n = n =
3. By integration by parts for n = e S = α Z dsA µ ( x ) ˙ x µ − α Z ds (cid:16) ∂ ν G µ (cid:17) ˙ x µ ˙ x ν , (2)and one can see that the first part of the action e S is similar to the electrodynamics action whereas the second part is similar tothe gravitational action. Indeed, from the generalized Euler-Lagrange equations (see [2–6]): d ds ∂ L ∂ ¨ x µ ! − dds ∂ L ∂ ˙ x µ ! + ∂ L ∂ x µ = , (3)with L = L + e L , we obtain mc η µν ¨ x ν − α ε µν ¨ x ν − α ∆ µνσ ˙ x ν ˙ x σ = − α F µν ˙ x ν , (4)where ε µν and ∆ µνσ are defined by ε µν ≡ ∂ µ G ν + ∂ ν G µ , (5a) ∆ µνσ ≡ ∂ ν ∂ σ G µ = ( ∂ ν ε µσ + ∂ σ ε µν − ∂ µ ε νσ ) , (5b)and where F µν ≡ ∂ µ A ν − ∂ ν A µ has a mathematical form that is similar to the Faraday tensor. From equations (4) and (5), onecan see the analogy with the equations of the motion of a charged particle in a gravitational field and in an electromagneticfield. However, the fixed metric (or background metric) is Minkowskian. Then, the quadrivector field G µ ( x ) can be seen as a displacement vector field and ε µν can be viewed as an strain tensor by analogy with the deformation theory of a continuousmedium [26], [27].Now, we take n = U (3) µ ≡ B µ . From the generalized Euler-Lagrange equations − d ds ∂ L ∂ ... x µ ! + d ds ∂ L ∂ ¨ x µ ! − dds ∂ L ∂ ˙ x µ ! + ∂ L ∂ x µ = , (6)we derive the following equations of motion mc η µν ¨ x ν − α ε µν ¨ x ν − α ∆ µνσ ˙ x ν ˙ x σ + α H µν ... x ν − α Υ µνσρ ˙ x ν ˙ x σ ˙ x ρ − α Σ µνσ ¨ x ν ˙ x σ = − α F µν ˙ x ν , (7)where H µν ≡ ∂ µ B ν − ∂ ν B µ Σ µνσ ≡ ∂ ν ∂ σ B µ Υ µνσρ ≡ ∂ ν ∂ σ ∂ ρ B µ . (8)We can see that this field generalizes the idea of the electromagnetic field because of H µν is antisymmetric. However, in (7) thereare some other fields, similar to ∆ µνσ , coupling with the combinations of the odd derivatives of x µ , i.e. ¨ x ν ˙ x σ and ˙ x ν ˙ x σ ˙ x ρ . B. Higher order Lagrangians
Now, we shall discuss the higher derivative terms. For n =
4, we denote the field K µ ( x ) ≡ U (4) µ ( x ). The dynamic equationshave a similar structure to the one we obtained for G µ ( x ) (i.e. for n = α x µ .... x µ is equivalent to this Lagrangian α ¨ x µ ¨ x µ and the quantity α ˙ x + α ¨ x µ ¨ x µ − α ˙ x µ ... x µ could be interpreted as a more general kinetic energy [6]. Similarly, the Lagrangian α K µ ( x ).... x µ is equivalent to α (cid:16) ∂ ν K µ (cid:17) ¨ x µ ¨ x ν + α (cid:16) ∂ σ ∂ ν K µ (cid:17) ¨ x µ ˙ x ν ˙ x σ , which has a more complicated expression than the one we obtained for the special case K µ ( x ) = x µ .To finish this section, let us now consider the generalized fields U ( n ) µ ( x ) , n ≥
1. From the generalized Euler-Lagrangeequations n X k = ( − k d k ds k ∂ L ∂ x ( k ) µ = , we find terms of the form ∂ µ · · · ∂ µ p U ( n ) , p = , · · · , n multiplied by the combination of the derivatives x ( l ) µ x ( l ) µ · · · x ( l p ) µ p , P pj = l j = n . The equations of motion shows that the “even” n -fields are analogous to the "gravitational field" while the “odd” n -fields are analogous to the "electromagnetic field" as the derivatives x ( n ) µ are multiplied by the symmetric (if n is even) / antisymmetric (if n is odd) part of the first derivative of the field:( ∂ µ U ν + ( − n ∂ ν U µ ) x ( n ) ν . We will see the consequences of this remark in the next section.
III. GENERAL FIELDS HYPOTHESIS
In this section, we propose to formulate a dynamical theory of the generalized vector fields introduced above. We introduce a generalized induction principle , analogous to the electromagnetic induction.
A. Construction of the n = In equation (1) for n = G µ is coupled with the acceleration of the particle as the field A µ is coupledwith the velocity of the particle. By analogy with the construction of the electromagnetic field theory, we suggest the followingfield equations: ∂ µ ε µν ( x ) = − κ j (2) ν ( x ) , (9)where the acceleration current density j (2) ν (generally non-conserved) is: j (2) ν ( x ) ≡ ρ m ( x ) c du ν ds , (10)where ρ m ( x ) is the density of particles and du ν ds is the 4-acceleration. This equation is the analog of the Gauss-Ampère law inelectrodynamics ∂ µ F µν = µ j (1) ν , where j (1) ν is the 4-electric current density. Notice that we can rewrite the coupling constant κ as follows κ = π G λ c where λ has the dimension of a length.To complete the system of field equations, we need ten additional equations: ∂ σ ∂ σ ε µν + ∂ µ ∂ ν ε σσ = ∂ µ ∂ σ ε σν + ∂ ν ∂ σ ε σν , (11)The equations (11) have the same form as the compatibility equations for the strain tensor in the three-dimensional non-relativistic theory of deformation of continuous media [26, 27]. There are also the analog of the Gauss-Faraday equations ∂ µ F νσ + ∂ ν F σµ + ∂ σ F µν .After combining the last two groups of equations (9)-(11), we can easily derive the wave equations ✷ ε νσ ( x ) + ∂ ν ∂ σ ε µµ ( x ) = − κξ (2) νσ ( x ) , (12)where ξ (2) νσ ( x ) ≡ ∂ σ j (2) ν ( x ) + ∂ ν j (2) σ ( x ) . Also, the trace of ε µν satisfies the equation ✷ ε µµ ( x ) = − κ∂ µ j (2) µ ( x ) , (13)this means that ε µµ is a non-massive scalar field. It is worth mentioning that the trace of the strain tensor is usually interpreted asthe contraction / dilation of the volume of the continuous medium [26]. Hence, from equation (13) we conclude that the relativisticdeformation of the volume of a four dimensional continuous medium is related to the non-conservation of the current j (2) . B. Generalization to the n-field equations
Now, we can generalize the construction of the general field theory for any n ≥
1. Following the rules mentioned above, werewrite the constants in (1) as α n = mc ( λ n ) n and α n − = q ( ξ n ) n − c n − , n ≥
1, where λ n and ξ n are fundamental constants that havethe dimension of length and G is the universal gravitational constant. We denote m and q the mass and the electric charge of theparticle, respectively.It comes naturally that for so-called gravitational-type fields U (2 n ) ≡ G ( n ) , n ≥
1, the coupling has the form: − π Gc ( λ n ) n c n G ( n ) µ ( x ) j (2 n ) µ ( x ) , (14)whereas for the so-called electromagnetic-type fields U (2 n − ≡ A ( n ) , n ≥
1, the coupling reads: µ ( ξ n ) n − c n − A ( n ) µ ( x ) j (2 n − µ ( x ) , (15)where A ( n ) µ has the dimension of N . A − (N is the Newton and A the Ampère), µ is the vacuum permeability ( µ = π × − N . A − ), and where G ( n ) µ has the dimension of a length. The generalized currents for n = , , , .. are constructed asfollows: j ( n ) ν ≡ ρ m ( x ) d n x ν d τ n , if n is even , (16a) j ( n ) ν ≡ ρ e ( x ) d n x ν d τ n , if n is odd , (16b)where ρ m ( x ) is the mass density and ρ e ( x ) the electric charge density. Similarly to equation (9), we construct an (2 n − ff erential theory to relate the sources and the fields: O ( n ) µ ( λ n ) ǫ µν ( n ) ( x ) = − π Gc λ nn c n j (2 n ) ν ( x ) , n ≥ , (17a) Q ( n ) µ ( ξ n ) f µν ( n ) ( x ) = − µ ξ n − n c n − j (2 n − ν ( x ) , n ≥ , (17b)where O ( n ) µ ( λ n ) and Q ( n ) µ ( ξ n ) are two (2 n − ff erential operators and where ε µν ( n ) ( x ) ≡ ∂ µ G ν ( n ) + ∂ ν G µ ( n ) and f µν ( n ) ( x ) ≡ ∂ µ A ν ( n ) − ∂ ν A µ ( n ) .From those rules we can obtain similar wave equations to (12) and (13) with higher order di ff erential operators ( λ n ) k ✷✷ · · · ✷ | {z } k times , k = , · · · , n . For example, for n =
4, we can take O (4) µ ( λ ) = ( λ ✷ + ∂ µ and then we get the wave equation for the trace of the tensor ζ µν ≡ ∂ µ K ν + ∂ ν K µ : (cid:16) λ ✷ + (cid:17) ✷ ζ µµ ( x ) = − π G λ c ∂ µ j (4) µ ( x ) , (18)and then ✷ ζ µµ is a massive scalar field. Similar equations have been studied in the context of generalized electrodynamics [14]and of higher derivative scalar field theories [16, 17]. The di ff erence here is that the source of the scalar field is related to thefourth-order general current j (4) µ ( x ), which is proportional to the fourth order derivative of the position (i.e., to the so-called snap ), see equation (16).Notice that for the electromagnetic-type fields, the choice of the field A ( n ) µ is not unique because the fields f µν ( n ) is antisymmetric.On the contrary, all of the components of the gravitational-like field G ( n ) µ are physical because the tensor ε ( n ) µν is symmetrical. C. Unitary fields
Physically, the generalized vector fields can be understood as a n − A (2) µ ≡ B µ ( x ) is the first order correction of the Minkowskian theory of Electromagnetismfield A (1) µ ≡ A µ . Hence, it becomes natural to unify the gravity type fields as well as the electromagnetic type fields. In order tounify the fields, we construct the dimensionless unification constants: γ jl = λ j λ l , θ jl = ξ j ξ l , j , l = , , , · · · where the constants λ n , n ≥ ξ n , n ≥
1, were introduced in the previous section.For example, if we suppose that A µ ( x ) = B µ ( x ), we get the coupling: µ A µ ( x ) j (1) µ ( x ) + ξ c j (3) µ ( x ) ! where we put ξ = ξ (we recall that α = α = ξ / c ). Phenomenologically, this means that for an electric circuit withan intensity of this type I ( t ) = I t /τ , where τ is a time constant, the third order time derivative of the vector position of theelectrons in the current is non-zero (this kinematic quantity is called the Jerk , see [29, 30]) and so that the electromagnetic fieldwould be modified by the jerk current j (3) . We mention that in the generalized theory of Electrodynamics [14] the relation withthe higher derivatives currents (c.f. eqs. (16b), (17b)) has not been suggested.Similarly, we can construct the unified coupling for the even fields n = n = − π Gc λ c G µ ( x ) j (2) µ ( x ) + γ λ c j (4) µ ( x ) ! assuming that G µ ( x ) = K µ ( x ) ≡ G (2) ( x ), where we put λ = λ, γ = γ , and where we introduce the e ff ective current e j (2) µ = j (2) µ + γ λ c j (4) µ . Consequently, the e ff ective strain tensor ǫ µν that deforms the Minkowski metric is also induced by the secondorder derivative of the acceleration (the so-called snap ) of the moving particles. IV. DISCUSSIONA. Microscopic Physics and generalized currents
The e ff ect of gravitation at the microscopic scale is not yet well known. It is also clear that the geometrical description ofgravity fails at the Planck scale [20]. Therefore, it is fair to ask whether the current density of accelerated masses j (2) plays asignificant role at this scale.Theoretically, it could be interesting to investigate the e ff ect of the generalized currents j ( n ) on the electromagnetic andgravitational fields. We can assume that the value of the parameters ξ and λ introduced in Section III C are very small, forthe physical e ff ects of generalized currents have never been observed. However, higher order derivatives should play a role atthe Planck scale [20, 31]. Hence, our theory of generalized fields and forces could be relevant in high-energy particle physics.Formulating a generalized quantum field theory is a challenging problem for future investigations. B. Strain and stress tensor in General Relativity
The special case n = α n ≥ =
0) has been recently discussed in [27, 28]. It consists of a modification of theEinstein general relativity theory (GR) that deserves to be mentioned in this paper. In this section we introduce the idea of acovariant strain / stress field theory in the framework of GR.Similarly to (5), it is clear that the strain tensor in Riemann spaces is given by ε µν = D µ G ν + D ν G µ where the operator D isthe covariant derivative. We construct a stress tensor σ µν ( x ) = α g µν ε γγ + βε µν (19)where α and β are the Lamé coe ffi cient, and where the trace ε γγ corresponds to the relative variation of the four-dimensionalvolume δ V / V of the space-time continuum. If one considers a hydrostatic fluid without shear stress (i.e., β = α corresponds to the bulk modulus of the medium.In this framework, the cosmological term Λ g µν in the standard model of cosmology can be interpreted as a deformation of a4-dimensional medium, where B = − ρ G c = Λ c π G corresponds to the bulk modulus for an isotropic medium. The stress tensor(19) is then added to the Einstein field equations of gravity: R µν − Rg µν = π Gc (cid:16) T µν + σ µν (cid:17) (20)and subsequently we obtain the relations: D µ σ µν ( x ) + D µ T µν ( x ) = T µν is the energy-momentum tensor in the Einstein field equation. This equation means that the total energy in the universeis conserved but that the ordinary matter-energy can be accelerated. Notice that for a general stress tensor σ µν ≡ C µνγδ ε γδ ,equation (21) generalizes (9) (where for the sake of simplicity we put α = β = κ − ). Gathering (20) and (21) with theequation of fields in T µν , we obtain an incomplete set of equations. Thus, there must exist an additional coupling between thematter-energy T µν and the strain field ε µν . The construction of this coupling has been discussed for Minkowski space in theprevious sections and generalized for GR in [27] and [28]. The idea is the following, the e ff ective strain tensor ε µν deforms(linearly) the Lagrangian in such a way that it introduces a coupling ε µν T µν as in (2). Hence, the e ff ect of this additional termis to modify the standard field equations by modifying the covariant derivative D by a linear deformation similar to the oneobtained in equations (4), (7) (deformations of η by ε ). To conclude the variation of inertia of the ordinary matter-energy (21) iscompensated by the divergence of the stress energy of the continuous medium. Somehow, this extension of GR revisits Mach’sprinciple as the inertia (of the matter-energy) also depends on the deformation of the curved space-time that is induced by thematter-energy contained therein. We refer the reader to the paper of Einstein on a related topic [32]. The relativistic theoryof an Aether was discussed several times, see for e.g. [33], [34]. In the present article, our hypothesis is di ff erent and gives arelativistic theory of the deformation of continuous media, for which the geometry is still described by the metric field whereasthe strain tensor is an additional field.Beyond n =
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