Application of Lagrange mechanics for analysis of the light-like particle motion in pseudo-Riemann space

We consider variation of energy of the light-like particle in the pseudo-Riemann space-time, find Lagrangian, canonical momenta and forces. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is compared with the Fermat's and geodesics principles. Equations for energy and momentum of the particle transferred to the gravity field are defined. Equations of the critical curve are solved for the metrics of Schwarzschild, FLRW model for the flat space and Goedel. The gravitation mass of the photon is found in central gravity field in the Newtonian limit.


I. INTRODUCTION
One of postulates of general relativity is claim that in gravity field in the absence of other forces the word lines of the material particles and light rays are geodesics. In differential geometry a geodesic line in case of not null path is defined as a curve, whose tangent vector is parallel propagated along itself [1]. Differential equations of geodesic, which is a path of extremal length, can be found also by the variation method with the aid of the virtual displacements of coordinates x i on a small quantity ω i . When we add variation to material particle coordinate, the time-like interval slow changes, though that leaves it time-like.
Finding of differential equations of the null geodesic, corresponding to the light ray motion, by calculus of variations is described in [2]. In space-time with metrical coefficients g ij it is considered variation of the first integral of these equations where µ is affine parameter. Deriving variation for extremum determination we must admit arbitrary small displacements of coordinates. The variation of integral of η expanded in multiple Taylor series is written as where µ 0 , µ 1 are meanings of affine parameter in points, which are linked by found geodesic. With finding geodesic equations the sum of terms containing variations ω i , dω i /dµ in first power is equated to null that gives the geodesic equations in form where Γ λ ij are Christoffel symbols: Here a comma denotes partial differentiation.
Other terms of series in (1.2), containing variations of coordinates and their derivatives by µ in more high powers or their products and being able to have nonzero values, don't take into account. Thus such method admits violation of condition η = 0, which means that with certain coordinates variations the interval a prior becomes time-like or space-like. Since this interval accords with the light ray motion, one leads to the Lorentz-invariance violation in locality, namely, anisotropies.
The possibility of Lorentz symmetry break for the photon in vacuum by effects from the Plank scale is studied in [3,4]. At the contrary, it is shown for the massive particle [5] that a fundamental space-time discreteness need not contradict Lorentz invariance, and causal set's discreteness is in fact locally Lorentz invariant. However, experiments [6] show exceptionally high precision of constancy of light speed confirmed a Lorentz symmetry in locality, and astrophysical tests don't detect isotropic Lorentz violation [4].
In the method of calculus of variations in the large [7] ones are considered as possible paths along the manifold disregarding kind of interval, not as the trajectories of physical particles. This approach exceeds the limits of classical variational principle in mechanics, according as which virtual motions of the system are compared with cinematically possible motions.
Approximating time-like interval conforming in general relativity to the material particle motion between fixed points to null leads in physical sense to unlimited increase of its momentum, and the space-like interval doesn't conform to move of any object. In this connection it should pay attention on speculation that discreteness at the Planck scale reveals maximum value of momentum for fundamental particles [8].
Geodesic line must be extremal [1], and the test particle moves along it only in the absence of non-gravity forces. Should photon have some rest mass variations of its path don't give different kinds of intervals, but this assumption doesn't confirm by experiments [9]. We examine choosing of energy so in order that application of variational principle to its integral for deriving of the isotropic critical curves equations would not lead to considering non-null paths.

II. DEFINITION OF ENERGY AND ITS VARIATION
The interval in Riemann space-time is written in form where ρ is some quantity, which is assumed to be equal 1. Putting down x 1 as time, coordinates with indexes k, q = 2, 3, 4 as space coordinates and considering ρ as energy of light-like particle with ds = 0 we present it as where σ is ±1. Indexes except k, q take values 1 to 4. With denotation of the velocity four-vector components as v i = dx i /dµ energy variation will be the partial derivatives with respect to coordinates are written as This expression is reduced to (2.6) The partial derivatives with respect to components of the velocity four-vector are For the particle, moving in empty space, lagrangian is taken in form and conforms to relation [10]: which is integral of the motion. Obtained derivatives give canonical momenta and forces (2.11) Components of the associated vector of canonical momenta are Units is chosen so that a light velocity constant is c = 1. Components of energy-momentum four-vector of photon in Minkowski space are proportional to four-velocities: where h is Planck constant and ν is frequency of photon. Normalized effective mass of light-like particle in Riemann's space-time is defined as coefficient of proportionality between components of the vectors of canonical momenta and four-velocities with raised indices. It becomes (2.14) With coefficient of normalization m ef f 0 it can be expressed in terms of effective mass as

III. EQUATIONS OF ISOTROPIC CRITICAL CURVE
Motion equations are found from variation of energy integral Energy ρ is non-zero, its variations leave interval to be light-like, and application of standard variational procedure yields Euler-Lagrange equations Critical curve equations are obtained by substitution of partial derivatives (2.6) and (2.7) in these equations. For derivative of the first component of four-velocity vector we have For finding of other three equations of motion the second term of (3.2) is presented in form d dµ Replacement of derivative dv 1 /dµ here on its expression, obtained from (3.3), and substitution found terms in Euler-Lagrange equations gives These equations contain accelerations corresponded to the space coordinates and coupled with (3.3) describe motion of the test light-like particle along critical curve. They don't coincide to usual form (1.3) of the null geodesics equations.

IV. PHOTON'S DYNAMICS IN SCHWARZSCHILD SPACE-TIME
Central symmetric gravity field in free space is described by the Schwarzschild metric. At spherical coordinates x i = (t, r, θ, ϕ) its line element is where α is constant. For this space we find equations of critical curve of integral energy ρ. Canonical momenta (2.10) for cyclic coordinates t, ϕ are constants of motion Equations (3.5) for coordinates r, θ give  Assuming that A = 1 and considering motion in plane θ = π/2 we write derivatives of cyclic coordinates Substituting these values in equation (4.6) with we find Found velocities coincide with solutions of standard null geodesic equations for the Schwarzschild space-time [2] to within parameter of differentiation where µ s corresponds with standard solution.
Canonical momenta (2.10) and forces (2.11) are (4.11) Nonzero components of the vector of canonical momenta with raised indices are It follows from Eq. (2.14) that normalized effective mass of photon in central gravity field changes as (4.14) A nonzero component of the vector of canonical forces with raised indices is In so far as Newtonian limit of gravity theory with gravitational constant G and mass M requires α = 2GM , the first term of F 2 yields twice Newton gravity force. One conforms to light deflection in central gravity field [11], which is twice value being given by Newton gravity theory. Substituting components of velocity four-vector in Eq. (4.4) we obtain radial acceleration A second Newton Law states that massive body under action of force − → F experiences acceleration − → a = − → F /m, where m is its inertial mass. By analogy normalized inertial mass of photon is found as ratio of canonical force F 2 to radial acceleration Thus excepting area, where signs of F 2 and dv 2 /dt coincide ore one of these quantities is equal to 0, inertial mass of photon is negative, i. e. it experiences antigravirational influence. One is not contradict to deviation of photon towards to center of gravity, because with approach to it the angular velocity (4.8) decreases relatively faster then radial velocity by comparison with motion in the absence of gravity field. Inertial mass of photon can be expressed in terms of effective mass. Euler-Lagrange equation Taking into account definition of normalized inertial mass of photon we obtain This equation yields value of inertial mass of photon in central gravity field, expressed in terms of its effective mass: With r ≫ α, h it turns out m in = −m ef f .

V. EXTREMAL ISOTROPIC CURVES IN FLRW SPACE-TIME
FLRW cosmological model for the flat space with rectangular coordinates x i = (t, x q ) is described by metric where a is length scale factor. Equation (3.3) gives where overdot denotes derivative with respect to time. Euler-Lagrange equations for the cyclic coordinates x q yield constants of motion Having extracted derivatives with respect to space-like coordinates from this equation and substituting them in (5.2) we obtain This equation has solution, which with denotation Π = p 2 q is written in form where B is constant. Substitution found first component of four-velocity vector in equation (5.3) gives Condition, following from Eq. (5.1): corresponds to isotropic curve. It yields B = 0 and components of four-velocity vector turn out to They conform to solution of standard equations of null geodesics for the FLRW space-time [2]. Canonical momenta of light-like particle are and constant p q . Canonical forces are Their associated values is written as where ω is constant. Canonical momenta (2.10) for cyclic coordinates t, y, z are constants of motion. They is written in form These equations with following from Eq. (6.1) condition They differ from solution of standart equations of null geodesics in Gödel's space-time [12]. With p 1 e √ 2ωr = 2p 3 takes place singularity.
Canonical momentum corresponding to coordinate r is Canonical forces have values Proposed form of energy allows applying of Lagrange's mechanics for analysis of light-like particle motion. Considered procedure of production of the motion equations by variation of the energy integral conforms to principles of the calculus of variations in classic mechanics in accordance with which the motion variations must be cinematically admissible for the system. Virtual displacements of coordinates retain path of the light-like particle to be null in Riemann space-time, i.e. not lead to Lorentz-invariance violation in locality. Solutions of extremal isotropic curve equations for metrics of Schwarzschild and Friedmann-Lemaitre-Robertson-Walker for the flat space coincide with solutions of standard null geodesics equations to within appropriate parameter. For the Gödel's space-time these solutions are different.
Normalized effective mass of light-like particle is defined as coefficient of proportionality between canonical momenta and components of four-velocity vector. Analog of Newton's inertial mass for photon in Schwarzschild's space-time has negative value for newtonian limit of gravity.