Regularization of electromagnetic field for self-force problem in de Sitter spacetime

The paper is concerned with the motion of a point electric charge in de Sitter spacetime. A point particle of mass m and charge q moving on a geodesic curve produces electromagnetic field that diverges at a particle’s position. The field is determined by the electromagnetic Green’s function by Higuchi and Lee (2008 Phys. Rev. D 78 084031). The self-force contains both divergent and finite terms, and the latter are responsible for the radiation reaction. Our derivation of an effective equations of motion is based on conservation laws corresponding to the group of isometry of de Sitter space. The Nöther quantities consist of particle’s individual characteristics and energy, momentum, and angular momentum carried by particle’s electromagnetic field. Following the Detweiler–Whiting concept that a charge’s motion should only be enforced by the regular component of its own field, we ignore the divergent terms in conservation laws. We assume that the divergencies are absorbed by particle’s individual characteristics within the renormalization procedure. Finite radiative terms together with kinematic particle’s characteristics constitute ten conserved quantities of closed particle plus field system. Their differential consequences yield the effective equations of motion of radiating charge in an external electromagnetic field and gravitation. Contributions to already renormalized particle’s four-momentum and its inertial mass originated from electromagnetic field and background gravity are also derived from ten balance equations.


Introduction
Recent years have been rapidly growing interest in the problems of electromagnetic and gravitational self-force on a point-like particle moving in curved spacetime. It was due to the development of gravitational wave astronomy [1] and the need to produce accurate gravitational waveforms [2] for future space-based gravitational interferometer LISA; rapid advances in laser technology which produces necessity of correct description of motion of ultrarelativivstic electron in intense electromagnetic fields [3][4][5][6][7]; growing interest in the theory of radiation in spacetime dimensions other than four [8,9]. The problems are intimately connected to each other.
Early studies of the electromagnetic self force is associated with Lorentz, Larmor, and Abraham. Lorentz [10] derived the self force by considering a finite size electron as a small continuously charged shell. Abraham [11,12] generalized the self-force expression to be valid for relativistic charged particle. This is especially interesting since he did it before the advent of special relativity. Having decomposed the particle's electromagnetic field into singular and regular parts, designed from the retarded and the advanced Liénard-Wiechert fields, Dirac [13] produced equation of motion of a point charged particle in flat spacetime. This is well-known Lorentz-Abraham-Dirac (LAD) equation which describes the dynamics of a point charge acted upon an external force as well as its own electromagnetic fieldF: Here u µ (s) = dz µ (s)/ds (µ = 0, 1, 2, 3) is a four-velocity in the position z(s) of the charge world line parameterized by the proper time s, a µ (s) = du µ (s)/ds is its 4-acceleration, andȧ µ (s) = da µ (s)/ds is the so-called 'jerk'. The self force is proportional to the small parameter τ 0 = 2q 2 /(3m) of the dimension of time (6.2664 × 10 −24 s for electron; in the dimension of length τ 0 = 1.8786 × 10 −15 m). The backreaction terms dominate in dynamics of high energy particles moving in extremely strong electromagnetic fields. Such a situation occurs in astrophysics, e.g. relativistic jets of radiation and particles from super-massive black holes in the centers of some active galaxies. In series of papers [14][15][16] the behavior of plasma behavior in a neutron star's magnetosphere is studied. The consideration is based on the LAD equation describing particle's motion in flat spacetime. In 1960 DeWitt and Brehme [17] generalized LAD to curved spacetime, and their calculations was corrected by Hobbs [18] in 1968: The covariant acceleration a µ (s) = Du µ (s)/ds is the covariant derivative of four-velocity u µ (s) along the world line; R ν λ is the Ricci tensor. The external force f µ ext = g µα (qF αβ u β ) is the covariant generalization of the Lorentz force. Expression in the right hand side of equation (1.2), i.e. the self-force, results from two types of particle's interactions with its own electromagnetic field. The first is a purely local and describes the radiation reaction depending on the current instant of time. The second is non-local one resulting from previously emitted radiation: Here G ret µν ′ is the retarded electromagnetic Green function in the curved spacetime [19, equation (1.33)], and ∇ µ is the covariant derivative in µth direction. The square brackets designate antisymmeterization of the indices. This interaction is due to the failure of Huygens principle: in curved spacetime electromagnetic waves propagate not just at the speed of light, but also at all speeds smaller than or equal to the speed of light. For this reason a massive particle 'fills' its own field, which acts on it just like the external one.
Having inspired by Dirac's seminal paper [13], Detweiler and Whiting [20] develop the regularization scheme based on the decomposition of Green's function in curved spacetime. The retarded field is splitted into singular and regular parts being the solutions to inhomogeneous and homogeneous Maxwell equations, respectively. Contrary to regular part, singular Green's function has no support inside the null cone and does not depend on the particle's past history. The regular Green's function had reasonable causal structure. It has no support on the future null cone. The function produces radiation field that detaches itself from the charge and leads an independent existence. The singular part of electromagnetic field is permanently attached to the charge and is carried along with it. It contributes to charge's inertia.
Having used this decomposition, Harte [21,22] investigates the dynamics of extended charged bodies interacting with external fields as well as their own electromagnetic fields. A body moves in an arbitrary curved spacetime. Harte's scheme is based on the concept of approximate space-time symmetries. In the sufficiently small neighborhood of a point on particle's world line the momentarily comoving reference frame is introduced where the region looks nearly flat. Action principle is proclaimed to be invariant with respect to transformations from generalized Poincaré group [21]. Harte performs volume integrations of convolution of the stress energy tensor and generalized Killing fields [22]. Singular component of self field contributes into both the linear and angular momenta of charged body. In approximation of small charge distribution with slow internal dynamics the equation of center-of-mass motion is obtained. In specific case of structureless point charge the equation becomes the well-known DeWitt-Brehme-Hobbs equation (1.2).
It is obvious that the de Sitter group can be substituted for Poincaré group in Harte's scheme [21,22]. In the standard theory of general relativity, de Sitter space is a highly symmetrical special vacuum solution. The isometry group of de Sitter space is the indefinite orthogonal group SO(4, 1) which contracts to Poincaré group for short distance kinematics. The choice of this slightly curved spacetime as the basic approximation of background gravity seems to be more adequate than the flat Minkowski space. The decomposition of the stress-energy tensor into singular and regular parts is necessary.
In the present paper, we study the self action problem of charged particle arbitrarily moving in de Sitter spacetime. We adopt the generalized Detweiler-Whiting axiom which states that the Lorentz force law involves the regular part of charge's self field only. We shall take the first step towards understanding how a gravity and a charge's electromagnetic potential energy change inertial properties of a charged particle, i.e. its individual momentum and its mass [22, equation (59)].
Topologically, de Sitter space is the direct product R × S 3 where R is time axis and S 3 is three-dimensional sphere [23]. The non-trivial topology makes electrodynamics very specific. A flux integral vanishes identically if there is no boundary surface in a compact manifold. The Gauss law says that the total outward flux of electromagnetic field through a closed surface is equal to the total charge of particles enclosed by this surface. So, the total charge in de Sitter space must be zero. These circumstances drastically change the method of usage of the retarded Green's function [24]. To reproduce the electromagnetic field, a set of initial data not only on a given point source is necessary, but also on a Cauchy surface consisting of points in the past of the field point. These circumstances make the self action problem highly nontrivial.
The question arises as how to deal with DeWitt-Brehme-Hobbs equation (1.2) in case of de Sitter metrics. One of our tasks is to figure it out.
Another task is to estimate the gravity correction for electron. We shall compare it with the small parameter τ 0 of LAD equation (1.1). Despite the fact of rapid development of experimental astrophysics, a man-maid machine only allows us to study the dynamics of radiating electron in very detail. The machine is multi-petawatt laser system [7]. Because of the third time derivative, LAD equation possesses pathological solutions [25][26][27], such as runaway solution (when acceleration grows exponentially with time) and preacceleration (when acceleration begins to increase prior the time at which the external force switches on). The reduction of order is necessary.
Landau and Lifshitz [28, § 76] replace the square of acceleration in the Larmor term by the negative scalar product (u ·ȧ) and substitute m −1 df µ ext /dτ for the problematic derivative of particle's four-acceleration ('jerk'). The Landau-Lifshitz (LL) equation is of Newtonian class which avoids the nonphysical solutions of the LAD equation. LL approximation of LAD equation serves as equation of motion of radiating electron in an intense laser pulse in most experiments [7,29,30]. Validity of this approximation depends on the magnitude of applied force as well as type of external electromagnetic field [3][4][5]7].
In [4] the series of the second order equations is investigated. They are obtained by successive iteration of order reduction procedure applied to LAD equation. The expansion parameter is τ 0 ω where τ 0 = 2q 2 /(3m) and ω is a typical field frequency scale. Since the parameter is extremely small, the procedure can be truncated at any order of τ 0 . The nth order approximation is denoted as LL n . The LL equation represents only the first of an infinite series of approximations to LAD. The authors [4] show that the perturbation expansion generated by reduction of order has zero radius of convergence. It is because the coefficients in series expansions are alternate in sign (see [4, equations (6) and (7)]). An infinite order approximation LL ∞ is also evaluated. The properties of the iterative procedure demonstrates that effects of nonzero curvature of Universe will be important at some iteration.
The paper is organized as follows. In section 2 we sketch the topology of de Sitter space in context of CPT-symmetric Universe [31]. We draw a Penrose diagram illustrating a compactified de Sitter space. We study geodesic curves on the one-sheeted hyperboloid modeling the space. In section 3 we analyze the electromagnetic field produced by a point charge +q placed at the so-called 'North pole', and a charge −q at the 'South pole' of de Sitter space. The form of Penrose diagram of CPT-symmetric de Sitter space allows us to apply the retarded Green's function in usual way. A Cauchy surface is not necessary. In section 4 we develop the regularization procedure based on the global conservation laws associated with ten linearly independent Killing vectors of de Sitter space. We derive the equation of motion of a point-like charge acted upon an external force as well as its own electromagnetic field and background gravity. In section 5, we summarize the main ideas and results.
As de Sitter space is the slightly curved Minkowski spacetime, it is reasonable to use the stereographic coordinates being 'a projection the de Sitter hypersurface into target Minkowski spacetime' [33][34][35]:
Analogously, when the chart (a) is considered, the conformal factor (2.9) does not vanish within the domain D a and coordinate transformation (2.12) is correctly defined. The segments χ ± τ = ±π belong to the domain D b . At these points the conformal factor (2.11) is equal to 1. According to equation (2.6), this implies (x · x) = 0. The light rays begin at point (τ = 0, χ = ±π) and end at infinities, either past or future. They are pictured in figure 1.
Causal structure of de Sitter space is induced by that in 'host' Minkowski spacetime of five dimensions [23,§ II.7]. The interval between two points, x and x ′ , in dS 4 is nothing but the projection of interval in five-dimensional flat spacetime on the hyperboloid (2.1): The interval is proportional to the standard one of special relativity. Thus two events, x and (2.14) Standard calculations yield Riemann tensor [32, equation (1.3.6)] It is easy to show that the tensors satisfy the source free Einstein field equation

Geodesics in de Sitter space
The full geodesic equation is where s is the proper time defined by the relation Here sign '−' stands for time-like geodesics while sign '+' defines space-like ones. Inserting the Christoffel symbols (2.14) we obtain the equation on geodesics in de Sitter spacë The 'dot' denotes the scalar product in flat spacetime of four dimensions, e.g. (x ·ẋ) = η αβ x αẋβ . According to equations (2.8) and (2.19), the squared four-velocity (ẋ ·ẋ) takes value either −Ω −2 for time-like geodesics or +Ω −2 for space-like ones. We overmultiply the equation on −(H 2 /2)Ω 2 (x)x µ . After some algebra we derive the nonhomogeneous second-order differential equation where parameter κ = Ω 2 (x)(ẋ ·ẋ) can takes three values: Rewriting the projection relation (2.5), we express the conformal factor Ω in terms of 4th coordinate of the 'host' Minkowski space of five dimensions: Inserting this in equation (2.21) we obtain homogeneous second-order differential equation on the fourth space coordinatë We deal with the manifestly covariant theory. It is reasonable to assume that the coordinates (2.4) satisfy analogous equation, namelÿ Let us prove that it is equivalent to geodesic equation (2.20). Substituting Ω(x)x µ for y µ we derive the equation Initial values satisfy the constraint (2.1) and its differential consequence: These yield the following relations on vectors A and B involved in equation (2.29): It is easy to show that the proper time condition Ω 2 (x)(ẋ ·ẋ) = −1 is equivalent to the constraint The initial five-velocity satisfies this relation too. It produces the relation Recall that the 'dot' denotes the scalar product in flat spacetime of four dimensions, e.g.
We expand the argument of hyperbolic cosine in powers of small parameter H. Restricting ourselves to the first nontrivial term, we obtain ρ(x, As would be expected, the approximated interval reduces to the difference of proper instants s and s ′ depicting two points on time-like geodesic line in flat Minkowski space of four dimensions. where constants B 1 = (H/2)y 4 (0) and Putting κ = +1 equation (2.25) we see that the coordinates y α also satisfy the oscillatory equations. Since y α = Ω(x)x α , the space-like geodesic curve ξ is spanned by coordinate functions It produces the relation Expanding the argument of cosine in powers of H we obtain ρ(x, x ′ ) = (s − s ′ ) 2 after passing to H → 0 limit. The approximated interval reduces to the difference of proper instants s and s ′ depicting two points on space-like geodesic line in flat Minkowski space of four dimensions.

Null geodesics.
In the domain D a the locus of points satisfying (x · x) = 0 are specified by the relations χ = ±τ where time τ ∈ [0, π/2). Inserting this in equation (2.10) we find out the coordinates of points of null cones in the standard form Taking into account equation (2.13), we see that points on null cones with vertex at point x ′ can be parameterized as x µ = x ′µ + rk µ where r is affine parameter and (k · k) = 0. This circumstance allows us to introduce the retarded coordinates in de Sitter space (see appendix C).

Electric charge in de Sitter space
The space-like geodesics in de Sitter space are expressed in terms of trigonometric functions which repeat their values at regular intervals. This reflects the circumstance that dS 4 is topologically R × S 3 where R is time axis. A flux integral vanishes identically if there is no boundary surface in a compact manifold. In [24] the electromagnetic field produced by a point charge +q placed at the so-called 'North pole', and a charge −q at the 'South pole' has been calculated. Both the particles are at rest.
Let us consider time-like geodesics of two opposite charges in CPT-symmetric Universe [31]. Gauss law is satisfied because to each charged particle in our Universe corresponds antiparticle in its mirror counterpart consisting of antimatter. Penrose diagram of CPT-symmetric Universe is pictured in figure 1. It is quite different from that in [24, figure 1(b)]. The world line of a static charge placed at North pole originates from Big Bang moment τ = 0, not from remote past τ = −π/2. As a consequence, the charge moves inside the causality region, not at its edge (cf figure 1 and [24, figure 1(b)]). For this reason, the electromagnetic potential is related to Green's function in usual way in CPT-symmetric de Sitter space.
De Sitter space is slightly curved spacetime in the absence of matter or energy. This residual curvature implies a positive cosmological constant Λ = 3H 2 determined by observation. The inverse Hubble length (7.2817 ± 0.0584) × 10 −27 m −1 corresponds to the Hubble constant H = 67.36 ± 0.54 (Km s −1 ) Mps −1 [36, equation (14)]. Let us imagine a proton that was born at the moment of Big Bang, i.e. T U = 13.7 billion light years ago. Dimensionless argument of hyperbolic cosine is HT U ≈ 0.9438. The zeroth coordinate x 0 (T U ) ≈ 12.7661 billion light years. If the dynamics of a charged particle in the processing chamber of a man-made machine is studied, the Big Bang looks as an event at the remote past. Another important point, the Horizon 2/H ≈ 29.03 billion light years, is the very distant future.
Do we have calculate time and distance from the Big Bang in de Sitter space? The answer is negative. De Sitter space is the quotient of the pseudo-orthogonal group SO(4, 1) and the Lorentz group L, namely dS 4 = SO(4, 1)/L [33]. Transformations from ten-parameter group SO(4, 1) do not change the interval (2.13). The group manifold is a bundle with de Sitter space as base space and the Lorentz group L as fiber. Similarly to Minkowski spacetime, we can choose the coordinate origin at an arbitrarily point and direct coordinate axes as we like a.s. For example, see [37] where well-known Twin Paradox in de Sitter Space is studied. Transitions between inertial frames is governed by the Lorentz group.
The first charge 'lives' in our Universe while the second one exists in its mirror counterpart consisting of antimatter. These worlds are not causally connected. With a great degree of accuracy we can put the interval s ∈ (−∞, +∞) instead of the time interval from Big Bang to Horizon in our Universe.
Let a point-like charge q moves along a world line ζ ⊂ U parameterized by four functions z α (s) of proper time parameter s. The charged particle produces current where u ν (s) = dz ν (s)/ds is the four-velocity. We take into account that in curved spacetime the δ-function definition involves the determinant of the metric tensor −g(x) (see [19, equation (13.2)]). Corresponding electromagnetic potential is where G ret µν ′ (x, x ′ ) is the retarded Green's function [24]. DeWitt-Brehme-Hobbs equation (1.2) is valid in de Sitter space. The equation adapted for de Sitter metric (2.8), Christoffel symbols (2.14), and Ricci tensor (2.16) does not distinguish between short-range and long-range radiations. To find corresponding expressions we apply Harte's scheme [22] to the self action problem of charged particle arbitrarily moving in de Sitter spacetime.

Maxwell equations in a curved spacetime
At each point of a four-dimensional Riemannian manifold (M, g), a given current densityĵ generates the electromagnetic field which satisfies the Maxwell equations where 4π is the area of unit three-dimensional sphere. For a given coordinate chart (U, x) on M, the field is determined by the Faraday 2-formF = 1 2 F αβ dx α ∧ dx β where the differentials of a set of coordinates serve as basic 1-forms. To define the Hodge dual 2-form ⋆F =ω(F) we construct two-vector F = 1 2 F µν e µ ∧ e ν with components and then find the dual of F with respect the volume 4-formω. Tensor g µα is the reciprocal of the metric tensor g αβ which induces this 4-form (3.10) We denote g(x) the determinant det |g αβ | of the metric tensor. So, the components of the Hodge dual 2-form ⋆F = 1 2 ⋆ F γδ dx γ ∧ dx δ heavily depends on metrics: Levi-Civita symbol ϵ γδµν is defined by 0 if any two labels are the same, +1 if γ, δ, µ, ν is an even permutation of 0, 1, 2, 3, and −1 if the set of labels is an odd permutation of these numbers. The volume form (3.10) is oriented as ϵ 0123 = +1. Hodge dual 3-form ⋆ĵ =ω(j) of a four-vector current j = j ν e ν is Routine calculations yield the second pair (3.8(b)) of Maxwell's equations in curved spacetime: Despite the use of partial derivatives, these equations are covariant under arbitrary curvilinear coordinate transformations. Thus if one replaces the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out [38]. Let us consider the first pair (3.8(a)) of Maxwell's equations in curved spacetime. We suppose that the Faraday 2-form is exact on coordinate chart (U, x) ⊂ M:F = dÂ. The electromagnetic field is a covariant antisymmetric tensor of degree 2 which can be defined in terms of the 1-form electromagnetic potential by where ∇ α denotes the covariant derivative. Inserting the second line of equation (3.14) into equation (3.13) and adopting the covariant Lorentz gauge one arrives at the well-known wave equation for the electromagnetic vector potential in a curved spacetime. Here □ = g µν ∇ µ ∇ ν is the covariant wave operator and R α β is the Ricci tensor. The Green's function technique is applied to solve the wave equation [19,20,24]. The retarded electromagnetic potential is given by equation (3.7) where G ret µν ′ (x, x ′ ) is the retarded Green's function of the equation (3.16). Inserting the square root of negative determinant √ −g = Ω 4 (x) and the reciprocal of de Sitter metric tensor g µα = Ω −2 (x)η µα into Maxwell's equation (3.13) we obtain

Electrodynamics in de Sitter space
It is of great importance that the conformal factors disappear in the left hand side of this equation. This circumstance motivates us to substitute ∂ α A β − ∂ β A α for F αβ (see the first line of equation (3.14)). We arrive at To simplify the equation we adopt here the non-covariant Lorentz gauge The potential satisfying this condition differs from that satisfying the covariant Lorentz gauge (3.15). Thus we use the notation A µ . Finally we obtain the wave equation where □ = η µα ∂ µ ∂ α is the standard d'Alembert operator in flat spacetime. It is of great importance that the left hand side of this equation is just the same as in Minkowski space. We apply the Green's function method (3.7) where we take into account that −g(x ′ ) = Ω 4 (x ′ ). The function G(x − x ′ ) gives a response to δ-shape excitation where □ is the standard d'Alembert operator in Minkowski space. There are two solutions: (causal) retarded Green's function and its advanced (acausal) counterpart. We restrict ourselves to the well-known retarded solution to equation (3.22) The retarded Green's function has support only on the past light cone of the field point x. We insert the function (3.23) and the δ-shape current (3.6) in equation (3.21). After some algebra we obtain the retarded potential in de Sitter space The potential is inversely proportional to the standard retarded distance in Minkowski space 25) i.e. to the scalar product on null linking vector K α = x α − z α (s ret (x)) and particle's fourvelocity, taken with opposite sign. For further convenience Greek indices are raised and lowered with the Minkowski metric tensorη = diag(−1, 1, 1, 1) in the present paper. With this in mind we rewrite the potential (3.24)

in the form
The denotation is not mathematically correct. Indeed, in de Sitter space we apply the covariant metric tensor g αβ = Ω 2 (x)η αβ in the operation of lowering indices. However, usage η αβ seems very natural because de Sitter metric tensor is proportional to Minkowski metric tensor.
In appendix A we demonstrate that the potential (3.26) in the non-covariant gauge (3.19) is gauge equivalent to the Higuchi-Lee potential (A.17) in the covariant Lorentz gauge. Recall that in terms of stereographic coordinates the line element (2.8) of de Sitter Universe satisfies the conformally flat space condition. Hobbs [40] proved that in spacetime with conformal metric the 'tail' term (1.3) vanishes (see also [41]). In appendix B we show that the integral (1.3) defining the nonlocal tail term of the radiation reaction force vanishes in de Sitter space.
In this specific case the scheme developed by Detweiler and Whiting is based on the symmetric Green's function [20, equation (6)] where tail function v(x, z) vanishes: We denote T = x 0 − z 0 and R i = x i − z i . The inhomogeneous field is the one-half sum of the retarded and the advanced solutions, while the homogeneous one becomes the one-half difference of these fields.

Electromagnetic field in de Sitter space.
The retarded electromagnetic field in de Sitter space F αβ = ∂ α A β − ∂ β A α can be derived from the potential (3.26) by using the differentiation rule The adjective 'normalized' means the property (k · u) = −1. It allows us to differentiate a function of x which contains an implicit reference to the retarded time s. In particular where (u · u) = η µν u µ u ν and (a · k) = η µν a µ k ν . It is worth noting that for time-like geodesics (u · u) = −Ω −2 (z) (see normalization relation (2.19)).
The rule can be derived as follows. Let us consider the interval (2.13) where x be a field point while x ′ is the source point z(s) ∈ ζ at which the world line ζ ⊂dS 4 intersects the past light cone of x. The points are linked by null geodesic: ρ(x, z(s)) = Ω(x)Ω(z(s))(x − z(s)) 2 = 0. We assume that neither field point nor source point lie on the surface where either Ω(x) or Ω(z(s)) vanishes. Therefore, in de Sitter space the null geodesics lie on the surface of Minkowski light cone (x − z(s)) 2 = 0.
Suppose that the field point x is shifted to the new point x + δx. This infinitesimal displacement yields corresponding change in the retarded time s + δs. New points, x + δx and z(s + δs), are related by null geodesic: ρ(x + δx, z(s + δs)) = 0. Expanding this in series of δx and δs, restricting ourselves to the first order in these small parameters, and using (x − z(s)) 2 = 0 we obtain K α δx α + rδs = 0, (3.30) where K α = x α − z α (s) is the linking null vector and r is the retarded distance (3.25). This immediately gives the differentiation rule (3.28). These allows us to derive the components of retarded electromagnetic field 2-form. After some algebra we obtain (3.31) All the particle's kinematic characteristics are referred to the retarded time s ret (x) associated with point x where the field strengths are measured. The centered dot designates the scalar product of two four-vectors produced by metric tensor of flat spacetime.

Radiation reaction force
For a point charge arbitrarily moving in flat spacetime the Harte's procedure [22] yields the flows of energy, momentum, and angular momentum through thin tube around particle's world line. At first the calculation of linear momentum flow has been performed by Teitelboim in his pioneering paper [42] on the derivation of the LAD equation (1.1) within the frame of retarded causality. The components of momentum four-vector carried by electromagnetic field of a point-like charge are where dσ µ is the vectorial surface element on a space-like surface σ(t). In [42, figure 1] the surface σ(t) is the section of future light cone and covariant hyperplane being orthogonal to particle's four-velocity. Poisson [43, figure 3] calculates flows through thin world tube r = const enclosing particle's world line. In [44] the hyperplane x 0 = const is used.
The stress-energy tensor is composed from the retarded Liénard-Wiechert field strengths (see equation (3.31) where Ω = 1.) It admits a natural decomposition into singular (bound) and regular (emitted) components: Each term has been labeled according to its dependence on the distance r. So, the subscript (−2) means that the radiative term is inversely proportional to the squared distance. Volume integration of the radiative partT rad =T (−2) of the stress-energy tensor (4.2) gives the Larmor relativistic rate of power of electromagnetic waves that detach themselves from the charge and lead an independent existence. Integration of the bound partT bnd =T (−4) +T (−3) yields singular terms inversely proportional to the distance. They constitute the bound part of electromagnetic radiation which is permanently attached to the charge and is carried along with it. It contributes to charge's inertia.
A point-like charge is the limiting case of an extended body. Teitelboim's decomposition is in line of the Harte's scheme. In order to calculate the electromagnetic field contribution in charge's inertia, the bound component T αβ bnd should be inserted in between the round brackets in [22, equation (35)]. The termT (−4) is composed from the velocity field [42, equation (2.3a)] which is inversely proportional to the squared distance. While the termT (−3) contains also the acceleration field [42, equation (2.3b)] scaled as 1/r. The singular field F S involved in [22, equation (35)] can not be extracted from the retarded Liénard-Wiechert field. According to [22], F S originates from new symmetric Green's function [20, equation (16)!]. In case of flat spacetime it is one-half of the sum of the retarded and advanced field strengths. The stressenergy tensor composed from F S is not equal toT bnd . Regular field F R becomes one-half difference of the retarded and advanced field strengths. The stress-energy tensor composed from F R is not equal toT rad . Their sum has no of true physical sense.
Similarly, the torque of the stress-energy tensor which defines the angular momentum tensor can be decomposed. The bound and radiative parts of integrand in equation (4.4) are as follows [44]: , where s is the retarded instant andT (−3) ,T (−4) denote the parts ofT bnd which are scaled as r −3 and r −4 , respectively.
In [44] the LAD equation (1.1) is derived via analysis of ten conserved quantities corresponding to Poincaré invariance of a closed system consisting of a point-like charge and its electromagnetic field: It is natural to assume that the divergences are absorbed by particle's individual characteristics.
They are proclaimed to be finite and measurable. The radiative terms which leave the charge and lead to an independent existence are valuable for conservation laws: (cf equation (4.7)). The system is then placed in an external field. Changes in particle's mechanical 4-momentum p α part and momentum p α rad carried by particle's electromagnetic field should be balanced by external force:ṗ α part +ṗ α rad = f α ext . Similarly, changes in angular momenta should be balanced by torque of the external force:Ṁ αβ part +Ṁ αβ rad = z α f β ext − z β f α ext . Careful analysis of ten algebraic equations yields the expression for particle's mechanical four-momentum and its time derivative dp µ part ds + 2q 2 3 (a · a) u µ = f µ ext . (4.10) The second term in equation (4.9) describes the electromagnetic field's contribution to particle's inertia. Substituting the left hand side of equation (4.9) for p µ part we obtain the LAD equation (1.1).

Radiation reaction force in de Sitter space
The dynamics of a free massive charge in the de Sitter space is governing by the Lagrangian where g αβ (z) = Ω 2 (z)η αβ . Here overdot denotes differentiation with respect to any time parameter τ . Variation of action based on this function yields the equation of motion dp α dτ where p α is particle's four-momentum and (ż ·ż) ≡ η αβż αżβ . The equation (4.12) can be rewritten as manifestly covariant one: z µ ∇ µ (p α ) = 0. In coordinate treatment we have dp α dτ − Γ γ αβ p γ dz β dτ = 0, (4.14) where Γ γ αβ are Christoffel symbols of the 2nd kind (2.14) (see also equation (9) in [34]). The equation, completely rewritten using proper time s satisfying the condition g αβ (z)ż αżβ = −1, takes the form The field enforces it to deviate from geodesic motion (cf equation (4.14)). In terms of kinematic quantities u(s) = dz(s)/ds and a(s) = du(s)/ds the equation of motion of test charge in de Sitter space takes the form where the external force f µ ext = g µα (qF ext αβ u β ). All tensors are evaluated at z(s), the current position of the particle on the world line. The equation governs the evolution of a test charged particle subject only to background gravity and external electromagnetism. Our next task is to calculate the self-force due to particle's own field.
We base our consideration on the balance of conserved quantities which, according to Nöther theorem, correspond to the symmetry of the problem. In [33][34][35]45] the idea of de Sitter invariant special relativity has been considered in very details. The group of isometry SO(4, 1) of de Sitter space involves the Lorentz group SO(3,1) as the subgroup. The space rotations describes isotropy of space while the boosts indicates the equivalence of inertial frames of reference. The set of generators of corresponding infinitesimal transformations [33, equation (14)] consists of the usual space rotations J k = ε k ij L ij and boosts K i = L i0 . The combination of usual space translations and proper conformal transformations constitute four non-commutative generators which are responsible for homogeneity. The generators [33, equation (15)] reduce to usual space-time translations in vanishing cosmological limit H → 0.
According to the Nöther theorem, the group of isometry SO(4, 1)of de Sitter spacetime produces ten conserved quantities: • de Sitter momentum [45, equations (27) and (28)] (4.20) • angular momentum [45, equation (29)] The tangent momentum four-vector p β is related to its cotangent counterpart (4.13) by means of the operation of rising indices. In the proper time parametrization p α = mΩ 2 (z)ż α and, therefore, p β = g βα (z)p α = mż β . When considering an isolated system of a point-like charge and its electromagnetic field, the conserved momentum is where the second term is the de Sitter momentum carried by the particle's electromagnetic field (3.31): By r β we denote the coordinate partial derivative (3.29) of the retarded distance (3.25). The outward-directed surface element (C.7) is derived in appendix C. Similarly, the total conserved angular momentum is with electromagnetic field's contribution The stress-energy tensor of de Sitter space is 4π where T αβ (x) is the stress-energy tensor (4.2) of Minkowski space. The retarded electromagnetic fieldF is given by equation (3.31).
In this section we derive the radiation reaction force from ten balance equations (4.22) and (4.24). Changes in total de Sitter momentum Π α and angular momentum Λ αβ should be balanced by the external Lorentz force and its torque: As the factor Ω −6 (x) in the electromagnetic stress-energy tensor (4.26) cancels the factor Ω 6 (x) in field's contributions to conserved quantities (4.23) and (4.25), the time derivatives of this quantities take the forṁ Angular integration can be handled via relations (D.8) presented in appendix D. Resulting expressions depend on kinematic quantities referred to the particle's current position z(s) on the world line. They consist of (short-range) divergent terms and distance-free radiative terms which leave the point source and lead to independent existence. In the pioneering spirit of Detweiler-Whiting [20] and Harte [22], it is natural to assume that the divergences are absorbed by particle's individual characteristics. Then they are proclaimed to be finite and measurable. The radiative terms which leave the charge and lead to an independent existence are valuable for balances of conservation laws only. Having performed the solid angle integration, we obtaiṅ We substitute they for electromagnetic field contributionsπ α em andλ αβ em in the balance equations (4.27) and (4.28), respectively. We obtain the system of ten linear algebraic equations on eight components π α part andπ α part . After some algebra we derive the following equation on π α part : where ω(z) = (2 − Ω −1 ) −1 . The solution is where M(s) is a scalar function which can be interpreted as inertial mass. It will be defined below. Assuming this we take into account that in curved spacetime 'm is not usually conserved, but rather varies according to' [22, equation (59)].

Conclusions
We derive the electromagnetic self force in de Sitter space via analysis of Nöther conserved quantities corresponding to the group of isometry SO (4,1). The derivation involves the volume integration of electromagnetic field's stress-energy tensor which has a singularity on a particle trajectory. The consideration is based on the splitting of electromagnetic field's stress-energy tensor and its torque into bound and radiative parts which contribute to the self force differently [42]. Volume integration of bound parts yields divergent terms which are permanently 'attached' to the charge and are carried along with it. They contribute in particle's mechanical momentum (4.35). In addition to the usual Schott term, the mixed term arises because of interaction of electromagnetic field and gravity. Volume integration of radiative parts produces long range electromagnetic radiation which dissipates in space. The amount of radiated energy, momentum, and angular momentum depends on all previous evolution of a source. The changes in energy-momentum and angular momentum carried by the electromagnetic field should be balanced by corresponding changes of particle's 4-momentum and angular momentum, respectively. Radiation reaction force is the consequence of these balance equations.
The intriguing phenomenon is discovered: a moving electric charge in expanding Universe changes its rest mass because of interaction of particle's electromagnetic field and gravitation (see equation (4.38)). This was previously established for a scalar charge [46,47]. A similar effect occurs in flat (2 + 1) Minkowski spacetime [48, equation (3.27)]. The mass change does not lead to dismission of energy conservation.
The Green's function of a point-like charge in de Sitter space derived by Higuchi and Lee [24] consists of terms being proportional to Dirac δ-function and Heaviside θ-function of Singe world function. The retarded electromagnetic field should spread not only along a light-cone surface, but also within its interior. Both the self-action and interaction between charged particles should be nonlocal in time, i.e. should depend on a whole particle history [19]. Nevertheless, we demonstrate that in de Sitter space a time-nonlocal term of the electromagnetic potential of point charge originated from Higuchi and Lee Green's function can be turned into the local expression by gauging out its nonlocal part. Moreover, this expression does not contribute to the self force. The self-action comes entirely from the basic local term of the electromagnetic potential and has form (3.26). It is a minimal generalization of the flat-space Lorentz-Abraham-Dirac expression to the de Sitter geometry.
As for the possible experiment with electron in multi-petawatt laser system, the order of gravity corrections is tiny: τ 0 H 2 ≈ 10 −67 m −1 . They are important for the fifth iteration of reduced LAD equation (1.1).
In the recent paper [6] the LL equation for a charged particle acted upon an external scalar field is investigated. The problem is reformulated in terms of curved spacetime with constant curvature. Coupling the scalar field potential Φ(x) with the particle's rest mass m, the authors form the position dependent mass M = m + Φ(x) which determines the conformal factor in the conformally flat space metric tensor. The dynamics of a point-like charge in the scalar field Φ(x) is reduced to the geodesic motion of a free particle in curved spacetime.
The special attention is paid [6] to the Nöther quantities arising from the symmetry of the problem with respect to the transformations from Poincaré group. The authors restrict themselves to analysis of evolution of the energy and momentum which correspond to the time and space translations, respectively. The other isometries, i.e. space rotations and boosts, are not considered. Subtle balance between bound Schott term and long-range Larmor contribution to the total self-force expression has been studied in very details. The authors conclude that the scalar self-force 'is not frictional, even in the ultra-relativistic limit, and can even be anti-damping. This leads to behavior that would not be expected from radiation reaction alone, such as a particle in a time-independent field gaining energy.'

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
Another result we shall need is the outward-directed surface element of a cylinder r = const in de Sitter space Σ α = Ω 4 (x)Ω 2 (z) ∂r ∂x α r 2 sin ϑ dsdϑdφ, (C.7) where coordinate derivative of the retarded distance is given by equation (3.29). and new evolution parameter dτ = Ω −1 ds. Conformal factor Ω is referred to z(s). These yield new four-acceleration Inserting these into equation (3.31) we rewrite the retarded electromagnetic field in terms of new variables where the Lorentz matrix takes usual form Our next aim is to derive the relations defining the solid angle integration of polynomials of components of null vector K defined by equation (D.6). The angular integration can be handled via the relations which involve spatial components of four-vector (C.4) Using the relation η αβ Λ α α ′ Λ β β ′ = η α ′ β ′ and taking into consideration that Λ α 0 ′ = U α , we derive the following relations 1 4π π 0 dϑ sin ϑ 2π 0 dφ K α = U α , In terms of new variables (D.1)-(D.4) the stress-energy tensor (4.2) decomposed into the bound and radiative components (4.3) the form Recall that we assume that r = const. There is the total time derivative in the right hand side of equation. Having integrated it, we finally obtain 14) The Coulomb-lire singularity should be coupled with particle's individual momentum (4.20): The singularity is absorbed by particle's momentum within the regularization procedure. The angular momentum (4.30) carried by electromagnetic field also consists of two terms, singular and regular: In flat spacetime one usually assume that the parameter m involving in particle's action term is the unphysical bare mass. It absorbs inevitable infinity within the regularization procedure and becomes the observable rest mass of the particle. The bare mass parameter is in Lagrangian (4.11) governing the dynamics of a free massive charge in the de Sitter space. In section 4.1, the expression (4.35) for particle's momentum is derived where the contribution due to its own electromagnetic field is taken into account. The time-dependent mass function (4.38) contains the observable mass m which absorbs the Coulomb-like divergency which arises in bound momentum (D.14) and angular momentum (D. 19).