Post-Newtonian limit: second-order Jefimenko equations

The purpose of this paper is to get second-order gravitational equations, a correction made to Jefimenko’s linear gravitational equations. These linear equations were first proposed by Oliver Heaviside in [1], making an analogy between the laws of electromagnetism and gravitation. To achieve our goal, we will use perturbation methods on Einstein field equations. It should be emphasized that the resulting system of equations can also be derived from Logunov’s non-linear gravitational equations, but with different physical interpretation, for while in the former gravitation is considered as a deformation of space-time as we can see in [2–5], in the latter gravitation is considered as a physical tensor field in the Minkowski space-time (as in [6–8]). In Jefimenko’s theory of gravitation, exposed in [9, 10], there are two kinds of gravitational fields, the ordinary gravitational field, due to the presence of masses, at rest, or in motion and other field called Heaviside field due to and acts only on moving masses. The Heaviside field is known in general relativity as Lense-Thirring effect or gravitomagnetism (The Heaviside field is the gravitational analogous of the magnetic field in the electromagnetic theory, its existence was proved employing the Gravity Probe B launched by NASA (See, for example, [11, 12]). It is a type of gravitational induction), interpreted as a distortion of space-time due to the motion of mass distributions, (see, for example [13, 14]). Here, we will present our second-order Jefimenko equations for gravitation and its solutions.


Introduction
In general relativity gravitational interaction is interpreted as deformation of space-time due to the presence and movement of masses. We can read in almost all books related to general relativity about the relationship existing between matter and space-time: "space-time tells matter how to move, matter a e-mail: dperezcarlos@gmail.com (corresponding author) tells to space-time how to curve" (see, [15]). This idea arose from the conclusion of Einstein that the field variable for the gravitational field must be the metric tensor of the Riemann space-time g μν , and that this quantity is determined by the distribution and motion of matter, this is the link between matter and geometry. General relativity theory is often mentioned by various authors as one of the most important theories developed in the last century. In words of the Nobel prize R. Feynman: "Einstein's gravitational theory, which is said to be the greatest single achievement of theoretical physics, resulted in beautiful relations connecting gravitational phenomena with the geometry of space; this is an exciting idea." [16]. Although this relationship between matter and geometry of space-time has been widely spread between the most of scientist, this was not the central result that Einstein wanted to highlight, this idea never was accepted by Einstein, instead of this, was the unification of inertia and gravity. In fact, the idea of the geometrization of gravity began with the work of Weyl. For a detailed discussion, see the paper, Why Einstein did not believe that general relativity geometrizes gravity [17].
To obtain the second-order Jefimenko equations, we will use Post-Newtonian approximation on the Einstein field equations. The same system of non-linear equations can be obtained from other approaches such as the system of equations obtained by Logunov.
The Jefimenko equations for gravitation were first obtained by Heaviside in [1], making use of Gibbs vector analysis, supposing the existence of a second gravitational field arising as a kind of gravitational induction. Years before Einstein wrote about this possibility in his not-so-well-known work Gibt es eine gravitationwirkung die der elektromagnetischen induktionswirkung analog ist? [18]. But also can be obtained from the approach developed by Arbab in [19] making use of Hamilton's quaternions, from the linearised field equations as were obtained by González Segura in [20]. Jefimenko has obtained this set of equations by postulating an analogy between his retarded solutions for the electromagnetic field and solutions for the gravitational field.
We will compare the resulting system of equations with those quantities obtained by Jefimenko and we will establish their respective solutions as a limit. Also, we will see that the Jefimenko equations give us the Newtonian theory for the limit v/c → 0, where c is the speed of propagation of plane wave solutions of the gravitational field in the absence of sources 1 .

Second-order Jefimenko equations
Jefimenko equations for gravitation are given in [9,10] by where g is the ordinary gravitational field due to the presence of masses and that acts on masses, whether they are moving or not, k is the Heaviside field (the field that is generated by the movement of masses and acts only on moving masses), ρ is the mass density, J = ρv is the mass current density with v the velocity of the mass, G is the universal constant of gravitation. The analogy is not perfect, owing to that in electromagnetic theory there are two kinds of electric charges, in Jefimenko's theory of gravitation, there is only one kind of mass. Whereas in the first the magnetic field is right-handed, in gravitational theory is left-handed. In the former, there is a flux of electrons inside a conductor, while in the gravitation, the mass current is the movement of the matter as a whole.
We will derive non-linear expressions for the Jefimenko equations from the Einstein field equations: where G μν = R μν − 1 2 g μν R is the Einstein tensor which determines the geometry of the space-time, due to the presence of the source, G is the gravitational constant, T μν is the energy-momentum tensor of matter. These equations can be written as where T = T μ μ is the trace of the tensor T μν , R μν is the Ricci tensor obtained from the contraction of the Riemann tensor R ρ μσ ν for ρ = σ , that is, and ρ μν are the Christoffel symbols, given in terms of the metric tensor of the Riemann space-time g μν as ρ μν = We will use the De Donder conditions which were first introduced by Fock in [22] and were considered as a privileged system of reference. He introduced these conditions when he was considering problems of the island type (also called isolated systems). Until there are not global Cartesian coordinates in Riemann space-time, the harmonic gauge conditions are valid on patches.
To solve Eqs. (6) and (9), it is necessary to construct the Riemannian manifold, which means to find the metric tensor of the Riemann space-time g μν (x).
The Post-Newtonian approximation is used in relatively weak gravitational fields and in processes where the speeds are small compared with c, the speed of propagation of plane wave solutions of the gravitational field in vacuum. For these reasons, Post-Newtonian formulation is enough to describe phenomena within our solar system and the experimental tests performed in it.
The parameter needed to build perturbation series is the quantity = v/c, and since we will consider c = 1 for simplicity in the calculations (at last, we will reintroduce the factors with this quantity), we can see that the speeds of the bodies in our solar system v are no greater than . This means that temporal and spatial derivatives are related to the next relationship: where i = 1, 2, 3. This relationship implies that all temporal variations are associated with the motion of matter. As a first step, we will expand the metric tensor g μν , where η i j is the spatial part of the Minkowski metric η μν , and (l) g μν with l = 2, 3, 4, ... are terms of the order l . If we consider the transformation t → −t, the sign of must be changed, too, and for this reason Eqs. (11) and (13) contain only even powers of , and (12) only odd powers. Eq. (12) does not contain (1) g 0 j because in the Newtonian approximation g 0 j cannot be lower than the second order in .
We can use Eqs. (11)-(13) to calculate g μν and its determinant g = det g μν g 00 = 1 + (2) g 00 + (4) g 00 + · · · , g 00 + (2) g 11 + g 33 − (4) g 00 + g 33 + (2) g 00 ( g 33 + (2) g 2 12 + (2) If we introduce the quantities g 00 (2) Equation (9) in Galilean coordinates are 1 2 (2) (2) g 2 00 − (4) g 00 − 1 2 (2) g 00 The expansion of the second rank Ricci tensor written in Cartesian coordinates is g 00,00 where we have used the harmonic gauge conditions (9) to ignore the next terms of Eqs. (24) and (25). Also, we will expand the energy-momentum tensor of matter, considering it as a perfect fluid, namely, where p is the specific isotropic pressure, ρ is the ideal liquid's density, is the specific self energy, also called internal energy per particle and is the general name for all energies with the except for the rest mass, and v μ is the four-velocity. The definition of is given in A first course of general relativity by B. Schutz in [23] = ρ where n is the number density, the number of particles per unit volume in a momentarily co-moving reference frame. Tensor (26) must satisfy covariant conservation law given by The density of the perfect fluid is invariant and must satisfy the covariant continuity equation, that is, We will expand T μν in the small parameter T 00 + (2) T 00 + (4) T 00 + · · · , (29) If gravitational forces are ignored, that is, in Newtonian approximation, the components temporal and spacial of v μ are (1) The expanded Einstein field equations (6) are η i j (4) g 00,i j + (2) g 00,00 − η ki η l j (2) g kl (2) g 00,i j − η i j (2) g 00,i T 00 + 2 (2) g 00 This set of equations allows us to write the tensor of the Riemann space-time in Newtonian and post-Newtonian approximations.
If we denote φ as the Newtonian potential, we can suppose that (2) then Eq. (36) can be written as which is the Poisson equation and = ∇ 2 is the Laplacian operator. We consider φ null at infinity, therefore, our solution is, where r = |x − x | is the distance between the source point and observation point (field point). From Eqs. (38) and (39), we get (2) If we substitute Eqs. (40), (41) and (44) in Eq. (37), we have (4) g 00 − 2φ 2 = −2φ ,00 + 8π G (0) Also, (4) g 00 must tend to zero at infinity, therefore from Eq. (45) we obtain (4) g 00 = 2φ 2 + 1 2π Eqs. (19) and (44) can be related by Substituting the expanded components of the energy-momentum tensor of matter given by Eqs. (33) and (34) in Eqs. (42) and (43), we get (2) g 00 = −2φ, where and In the lowest order of approximation, the metric coefficients of the Riemann space-time are We can use our last results in the covariant conservation law (27) and the covariant continuity equation (28) to find the next approximation for the components of the energy-momentum tensor of matter. We need for this task the Christoffel symbols α βγ , √ −g and u 0 in the zero-order approximation, Thus, covariant conservation law stays as (2) T 00 ,0 + (2) the covariant continuity equation will be and the equation of motion of an ideal liquid is given bŷ whereρ = √ −gρu 0 , which is the conserved mass density, which can be written as follows in our approximation therefore we can replaceρ by ρ which is invariant density. So, we have the solutions for energy-momentum tensor of matter in the required expansion expressed as (2) where we have explicitly written the dependence of inverse factors on c, and we have used the identity demonstrated in [24]. We can write our system of Eqs. (36)-(39) as follows where = ψ − 2φ 2 and ψ = (4) g 00 .
Condition (47) is written as From the system (65)-(69), we have a solution for φ, the ordinary Newtonian scalar potential given by Eq. (49) The Heaviside vector potential whose components are given by Eq. (50) is written in vector form as follows We define the potentials and χ to write a simplified solution to Eq. (67) and therefore the solution to Eq. (67) is We define the total contribution of the scalar and vector potentials φ t and q t as where, for convenience, we have introduced = 1 2 and q t = 1 4 (2) In both equations, we have reduced the order of the potentials because we have multiplied Eq. (74) by c 2 and Eq. (75) by c in the given definitions. Then, we can define the ordinary gravitational vector g and the Heaviside field vector k as functions of the scalar and vector potentials φ t and q t and k = ∇ × q t .
So, in this way, we can obtain the second-order Maxwelllike formulation of the gravitational field equations, once we have reintroduced the factors c −2 , ∇ · g = −4π Gρ 1 − 2 c 2 2φ + + 2v 2 + 3 p ρ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .