Non-wave Solutions of the Maxwell-Einstein Equations

This article is devoted to treating of non-wave, i


INTRODUCTION
Gravitational instantons attract attention, starting from [1].We recall the definition.Instantons are known as topologically nontrivial localized solutions of the classical pseudo-Euclidean field equations characterized by a finite action and connecting two different vacuums of the theory [2].Euclidean version of the theory is introduced by replacing the Minkowski metric g ν (g 00 = 1, g ij =δ ij , i,j=1,2,3) to the Euclidean metric δ ν .Formally, the transition from the description in Minkowski space to a description in pseudo-Euclidean space is performed by replacing the time coordinate x 0 in Minkowski space to coordinate y 0 = ix 0 in pseudo-

Original Research Article
Euclidean space, while introducing a pseudo-Euclidean action Λ, associating it with the action in Minkowski space S by expression Λ = iS, i = (-1) ½ .
Instantons of classical field equations in Minkowski space describe in the semiclassical approximation quantum tunneling process between degenerate classical states located near different classical vacuums.In the theory of Maxwell-Einstein (M-E) equations these degenerate states are states in which there is convergent (divergent) electromagnetic wave at spatial infinity, which represent the two degenerate vacuums of the theory.As was shown in [3] classical transition between these states is impossible.Indeed, if we consider the vacuum in which there is a convergent spherical electromagnetic wave (SEMW), and taking into account the curvature of space-time due to the wave then almost all the rays corresponding to small portions of the wave front will capture by curvature of metric and do not give a contribution to the outgoing wave 1 .Therefore, the role of the instanton of the M-E equations is extremely important in description of such an intuitive and "simple" phenomenon, which seems to be the process of transformation a convergent SEMW to a divergent one.Another important application of the instanton of the M-E equations is development a physical theory of electromagnetic resonators, which eliminates the unphysical singularities of fields, for example, in a spherical cavity [5].And at last, an important application of the theory developed is cosmology, because the process of transformation of a convergent to a divergent SEMW is one of the main processes in the universe.
A brief scope of the present results is published in the Internet report [6].

BASIC EQUATIONS
As the initial equations we choose the Einstein's gravitational equations and the equations of the electromagnetic field in vacuum (Maxwell's equations) associated with each other [7]: Here R = R i itrace of Ricci's tensor R i k :, g ikmetric tensor; T ik and F iktensor of energymomentum and electromagnetic one; Г i kl -Christoffel's symbols; с -light speed in vacuum, Kgravitation constant; indices i, k, l take values 0, 1, 2, 3; repeated indices mean summation; comma means usual, i.e. non-covariant derivative [8].Let us find a solution of (1) which corresponds to existence of spherical light wave at r → ∞.For this we use an expression for interval just as in well-known Schwarzschild problem [8]: is characterized by frequency ω and angular moment vector J .Let us choose z -axis of the co-ordinate system in direction perpendicular to J .It simplifies a treating because the dependence of azimuth angle φ in (1) may be omitted.
Energy-momentum tensor's components T k i may be expressed by the components of metric tensor g k i with the help of Einstein's equation of gravity [8]: where δ k i -unit 4-tensor, and R -is a trace of tensor R i k .The details of calculations one can find in [8], for example.Besides Christoffel's symbols presented in [8], we need some additional ones; a symbol ~ (tilde) means a derivative by the angle θ: A result looks as follows: We also express the components of energy-momentum tensor T k i through solution of Maxwell's equations for In accordance with [7] right sides of equations ( 9) is averaged over the angle θ.In addition, for the wave solutions they are averaged over time [7].For the non-wave solutions the procedure of averaging over time has no meaning.Consider first the last three equations: for the Т 0 1 , Т 0 2 and Т 1 2 .Note, that their right hand sides, except the equation for the Т 0 1 are of the order of value ~ r s 2 /r 2 <<1, where r s 2 = K< f > 2 /2c 4 , <f>-solution's order of value, f = r 2 Ψ.Therefore, at distances of the order of the wavelength of light right hand sides of equations can be omitted 3 .This is consistent with the equations forТ 0 2 and Т 1 2 (5), if 0 ~= β .The equation for Т 0 1 in (9) we will use to find α.

TREATMENT THE EQUATIONS
Subtracting the second equation in ( 9) from the first one and equating the result with the similar operation which is done with equations (8) we receive: A is a constant.The solutions of ( 6), corresponding to the equation ( 10) one can treat in pseudo-Euclidean space which metric follows from the Minkowski space's metric with substitution time co-ordinate x 0 to "imaginary time" co-ordinate -iy 0 in pseudo-Euclidean space.At the same time one can introduce pseudo-Euclead action Λ, which is connected with the action S in Minkowski space as follows Λ = iS, i = (-1) ½ .It is known [2] that localized solutions of Euclidean field equations with finite Euclidean action are instantons.Instantons of classical field equations in Minkowski space describe in quasi-classical limit tunneling between degenerate classical states, which contain convergent or divergent SEMW.This procedure turns second hyperbolic equation in (6) to the elliptic one.If one suppose its finiteness at ∞ → r then he receives a condition А = 0 from the equation (10).This provides second equation ( 6) looks as follows: Here prime still means a derivative on x 1 = r, and point -on y 0 =cτ.
Signs ± hereafter correspond to different vacuums of the theory, located at τ → ± ∞.Using (11) we can rewrite the equations for instantons (8) and ( 9) in the form: Einstein equations: Maxwell equations: Consider the equations (12).They are compatible if the following condition is true: Condition ( 14) allows us to rewrite equations (12) in the form: Treating these equations is very difficult even numerically.Therefore, we are interested mainly in the asymptotic behavior of their solutions at distances r ≥ λ, λ is a wavelength of light.Note that their right sides are of the order ~ r s 2 /r 2 <<1 and they can be omitted with the adopted accuracy.Then Einstein's equations reduce to a single equation.Its solution gives the metric in a space free of matter which is well known.
Here the value of the constant const is to be determined.Furthermore, this asymptotic equation has auto-scale solutions, depending on z = cτ /r.For such solutions instead of ( 14) we obtain the equation Equation ( 17) is easily integrated and leads to the expression , where Eiintegral exponent.Using the well-known series expansion [10]: we can get the expression for the metric for large values of z: Formula (18) describes the transition between the vacuum states with flat metric corresponding to the presence of at z → -∞ convergent SEMW, and at z → + ∞ divergent one.This transition is localized with respect to τ, the size of the region of localization is a ~ r/c.Einstein's equations are also satisfied because "time" τ does not appear in them, and the equation ( 17) has a solution that can be represented for small z (large r) as a series expansion

PSEUDO-EUCLIDEAN ACTION
Let us calculate the action in curved space-time [7]  Turning to action in pseudo-Euclidean space Λ = iS f , dx 0 = -icdτ, and using (11) and the normalization condition for Φ(θ) [9], we receive (if In ( 21) is also taken into account that A θ can be expressed through A r [7].Let us treat extremes of Λ.To do this we calculate the variation Λ on A r provided condition δA r = 0 on the boundaries of integration and equate it to zero.As a result, we obtain: Because of the arbitrariness δA r integrand in ( 22) is equal zero, which gives the equation of the instanton: It reduces to the equation: The solution of this equation has the form: Pseudo-Euclidean action must be calculated for a classical trajectory which beginning and end (τ→± ∞) lie in the regions where the space-time is not curved, i.e. at

∞ → r
, where the field of SEMW tends to zero.Among the set of solutions of equation ( 23) satisfying this condition, we choose the solution: where E -is a constant with the dimension of the electrical field.Its value is related to the socalled topological charge of the instanton In evaluating the integral in (28) we use the expression (27) 4 .An important feature of the solution ( 27) is that it field decreases at r → 0, which is consistent with the tunneling nature of the instanton.
In calculating the pseudo-Euclidean action for the solution (27) to avoid divergence of the integral in (26) we cut off the integral over dr in the upper limit at the distance r 0 , having a sense of the size of the instanton, which will be defined below.With this in mind, the result of calculations (26) looks as follows: [ ] Recall that the action determines the probability w of transition of convergent SEMW to divergent one: The value r 0 we will find from the condition of matching metrics inside and outside the instanton.In the outer region metric is given by [3].
Metric in the inner region is found from formula (11), where we substitute the solution (27), given that f = ir 2 /c∂A r /∂τ.The result looks as follows:

Qualitative behavior of W (x) is stood the same for any l
The calculation shows that near the boundary of the instanton and wave, i.e. for x ≈ x 0 (r ≈ r 0 ) energy density is negative.The calculation shows also that the magnitude R 00 < 0 in a sufficiently large vicinity of x = x 0 for all l.The latter circumstance is essentially for the proof of the presence (or rather lack of it) of singularities, which, as is known, is based on the fact R αβ ξ α ξ β > 0, where ξ -is any non space-like 4 -vector5 [12].Absence of singularities associated with horizons of the metric (30), can be seen both from the expression (31) and from Fig. 1.The only fatal singularity is a singularity at x = 0, where W(x) ~ x -4 .

PROBLEM OF INSTANTONS FROM ENERGETIC POINT OF VIEW
During the propagation of SEMW part of its energy converts into other energy forms, such as energy of the gravitational waves.This issue was left outside the scope of the work (see [3,5,7]. In the literature there are different points of view on the question on interaction of EMW and gravitational waves.In [13] the author argues that the processes of transformation of the two photons in the graviton (and back) are prohibited by the conservation laws.At the same time, Wheeler did not rule out such a possibility [14].In [15], these processes are considered without any discussion.These differences can be overcome, if we consider the photongraviton processes in the presence of a static gravitational field created by SEMW, which removes the restrictions imposed by the conservation laws 6 .Leaving this issue for further discussion, make the following remark.Consider the first relation from (11).
for instantons, and a similar relation for waves [7] binding metrics and fields of instanton or electromagnetic waves.For definiteness we take in (33) and (33a) "+" sign.Using Maxwell's equations in curved space-time [8] one can write them in the form, respectively.H E , -electric and magnetic fields of the instanton or SEMW.The magnitude of the integrand is proportional to the conductivity (the role of the current density plays displacement current density), the value for which is real for wave and imaginary for instanton.The first means that the energy irreversibly transfers from SEMW to some other form, which is most likely connected with gravitational waves.The second indicates the reversible transfer of energy from the electromagnetic wave to the instanton, with subsequent return to the SEMW.
A question of interest is that, at what stage of the study was the neglect of gravitational waves, and what role they play in the problem.If we argue by analogy with the problem of the gravitational collapse of a non-spherical body, it can be assumed that the emission of gravitational waves will accompany the propagation of a spherical electromagnetic wave with a nonzero l, that, in the end of ends allow to speak about a spherically symmetric metric for l ≠ 0. Thus, used in this work, as well as in [3,5,7], averaging tensor T i k (9) on the angle θ as a consequence led to the fact that gravitational waves have been left out of consideration.

CONCLUSION
This article is devoted to treating the role of instantons in considering the dynamics of spherical electromagnetic waves by means of Maxwell-Einstein equations.Due to instantons convergent wave can be transformed into a divergent one what allow transmission of information from the past to the future.This article discusses the two different solutions for instantons -an auto-scaled one depending on z = cτ / r (25) which does not have a finite pseudo-Euclidean action, and the solution (27) with the finite one Λ(A r I ) (29).A special feature of the first solution is that in a world where it could be realized, the past is separated from the future with an infinite barrier, i.e. there is no flow of time in this world.The second solution is more consistent with the state of affairs in the real world -past goes to the future with some finite probability.The result obtained above, consisting in violation by instantons of the so-called "weak energy condition" T αβ ξ α ξ β > 0, where ξ -is any non space-like 4 -vector is important in research of the space-time singularities [13].
Note that most of the work on gravitational instantons available on the resource [17], are devoted to the classification of instanton solutions of Maxwell-Einstein in multidimensional Riemannian manifolds and their applications to the physics of black holes.
below means derivative y 0 .

Z 1 -
cylindrical function.Below, however, we will use another solution of equation (24), since the solution Y(z) does not have finite action.Calculating the pseudo-Euclidean action for the instanton that r s <<r c and given in equation e α in = e α out most significant members we get (up to terms ~ [l(l+1)] 8) for the metric (31) where T 00 = Wfield energy density of the instanton.
calculation is shown in Fig.1