Time Dilation as Field

It is proved, there is no aether and time-space is the only medium for electromagnetic wave. However, considering time-space as the medium we may expect, there should exist field equations, describing electromagnetic wave as disturbance in time-space structure propagating in the time-space. I derive such field equations and show that gravitational field as well as electromagnetic field may be considered through one phenomena-time dilation.


Introduction
One of the main problems of the contemporary theoretical physics is Quantum Gravity (Bertfried Fauser, Jürgen Tolksdorf, Eberhard Zeidler, [1]).
The motivation to create this paper is conviction, that reformulation of the concept of fields by emphasis on itsrelationship with time dilatation factor and time-space structure may support to efforts to field unification.
Searching for Higgs boson or considering possible alternatives to Standard Model we try to explain issues, sort of:  The nature of the elementary particle rest mass,  The nature of the photon energy,  Photon's behavior on Planck's energy scales.
The aim of this paper is tosupport issues mentioned above, by redefining electromagnetic field equations and stress similarity to Schwarzschild solution, what may open new ways for the quest for quantum gravity and the unified field theory.
Almost a hundred years have passed since 1908 when Hermann Minkowski gave a four-dimensional formulation of special relativity according to which space and time are united into an inseparable four-dimensional entitynow called Minkowski space or simply spacetime-and macroscopic bodies are represented by four-dimensional worldtubes.But so far physicists have not addressed the question of the reality of these worldtubes and spacetime itself" (Vesselin Petkov, page 1, [2]).
In this paper I reformulate Schwarzschild and Minkowski metrics and explain these metrics as consequence of introduced electromagnetic field description.
In first section I recall that one may consider curved time-space as collection of locally flat parts of Riemannian manifolds with assigned stationary observers.These infinite small flat fragments of time-space, according to transformed Schwarzschild solution and Rindler's transformation appears to be accelerated.This approach allows us to define important reference frame, that may be used farther.
In second section I use above approach and introduce some fields, that binds together time flow and motion in d'Alembertians.Derived wave equation express disturbance in time-space structure propagating in time-space that may be explained as light.
In this paper we also refer to Max Planck's Natural Units introduced in 1899.Let us then denote following designnations: Farther I will deal with relativistic dynamics and show, that adding axis to Hamiltonian and Lagrangian we may obtain proper Lagrangian and Hamiltonian for gravitational field that one may understand as reformulation of the field interaction phenomena.This way we develop farther the idea presented by Alex-ander Gersten: "(…) we have shown that thenon-relativistic formalism can be used provided the momenta and Hamiltonian belong to the same 4-vector."(Alexander Gersten, page 10, [3]).
We will start with reference to the main equation of the General Theory of Relativity.We will narrow down our discussion to a spherically symmetrical mass to apply the Schwarzschild solution and then we will generalize above thanks to Rindler's transformation.

Schwarzschild Metric and Time Dilation
Let us start with recalling Schwarzschild metric (R. Aldrovandi and J. G. Pereira, page 111, [4]) and consider relation between gravitational potential and time dilation.To simplify calculations, in whole section we are assuming c = 1.For body orbiting at one plane around non-rotating big mass, we may write metric in form of: We assume:  τ is the proper time of observer's reference frame;  t is the time coordinate (measured by a stationary clock at infinity);  r is the radial coordinate;  φ is the colatitude angle;  r s is the Schwarzschild radius.
According to this solution, the Schwarzschild's radius and mass formulas are: We introduce relativistic gamma factor: were call, that Schwarzschild's solution drives to gravitational acceleration in "r" distance equal to: As we may easy calculate: Above formula drives us to conclusion that relativistic gamma acts here as it would be scalar field.Let us explain above and its wide consequences in At first step, let us rewrite Schwarzschild metric for so  few steps.me new reference frame.We start using formula (5): observer, hanging at some point at distance "r" to source of gravitational force (such observer has to force to keep his position).We will denote such observer pr use some oper time as τ obs : Now, we might rewrite Schwarzschild metric (8) referring to some local, chosen stationary observer reference frame and its proper time.If we will note above for geodesics we obtain: Above formula will be useful soon.we recall that Riemannian manifolds are loca considered time-space into spheres w di i metric with sl  Using such stationary observer reference frame lly flat.If we shrink ith chosen "r" raus we obtain spherical, anisotropic Minkowsk ower coordinate light speed according to (11).
If we shrink it more, we consider infinitive small, local, part of chosen sphere, where photon meets Stationary Observer.
At second step, let us introduce velocity "v r ": We recognize velocity "v r " as Escape velocity and Free-falling velocity, thus we introduce some related spatial increment dx obs : and then derive from (9) below formula: It is easy to notice, that above formula acts would be Minkowski for free-falling velo dler's transformaacefor body movg with acceleration "a", achieving velocity "v".We may consider such body using co-moving obs pt.We will denote its proper time as "τ" and note: Copyright © 2012 SciRes.JMP Now, we may consid acceleration "g r ", achieving velocity "v r " with proper time "τ ".
(17) er some hypothetical body with Let us perform following transformation: ce spatial increment as it w inplane Minkowski metric.We will note this inc in polar coordinates: Let us also introdu ould be rement Let us note Minkowski metric for co-moving body: At the end, by substituting (21) we obtain: Comparing above to (11) we recognize g t our Rinin next section that st mass existence is not necessary to consider acceleration for light.
We may also easy transform (18) to form of: Joining above with (7) we may explain acceleration by: cceleration r may be then expressed by just introduced imaginary proper time τ r and velocity v r .
Recalling (14) we should conc Schwarzschild metrics may be explained (besides classical explanation) as combination of two Minkowski met-ric fal ing to above conclusions we will introduce (in Se new description of field inte

Vector Fields for Minkowski Time-Space
As we know there is no ether and the medium for electromagnetic wave is time-space.We should expect, then, there must tromagnetic wave as disturbance in time-space struct (structure of the medium) distributing in the time-spac Let us prepare to such electromagnetic field description, describing at first some regular rotation of Planck's mass m P , with line velocity v r , on the circle with radius R. We will define velocity as function of R equal to: where R co is some defined constant.Related gamma factor will be equal to: Non-relativistic angular momentum we may denote as: (30 Non-relativistic radial acceleration we denote as: Maxwell has defined electrom by sremains [5].Let us do the same, but eliminating test body while motion and time flow remains. We will construct vector fields to describe whole class of   (33) agnetic field phenomena eliminating particles from equations while field just introduced rotations defined for any place in space.Rest mass we may understand as parameter.
Let us define at first three versors n R , n x , n y .For any conductive vector R we define: Please, note similarity in Schwarzschild metric soon.As we can easy show: of above A field to acceleration noted in (7) what will be useful because : Let us define scalar field equal to Rv (related to angular momentum) and t lia r ry vector fields U and Ω. wo auxi show, that: Using (38) we obtain: d above we der r From (44) an ive two d'Alembertians: Let us note the d'Alembertians in form of: ove e-space.This way in local reference frame light has always "c" speed.
We may also expect now, that de act similar to derivative by proper-time.Let us perform some hypothetical calculation to prove it:  Ab form of d'Alembertians describes move with "c" speed in infinite small, local part of tim rivative by R should Now, let us analyze consequences of derived fie tions and define Lagrangian and Hamiltonian fo defined this way.

Lagrangian and Hamiltonian
Le ld equar fields t us show how we may describe mechanics when we assign potential with gamma factor denoted as r  and defined with formula (29).d with gene-Let us first recall a Hamiltonian expresse ral variables: where L is Lagrangian.
For three dimensional space we denote: Now, let us add extra zero-dimension an locity and momentum for such dimension.L to defined previously rotation velocity (28) but here deex: Summing expression for indexes 0 miltonian in form of: Now we define Lagrangian in form of: Copyright © 2012 SciRes.JMP y easy see, in infinity above Hamiltonian express relativistic kinetic energy just as we shoul s, introduced Lagrang Hamiltonian drives to the same results then we may derive from GR for gravity.

Let us now check if introduced Lagrangi
At the beginning we should notice, th as we ma d expect: as we will show in few step ian and

Test for Lagrangian
an pass the tests.at from (51) we obtain: (64)  ian we should then rewrite : Test condition for Lagrang as true only locally in form of First derivative in LHS introduce momentum: Next derivative result with rela by gamma: vative drives to: have obtained expected Substituting Schwarzschild R co and using (4) we could transform above to form of: (69) Let us compare (6) with obtained acceleration to notice, that we acceleration.radius in place of constant We have obtained the force expected for introduced field.W R e will denote above force using "r" index: Now, using above and (68) we rewrite formula (66) as: ace is equal to force caused by field.Above is true for known fields.

Le (72)
As we may conclude, just derived relativistic force acting on body in every particular place in sp t us calculate divergence for line velocity: Now we calculate divergence for acceleration: Now we multiply and div ide RHS by constants: Substituting Schwarzschild radius in place of constant R co and using (4) we could transform above to form Let us denote mass density as: Let us express pulsation using rotating versor n.W Using (51) we obtain: Now, using above to construct 4tive vector we obtain relation bet time-space curvature the same as main equation of General Relativity.

Rest Energy Formula
Let us consider Hamiltonian for empt test body's rest energy let us use Pl m dimentional conducween mass density and   The smallest R we can substitute is Planck's length.
  For very small constants we could rewrite eq Maclaurin's expansion: Let us notice similarity to rest energy f R co acts like Schwarzschild radius.We may conclude, that Schwarzschild radius might be explained as approximation of composition of set of some smaller particles w gy as follows: ormula, where ith R co  l P. Let us then define two variables for hy- pothetical mass and ener Using Maclaurin's expansions cities and large scales we obtain: Now, we may easy show that introduced Hamiltonian approximates for small velocities and large distance Mechanical Energy defined by Newton.Let us transform (62) to form of: of above for small velo- As we may see, above formula acts the s Newton's Mechanical Energy formula.

Photon Energy
Let us notice, that pulsation described in (30) is equal to: ame way that

Photon and Electrostatic
Non-relativistic angular momentum for such move is equal to: as we may notice, there is some Radius causing that angular momentum become equal to smallest action-re- Now we may introduce hypothetical gamm rotation velocity with "E" index: a factor and Kinetic Energy for such rotation is equal to: express pulsation in reference frame assigned to rotating frame.On a circle with radius R for line velocity c  we may note as follows: Let then denote inverse of Radius as pulsation and write down: If we consider two twisted vectors of rotating field making double Helix, we will obtain we mula: ll recognized for- We may suppose that E  describes electromagnetic field and the above quantum of energy may be assigned to photon.This way we may treat (50) and (51) as equi-Copyright © 2012 SciRes.JMP P. OGONOWSKI 206 valent versions of Maxwell's Equations [6] Pair production phenomena might be thu .s rewritten as: d after Maclaurin's approximation: 4 4 s introduce auxiliary constan Let u t "ε" and variables: 1 ve as equivalent of (85).Now, let us assume that expression (104) is Maclaurin's approximation time dilation factor We may understand abo for R  l P of interaction based on  .
Therefore, it should follow below formula:   1 2 as we can easy derive:   lim 1 4   P 1 4 One may easy derive, that el elementary charges may be expressed as: d 4

Summary
As we have just shown, the same field equations may describe as well electromagnetism as well gravity Field "A" defined in (37) may enter in place of:  Field "T" (36) and related scalar potential may act as:  Electrostatic potential,  Gravitational potential.
third step, let us recall Rin tion in some plane Minkowski time-sp in erver conce 2 d obs x  metric.Thus we must conclude tha dler transformationmight be done for accelerated light… To support above claim we will show re (25) just time dilation around rotation center.
y space.In place of anck's Energy with eaning of unit "one".
s action.Let us note such Radius with R ω and define as: 's units, can be noted as: e Electrostatic potential of two elementary charges, 2 2 obtained epsilon close to 2π.It brings to mind De Broglie condition the orbit.Let us follow this indication.usbeing limit for R in formula (106) using present knowledge.Now, we define auxiliary radius such way, to obtain us redefine (106) and (107) as follows: from r critique, asked me crucial lectures he has pointed to me.al Wierzbicki from Warsaw his time and books he had om CERN for his interthe article.rs from BAUT forum: cave-Ji ar article to my Joanna, for the fact that I co .Electromagnetic field,  Gravitational acceleration. Magnetic rotat  Time-space cu