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Time Dilation As Field Journal of Modern Physics, 2012, 3, 200-207 http://dx.doi.org/10.4236/jmp.2012.32027 Published Online February 2012 (http://www.SciRP.org/journal/jmp) Time Dilation as Field Piotr Ogonowski Warsaw, Poland Email: [email protected] Received November 29, 2011; revised January 6, 2012; accepted January 21, 2012 ABSTRACT 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. Keywords: Relativity; Maxwell Equations; Dilation; Lagrangian Mechanics 1. Introduction infinite small flat fragments of time-space, according to transformed Schwarzschild solution and Rindler’s trans- One of the main problems of the contemporary theoreti- formation appears to be accelerated. This approach allows cal physics is Quantum Gravity (Bertfried Fauser, Jürgen us to define important reference frame, that may be used Tolksdorf, Eberhard Zeidler, [1]). farther. The motivation to create this paper is conviction, that In second section I use above approach and introduce reformulation of the concept of fields by emphasis on its- some fields, that binds together time flow and motion in relationship with time dilatation factor and time-space struc- d’Alembertians. Derived wave equation express distur- ture may support to efforts to field unification. bance in time-space structure propagating in time-space Searching for Higgs boson or considering possible al- that may be explained as light. ternatives to Standard Model we try to explain issues, sort In this paper we also refer to Max Planck’s Natural Units of: introduced in 1899. Let us then denote following design- The nature of the elementary particle rest mass, nations: The nature of the photon energy, , Photon’s behavior on Planck’s energy scales. tP Planck s time The aim of this paper is tosupport issues mentioned above, lctPP by redefining electromagnetic field equations and stress 4 similarity to Schwarzschild solution, what may open new lcP , EP Planck s Energy ways for the quest for quantum gravity and the unified tGP field theory. h Etreduced Planck, s action Almost a hundred years have passed since 1908 when 2 PP Hermann Minkowski gave a four-dimensional formula- 2 (1) ElcPP , tion of special relativity according to which space and time mP 2 Planck s mass are united into an inseparable four-dimensional entity— c G now called Minkowski space or simply spacetime—and qe , qcP 40 Planck s charge macroscopic bodies are represented by four-dimensional worldtubes. But so far physicists have not addressed the fine structure constant question of the reality of these worldtubes and spacetime q elementary charge itself” (Vesselin Petkov, page 1, [2]). e In this paper I reformulate Schwarzschild and Min- Farther I will deal with relativistic dynamics and show, kowski metrics and explain these metrics as consequence that adding axis to Hamiltonian and Lagrangian we may of introduced electromagnetic field description. obtain proper Lagrangian and Hamiltonian for gravita- In first section I recall that one may consider curved tional field that one may understand as reformulation of time-space as collection of locally flat parts of Rieman- the field interaction phenomena. nian manifolds with assigned stationary observers. These This way we develop farther the idea presented by Alex- Copyright © 2012 SciRes. JMP P. OGONOWSKI 201 ander Gersten: “(…) we have shown that thenon-rela- Above formula drives us to conclusion that relativistic tivistic formalism can be used provided the momenta and gamma acts here as it would be scalar field. Let us ex- Hamiltonian belong to the same 4-vector.” (Alexander plain above and its wide consequences in few steps. Gersten, page 10, [3]). At first step, let us rewrite Schwarzschild metric for We will start with reference to the main equation of some new reference frame. We start using formula (5): the General Theory of Relativity. We will narrow down 1 our discussion to a spherically symmetrical mass to apply dd 22trr d 2222 d (8) 2 r the Schwarzschild solution and then we will generalize r above thanks to Rindler’s transformation. We may consider Schwarzschild metric for stationary observer, hanging at some point at distance “r” to source 2. Time Dilation as Field of gravitational force (such observer has to use some 2.1. Schwarzschild Metric and Time Dilation force to keep his position). We will denote such observer proper time as τobs: Let us start with recalling Schwarzschild metric (R. Al- 1 dd t (9) drovandi and J. G. Pereira, page 111, [4]) and consider obs relation between gravitational potential and time dilation. r To simplify calculations, in whole section we are assum- Now, we might rewrite Schwarzschild metric (8) refer- ing c = 1. For body orbiting at one plane around non-ro- ring to some local, chosen stationary observer reference tating big mass, we may write metric in form of: frame and its proper time. 22 2222 2 dd obs drr r d (10) 22rs dr 22 d1d tr d (2) r rs If we will note above for geodesics we obtain: 1 r 22222 dd obs rr rd (11) We assume: Above formula will be useful soon. τ is the proper time of observer’s reference frame; Using such stationary observer reference frame we re- t is the time coordinate (measured by a stationary clock call that Riemannian manifolds are locally flat. If we shrink at infinity); considered time-space into spheres with chosen “r” ra- r is the radial coordinate; dius we obtain spherical, anisotropic Minkowski metric with φ is the colatitude angle; slower coordinate light speed according to (11). rs is the Schwarzschild radius. If we shrink it more, we consider infinitive small, local, According to this solution, the Schwarzschild’s radius part of chosen sphere, where photon meets Stationary Ob- and mass formulas are: server. 2GM At second step, let us introduce velocity “vr”: rs 2 (3) c r vc s (12) cr2 r r M s (4) 2G We recognize velocity “vr” as Escape velocity and We introduce relativistic gamma factor: Free-falling velocity, thus we introduce some related spa- tial increment dx : 1 obs r (5) dx r v obs (13) 1 s r dt r and then derive from (9) below formula: were call, that Schwarzschild’s solution drives to gravita- 22 2 tional acceleration in “r” distance equal to: ddt obs dxobs (14) r GM s It is easy to notice, that above formula acts just like it grr2 (6) r 2r would be Minkowski for free-falling velocity. r 2 1 s r Now, at third step, let us recall Rindler’s transforma- tion in some plane Minkowski time-spacefor body mov- As we may easy calculate: ing with acceleration “a”, achieving velocity “v”. We r 1 may consider such body using co-moving observer con- d1 s d cept. We will denote its proper time as “τ” and note: g r r (7) r ddrr vat (15) Copyright © 2012 SciRes. JMP 202 P. OGONOWSKI tv rics for: (16) a (14) stationary observer moving against accelerated, free- falling surroundings (light), v (17) (11) free-falling surroundings (light) considered in re- a lation to stationary observer proper time. Now, we may consider some hypothetical body with Referring to above conclusions we will introduce (in Section 3) reformulation of Lagrangian and Hamiltonian acceleration “gr”, achieving velocity “vr” with proper what might be understood as new description of field in- time “τr”. v 2r teractions. r (18) r g v rrr 2.2. Vector Fields for Minkowski Time-Space Let us perform following transformation: As we know there is no ether and the medium for elec- 2r tromagnetic wave is time-space. We should expect, then, d there must exists some field equations explaining elec- d r vrr vrr (19) tromagnetic wave as disturbance in time-space structure ddrr (structure of the medium) distributing in the time-space. 1 Let us prepare to such electromagnetic field descrip- dd 2222rvd1r2 2 (20) rrr2 r tion, describing at first some regular rotation of Planck’s r mass mP, with line velocity vr, on the circle with radius R. 2222 ddrrrr d (21) We will define velocity as function of R equal to: Let us also introduce spatial increment as it would be R vc co (28) inplane Minkowski metric. We will note this increment r R in polar coordinates: where Rco is some defined constant. ddrr222 (22) Related gamma factor will be equal to: Let us note Minkowski metric for co-moving body: d1t r (29) 22222 d Rco d obs dr ddrr (23) 1 R At the end, by substituting (21) we obtain: 2222222 Angular velocity for rotating body we will denote as: dd obs rrrrr ddd (24) vr 22222 (30) dd obs rrr d (25) R Comparing above to (11) we recognize geodesics in Non-relativistic angular momentum we may denote as: Schwarzschild metric. Thus we must conclude that our Rin- L Rv mP r (31) dler transformationmight be done for accelerated light… To support above claim we will show in next section that Lm P Rvr (32) rest mass existence is not necessary to consider accelera- tion for light. Non-relativistic radial acceleration we denote as: We may also easy transform (18) to form of: dv d aRr R 2 (33) ddtt dvr gr (26) d r Maxwell has defined electromagnetic field phenomena by eliminating particles from equations while field sre- Joining above with (7) we may explain acceleration by: mains [5].
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