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  40 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 22trr 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) 22rs dr 22 d1d tr  d (2) r rs If we will note above for geodesics we obtain: 1 r 22222 dd obs rr 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 spatial 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 grr2  (6) r 2r would be Minkowski for free-falling velocity. r 2 1 s r Now, at third step, let us recall Rindler’s transformation 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)

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 2222rvd1r2 2 (20) rrr2 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 ddrrrr 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 dr ddrr (23) 1 R At the end, by substituting (21) we obtain:

2222222 Angular velocity for rotating body we will denote as: dd obs rrrrr ddd (24) vr 22222   (30) dd obs rrr 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 m (31) dler transformationmight be done for accelerated light… P r

To support above claim we will show in next section that Lm P  Rvr (32) rest mass existence is not necessary to consider acceleration for light. Non-relativistic radial acceleration we denote as: We may also easy transform (18) to form of: dv d aRr 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]. Let us do the same, but eliminating test body 1 d while motion and time flow remains. dvr r We will construct vector fields to describe whole class gr  (27) d r dr of just introduced rotations defined for any place in space. Rest mass we may understand as parameter. Gravitational acceleration g may be then expressed by r Let us define at first three versors n , n , n . For any just introduced imaginary proper time τ and velocity v . R x y r r conductive vector R we define: Recalling (14) we should conclude, that geodesics in R Schwarzschild metrics may be explained (besides classi- n  (34) cal explanation) as combination of two Minkowski met- R R

nnR xy n (35) Let us note the d’Alembertians in form of: c 1d2 Let us define scalar field and two related vector  2 0 (50)  22 fields: r c d c Tn (36) 2 y 1  A 2  r  A 0 (51) c22 c  v2 An  rr   n (37) Above form of d’Alembertians describes move with  y 2cR x r “c” speed in infinite small, local part of time-space. This Please, note similarity of above A field to acceleration way in local reference frame light has always “c” speed. in Schwarzschild metric noted in (7) what will be useful We may also expect now, that derivative by R should soon. As we can easy show: act similar to derivative by proper-time. Let us perform some hypothetical calculation to prove it: TA because : (38) ddRTT   dR cc  RRTAT (52) dddRRR  nny   y (39) rr dRvr  dvrrdR v  R vnvryr (53) Let us define scalar field equal to Rvr (related to an- ddRRd2R gular momentum) and two auxi liary vector fields U and Ω. dRv  v r  U R  (54) URv nn r  (40) dR rx2 y U  (41) Now, let us analyze consequences of derived field equations and define Lagrangian and Hamiltonian for fields Let us notice, that: defined this way. UvRR (42) r 3. Reformulated Lagrangian and rr2 d  r Hamiltonian ARRR   a (43) ccdtc 3.1. Lagrangian and Hamiltonian From above we derive: Let us show how we may describe mechanics when we  d A r  (44) assign potential with gamma factor denoted as  r and ctd defined with formula (29). Let us also show, that: Let us first recall a Hamiltonian expressed with general variables: c L d  ny H  xLxp L (55) dn iii rrdT  r rc y x   (45) iii ctcddt c d t r where L is Lagrangian.  dT dn For three dimensional space we denote: r y  (46) 3 ctdd t 2  xiip v mv  mv  (56) Using (38) we obtain: i1  dA Now, let us add extra zero-dimension and define ve- r  (47) ctd locity and momentum for such dimension. Let it be equal to defined previously rotation velocity (28) but here de- From (44) and above we derive two d’Alembertians: noted with index “r” index:

2 2 2  r d  2 x p v  mv  mv  (57)  0 (48) 00 rrrrr ct22d Summing expression for indexes 03 we define Ha-  2 2 A miltonian in form of: r 2 A 0 22 (49) 3 ct 22 H   xpii L mv rr  mv  L (58) Above d’Alembertians describes wave with line velo- i0 city c  r or just time dilation around rotation center. Now we define Lagrangian in form of:

11 LG2 MmM LE00E (59) rrmc 22G 2 (70)  r  R cR R Thus: We have obtained the force expected for introduced field. 2211 We will denote above force using “r” index: HmvE00mvErr (60)   r L   rrF (71) 22 R vv1r 1 HE00Er  (61) 22 22 Now, using above and (68) we rewrite formula (66) as: cc  r

F  Fr (72) HE00 E r (62) as we may easy see, in infinity above Hamiltonian ex- As we may conclude, just derived relativistic force press relativistic kinetic energy just as we should expect: acting on body in every particular place in space is equal to force caused by field. Above is true for known fields. lim HE000EE 1 (63) R 3.3. Time-Space Curvature as we will show in few steps, introduced Lagrangian and Hamiltonian drives to the same results then we may de- Let us calculate divergence for line velocity: rive from GR for gravity.  vr RR 0 (73) 3.2. Test for Lagrangian Now we calculate divergence for acceleration: Let us now check if introduced Lagrangian pass the tests. d dv  r (74) At the beginning we should notice, that from (51) we ddtt obtain: d 1  a (75)  (64) dt  Rc obs 22Rco Test condition for Lagrangian we should then rewrite 4πRc 2 2 d R 4π Rcco as true only locally in form of:   (76) dtV V L d Now we multiply and divide RHS by constants: v L 0 (65) 22 d obs R cRco cRco d 22GG L 42 GG   8   (77) d dtVV v L  r  (66) Substituting Schwarzschild radius in place of constant dtR Rco and using (4) we could transform above to form of: First derivative in LHS introduce momentum: d M  8 G  (78) 1  dtV E0  L  mv p (67) Let us denote mass density as: vv M   (79) Next derivative result with relativistic force multiplied V by gamma: dp Let us express pulsation using rotating versor n. We  F (68) rrdt obtain: d8nG For RHS derivative drives to:  c2 (80) dtc22 R L co 2 E0  r (69) d8nG  R 2R2 r  c22 (81) dtc22 c4 r Let us compare (6) with obtained acceleration to notice, that we have obtained expected acceleration. Using (51) we obtain: Substituting Schwarzschild radius in place of constant 228G 2 nc4 r (82) Rco and using (4) we could transform above to form of: c

Now, using above to construct 4-dimentional conduc- Rco tive vector we obtain relation between mass density and   c R3 time-space curvature the same as main equation of Gen- eral Relativity. Non-relativistic angular momentum for such move is equal to:

3.4. Hamiltonian and Rest Energy Formula RRco RRco LmclPP 22  (92) Let us consider Hamiltonian for empty space. In place of llPP test body’s rest energy let us use Planck’s Energy with as we may notice, there is some Radius causing that an- meaning of unit “one”. gular momentum become equal to smallest action-re-

HEPPrPr E  E1 (83) duced Planck’s action. Let us note such Radius with Rω and define as: The smallest R we can substitute is Planck’s length. l R P  co (93)  Rl   P 1 lim EE 1 (84) Now we may introduce hypothetical gamma factor and P  Rl P R rotation velocity with “E” index: co 1 lP  lP vcE  (94) For very small constants we could rewrite equation using R Maclaurin’s expansion: 11 4  E  (95) 2 RcRco co l P vE lim EEP  (85) 1 Rl 1 2 P 22lGP R c Let us notice similarity to rest energy formula, where Kinetic Energy for such rotation is equal to: Rco acts like Schwarzschild radius. We may conclude,  that Schwarzschild radius might be explained as appro- 11l 1 P ximation of composition of set of some smaller particles EEkin P 1 EP   (96) l 22R R with Rco  lP. Let us then define two variables for hy- 1 P pothetical mass and energy as follows: R c cR2 c M  co (86) Now let us see, that express pulsation in reference co 2G R frame assigned to rotating frame. On a circle with radius EMc2 (87) co co c R for line velocity we may note as follows: Now, we may easy show that introduced Hamiltonian  approximates for small velocities and large distance Me- c chanical Energy defined by Newton. Let us transform (62) TR2 (97)  to form of: HE11E  (88) c 22 00 r    (98) RT T' Using Maclaurin’s expansions of above for small velocities and large scales we obtain: Let then denote inverse of Radius as pulsation and write mv2mv2 mv2R down: Hmr c2 co (89) 111 222 2R E   (99) kin 22R mv2 mM  HG co (90) c 2 R If we consider two twisted vectors of rotating field ma- As we may see, above formula acts the same way that king double Helix, we will obtain well recognized for- Newton’s Mechanical Energy formula. mula: EE 2   (100) 4. Photon and Electrostatics Hypothesis kin We may suppose that  describes electromagnetic 4.1. Photon Energy E field and the above quantum of energy may be assigned Let us notice, that pulsation described in (30) is equal to: to photon. This way we may treat (50) and (51) as equi-

valent versions of Maxwell’s Equations [6]. RlQP 0.98890 (111) Pair production phenomena might be thus rewritten as: Let us redefine (106) and (107) as follows:  (112)   EEQPQP lim  1 E  4  11 (101) RR Q 2121EEPP  lR ll 1 Pc1 o P P   VEQQQ1EEQ P (113) R lP  4RR and after Maclaurin’s approximation: One may easy derive, that electrostatic force for two  2 mc2 (102) elementary charges may be expressed as: 1 d 4.2. Electrostatics  Q llP P FEQQ EEQQP22 (114) The Electrostatic potential of two elementary charges, dR 4RR expressed in Planck’s units, can be noted as: qq22q2 5. Summary Vceece (103) Q 2 As we have just shown, the same field equations may de- 4400Rc RqRP scribe as well electromagnetism as well gravity. lP Field “A” defined in (37) may enter in place of: VEQP (104) R  Electromagnetic field, Let us introduce auxiliary constant “ε” and variables:  Gravitational acceleration. 1 Field “T” (36) and related scalar potential may act as:   (105)  Electrostatic potential, l 1 P  Gravitational potential.  R Field “Ω” (41) may act as:  Magnetic rotation, EEQPlim 1 (106) Rl P  Time-space curvature factor. We may understand above as equivalent of (85). Now, let us assume that expression (104) is Maclau- 6. Acknowledgements rin’s approximation for R  lP of interaction based on I would like to thank prof. Iwo Bialynicki-Birula from time dilation factor  . Polish Academy of Science for critique, asked me crucial Therefore, it should follow below formula: questions and valuable time he had dedicated to me. ll I would like to thank Dr. Rafal Suszek from Warsaw VE1 EP EP (107) QQ Q2 RRP University for his time and lectures he has pointed to me. I also thank to Dr. inz. Michal Wierzbicki from Warsaw as we can easy derive: University of Technology for his time and books he had  borrowed to me.  I thank to Markus Nordberg from CERN for his inter- 11 1  (108) est and hints on early stage of the article. 2 1 I would like to thank posters from BAUT forum: cave- 1  man 1917, Celestial Mechanic, Kuroneko, Paul Logan,   6.245919 (109) Tensor, Shaula, tusenfem, Garrison, macaw, grapes, slang, Strange, amazeofdeath, Geo Kaplan, pzkpfw, Eigen State, What astonishing, we have obtained epsilon close to Jim, czeslaw, captain swoop, starcanuck 64 and Luckmei- 2π. It brings to mind De Broglie condition for length of ster. Thank you for your time, critique and questions. This the orbit. Let us follow this indication. article would never have matured without you. Especially Let us redefine radius being limit for R in formula I would like to thank to Celestial Mechanic for hope. (106) using present knowledge. Now, we define auxiliary I dedicate this article to my Joanna, for the fact that I radius such way, to obtain   2 . could work using the time, that should belong to her. Let us introduce: 1  Q  (110) REFERENCES l 1 P [1] B. Fauser, J. Tolksdorf and E. Zeidler, “Quantum Gravity: 2R Mathematical Models and Experimental Bounds,” Springer,

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