Time Dilation as Field

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

One of the main problems of the contemporary theoreti-cal 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 its- relationship with time dilatation factor and time-space struc- ture may support to efforts to field unification.

Searching for Higgs boson or considering possible al- ternatives 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 formula- tion of special relativity according to which space and time are united into an inseparable four-dimensional entity— now 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 Min- kowski 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 Rieman- nian manifolds with assigned stationary observers. These

infinite small flat fragments of time-space, according to transformed Schwarzschild solution and Rindler’s trans- formation 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 distur- bance 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 design- nations:

4

2

2

0

, Planck s time

,

Planck s Energy

, reduced Planck s action 2

, Planck s mass

,

4 Planck s charge

fine structure constant elementary charge P

P P

P P

P

P P

P P

P

e P

e

t l ct

l c E

t G

h _{E t}

E l c m

G c

q

q c

q

  





 

  

 

 

  







(1)

Farther I will deal with relativistic dynamics and show, that adding axis to Hamiltonian and Lagrangian we may obtain proper Lagrangian and Hamiltonian for gravita- tional field that one may understand as reformulation of the field interaction phenomena.

ander Gersten: “(…) we have shown that thenon-rela- tivistic 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.

2. Time Dilation as Field

2.1. Schwarzschild Metric and Time Dilation

Let us start with recalling Schwarzschild metric (R. Al-drovandi and J. G. Pereira, page 111, [4]) and consider relation between gravitational potential and time dilation. To simplify calculations, in whole section we are assum-ing c = 1. For body orbiting at one plane around non-ro- tating big mass, we may write metric in form of:

2

2_d 2

s

r

t r

r r

2 2 d

d 1 d

1 s

r r

  

 _ 

 

 

 

 _{ } 

  (2)

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;  rs is the Schwarzschild radius.

According to this solution, the Schwarzschild’s radius and mass formulas are:

2

2

s

GM r

c

 (3)

2

2 s

c r M

G

 (4)

We introduce relativistic gamma factor: 1

1 r

s

r r  



(5)

were call, that Schwarzschild’s solution drives to gravita-tional acceleration in “r” distance equal to:

2

2 s

r r

s

r r r

r 2 ₁

GM g

r



 

  (6)

As we may easy calculate:

1 d

d d

s

r r

r

r r

d 1

r

g 

 

 

 

  (7)

Above formula drives us to conclusion that relativistic gamma acts here as it would be scalar field. Let us ex- plain 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):

2 2 2 2 2 2

2

1

d dt dr _r r d

r

       (



We may consider Schwarzschild metric for stationary 8)

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 τobs:

1

d obs d

r t 



  (9)

Now, we might rewrite Schwarzschild metric (8) refer-ring to some local, chosen stationary observer reference frame and its proper time.

2 2 2 2 2 2

d dobs dr r r d (10)

2 2_d 2

r r

If we will note above for geodesics we obtain:

2 2

dobs dr    (11) Above formula will be useful soon.

we re-call 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” ra-us 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 Ob- server.

At second step, let us introduce velocity “vr”:

s r

r v c

r

 (12)

We recognize velocity “vr” as Escape velocity and

Free-falling velocity, thus we introduce some related spa- tial increment dxobs:

d d

obs r

x v

t

 (13)

and then derive from (9) below formula:

2 2

dt d_obs

just like it city.

Now, at third step, let us recall Rin tion in some plane Minkowski time-sp in

erver con- ce

2

dx_obs

 ₍₁₄₎

It is easy to notice, that above formula acts would be Minkowski for free-falling velo

dler’s transforma- acefor body mov- g 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:

t v a

  (16) v

a  

Now, we may consid

acceleration “gr”, achieving velocity “vr” with proper

time “τ”.

(17)

er some hypothetical body with

r

2r r v r

r r r

g v





  (18)

Let us perform following transformation: 2

r r r

r r r

v v   d

d

dr dr 

 

 

 

  (19)

2 2 2 2

d r dr vr r 2 2 2

1

dr 1 r

r

  



 

 _{ }

  (20)

2 2 2 2

d_r dr _r dr (21)

ce spatial increment as it w inplane Minkowski metric. We will note this inc in polar coordinates:

2 2 2

dr r d

  

Let us also introdu ould be

rement





2 _dr2 _r2_d 2

(22) Let us note Minkowski metric for co-moving body:

2

d_obs d_r    (23)

At the end, by substituting (21) we obtain:

2 2 2 2 2 2 2

d_obs dr_r dr dr r d (24)

2_d 2

r

r r

2 2 2

dobs d   

eodesics in Schwarzschild metric. Thus we must conclude tha

dler transformationmight be done for accelerated light… To support above claim we will show

re

(25) Comparing above to (11) we recognize g

t our Rin-

in next section that st mass existence is not necessary to consider accelera-tion for light.

We may also easy transform (18) to form of: d

d

r r

r v g



 (26)

Joining above with (7) we may explain acceleration by: 1

d d d

r

v r

d

r r

g

r  

 

cceleration rmay be then expressed by

just introduced imaginary proper time τrand velocity vr.

Recalling (14) we should conc

Schwarzschild metrics may be explained (besides classi-cal explanation) as combination of two Minkowski

met-ric

- fal

ing to above conclusions we will introduce (in Se

new description of field in- te

2.

exists some field equations explaining elec- ure e.

(27)

Gravitational a g

lude, that geodesics in

s for:

(14) stationary observer moving against accelerated, free ling surroundings (light),

(11) free-falling surroundings (light) considered in re- lation to stationary observer proper time.

Referr

ction 3) reformulation of Lagrangian and Hamiltonian what might be understood as

ractions.

2. Vector Fields for Minkowski Time-Space

As we know there is no ether and the medium for elec-tromagnetic 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 descrip-tion, describing at first some regular rotation of Planck’s mass mP, with line velocity vr, on the circle with radius R.

We will define velocity as function of R equal to:

co r

R v c

R

 (28)

where Rcois some defined constant.

Related gamma factor will be equal to:

d 1

d 1 r

co

t

R 



 



ill denote as:

R (29)

Angular velocity for rotating body we w

r

v R

 

Non-relativistic angular momentum we may denote as: (30)

P r m

 

L R v (31)

r Rv P

L m  (32) Non-relativistic radial acceleration we denote as:

2

d d

d d

r _R

t t 

 

v

a R

Maxwell has defined electrom

by

sre-mains [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 nR,nx,ny.For any

conductive vector R we define:

R _R R

R x  y

n n n (35)

Let us define scalar field

r c

 and two related vector

fields:

y r



 

T c n (36)

2

2

r r

y x

r R



     

A c n  v n (37)

c

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 :



T A

   (38)

y y

r r

c c

 

 

 _  _  

 n  n (39)

Let us define scalar field equal to Rv (related to an- gular momentum) and two auxiliary vector fields U and r Ω.





2

r

r x

v

n  n

  U  (41)

r

R R

   

U  v (42) y (40) Rv

 

U

Let us notice, that:



 







2 d

d

c  c t c

r r r

R   R  

 A   R

ve:

    a (43)

From above we deri d

d

r c t 

 

A 

show, that:

 

Let us also

(44)

d

d d

d

y

d d

y

r r

r

t  c   t (45)

r r

c

c

c t c



  

 



 

 

  

n

n T

d

d y

r



 T  n

d d

c t t   (46)

Using (38) we obtain: d

d

c t



  A  ₍₄₇₎

d above we der

r

From (44) an ive two d’Alembertians:

2 2

2

d r



  

2 _d 2 0

c t  (48) 2 2

2 r

  A

)

Above d’Alembertians describes wave with line velo- city

2 2 0

c  t   A (49

or just time dilation around rotation center.

Let us note the d’Alembertians in form of:

2 2

2 2

1 d

0 d

c    

 _

(50)



2 2

1

0



2 2

c     

A

A (51)

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

 

 

r c

d R d dR

R R

dR  dR  dR  

T T

T A T (52)





d r d r d r

r y r

R R v

R

dR  dR  dR  2 

v v

v n v (53)





d

d r

R

R R   

v

U  (54)

Now, let us analyze consequences of derived fie tions and define Lagrangian and Hamiltonian fo defined this way.

3. Reformulated Lagrangian and

Hamiltonian

3.1. Lagrangian and Hamiltonian

Le

ld equa- r fields

t us show how we may describe mechanics when we assign potential with gamma factor denoted as rand

defined with formula (29).

d with gene- Let us first recall a Hamiltonian expresse

ral variables:

i i i

i i i

L

H x L x p

x L



  











  (55)

where L is Lagrangian.

For three dimensional space we denote:

3

2

1 i i

i

x p v mv mv 



  



 (56)

Now, let us add extra zero-dimension an locity and momentum for such dimension. L

to defined previously rotation velocity (28) but here de-ex:

0 0 r r r r r

d define ve-et it be equal

noted with index “r” ind

2

x p   v mv  mv (57) Summing expression for indexes 0

miltonian in form of:

3 we define Ha-

3

2 2

0 i i r r

i

H x p L mv  mv  L







      (58)

0 0

1 1

E

r

L E

 

Thus:

  (59)

2 2

0 0

1 1

r r r

mv  E H mv  E

   

  (60)

2

0 0

1

v v

H E  _{2 2} E  r2₂ 1₂ r

c  c r 

  

      

  (61)

0 0 r

H E E  (62) y easy see, in infinity above Hamiltonian ex- press relativistic kinetic energy just as we shoul





0 E0 E0  1 (63)

s, introduced Lagrang Hamiltonian drives to the same results then we may de-rive from GR for gravity.

Let us now check if introduced Lagrangi At the beginning we should notice, th as we ma

d expect: lim

RH E 

as we will show in few step ian and

3.2. Test for Lagrangian

an pass the tests. at from (51) we obtain:

1

R c obs

 _ 

  (64)

ian we should then rewrite :

Test condition for Lagrang as true only locally in form of

d L

L v



 

_{  }

  ₀

dobs R

 

 (65)

d L

R



 

d

r

L v t     

 (66)

First derivative in LHS introduce momentum:

0

1

E L

v v





mv p

 _

 _{ }

  

  (67)

Next derivative result with rela by gamma:

 

tivistic force multiplied

d d

r r

p _F t

  (68)

vative drives to: For RHS deri

0 2

co r

R L

E R 2R 

 _{ }



have obtained expected Substituting Schwarzschild

Rco and using (4) we could transform above to form of:

(69)

Let us compare (6) with obtained acceleration to no-

tice, that we acceleration.

radius in place of constant

2

L G

mc

 

 _{ }

2 2 2

r r

M mM

G

R c R  

 (70)

We have obtained the force expected for introduced field. W

R

e will denote above force using “r” index:

r r

L F R 

 

 (71)

Now, using above and (68) we rewrite formula (66) as:

r

FF

ace is equal to force caused by field. Above is true for known fields.

3.3. Time-Space Curvature

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:



R



R 0

r

v          (73)



Now we calculate divergence for acceleration: d

d _{  } vr ₍₇₄₎

dt dt

d

dt 



  a (75)

2 2

2

d

co

R

t V V (76)

Now we multiply and div

2

4π _4π

d R R c Rco c

 _{ }

_{   }

ide RHS by constants:

2 2

d _{4 2 8}

dt G V G V



        (77)

Substituting Schwarzschild radius in place of constant

Rco and using (4) we could transform above to form

2 2

co co

c R c R

G G

of: d

8 d

M G

t V



    (78)

Let us denote mass density as:

M V

Let us express pulsation using rotating versor n. W

 (79)

e obtain:

2

c

2 2

d 8

d

n G

t c 



   (80)

2

 2 2

2 2

d 8

d r

n G

t c 4 c r

c  

    (81)

Using (51) we obtain:

2 8 G 2 2

4 r

n c

c  



Now, using above to construct 4-tive vector we obtain relation bet

time-space curvature the same as main equation of Gen-eral Relativity.

3. Rest Energy Formula

Let us consider Hamiltonian for empt test body’s rest energy let us use Pl m

dimentional conduc-

ween mass density and 3

co R c

R 

4. Hamiltonian and

y space. In place of anck’s Energy with eaning of unit “one”.



1



P P r P r

H E E   E   (83)

The smallest R we can substitute is Planck’s length.

1

 

 

lim 1

1

P P

R l

co P

E E

R l

     

  

 

 

(84)

 

For very small constants we could rewrite eq Maclaurin’s expansion:

uation using

4

R c R

lim

2 2

P

co co

P R l

P

E E

l G

     (85)

Let us notice similarity to rest energy f

Rco acts like Schwarzschild radius. We may conclude,

that Schwarzschild radius might be explained as appro- ximation of composition of set of some smaller particles w

gy as follows:

ormula, where

ith Rco  lP. Let us then define two variables for hy-

pothetical mass and ener

2

2 co co

c R M

G







0  1 E0 r1 (88) Using Maclaurin’s expansions

cities and large scales we obtain:

(86)

2

co co

E M c (87) Now, we may easy show that introduced Hamiltonian approximates for small velocities and large distance Me-chanical Energy defined by Newton. Let us transform (62) to form of:





HE 

of above for small velo-

2

2 _mv2 _R

2 m v

m v

2 2 2 2

co r

H mc

R

     (89)

2

2

co

mM mv

H G

R

  (90)

As we may see, above formula acts the s Newton’s Mechanical Energy formula.

s Hypothesis

4.1. Photon Energy

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

4. Photon and Electrostatic

Non-relativistic angular momentum for such move is equal to:

2 2

co co

P P

R R R R

L m c l    

P P

l l (92)

duced Planck’s action. Let us note such Radius with Rω

and define as:

as we may notice, there is some Radius causing that an- gular momentum become equal to smallest action-re-

co P

P R l

 (93)

R_ l

Now we may introduce hypothetical gamm rotation velocity with “E” index:

a factor and

P E

l

v c

R

 (94)

2

1

E

2

1

P

1 1

E l

R

  



(95)

v c



Kinetic Energy for such rotation is equal to:

1 1 1

1 1

kin P

P

l

E E

l

c R



2 2

P P

E

R R



 

 

 

 

 

 

Now let us see, that

  

  (96)

c

R express pulsation in reference

frame assigned to rotating frame. On a circle with radius

R for line velocity c

 we may note as follows:

2

c _{T R}

    (97)

2 2

c 

'

R T T 

 

  (98)



Let then denote inverse of Radius as pulsation and write down:

1 1 1

2 2

kin

E

R_ c



    (99)

If we consider two twisted vectors of rotating field ma- king double Helix, we will obtain we

mula:

ll recognized

for-2 kin

valent versions of Maxwell’s Equations [6] Pair production phenomena might be thu .

s rewritten as:

1 1

2EP 1 2 EP 1

l R

   

 _{  }  _ 

    (101)

1 P 1 co

R_

   

     

 

 

an

P

l

 

 

d after Maclaurin’s approximation:

2

2 mc

 

 (102)

4.2. Electrostatics

Th

expressed in Planck’s units, can be noted as:

e Electrostatic potential of two elementary charges,

2 2

4 4

e e

Q

q q

V c

R c

 

 

   

2

2

0 0

e P

q c

R q R (103)

P

Q P

V E

R 

  l (104)

s introduce auxiliary constan

Let u t “ε” and variables:

1

l

1 P

R





 (105)

P

Q _{R l} P (106)

ve as equivalent of (85). Now, let us assume that expression (104) is Maclau-rin’s approximation

time dilation factor







lim 1

E  E  

We may understand abo

for R  lP of interaction based on .

Therefore, it should follow below formula:



1



2

P P

P

l E 

  (107)

Q Q Q

l

V E E

R R





  

as we can easy derive:

1 1

1 2 1 1

  

 



 

 

 

 

 

(108)

 _{ } _

6.245919

 (109) , we have obtained epsilon close to 2π. It brings to mind De Broglie condition

the orbit. Let us follow this indication.

us being limit for R in formula (106) using present knowledge. Now, we define auxiliary radius such way, to obtain 2

What astonishing

for length of

Let us redefine radi

 . Let us introduce:

1

1 2

P

l R

 



(110)

Q 

0.98890

Q P

R  l (111) Let us redefine (106) and (107) as follows:





lim 1 4

Q

Q _{R R} P Q P

E E  E 



     (112)





₁ P

4

P

Q P

l l

E E

R   R

 (113)

One may easy derive, that el

elementary charges may be expressed as:

Q Q Q

V E   

ectrostatic force for two

1 d

Q

2 2

d 4

P P

l l

E    E (114)

Q Q Q Q P

R R

F E

R 

  

5. Summary

As we have just shown, the same field equations may de- scribe 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. Field “Ω” (41) may act as:

ion,

rvature factor.

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 the 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

6. Acknowledgements

I

Polish Academy of Scienc

would like to thank prof. Iwo Bialynicki-Birula e fo

questions and valuable time he had dedicated to me. I would like to thank Dr. Rafal Suszek from Warsaw University for his time and

I also thank to Dr. inz. Mich niversity of Technology for U

borrowed to me.

I thank to Markus Nordberg fr est and hints on early stage of

I would like to thank poste

man 1917, Celestial Mechanic, Kuroneko, Paul Logan, Tensor, Shaula, tusenfem, Garrison, macaw, grapes, slang, Strange, amazeofdeath, Geo Kaplan, pzkpfw, Eigen State, m, czeslaw, captain swoop, starcanuck 64 and Luckmei- ster. Thank you for your time, critique and questions. This ticle would never have matured without you. Especially I would like to thank to Celestial Mechanic for hope.

I dedicate this

uld work using the time, that should belong to her.

REFERENCES

Berlin, 2007.

[2] V. Petkov, “On the Reality of Minkowski Space,” Foun- dations of Physics, Vol. 37, No. 10, 2007, pp. 1499-1502. doi:10.1007/s10701-007-9178-9

[3] A. Gersten, “Tensor Lagrangians, Lagrangians Equivalent to the Hamilton-Jacobi Equation and Relativistic Dynamics,” Foundations of Physics, Vol. 41, No. 1, 2011, pp. 88-98. doi:10.1007/s10701-009-9352-3

[4] R. Aldrovandi and J. G. Pere ction to Ge-neral Relativit Te´orica, Universi-

ype Approach Legacy,” In-

ematically Equi-

ira, “An Introdu y,” Instituto de F´ısica

val

dade Estadual Paulista, São Paulo, 2004.

[5] A. K. Prykarpatsky and N. N. Bogolubov Jr., “The Max- well Electromagnetic Equations and the Lorentz T Force Derivation—The Feynman

ternational Journal of Theoretic Physics, Vol. 51, No. 5, 2011, pp. 237-245.

[6] T. L. Gill and W. W. Zachary, “Two Math

ent Versions of Maxwell’s Equations,” Foundations of Physics, Vol. 41, No. 1, 2011, pp. 99-128.