Time Dilation as Field

Transcription

1 Jounal of Moden hysis ublished Online Febuay 0 ( Time Dilation as Field iot Ogonowski Wasaw oland piot.ogonowski@piotogonowski.pl eeied Noembe 9 0; eised Januay 6 0; aepted Januay 0 ABSTACT It is poed thee is no aethe and time-spae is the only medium fo eletomagneti wae. Howee onsideing time-spae as the medium we may expet thee should exist field equations desibing eletomagneti wae as distubane in time-spae stutue popagating in the time-spae. I deie suh field equations and show that gaitational field as well as eletomagneti field may be onsideed though one phenomena-time dilation. Keywods: elatiity; Maxwell Equations; Dilation; Lagangian Mehanis. Intodution One of the main poblems of the ontempoay theoetial physis is Quantum Gaity (Betfied Fause Jügen Tolksdof Ebehad Zeidle []). The motiation to eate this pape is onition that efomulation of the onept of fields by emphasis on itselationship with time dilatation fato and time-spae stutue may suppot to effots to field unifiation. Seahing fo Higgs boson o onsideing possible altenaties to Standad Model we ty to explain issues sot of: The natue of the elementay patile est mass The natue of the photon enegy hoton s behaio on lank s enegy sales. The aim of this pape is tosuppot issues mentioned aboe by edefining eletomagneti field equations and stess similaity to Shwazshild solution what may open new ways fo the quest fo quantum gaity and the unified field theoy. Almost a hunded yeas hae passed sine 908 when Hemann Minkowski gae a fou-dimensional fomulation of speial elatiity aoding to whih spae and time ae united into an insepaable fou-dimensional entity now alled Minkowski spae o simply spaetime and maosopi bodies ae epesented by fou-dimensional wolubes. But so fa physiists hae not addessed the question of the eality of these wolubes and spaetime itself (Vesselin etko page []). In this pape I efomulate Shwazshild and Minkowski metis and explain these metis as onsequene of intodued eletomagneti field desiption. In fist setion I eall that one may onside ued time-spae as olletion of loally flat pats of iemannian manifolds with assigned stationay ees. These infinite small flat fagments of time-spae aoding to tansfomed Shwazshild solution and indle s tansfomation appeas to be aeleated. This appoah allows us to define impotant efeene fame that may be used fathe. In seond setion I use aboe appoah and intodue some fields that binds togethe time flow and motion in d Alembetians. Deied wae equation expess distubane in time-spae stutue popagating in time-spae that may be explained as light. In this pape we also efe to Max lank s Natual Units intodued in 899. Let us then denote following designnations: t lank s time l t 4 l E lank s Enegy t G h E edued lank t s ation () E l m lank s mass G qe q 4 0 lank s hage fine stutue onstant q elementay hage e Fathe I will deal with elatiisti dynamis and show that adding axis to Hamiltonian and Lagangian we may obtain pope Lagangian and Hamiltonian fo gaitational field that one may undestand as efomulation of the field inteation phenomena. This way we deelop fathe the idea pesented by Alex- Copyight 0 Sies. JM

2 . OGONOWSKI 0 ande Gesten: ( ) we hae shown that thenon-elatiisti fomalism an be used poided the momenta and Hamiltonian belong to the same 4-eto. (Alexande Gesten page 0 [3]). We will stat with efeene to the main equation of the Geneal Theoy of elatiity. We will naow down ou disussion to a spheially symmetial mass to apply the Shwazshild solution and then we will genealize aboe thanks to indle s tansfomation.. Time Dilation as Field.. Shwazshild Meti and Time Dilation Let us stat with ealling Shwazshild meti (. Aldoandi and J. G. eeia page [4]) and onside elation between gaitational potential and time dilation. To simplify alulations in whole setion we ae assuming =. Fo body obiting at one plane aound non-otating big mass we may wite meti in fom of: s d d d s We assume: τ is the pope time of ee s efeene fame; t is the time oodinate (measued by a stationay lok at infinity); is the adial oodinate; φ is the olatitude angle; s is the Shwazshild adius. Aoding to this solution the Shwazshild s adius and mass fomulas ae: () GM s (3) s M (4) G We intodue elatiisti gamma fato: s (5) wee all that Shwazshild s solution dies to gaitational aeleation in distane equal to: g GM As we may easy alulate: g s s (6) s d d (7) d d Aboe fomula dies us to onlusion that elatiisti gamma ats hee as it would be sala field. Let us explain aboe and its wide onsequenes in few steps. At fist step let us ewite Shwazshild meti fo some new efeene fame. We stat using fomula (5): d d d ( 8) We may onside Shwazshild meti fo stationay ee hanging at some point at distane to soue of gaitational foe (suh ee has to use some foe to keep his position). We will denote suh ee pope time as τ : d (9) Now we might ewite Shwazshild meti (8) efeing to some loal hosen stationay ee efeene fame and its pope time. d d d d (0) If we will note aboe fo geodesis we obtain: d d d () Aboe fomula will be useful soon. Using suh stationay ee efeene fame we eall that iemannian manifolds ae loally flat. If we shink onsideed time-spae into sphees with hosen a- we obtain spheial anisotopi Minkowski meti with dius slowe oodinate light speed aoding to (). If we shink it moe we onside infinitie small loal pat of hosen sphee whee photon meets Stationay Obsee. At seond step let us intodue eloity : s () We eognize eloity as Esape eloity and Fee-falling eloity thus we intodue some elated spatial inement dx : dx (3) and then deie fom (9) below fomula: d dx (4) It is easy to notie that aboe fomula ats just like it would be Minkowski fo fee-falling eloity. Now at thid step let us eall indle s tansfomation in some plane Minkowski time-spaefo body moing with aeleation a ahieing eloity. We may onside suh body using o-moing ee onept. We will denote its pope time as τ and note: at (5) Copyight 0 Sies. JM

3 0. OGONOWSKI t a (6) a (7) Now we may onside some hypothetial body with aeleation g ahieing eloity with pope time τ. g (8) Let us pefom following tansfomation: d d (9) d d d d d (0) d d d () Let us also intodue spatial inement as it would be inplane Minkowski meti. We will note this inement in pola oodinates: d d Let us note Minkowski meti fo o-moing body: d d d d () (3) At the end by substituting () we obtain: d d d d d (4) d d d (5) Compaing aboe to () we eognize geodesis in Shwazshild meti. Thus we must onlude that ou indle tansfomationmight be done fo aeleated light To suppot aboe laim we will show in next setion that est mass existene is not neessay to onside aeleation fo light. We may also easy tansfom (8) to fom of: d g (6) d Joining aboe with (7) we may explain aeleation by: d d g (7) d d Gaitational aeleation g may be then expessed by just intodued imaginay pope time τ and eloity. ealling (4) we should onlude that geodesis in Shwazshild metis may be explained (besides lassial explanation) as ombination of two Minkowski metis fo: (4) stationay ee moing against aeleated fee- falling suoundings (light) () fee-falling suoundings (light) onsideed in elation to stationay ee pope time. efeing to aboe onlusions we will intodue (in Setion 3) efomulation of Lagangian and Hamiltonian what might be undestood as new desiption of field inteations... Veto Fields fo Minkowski Time-Spae As we know thee is no ethe and the medium fo eletomagneti wae is time-spae. We should expet then thee must exists some field equations explaining eletomagneti wae as distubane in time-spae stutue (stutue of the medium) distibuting in the time-spae. Let us pepae to suh eletomagneti field desiption desibing at fist some egula otation of lank s mass m with line eloity on the ile with adius. We will define eloity as funtion of equal to: o (8) whee o is some defined onstant. elated gamma fato will be equal to: d o (9) Angula eloity fo otating body we will denote as: (30) Non-elatiisti angula momentum we may denote as: L m (3) L m (3) Non-elatiisti adial aeleation we denote as: d d a (33) Maxwell has defined eletomagneti field phenomena by eliminating patiles fom equations while field semains [5]. Let us do the same but eliminating test body while motion and time flow emains. We will onstut eto fields to desibe whole lass of just intodued otations defined fo any plae in spae. est mass we may undestand as paamete. Let us define at fist thee esos n n x n y. Fo any ondutie eto we define: n (34) Copyight 0 Sies. JM

4 . OGONOWSKI 03 n nx n y (35) Let us define sala field and two elated eto fields: T n (36) y A ny n x (37) lease note similaity of aboe A field to aeleation in Shwazshild meti noted in (7) what will be useful soon. As we an easy show: T A beause : (38) ny n y (39) Let us define sala field equal to (elated to angula momentum) and two auxiliay eto fields U and Ω. U nx ny (40) U (4) Let us notie that: U (4) d A Fom aboe we deie: d A Let us also show that: a (43) (44) d ny d d T ny (45) d d T n Using (38) we obtain: y (46) da (47) Fom (44) and aboe we deie two d Alembetians: d 0 A t 0 (48) A (49) Aboe d Alembetians desibes wae with line eloity o just time dilation aound otation ente. Let us note the d Alembetians in fom of: d 0 (50) d A A 0 (5) Aboe fom of d Alembetians desibes moe with speed in infinite small loal pat of time-spae. This way in loal efeene fame light has always speed. We may also expet now that deiatie by should at simila to deiatie by pope-time. Let us pefom some hypothetial alulation to poe it: dt dt d T AT (5) d d d d d d ny (53) d d d d U (54) d Now let us analyze onsequenes of deied field equations and define Lagangian and Hamiltonian fo fields defined this way. 3. efomulated Lagangian and Hamiltonian 3.. Lagangian and Hamiltonian Let us show how we may desibe mehanis when we assign potential with gamma fato denoted as and defined with fomula (9). Let us fist eall a Hamiltonian expessed with geneal aiables: L H xi L xipi L (55) x i whee L is Lagangian. Fo thee dimensional spae we denote: i i 3 x ipi m m (56) i Now let us add exta zeo-dimension and define e- it be equal loity and momentum fo suh dimension. Let to defined peiously otation eloity (8) but hee de- noted with index index: x p m m (57) 0 0 Summing expession fo indexes 03 we define Hamiltonian in fom of: 3 i i (58) i0 H xplm m L Now we define Lagangian in fom of: Copyight 0 Sies. JM

5 04. OGONOWSKI Thus: L E E (59) 0 0 H m E0 m E0 (60) H E0 E0 0 0 (6) H E E (6) as we may easy see in infinity aboe Hamiltonian expess elatiisti kineti enegy just as we should expet: lim H E 0 E0 E0 (63) as we will show in few steps intodued Lagangian and Hamiltonian dies to the same esults then we may deie fom G fo gaity. 3.. Test fo Lagangian Let us now hek if intodued Lagangian pass the tests. At the beginning we should notie that fom (5) we obtain: (64) Test ondition fo Lagangian we should then ewite as tue only loally in fom of: L d L 0 d (65) L d L (66) Fist deiatie in LHS intodue momentum: E0 L m p (67) Next deiatie esult with elatiisti foe multiplied by gamma: dp F (68) Fo HS deiatie dies to: L o E0 (69) Let us ompae (6) with obtained aeleation to notie that we hae obtained expeted aeleation. Substituting Shwazshild adius in plae of onstant o and using (4) we ould tansfom aboe to fom of: L GM mm m G (70) We hae obtained the foe expeted fo intodued field. We will denote aboe foe using index: L F (7) Now using aboe and (68) we ewite fomula (66) as: F F As we may onlude just deied elatiisti foe ating on body in eey patiula plae in spae is equal to foe aused by field. Aboe is tue fo known fields Time-Spae Cuatue Let us alulate diegene fo line eloity: 0 (7) (73) Now we alulate diegene fo aeleation: d d (74) d a (75) o 4π d 4π V V Now we multiply and diide HS by onstants: o (76) d o o 4 G G 8 G G (77) V V Substituting Shwazshild adius in plae of onstant o and using (4) we ould tansfom aboe to fom of: d M 8 G (78) V Let us denote mass density as: M (79) V Let us expess pulsation using otating eso n. We obtain: dn 8G (80) G 4 dn 8 (8) Using (5) we obtain: 8G n 4 (8) Copyight 0 Sies. JM

6 . OGONOWSKI 05 Now using aboe to onstut 4-dimentional ondu- tie eto we obtain elation between mass density and o 3 time-spae uatue the same as main equation of Geneal elatiity. Non-elatiisti angula momentum fo suh moe is equal to: Hamiltonian and est Enegy Fomula o o L ml (9) Let us onside Hamiltonian fo empty spae. In plae of l l test body s est enegy let us use lank s Enegy with as we may notie thee is some adius ausing that anmeaning of unit one. gula momentum beome equal to smallest ation-e- H E E E (83) dued lank s ation. Let us note suh adius with ω and define as: The smallest we an substitute is lank s length. l o (93) l lim E E (84) Now we may intodue hypothetial gamma fato and l o otation eloity with E index: l l E (94) Fo ey small onstants we ould ewite equation using Malauin s expansion: 4 E (95) o o l lim E E (85) E l l G Let us notie similaity to est enegy fomula whee Kineti Enegy fo suh otation is equal to: o ats like Shwazshild adius. We may onlude that Shwazshild adius might be explained as appo- l ximation of omposition of set of some smalle patiles Ekin E E (96) with o l. Let us then define two aiables fo hy- l pothetial mass and enegy as follows: M o o G E M 0 0 Using Malauin s expansions of aboe fo small eloities and lage sales we obtain: m m m o H m (89) m mm H G o (90) As we may see aboe fomula ats the same way that Newton s Mehanial Enegy fomula. (86) o o (87) Now we may easy show that intodued Hamiltonian appoximates fo small eloities and lage distane Mehanial Enegy defined by Newton. Let us tansfom (6) to fom of: H E E (88) 4. hoton and Eletostatis Hypothesis 4.. hoton Enegy Let us notie that pulsation desibed in (30) is equal to: Now let us see that expess pulsation in efeene fame assigned to otating fame. On a ile with adius fo line eloity we may note as follows: T (97) (98) T T' Let then denote inese of adius as pulsation and wite down: E kin (99) If we onside two twisted etos of otating field making double Helix we will obtain well eognized fomula: E E kin (00) We may suppose that E desibes eletomagneti field and the aboe quantum of enegy may be assigned to photon. This way we may teat (50) and (5) as equi- Copyight 0 Sies. JM

7 06. OGONOWSKI alent esions of Maxwell s Equations [6]. ai podution phenomena might be thus ewitten as: E E l (0) o l and afte Malauin s appoximation: 4.. Eletostatis m (0) The Eletostati potential of two elementay hages expessed in lank s units an be noted as: qe qe VQ 4 q e (03) 4 q VQ E 0 0 l (04) Let us intodue auxiliay onstant ε and aiables: (05) l E lim E Q l (06) We may undestand aboe as equialent of (85). Now let us assume that expession (04) is Malauin s appoximation fo l of inteation based on time dilation fato. Theefoe it should follow below fomula: l l VQ EQ EQ E (07) as we an easy deie: (08) (09) What astonishing we hae obtained epsilon lose to π. It bings to mind De Boglie ondition fo length of the obit. Let us follow this indiation. Let us edefine adius being limit fo in fomula (06) using pesent knowledge. Now we define auxiliay adius suh way to obtain. Let us intodue: Q l (0) Q l () Let us edefine (06) and (07) as follows: E lim E E 4 () Q Q Q V E E Q Q Q Q l 4 E l (3) One may easy deie that eletostati foe fo two elementay hages may be expessed as: 5. Summay d Q l l FQ EQ EQ E (4) Q d 4 As we hae just shown the same field equations may desibe as well eletomagnetism as well gaity. Field A defined in (37) may ente in plae of: Eletomagneti field Gaitational aeleation. Field T (36) and elated sala potential may at as: Eletostati potential Gaitational potential. Field Ω (4) may at as: Magneti otation Time-spae uatue fato. 6. Aknowledgements I would like to thank pof. Iwo Bialyniki-Biula fom olish Aademy of Siene fo itique asked me uial questions and aluable time he had dediated to me. I would like to thank D. afal Suszek fom Wasaw Uniesity fo his time and letues he has pointed to me. I also thank to D. inz. Mihal Wiezbiki fom Wasaw Uniesity of Tehnology fo his time and books he had boowed to me. I thank to Makus Nodbeg fom CEN fo his intethe atile. est and hints on ealy stage of I would like to thank postes fom BAUT foum: aeman 97 Celestial Mehani Kuoneko aul Logan Tenso Shaula tusenfem Gaison maaw gapes slang Stange amazeofdeath Geo Kaplan pzkpfw Eigen State Jim zeslaw aptain swoop staanuk 64 and Lukmeiste. Thank you fo you time itique and questions. This atile would nee hae matued without you. Espeially I would like to thank to Celestial Mehani fo hope. I dediate this atile to my Joanna fo the fat that I ould wok using the time that should belong to he. EFEENCES [] B. Fause J. Tolksdof and E. Zeidle Quantum Gaity: Mathematial Models and Expeimental Bounds Spinge Copyight 0 Sies. JM

8 . OGONOWSKI 07 Belin 007. [] V. etko On the eality of Minkowski Spae Foundations of hysis Vol. 37 No pp doi:0.007/s [3] A. Gesten Tenso Lagangians Lagangians Equialent to the Hamilton-Jaobi Equation and elatiisti Dynamis Foundations of hysis Vol. 4 No. 0 pp doi:0.007/s [4]. Aldoandi and J. G. eeia An Intodution to Geneal elatiity Instituto de F ısia Te oia Uniesi- dade Estadual aulista São aulo 004. [5] A. K. ykapatsky and N. N. Bogolubo J. The Maxwell Eletomagneti Equations and the Loentz Type Foe Deiation The Feynman Appoah Legay Intenational Jounal of Theoeti hysis Vol. 5 No. 5 0 pp [6] T. L. Gill and W. W. Zahay Two Mathematially Equialent Vesions of Maxwell s Equations Foundations of hysis Vol. 4 No. 0 pp doi:0.007/s Copyight 0 Sies. JM