The recoil nature of electrostatic and gravitational forces

Transcription

1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 55, No. 4, 7 Th roil natur of ltrostati and gravitational fors S.L. HAHN Polish Aadmy of Sins, 1 Dfilad Sq., -91 Warszawa, Poland Abstrat. Th papr xplains th for indud by th ltrostati fild on th ltron as a roil for. Th starting point is th hypothsis that in th dynami quilibrium with th vauum, th ltron simultanously absorbs and mitts nrgy. With no xtrnal ltrostati fild th radiation pattrns of absorption and mission ar assumd to b isotropi. Th xtrnal ltrostati fild indus anisotropy of th mission rsulting in a roil for. Th papr prsnts a thortial dsription of this for using a modl of th angular powr dnsity pattrn of th mission in th form of an llipsoid. Calulations show that th total radiatd powr is xtrmly high. This radiation is ompard with th ltromagnti radiation of th ltron on th Bohr orbit in th idalizd hydrogn atom. An analogous problm for gravitational fors is prsntd. 1. Introdution This papr is a modifid vrsion of a similar papr writtn by th author in 1976 [1]. W didd to writ th modifiation for two rasons: Firstly, th papr [1] was writtn in Grman and printd in Klinhubahr Briht whih has a limitd numbr of radrs. Sondly, in rnt yars w obsrvd a bttr undrstanding of th natur of quantum vauum as a mdium of xtrmly high nrgy dnsity. Not, that th papr [1] was basd on idas onrning th natur of th vauum prsntd by Vallé in [].. Prlminaris For onvnin, th dfinitions and notations of all notions usd in this papr ar prsntd in th Appndix in Tabl 1. Considr a uniform ltrostati fild E x = E x, whr x is a vrsor dirtd along th x axis in th 3-D spa R. Lt us assum that a singl ltron an b rprsntd by a point harg. Th xtrnal ltrostati fild E x indus a for F x = E x. (1) Lt us assum that this quation holds for finit dimnsions of th ltron. An analogous formula for a point mass m and th gravitational fild G x = G x is F x = m G x. () Th nrgy dnsity of th ltrostati fild is E E =.5ε E (3) whr ε is th prmittivity of th fr spa. Th analogous nrgy dnsity of th gravitational fild is E G =.5γG (4) whr γ is th gravitational prmittivity of th fr spa (s Tabl 1). Not th minus sign whih indiats that th nrgy lvl of th gravitational fild is lowr than th nrgy of th zro-fild vauum. This fat has bn dsribd in []. A simpl vidn is givn is Appndix. 3. Enrgy dnsity of th quantum vauum Th nrgy dnsity of th quantum vauum an b drivd using th Plank s formula ρ (f, T) = 8πf 3 [ hf hf kt 1 + hf whr h is th Plank onstant, k th Boltzmann onstant, T th absolut tmpratur and f th frquny of th radiation. For T = w gt ] (5) ρ (f, T = )) = 4πhf3 3. (6) Th trm hf/ rprsnts th zro-point flutuations of th quantum vauum. Th quantum vauum is not mpty (void) but is a mdium of xtrmly high nrgy dnsity. Th total nrgy dnsity in th frquny band from f 1 to f is ρ = f f 1 ρ (f)df = πh ( f 4 3 f1 4 ). (7) Exampls. 1. Plank suggstd that th highst frquny of th quantum vauum is dfind by th formula f max = f = 5 hg [Hz]. (8) -mail: [email protected] 45

2 S.L. Hahn Tabl 1 Notations, dfinitions, valus and dimnsions (in SI) of sltd physial quantitis Notation Nam Dfinition Valu SI Units µ Prmability of fr spa Dfind valu 4π 1 7 Hy/m Spd of light in fr spa = 1/ ε µ M/s ε Prmittivity of fr spa ε = 1/ ( µ ) F/m h Plank s onstant Js Modifid Plank s onstant = h/π Js Elmntary harg C m Rst mass of th ltron kg m p Rst mass of th proton kg Z Impdan of fr spa Z = µ /ε Ω R H Quantizd Hall rsistan R H = h/ Ω r B Bohr radius r B = [ε /(πm )] (h/) m α Fin strutur onstant α = Z /(R H ) α 1 Riproal of α λ Compton wavlngth of th ltron λ = h m m f Compton frquny of th ltron f = = m λ h Hz f p Compton frquny of th proton f p = = mp λ p h Hz r lass. Classi radius of th ltron r lass. = µ 4πm m r Compton radius of th ltron r = λ 4π = h 4πm m λ p Compton wavlngth of th proton λ p = h m p m 4πε r E max Th maximum possibl valu of th ltri E max = V/m filds dfind by th ylindrial modl of th ltron = 4π ( m ) ( /h ) µ B Bohr magnton µ B = h/(4πm ) Am or JT 1 λ Gravitational prmittivity of fr spa γ = 1 4πG kg s /m 3 G Gravitational onstant Masurd m 3 /kgs η Gravitational prmability of fr spa η = 16πG m/kg This valu of f with f 1 = yilds ρ max = πhf4 max [J/m 3 ]. (9) W may assum, that this total nrgy dnsity of th quantum vauum is infinit.. Considr th frquny band of visibl light with f 1 = [Hz] and f = [Hz]. W gt ρ = [J/m 3 ] In th nxt two xampls w alulat th valu of th frquny band loatd around th Compton frquny of th partil whih yilds th nrgy dnsity of th quantum vauum qual to th nrgy dnsity dfind by th division of th Einstin s nrgy E = m by th volum of th partil. Fig. 1. Th ylindrial modl of th ltron of rfrn aftr Rf. 3. Th volum of th ylindr modl of th ltron is (s Fig. 1) is V = πr 3 = π h 3 (4πm ) 3 = [m 3 ] (1) This yilds th nrgy dnsity dfind by th quation ρ = m V = 3π m 4 5 h 3 = [J/m 3 ]. (11) In this xampl w insrt in Eq.(7) f = af and f 1 = gtting th nrgy dnsity (f s Tabl 1) E = πh 3 a4 f 4 = πa4 m 4 5 h 3. (1) Equations (11) and (1) yild a 4 = 4π, i.., a Conluding, th Einstin s nrgy of th ltron orrsponds to th nrgy of th vauum in a wid frquny band xtnding from zro to about th Compton frquny of th ltron. 4. Lt us prsnt a similar alulation for th proton. W start with a masurd valu of th radius of th proton r p = [m] and th mass m p = [kg]. Assuming a sphrial modl of th proton, th orrsponding volum is V p = (4/3)πr 3 p = [m 3 ]. This yilds th nrgy dnsity ρ p = m p V p = [J/m 3 ] (13) 46 Bull. Pol. A.: Th. 55(4) 7

3 Th roil natur of ltrostati and gravitational fors In this xampl w insrt in (7) f = f p (1 + a) and f 1 = f p (1 a), whr th Compton frquny of th proton is 8f 4 p yilds f p = m p = = (14) h λ p [ W hav f 4 f4 1 = fp 4 (1 + a) 4 (1 a) 4] = ( ) a + a 3. Th insrtion in (7) and quating with (13) 8πh 3 [ a + a 3 ] = (15) W gt (a + a 3 ) =.43 giving a.49. W obsrv that diffrntly to th as of ltron, th frquny band around th Compton frquny of th proton yilding th Einstin s nrgy dnsity is narrow and quals to f =.858f p Th modl of th ltron usd in [] to alulat th highst possibl valu of th ltri fild strngth. Th author of [] proposd a ylindrial modl of th ltron shown in Fig. 1. Th radius of th bas quals to r = λ 4π = h 4πm = [m], (16) whr λ is th Compton wav-lngth of th ltron. Th hight of th ylindr quals r. Th author of [] assumd, that th lmntary harg is uniformly distributd on th surfa S = 4πr of th sid wall of th ylindr. W hav th rlation = Sε E max (17) whr E max is th fild strngth at th bordr of th sid-wall of th ylindr. W gt E max = ε S = 4πε r = 4πµ f = [V/m]. (18) If w apply instad of th ylindr a sphr of radius r s and assum a uniform distribution of th harg on th surfa of this sphr, thn E max = 4πε rs (19) and w gt r s = r, i.., th sam as th radius of th ylindr. Sin th nrgy dnsity of th ltrostati fild as a funtion of th radius r of a sphr is ρ E =.5ε E (r), th total nrgy outsid th sphr is r 4πr ρ (r) dr = 8πε r. () Lt us ompar this nrgy with km ( < k < 1). W gt r = 1 k µ 8πm. (1) For k =.5 w gt r = r, whr r is th lassial radius of th ltron. For a sphrial modl of th ltron with lassi radius, th nrgy of th xtrnal fild quals on half of m. It an b shown, that r = 1 α 1 r (α fin strutur onstant), i.., th radius of th ylindrial modl proposd in [] is about α 1 / 68.5 tims longr than th lassial radius. As wll, th ltri fild strngth at th bordr of th sphr of lassial radius would b α /4 gratr with rspt to E max dfind by (18). Th insrtion in (1) of r = r yilds k = k = α = () This alulation holds for th sphrial modl and with a good approximation for th ylindrial modl. For ths modls, th nrgy of th xtrnal ltrostati fild is ngligibl in omparison to th Einstin s rst nrgy of th ltron. Howvr, (16) is drivd assuming, that th nrgy insid th ylindrial modl xatly quals m. 4. Th hypothsis about th roil natur of ltrostati fors Lt us prsnt drivations dsribing th ltrostati for (1) as a roil for. Th drivations ar basd on th hypothsis that th ltron is in a dynami quilibrium with th nrgti mdium of th quantum vauum. It is assumd that th ltron ontinuously absorbs and mits radiation. In absn of any xtrnal ltrostati fild, it is assumd that th dirtional pattrns of absorption and mission ar isotropi (for th sphrial modl) or irularly symmtri (for th ylindrial modl). It is assumd that th xtrnal ltrostati fild indus anisotropy of th mission whil th absorption rmains isotropi. In that as. th anisotropy of th mission indus a roil for givn by th intgral v σ Ω n dω (3) max 4π whr σ Ω [W/Str] is th angular powr dnsity of th radiation (powr pr unit solid angl), v th vloity of th radiation and n a vrsor, whih yilds th maximum valu of th intgral. Lt us invstigat th as for th powr dnsity diagram givn by th llipsoid (s Fig.) σ Ω = σ max 1 ε 1 + ε os(ϕ) (4) whr ε is th ntriity of th llipsoid. If ε 1, th drivation prsntd in th Appndix 1 yilds v Pε, (5) 3 whr P is th total radiatd powr Th for (5) should b qual to th ltrostati for E. This yilds th following xprssion for th powr P Bull. Pol. A.: Th. 55(4) 7 47

4 S.L. Hahn E v Pε 3 E 3 P =. (6) v ε Fig.. Th rotation of th llips around th major x axis dfins th llipsoid In th drivation in th Appndix 1 w suggstd that th ntriity ε E E. Thrfor, assuming [ v) =, w gt max P = 3 E max = 1π fz =, [W]. (7) With no ompnsation of th mission by th absorption, th ltron should day in a tim t = m P = [s]. (8) W hav shown, that th powr simultanously absorbd and mittd in th dynami quilibrium with th quantum vauum is xtrmly high. Not that in th prsn of an ltrostati fild, th xtrnal nrgy dnsity is diffrnt at th two sids of th ltron. Th ltron radiats mor nrgy towards th lowr nrgy dnsity rgion w.r.t. th othr rgion with highr nrgy dnsity. This xplains th natur of th roil for. 5. Th hypothsis about th roil natur of gravitational fors Lt us invstigat, whthr th abov dsribd natur of th ltrostati for as a roil for applis also for gravitational fors. Th attration of two point masss orrsponds th rpl of two point hargs of th sam sign. This diffrn is ausd by th fat, that th nrgy lvl of th vauum in prsn of th ltrostati fild is highr than th nrgy lvl of th zro-fild vauum, whil th gravitational fild lowrs th nrgy lvl of th vauum. (s Appndix 3). Howvr, th appliation of th llipsoid modl for th dirtivity pattrn of th mission in th gravitational as rquirs th knowldg of th ntriity of th llipsoid. In th ltrostati as it was assumd that th ntriity is ε E / E max, whr E max is th maximum possibl valu of th ltrostati fild strngth postulatd in []. Th xtrnal fild is always lowr than E max, and usually ε 1. Diffrntly, th magnitud of ign gravitational fild at th bordr of th ltron or any othr lmntary partil is vry small and may b ngligibl in omparison to th magnitud of th xtrnal gravitational fild. Th maximum valu of th ltrostati fild is dfind mirosopiaaly as a fild at th bordr of a hargd partil. Diffrntly, th vntual maximum valu of th gravitational fild is dfind marosopially, for xampl at th bordr of a nutron star or insid a blak hol. Lt us hav an xampl with th nutron star PSR B (a mmbr of a twin star). It has th mass m PSR = th mass of th sun [kg] and a radius [m]. This yilds th immns mass dnsity g = [kg/m 3 ]. Th orrsponding magnitud of th gravitational fild at th sur- fa of th star is grp SR PSR = 3γ = [m/s ]. A blak hol having th mass qual to th arth-mass, should hav a mass dnsity 1 3 [kg/m 3 ] and th Shwartzshild radius r S [m]. This yilds th gravitational fild on th surfa grs 3γ = [m/s ]. W may assum that th vntual maximum possibl valu max has th sam ordr. Th gravitational form of th Eq.(6) taks th form m v Pε 3m 3 Pε = v. (9) Th alulation of th valu of th powr P would b possibl if th ntriity ε ould b alulatd. Atually w hav not found a mthod nabling th dtrmination of th valu of ε. For a mass m = 1 [kg] loatd at th arth surfa w gt, using 9.81 [m/s ] and v, Pε [W]. Of ours, th powr P is svral ordr of largr, sin rtainly ε 1. Not, that th powr pr a singl partil is lowr. 6. Th radiation of th ltron on th Bohr orbit Last tim th author found in rfrn [3 5] a short information, that th ltron in th Bohr modl of th hydrogn atom radiats ltromagnti nrgy and that this nrgy is simultanously ompnsatd by th vauum. Howvr, th author of [3 5] has not prsntd any vidn or alulations. As wll h gav not a rfrn to th papr [1]. Lt us quot a statmnt from th intrnt part of rfrn [3 5]. Thr it is shown that th ltron an b sn as ontinually radiating away its nrgy as prditd by lassial thory, but simultanously absorbing a ompnsating amount of nrgy from th vr-prsnt sa of zro-point nrgy in whih th atom is immrsd, and an assumd quilibrium btwn ths two prosss lads to th orrt valus for th 48 Bull. Pol. A.: Th. 55(4) 7

5 Th roil natur of ltrostati and gravitational fors paramtrs known to dfin th ground-stat orbit. End of itation. In fat, thrading of th rfrn [3 5] supportd our dision to writ this papr. For ompltnss lt us alulat th powr of th ltromagnti radiation of th ltron moving with a onstant vloity a (α fin strutur onstant) on th irular orbit of th Bohr modl of th hydrogn atom. In this xampl, thr is no nd to introdu orrtions du to th finit mass of th proton and rlativisti mass of th ltron. With ths assumptions th Bohr radius is givn by th formula r B = ε πm Th angular vloity of th ltron is ( ) h. (3) ω = πm 4 ε. (31) h3 This yilds th priod of a singl rvolution T = 4ε h3 m 4. (3) Th instantanous powr radiatd by a harg moving along a path s(t) with th alration s (t) is drivd in [6] P (t) = 6πε 3 ( s (t)). (33) On th irular Bohr orbit w hav a ntrifugal tim indpndnt alration s = r B ω = πm 6 4ε 3. (34) h4 Th insrtion of (33) in (3) yilds th following powr radiatd by th ltron irulating on th Bohr orbit P = πm 14 96ε 7 3 h 8 = 4π 3 m 4 h α7 = [W], (35) and th nrgy radiatd during a singl rvolution is E T = PT = 4π 3 m α 5. (36) W obsrv, that th powr of th ltromagnti radiation of th ltron in th Bohr modl of th hydrogn atom is vry small but finit. It is ngligibl w.r.t. th powr du to th dynami quilibrium of th ltron with th quantum vauum. Lt us say, that w ar awar that th Bohr modl may b far from physial rality. For xampl, th author of [7] prsntd anothr modl bliving it is losr to th physial rality. Appndix 1. Drivation of th roil for for an llipsoidal powr radiation pattrn W assum, that th angular powr radiation pattrn (powr dnsity pr unit solid angl) is givn by th rotation around th major axis of th llips 1 ε σ Ω = σ max [W/Str] (A1) 1 + ε os(ϕ) whr ε is th ntriity of th llips. This formula uss th polar oordinats ntrd in th fous of th llipsoid. Th roil for is givn by th intgral v σ Ω n dω (A) 4π whr v is th vloity of radiation and n a unit vtor dirtd along th major axis of th llips. Th insrtion of (A1) and using th projtion of th radius ntrd in th fous on th major axis (os(ϕ)) yilds v ( ) 1 ε os(ϕ) dω. (A3) 1 + ε os(ϕ) W gt σ max π π 4π ( ) 1 ε os(ϕ) sin(ϕ) dϕdψ. 1 + ε os(ϕ) Th valuation of th intgral yilds σ max f 1 (ε), whr f 1 (ε)= [ 1 ε ε (A4) (A5) log ( 1 ε ) + ( ] 1 ε ) ε n 1 π. n (n 1) n=1 (A6) Howvr, σ max should b normalizd to kp th total powr P indpndnt on ε. Th powr gain of th llipsoid is givn by th formula G = 4π B whr B is th quivalnt solid angl with B = π π (A7) 1 ε 1 + ε os(ϕ) sin (ϕ) dϕdψ = f (ε) (A8) [ (1 f (ε) = ε ) ( )] 1 + ε log π. (A9) 1 ε Sin σ max = PG w gt σ max = P f (ε). (A1) Bull. Pol. A.: Th. 55(4) 7 49

6 S.L. Hahn Th insrtion of (A1) in (A5) yilds P f 1 (ε) f (ε). (A11) If ε << 1, th ratio f 1 (ε)/f (ε) ε/3. Th vrsor n is in th dirtion of th x axis. Th angular dirtional pattrn is dfind w.r.t. th right fous. Th roil for has th dirtion opposit to th x axis. Appndix. Th ngativ sign of th nrgy dnsity of th gravitational fild In th fram of th analogis btwn ltromagnti and gravitation whih apply for linarizd Einstin s quations, th nrgy dnsity of th gravitational fild is givn by th quation E G =.5γ [J/m 3 ], whr G [m/s ] is th gravitational fild and γ = [kg s /m 3 ] is th gravitational prmittivity of fr spa [8]. Lt us driv why th gravitational fild lowrs th nrgy dnsity of th vauum. Considr two paralll infinit plans ah ovrd by a mass dnsity ρ m [kg/m ] (Fig. A1). Th gravitational fild insid th plans quals zro and its magnitud outsid th plats is ρ m /γ. Th nrgy dnsity outsid th plans is E G =.5γ ρ m /γ [kg/ms ]. Imagin that th distan btwn th plans is nlargd by z. Th gravitational fild in th volum dfind by z is anlld. Not that nrgy dnsity and prssur hav th sam dimnsions. Sin th plans attrat, th nlargmnt is a shift against th prssur and orrsponds to th input of a positiv nrgy. Thrfor, th anllation of th gravitational fild rquirs a positiv nrgy. In onsqun, th nrgy of th gravitational fild is ngativ. If w apt th hypothsis that nrgy is a positiv dfind quantity, th abov ngativ nrgy orrsponds to a lowring of th positiv nrgy of th vauum. Fig. A1. Two paralll plans ovrd by a mass dnsity ρ m REFERENCES [1] S. Hahn, Di Dutung dr Kraft auf in Elktron im Elktrostatishn Fld als Rűkstasskraft, Klinhubahr Briht, (1976). [] R.L. Vallé, L Énrgi Éltromagntiqu Matérill t Gravitationll, Masson Ci, Paris, [3] H.E. Puthoff, Ground stat of th hydrogn atom as a zropoint flutuation stat, Phys.Rv. 35 (1), (1987). [4] H.E. Puthoff, Why atoms don t ollaps, Nw Sintist 1, 6 (1987). [5] H.E. Puthoff, Quantum vauum flutuations: a nw Rostta ston of physis, (7). [6] K. Simonyi, Thortish Elktrothnik, VEB Vrlag dr Wissnshaftn, Brlin, [7] M. Gryziński, Th qustion of th atom, Homo Sapins, Warszawa, 1, (in Polish). [8] R.L. Forward, Gnral rlativity for th xprimntalist, Pro. IRE, (1961). 41 Bull. Pol. A.: Th. 55(4) 7