Gravitational Equivalent Frequency, the Planck Length and Matter Waves

Transcription

1 Gaitational Equialent Fequeny, te Plank Lengt and Matte Waes by Roge Ellman Abstat Analysis of gaitation and of matte waes disloses a geate signifiane tan eetofoe eognized fo te fequeny, wae, osillation aspet of mass and matte as ompaed to te pesent empasis on disete patiles. Te mass equialeny m = f applies to gaitational mass just as to inetial mass. Te gaitational mass as a oesponding equialent fequeny, f. Wit tat te signifiane of te Plank Lengt, l Pl, laifies; te Plank Lengt is fundamental to gaitation and, in effet, supesedes te Newtonian Gaitational onstant, G, in tat ole. Tee is opeational meanial signifiane to te ole of te Plank Lengt in gaitation weeas G is only a onstant of popotionality. Te natue of elatiisti kineti enegy and its signifiane fo matte waes is deeloped. Teoetial impliations ae pesented and would appea to imply a geate signifiane fo te fequeny, tat is te wae, aspet of mass, matte, and patiles in geneal tan eetofoe eognized. Roge Ellman, Te-Oigin Foundation, In. 3 Gemma Cile, Santa Rosa, CA 9544, USA RogeEllman@Te-Oigin.og ttp://

2 Gaitational Equialent Fequeny, te Plank Lengt and Matte Waes Gaitational Equialent Fequeny by Roge Ellman Conside a small indiidual patile su as a poton o a neuton. Te gaitational ation of a massie body is te olletie effet of te indiidual ation, below, in ea of its su patile omponents. Newton's law of gaitation expessed in tems of m soue and m ated-on and wit bot sides of te equation diided by m ated-on is, of ouse, msoue () a ga = G d stating tat gaitation is a popety of a body's mass. Howee, mass and enegy ae equialent, so tat a mass, m, is popotional to a fequeny, f, tat is aateisti of tat mass. Tat is () m = f o f = [ ] m so tat te m soue of equation () as a oesponding equialent fequeny, f soue. Tat being te ase, te gaitational aeleation, a ga, an be expessed in tems of tat fequeny as te ange,, in te eloity,, of te attated mass pe time peiod, T soue, of te osillation at te oesponding fequeny, f soue, as follows. (3) a ga = / Tsoue = f soue Gaitation and te Plank Lengt It an ten be easoned using equation (3) = equation () as follows. msoue (4) a ga = f soue = G d Equation (5), below, is obtained by using tat fequeny is popotional to mass so tat wit f p and m p as te poton fequeny and mass ten f soue = [m soue / m p ] f p. m soue (5) f p = G mp m d soue

3 Reaanging and aneling m soue on bot sides of te equation, (6) = (7) = G m d f p p pe yle of f soue. Ten substituting, pe equation (), m p =[ f p ] /, = (8) l Pl G d f G d p f p pe yle of f soue. Te Plank Lengt, l Pl, is defined as G 3 so tat G = 3 l Pl Substituting G as a funtion of te Plank Lengt fom equation (8)into G as it is in equation (7), te following is obtained. (9) = l Pl 3 Pl d l = pe yle of f soue. d Tis esult states tat: - te eloity ange due to gaitation,, - pe yle of te attating mass's equialent fequeny, f soue, wi quantity, f soue, is te gaitational aeleation, a ga, - is a speifi fation of te speed of ligt,, namely te atio of: - times te Plank Lengt squaed, l Pl, to - te squaed sepaation distane of te masses, d. Tat squaed atio is, of ouse, te usual inese squae beaio. Tis also means tat at distane d = l Pl fom te ente of te soue, attating mass, te aeleation,, pe yle of tat attating mass's equialent fequeny, f soue, is equal to te full speed of ligt,, te most tat it is possible to be. In ote wods, at tat [quite lose] distane fom te soue mass te maximum possible gaitational aeleation ous. Tat is te signifiane, te pysial meaning, of l Pl o, ate, of l Pl. Te pysial signifiane of l Pl is tat it sets a limit on te minimum sepaation distane in gaitational inteations and it implies tat a "oe" of tat adius is at te ente of fundamental patiles aing est mass. Tat is, equation (9) lealy implies tat it is not possible fo a patile aing est mass to appoa anote su patile lose tan tat distane. 3

4 Tat pysial signifiane of l Pl, is so fundamental to gaitation and appaently to patile stutue, tat it moe tuly epesents a fundamental onstant tan does l Pl. Fo tose easons tat lengt sould eplae l Pl as a fundamental onstant of natue as follows. () Te fundamental distane onstant, l Pl = metes Equation (9) ten beomes equation (). () pe yle of f soue. d [6 CODATA Bulletin] a quite pue, peise and diet statement of te opeation of gaitation. It states tat gaitation is a funtion of te speed of ligt,, and te inese squae law, in te ontext of te osillation fequeny, f soue, oesponding to te attating, soue body's mass. Te Wae Aspet of Gaitation and Patiles Tee is an impliation in all of tis tat gaitation and te gaitational field inole someting osillatoy in natue, taeling o popagating at wile osillating at f soue. Essentially te same desiption an be made of ligt and of all eleto-magneti adiation. It would seem somewat absud fo mateial eality to inole two diffeent, oelapping su popagations. Rate, tee must be one simple su undelying fom fo bot effets, gaitational and eleto-magneti. If te oiginal definition of l Pl ad been in tems of, not -ba = /, te distintion wit egad to would not now be neessay. Te is a gatuitous addition to te statement of te Plank Lengt, and pobably ame about fom te deeming of te ause of te Hydogen atom's stable obits as being quantization of obital angula momentum. Te statement tat te obital eleton's obital angula momentum is quantized, as in () m R = n [n =,, ] is a mis-aangement of (3) R = n m = n mw [n =,, ] te statement tat te obital pat, R, must be an integal numbe of matte waelengts, mw, long. Tat mis-aangement may ae esulted fom a lak of onfidene in te fundamental signifiane of matte waes beause of te failue to deelop teoy tat podued aeptable, alid, matte wae fequenies, to gie pope esults in te obious neessity tat (4) f mw mw = patile eloity. A e-examination of Einstein's deiation of elatiisti kineti enegy (wi podued is famous E = m ) leads to a alid matte wae fequeny as alled fo in equation (4), as follows. 4

5 Einstein's Deiation of Relatiisti Kineti Enegy Te elatiisti kineti enegy, KE, lassially o non-elatiistily = ½ m, of a mass at some eloity,, is equal to te wok done by te foe, f, ating on te patile o objet of mass, m, oe te distane tat te foe ats, s, to podue te eloity and is alulated by integating te ation oe diffeential distanes as in Figue, below, but using elatiisti mass, m, tat is est mass Loentz ontated by its eloity. Te esult is ten sligtly poessed as in Figue ending in stating, fo tat deelopment, Einstein s esult tat: (5) KE = m m = [Total Enegy] [Rest Enegy] [m is total mass at = ; m is total mass at ]. S KE f ds (definition) S d(m ) ds dt (m ) ds d(m ) dt (Newton s nd law) (Reaangement of fom) (m ) d(m ) ( ds ) dt d m mismloentz ontated by m is est mass d m m m KE m m Figue Fist Pat integation by pats integation A nd of tem 5

6 m KE m m A pe figue m m m m plae nd tem oe st tem denominato m m m m expand tem witin bakets m m simplify KE m m Figue Seond Pat [Te appeaane in tis esult tat te enegies ae te podut of te masses times, te speed of ligt squaed, was te oigination of Einstein s famous E = m. Te onept falls out natually fom applying te Loentz tansfoms to te lassial definition of kineti enegy. [It is somewat supising tat Einstein was te fist to do tat inasmu as it was Loentz wo deeloped te Loentz tansfoms and te Loentz ontations essential to te deelopment.] Altenatie Teatment of te Same Deiation If in te aboe oiginal deiation one poeeds diffeently fom Step [A] of figue ten tee is a somewat diffeent esult. Tis is done by moing te tem -m to te left side of te equation, as +m, ten ealuating te tee tems of te esulting equation as in Figue 3. Te esult is equialent to (6) Total Enegy in Enegy in Enegy Kineti Fom Rest Fom m m m 6

7 m KE m m A as te oiginal Ealuating: m KE m m KE m kineti plus est enegies total enegy m m moed m A elatiistially ineased enegy of motion at eloity, wi inease is zeo wen m 3 m A elatiistially edued "est enegy" wi is te at est enegy wen m m Figue 3 Altenatie Teatment Te esulting enegy equation is equation (6) epeated below. (6) Total Enegy in Enegy in Enegy Kineti Fom Rest Fom m m m Diiding tat by to obtain an equation in mass, te esult is equialent to (7) Total Mass in Mass in Mass Kineti Fom Rest Fom m m m. 7

8 Wy is te fomulation fo lassial Kineti Enegy, KE = ½ m, but Enegy in Kineti Fom is simply m witout te ½? Wen dealing wit quite small eloities wee elatiisti effets ae nil ( ey small elatie to ), te exusion of total enegy aboe est enegy and te exusion of enegy in est fom below est enegy ae bot minute and essentially linea. In tat ase te exusion aboe te est ase is essentially alf of te total of te exusions aboe and below. Te lassial kineti enegy is ten alf, ½ m, of te total enegy in kineti fom, m. Te Coet Matte Wae Fequeny Tus te taditional iew of kineti enegy as te enegy inease due to motion may not be alid as a desiption of te poesses taking plae. Befoe te enounteing of te elatiisti ange in mass wit eloity te taditional iew did not lead to te poblems tat now appea wen elatiity is taken into aount. Using mass-in-kineti-fom and enegy-in-kineti-fom to obtain a oet fequeny of te matte wae poeeds as follows. Te matte wae waelengt, λ mw, as as been expeimentally eified is (8) mw. patile momentum m Using taditional kineti enegy to obtain te matte wae fequeny, f mw, pe equation () podues a matte wae eloity alf te patile eloity, as equation (9). (9) kineti enegy m fmw m mw mw f mw m Using enegy-in-kineti-fom to obtain te matte wae fequeny esoles te poblem. () enegy in kineti fom m fmw m mw mw f mw m and te matte wae is taeling wit and as te patile. Beause enegy in kineti fom podues te oet matte wae fequeny weeas kineti enegy does not alidates te oetness of enegy in kineti fom and its dietly elated enegy in est fom. On tat basis te wae aspet of matte is ten establised bot expeimentally (Daison and Geme and tei suessos) and teoetially (te aboe deelopment). Tat gies new signifiane to te fat, obseed at te time of Bo's deelopment of te elationsip between atomi line speta and atomi obital stutue, tat te stable obits of atomi eletons ae an intege multiple of te obital eleton's matte wae lengt. Te fat of te stable obits as long been aepted witout a speifi eason, a speifi opeatie ause, fo tose obits and only tose obits being stable. Te matte wae of te obiting eleton now poides an opeatie eason, as follows. 8

9 Fo te obit to be stable it must be te same fo ea pass, pass afte pass. If ea pass inludes exatly an intege numbe of te obital eleton's matte wae lengts ten ea pass is idential to te otes. But if, fo example, te obital pat lengt ontains only 9 / of a matte wae lengt, tat is 9 / of te matte wae peiod, ten te next pass will ontain te missing / of its pio matte wae lengt o wae peiod plus 8 / of te next, and so on. Te matte wae being sinusoidal in fom, te suessie obital passes will all diffe fom ea ote. Obital angula momentum as noting to do wit obit stability wit egad to wi it as no opeatie ause no any opeatie effet. Conlusion Analysis of an aspet of gaitation disloses tat tee is a fundamental distane onstant,, tat plays a key ole in gaitation. Te analysis also disloses tat gaitational attation is ommuniated fom one gaitational mass [te attating mass] to anote gaitational mass [te attated mass] by an osillatoy popagation, i.e. a wae popagation, at te speed of ligt,. Te eloity of te attated mass towad te attating mass is ineased ea yle of te wae popagation by tat fation of tat is te squaed atio of te fundamental distane onstant,, to te sepaation distane between te two masses. Tat effet taking plae ea yle of te attating popagation, ten te oeall aeleation podued is dietly popotional to te fequeny of te attating wae, wi fequeny is dietly popotional to te attating mass pe te mass enegy elationsip: () epeated m = f wit wi te analysis began. In addition te analysis as eised te signifiane of matte waes by esoling te poblem of te matte wae fequeny, a esult of te oeted alulation of elatiisti kineti enegy. Te oeted status of matte waes ten suppots tei appliation in esoling te opeational ause of te atomi obital eletons stable obits. Tese esults and fatos togete would indiate a geate signifiane fo te fequeny, wae, osillation aspet of mass and matte as ompaed to te pesent empasis on disete patiles in tat egad. Refeenes. R. Ellman, Te Oigin and Its Meaning (Te-Oigin Foundation, In., 4, seond edition) (aailable at ttp:// 9