A Classical Reconstruction of Relativity

Transcription

1 A Classial Reonstrution o Relatiity Abstrat Delan Traill B.S July 5, By inerting a key assumption o Relatiity Theory, one an understand its predited odd eets o time dilation, length ontration and mass inrease in terms o Classial Physis. The belie that must be suspended is that ight always traels at onstant speed. The alternatie premise is that ight and matter waes trael through a ield generated by mass, at a ariable speed determined by the ield s intensity. This new premise also leads to a Classial explanation or the attration o Graity. Introdution One o the most exiting, proound, yet hard-to-athom theories in Physis is Einstein s Theory o Relatiity. The theory predits seeral ounter-intuitie, bizarre eets suh as time dilation, length ontration and mass inrease. These eets our most notieably to objets that trael at ery high speeds, or are subjeted to high aelerations, as in intense graitational ields. These eets are real - not just theory, or thought experiments - and hae been eriied, by areully-perormed experiments, to a high degree o auray. Een when presented with these experimental proos, many people hae great diiulty belieing that the eets atually our. I the ause o the eets ould be isualized, and explained in terms o Classial Physis onepts, they would be muh easier to understand, and beliee. Einstein s Relatiity omprises two theories: Speial Relatiity (95), and General Relatiity (95). Speial Relatiity desribes the eets on a body that has high speed motion, and General Relatiity desribes the eets on a body due to a graitational ield. Both theories proide equations or alulating the hange in the rate o time ( time dilation ) that ours, either as a result o the objet s speed, or its graitational enironment. One o the ore assumptions o Relatiity is that light always traels at a onstant speed, and the laim made by Relatiity is that this leads to eets suh as length ontration and time dilation. It is interesting that two dierent situations, ery high speed, and strong graitational ields, yield the same eet o time dilation. In both situations, time slows down or the objets onerned. Gien the same undamental hange to the physis o an objet, what i the same underlying priniple were ausing the eet in both ases? This essay will demonstrate that both o these theories (Speial and General Relatiity), and their equations, desribe the eets o a ommon ausal ator. This ator is an energy ield generated by matter, whih ills spae, and an be onsidered the root ause o the strange eets. By suspending Relatiity s assumption that light speed is onstant, and by instead positing that light and matter waes low through an energy ield, at a rate determined by the ield s intensity, I will show how all o the odd eets o Relatiity an be explained in terms o Classial Physis Field Theory. The proposal: (a) That spae is illed with an energy ield that is generated by mass and, the ield is proportional to the graitational potential [,, 4]. (b) The ield at any point in spae is the sum o all the ield ontributions made by all masses in the ausally onneted Unierse. () That matter waes and light waes are transmitted through the ield at a speed that depends on the intensity o the ield. Thus, waes trael through the ield more slowly where it is intense (suh as in the spae near a star). (d) Matter waes and light waes low through the ield muh like water waes low through water - so that they an be lowing 'upstream' or 'downstream' with respet to the ield. The graitational potential is a salar quantity that expresses the graitational potential o a single body generated by mass/energy: r Gm adr where a is the aeleration due to the graitational ore ating on a body that has unitary mass. I propose that a (positiely signed) salar ield exists that is proportional to, and is deined as ollows: et be a salar ield, suh that: r Jkg () a dr is the integration onstant, that is, the magnitude the ield has in the absene o the body being onsidered (i.e. i Jkg () where ). The ield is isualized as a ield extending into spae around all bodies with mass/energy. It is known that the priniple o superposition applies to the graitational potential ield, so the alue o the ield at a point in spae is the sum o all the ield ontributions made by all the masses in the system [7] n Jkg or equialent m s (3) Now, suppose that the onstany o the speed o light were expressed in dierent terms - suh that it is determined by the alue o the ield. In a high intensity ield, light s speed dereases. Howeer, the high intensity ield also slows all other physial proesses equally, suh that the rate o time within that reerene rame slows too. Eerything in the Unierse is omposed o waes. Ultimately the rate at whih physial proesses our is determined by the speed at whih these waes propagate. The speed o light s apparent onstany then results rom the time dilation that aompanies light s hange in speed. The quantity is the speed o light in the ield, and is the remains onstant, where Page o

2 General relatiisti time dilation ator. To an outside obserer (in a weaker ield) obsering the reerene rame, both the rate o time and the speed o light o the obsered rame are slower. The ollowing deinitions ollow rom the aboe disussion, and will be used through the rest o this essay: Jkg The deinition o the ield near a mass m (4) m se The speed o light in a ield o magnitude (5) ight is a Classial Wae ight is treated as a dierent sort o wae than other types o waes, suh as sound waes or water waes. This is indiated by the at that Doppler requeny shit equations used or sound or water waes do not hold or light waes. Howeer, light waes an be treated as normal Classial waes i one takes the Doppler-shit equation or light and splits it into two omponent parts: () The usual Doppler-shit attributed to other types o waes. () A relatiisti requeny shit o the emitted light at its soure. I the requeny o the emitted light is requeny-orreted due to the soure s motion prior to applying the normal Doppler-shit equation, then the resulting requeny shit is the same as that gien by the equation normally used or alulating the Doppler-shit or light. Doppler-shit equation or normal waes (6): Doppler-shit equation or light (7): (6) Where: is the transmission speed o the wae. is the reession speed o the soure. For a light soure moing at speed, Relatiity states that the requeny o the emitted light will be: (7) emitted (8) So, treating light as a normal wae, the total Doppler-shit should be : emitted (9) () To proe that this treatment is orret, it is neessary to show that (7) and () are equialent. This proo ollows: Using () gies: thus so... ' Page o whih is the same as (7). Q.E.D. The Doppler equation or sound waes an be applied to light waes, also, i one splits the Doppler equation or light into its two omponent parts. Thus the same general Doppler equation an be used or all types o waes General Relatiity Considered Graitational Aeleration/Potential: The aepted time dilation due to General Relatiity [6] is: Gm r () where is the graitational potential dierene between the soure o the photon and the detetor. Equation () an be deried rom ield onsiderations alone: ()

3 et: ield intensity o the soure o the photon. ield intensity at the detetor o the photon. We an see that the potential dierene between the soure and detetor is: (3) The ield at the detetor is the ration k times more intense than the ield at the soure. k (4) So... Substituting (3) into (5) gies: k (5) k (6) So the requeny o the photon at the detetor will be less than the requeny at the emitter by the quantity k : k k (7) then we hae: Substituting (6) into (7) gies: I we let Thus the ull deinition o the ield is: (8) whih is the same as equation (). Q.E.D. (9) So the ield is ompletely deined, and an aount or the time dilation due to General Relatiity. The alue o an be understood as the ield ontribution rom the whole Unierse. Perorming a alulation o up this inding [8]: GM, using alues or the Unierse, yields. Other researh baks R The well known solution or here is just the sum o the ontributions to the potential due to all o the matter in the ausally onneted part o the Unierse (that is, within the partile horizon in the parlane o osmologists). When alulated, this turns out to be roughly GM R, where M is the mass o the Unierse and R is about times the age o the Unierse. Using reasonable alues or M and R, GM R omputes to a alue o about GM R hae roughly the numerial alue o. Not only does too. This seems to suggest a deep onnetion between and., it has the same dimensions This ield deinition embodies Mah s Priniple in the way it inludes the ontributions rom all the matter in the Unierse. Thus represents the total potential o a body : Total Potential Global potential ( ) + oal Potential ( ) Jkg () and the quantity m expresses the total energy required to remoe a body rom this potential. Energy Global potential energy ( m ) + oal Potential energy ( m ) Energy m E m m Speial Relatiity Considered Part A - ight moing perpendiular to the diretion o Motion: Consider the ollowing: (a) A reerene rame at rest in a region o spae illed with a ield o magnitude. Page 3 o Joules () (b) An idential reerene rame to (a), moing with a eloity through the same ield with magnitude. Please reer to Figure () below. A pulse o laser light (depited as the dashed arrow) is sent aross the reerene rame (rom a soure onneted to the reerene rame) perpendiular to the diretion o motion through the ield. In the stationary rame, the path length taken by the light is as expeted; but in the moing rame the path taken is longer ( at that the light moes with a ertain eloity with respet to the ield, rather than the moing reerene rame. ) due to the onstant low o the ield through the rame, and the

4 A moing laser will emit a beam that ollows the path gien by. This an be demonstrated with a Huygens onstrution [] o the waelets omprising the beam as it is being emitted. Stationary Frame Moing Frame In this example, the so t : () t (4) t (3) Using () and (4) we hae: Using (3) gies: Soling or t gies: The orentz ator [7] : t t (5) t t t (6) t t t t (7) Equation (8) is the aepted (and eriied) equation or alulating the time dilation due to relatie motion. Part B - ight moing parallel to the diretion o Motion: Please reer to Figure () below. Now onsider the same situation as depited in Fig (b), but with a light pulse sent aross the reerene rame parallel to the diretion o motion. Consider the light s journey both in the diretion o trael, Fig (a), and in the opposite diretion, Fig (b), as separate ases, then ombine the results to gie an oerall, round-trip result. The reerene rame traels dierent distanes in eah ase as t t. This means that the atual time dilation is dierent in eah diretion, but it an be demonstrated that or a round trip the total time dilation during the trip is the same as it was in Fig (b) where the light traelled perpendiular to the diretion o motion. (8) In this example, the ield has a alue o so t Soling or t and t gies :, giing: t t (3) (9) t t (3) t (3) Page 4 o

5 The round trip time is deined as : t a t t (33) et ta be the total time taken by the light pulses to omplete the round trip in reerene rame A that is traeling at speed through the t be the total time taken by similar light pulses in reerene rame B that is stationary in the ield. I the soure o the ield is ield, and b known, and we are able to determine eah rame's relatie motion with respet to it (indiating that rame A is the one moing relatie to that ield and rame B is not), then we expet that ta tb beause in Frame A the light has had to trael urther than the light in Frame B had to. a So the time dilation ator parallel by deinition (34) tb For Frame B, the time taken by the light pulse in his reerene rame is simply: t b t (35) For Frame A, the upstream & downstream times must be onsidered separately, and then summed : Using equations (3) (3) and (33) gies : ( ) t a Then using (34) (35) and (36) we are able to alulate parallel (37) So parallel : parallel Page 5 o (36) (38) Also the length is shorter by an amount equal to the orentz ator. So the length o the moing reerene rame in the preious alulation is rather than, where : (39) I this new length is used in the alulation or equation (36), we hae: t a ( ) Then using (34) (35) and (36) we are able to re-alulate parallel : parallel Substituting (37) into (4) : parallel (4) Thus, we an see that the times taken or a light pulse to trael in the perpendiular and parallel diretions are equal, despite the motion o the experimental apparatus and the obserer through the ield. This is the same outome as predited by Speial Relatiity theory. Modeling a laser moing at Relatiisti speed: We hae seen how the timing o the light pulses an be explained and mathes the experimental results, but what about the requeny & phase o the waes? A laser s resonating aity proides a good experimental test-bed or these onsiderations, as it ontains a standing wae whih an be thought o as omprising two sinusoidal waes traeling at the same speed, but in opposite diretions, and with mathing requenies. Thus, aording to my proposal, i the operating laser is brought up to relatiisti speed through spae, one o the waes omposing the standing wae is traeling upstream and the other wae is traeling downstream. To an obserer moing with the laser aity there should not be any detetable dierene in the laser s operation, or the struture o the standing wae ontained within it, when it is moing ompared to when it is (4) (4)

6 stationary. O ourse, i he looks to another (stationary) reerene rame, he will disoer the time dilation that exists in his reerene rame and thereore be able to dedue that the laser is in atual at running more slowly too. In modeling the moing laser aity, seeral dierent eets must be onsidered at the same time. The path length taken between the releting mirrors o the laser aity by the upstream & downstream waes will be dierent, sine they are traeling at the loal speed o light through the ield that ills spae. Due to the time dilation that exists in the atoms o gas inside the moing laser (that are moing with the laser), the initial requeny o the light emitted into the aity will be lower than or an equialent stationary laser. The requenies o the upstream and downstream waes will also be Doppler-shited due to the relatie motion o the laser's mirrors through the spae-illing ield (higher or the upstream & lower or the downstream), and as shown earlier, the length o the aity will be ontrated. I hae written a omputer program to model all o these arious eets simultaneously. This model learly shows how the atual eletromagneti waeorm hanges in spae due to the relatiisti motion, and yet to the obserer traeling with the aity, the waeorm appears to be unhanged rom its appearane when at rest. Figure (3) is a sreen shot o the output o this program, modeling a laser aity traeling at 4% the speed o light. Figure 3 Box depits the laser aity as it operates when stationary in the spae-illing ield. Box depits the laser aity as it operates when traeling at 4% o the loal speed o light. Box 3 depits the operation o the aity as measured by an obserer traeling with the laser. In boxes and, the two waeorms on the let are the upstream and downstream waes (respetiely) as they exist in the spae between the two reletors o the laser aity. The waeorm on the right is the sum o the upstream and downstream waes, and thus represents the atual eletromagneti wae that exists in the spae inside the aity. The blak ertial lines interseting the waeorms indiate points where the eletri ield is zero (the nodes o the standing wae, or example). As I showed earlier, light is a Classial wae, so the normal Doppler-shit equation an be used. The upstream wae will be blue-shited in the spae inside the aity, but will arrie at the upstream reletor with the same apparent requeny as when it was emitted (sine the destination mirror is moing at the same speed as the soure o the photons). Similarly, the downstream wae will be red-shited in the spae inside the aity, but will arrie at the downstream reletor with the same apparent requeny as when it was emitted. I is the requeny o both the upstream and downstream waes in the laser aity when it is at rest, and e is the requeny o the light emitted into the laser aity when it is moing, then: Page 6 o

7 e is the orentz Fator. (43) Using the Doppler equation or a moing soure, the requenies o the upstream up e (44) s Also, the apparent speeds o the waes traersing the aity an be expressed : s up down up s (46) s and downstream down waes are: e (45) s s down (47) The obserer knows how ar down the aity eah sensor is, and expets that the time taken by the signal rom that sensor to reah the point at the end o the aity to be proportional to the sensor s distane rom the end o the aity. In order to get a piture o the eletri ield inside the aity, the obserer needs to onstrut a proile o the waeorm using these signals. To determine what the proile o the eletri ield in the laser aity looks like at a gien point in time, one must apply a time orretion to the reeied signals, suh that a set o signals rom all the sensors atually orrespond to the same moment in time. In applying this orretion, the obserer will naturally assume that the time taken by the signal is simply the distane that the sensor is rom the end o the aity x diided by the speed o light. x t (48) So the time that a partiular signal was emitted at its soure is gien by : t t t (49) emitted measured This time orretion will always be applied by the obserer on the signals he reeies, regardless o the speed o the laser (and the obserer), beause one always measures light to trael at speed regardless o one's speed through spae. So substituting (64) into (65) gies: t emitted x tmeasured (5) This is the equation to be used or a laser that is at rest with respet to the spae-illing ield. Howeer, when the obserer is traeling at speed, the time t measured (the time at whih the signal rom a sensor reahes the end o the laser aity) will atually be dierent than the alue it has or a stationary laser. It will also be a dierent alue depending on whih end o the laser aity the signals are taken to. For a moing laser, i the signals are taken to the upstream end o the laser aity, the time taken or the signal to reah that point will be: x tup (5) x (5) sup where x is the length-ontrated distane to the sensor. Thus or the upstream diretion, substituting (46) and (5) into (5) gies: t up x s For a moing laser, an obserer will alulate t emitted to be : emitted measured up So substituting (53) and (48) into (54) gies: x (53) t t t t (54) t emitted x x tmeasured s I a plot is then made o all the signals rom the sensors that hae the same alue o t emitted then that plot represents a proile o the eletri ield as measured by the obserer moing with the laser. As a result o the orretion equation (55), the waeorm in Box is transormed into the waeorm shown in Box 3. The waeorm in Box 3 is a standing wae just like that in Box, but it osillates more slowly as a result o the time dilation that aompanies the laser s motion. Its osillation is slower by a ator o. For the downstream diretion, the equation is: t emitted x x tmeasured s Also to be noted rom this model is that when the upstream and downstream waes are summed, the distane between the nodes o the standing wae are shorter by the exat amount required by relatiisti length shortening. Thereore, we an now understand how the length ontration ours: rom the summation o the higher & lower requeny waes. Relatiisti Mass Inrease The key to understanding mass inrease is the understanding that solid matter is atually omposed o standing waes that an be thought o as being the sum o an upstream and a downstream wae. Eah o the two waes that omprise a partile has a ertain energy assoiated with it (depending on that wae s requeny), and the total energy o the partile is the sum o the energies o the upstream and downstream waes. One this total energy has been alulated, then the mass equialent or that energy an be alulated using the usual equation: E m (57) (55) (56) t Page 7 o

8 We an use the standing wae inside a laser s resonating aity as a model or a partile, beause it is omposed o an upstream and downstream wae that are summed, resulting in a standing wae. The proo that this approah an work mathematially ollows: et e energy per unit length o the upstream/downstream wae. Sine there are two waes inside the laser s aity (upstream & downstream), the total energy o the waes inside the aity is: Estationary e is the length o the laser s aity. (58) Where For a laser (or partile) that is moing at Relatiisti speed, the ollowing equations apply: et: e up energy per unit length o the upstream wae. e down energy per unit length o the downstream wae. The requeny o the upstream wae and downstream wae are: up (59) down (6) Where is the requeny o the upstream/downstream waes in the stationary laser. Sine the energy per unit length o a wae is proportional to the wae s requeny ( e ), then (59) and (6) an be rewritten as : The energy per unit length o the upstream wae is: up The energy per unit length o the downstream wae is: down e e e (6) e (6) So the total energy o the waes inside the moing laser s aity (or omprising a moing partile) is expressed as : E e e (63) moing up down where is the ontrated length o the laser aity due to the laser s Relatiisti motion: (64) and is the orentz ator due to Speial Relatiity : So by substituting (6), (6) and (64) into (63) we hae the ollowing : Emoing e e E moing Then by substituting (65) into (67) : (66) e ( ) e e (67) e Emoing e (68) E E (69) Finally, by substituting (58) into (68) we hae : moing stationary and by onerting Energy into mass equialent (equation (57) ) : moing stationary m (65) m (7) Q.E.D. The exat mass inrease predited by Speial Relatiity an be explained and alulated using this new model, where a moing mass an be modelled as omprising an upstream and a downstream wae, eah with dierent requenies, hene energies. When the mass equialent o these waes energies are summed, the orret mass inrease is obtained. An Explanation or Graitational Aeleration A partile an be modeled as a standing wae omposed o inward and outward traeling spherial waes [9] that eah relet at the ombined standing wae s nodes thus an inward wae beomes an outward wae and ie ersa at eah reletion. A balane is ahieed in the distribution o the inward and outward waes amplitude (and hene momentum) suh that the natural shape or a partile in ree spae is spherial. As the outward waes trael away rom the enter o the partile and into spae, they derease in amplitude and energy density; but they are peretly balaned by the inward waes that inrease in amplitude and energy density as they onerge towards the enter o the partile. Page 8 o

9 To see how suh a wae struture behaes in a graitational ield, we must onsider how eah o the omponent waes are aeted by the graitational ield [3]. The graitational ield itsel is a ield o arying time dilation. The loser one gets to a mass where the graitational potential is greater, the more time is dilated (running slowly). The eet is ery small, but when it has an eet on waes that are traeling ery ast, and whih remain in the ield or a onsiderable period o time, the result is graity, as we know it. Normally, light waes do not stay within the graitational ield or ery long beause their speed is so high, so only a slight bending o the wae ours toward the mass. The waes that make up matter are aeted in the same way, but they remain loalized in the ield or muh longer, so the eets are muh more notieable. To see what the eet on the waes o a partile would be, we need to do the ollowing math. From my earlier analysis o moing standing waes, we saw that or a standing wae (partile) that is traeling along at speed : high (7) Page 9 o low (7) ( high and low reer to the requenies o the waes ompared to the original requeny ) And rom General Relatiity [5] we know that when a wae traels rom one graitational potential to another, its requeny (whih equates to energy) hanges. Thus or a standing wae (partile) that is plaed in a graitational ield with aeleration a : high ah (73) The time taken or light to trael the ertial distane h is gien by: low t ah (74) h (75) I a partile is held stationary in a graitational ield by an upward ore, suh as rom a table top, the waes at the bottom o the partile trael ery slightly slower than the waes on the top (due to the greater graitational potential at the bottom), so in order or the standing wae to remain a ontinuous waeorm, the waes at the bottom must bunh up (get loser together) so that the same number o wae rests trael rom top to bottom as rom bottom to top. As the downward waes slow down and bunh up as they moe into a region o higher graitational potential, their requeny inreases, as does the momentum they arry, thus the ore they impart on the table on whih they are resting inreases. I the table is suddenly remoed, the irst thing that happens is that the opposing ore rom the waes omprising the table is remoed, so the bunhed up waes at the bottom o the partile spread out to restore the partile's normal spherial geometry. One this ours, howeer, the number o wae-rests propagating rom the bottom o the partile to the top dereases. Similarly, the upward wae releting at the node to orm the downward wae will arrie at the bottom as a slightly higher requeny wae beause it is releted and Doppler-shited at a node that is moing downwards. By this method the hange is transmitted rom one node to the next and aets the whole partile s wae struture. Consequently, a moment ater the table is remoed, the partile beomes a standing wae omposed o a higher requeny down wae and lower requeny up wae. As a result, the partile will gain more momentum in the downward diretion, and the standing wae s nodes (and thereore the whole partile) will attain a downward speed ( ). This is the oniguration or a partile in motion. So to alulate this speed ( ) or a partile aelerating in a graitational ield or the period o time ( t ), we an perorm the ollowing operations on the aboe equations: Substitute (7) and (75) into (73): at so: at and... at at at so: at (76) Then substitute (7) and (75) into (74): at so: at and... at at at so: at (77) Then substitute (76) into (77) : at at so: at thus... at Q.E.D. (78) I hae deried the lassial eloity that a partile ahiees when it is aelerated by graity or a period o time ( t ) by onsidering only the time dilation eet in the graitational ield and its eet on the requenies o the upward and downward waes. It appears that the aeleration due to graity is explained by the ollowing our-step proess: The Primary Cause: () An inrease in requeny (hene momentum) o the downward omponent o the partile's standing wae, due to the slowing o waes in a higher graitational potential. Similarly, a derease in requeny o the upward omponent wae, due to a speeding-up o waes in a lower graitational potential. Seondary Considerations: () The higher momentum downward wae then pushes the standing wae s nodes downward. (3) The releted upward wae is then Doppler-shited to a slightly lower requeny. (4) The upward/downward waes then ontinue to relet bakwards and orwards between the nodes, onstantly undergoing small Dopplershits, ausing momentum to build in the downward diretion.

10 REFERENCES [] D. Traill, On the Quantum-Wae Nature o Relatiisti Time Dilation and ength Contration, p. 57 (Galilean Eletrodynamis, Summer (Vol., Speial Issues No. 3). [] Delan Traill, Relatiely Simple? An Introdution to Energy Field Theory, ast modiied May, 8. [3] Delan Traill, How a Standing Wae (Partile) Behaes in a Graitational Field, ast modiied 9 May, 8. [4] Robert. R. Traill. An Epistemologial Re-Assessment o Einstein s Speial Relatiity Theory, and o the Coneiable Alternaties to it. (978) [5] "Graitational time dilation", ast modiied 8 July,. [6] Einstein (95) Relatiity The Speial and General Theory 3. Uniersity Paperbaks. Methuen & Co td [7] Halliday & Resnik. (988) Fundamentals O Physis 34, 958. Wiley [8] James F. Woodward Transient Mass Flutuations 6, ast modiied 3 Noember, 5. [9] Milo Wol The Physial Origin o Eletron Spin 4, Aessed 3 August [] F.A.Jenkins & H.E.White (95) Fundamentals O Optis 393 MGraw-Hill: NY Page o