Einstein s Road Not Taken

Transcription

1 Einstein s Road Not Taken Robert D. Bok R-DEX Systems, In. May 25, 2017 Abstrat When onfronted with the hallenge of defining distant simultaneity Einstein looked down two roads that seemingly diverged. One road led to a theory based on bakward null one simultaneity and the other road led to a theory based on standard simultaneity. He felt that alone he ould not travel both. After areful onsideration he looked down the former and then took the latter. Sadly, years hene, he did not return to the first. In the following we investigate Einstein s road not taken, i.e., the road that leads to a theory based on bakward null one simultaneity. We show that both roads must be traveled to develop a onsistent quantum theory of gravity and also to understand the relationship between the gravitational and eletromagneti fields. robert at r-dex dot om 1

2 The Road Not Taken Two roads diverged in a yellow wood, And sorry I ould not travel both And be one traveler, long I stood And looked down one as far as I ould To where it bent in the undergrowth; Then took the other, as just as fair, And having perhaps the better laim, Beause it was grassy and wanted wear; Though as for that the passing there Had worn them really about the same, And both that morning equally lay In leaves no step had trodden blak. Oh, I kept the first for another day! Yet knowing how way leads on to way, I doubted if I should ever ome bak. I shall be telling this with a sigh Somewhere ages and ages hene: Two roads diverged in a wood, and I? I took the one less traveled by, And that has made all the differene. Robert Frost [1] In his groundbreaking paper on speial relativity, Einstein [2] argued that a pratial definition of distant simultaneity an be ahieved by synhronizing loks using light signals and assuming that the one-way speed of light is idential to the experimentally measured two-way speed of light. This method of synhronizing distant loks results in a simultaneity relation now ommonly referred to as standard simultaneity. It is less well known that Einstein also ontemplated another method of synhronizing distant loks, namely, the method that leads to a simultaneity relation that is now alled bakward null one simultaneity [3, 4]. Indeed, Einstein wrote in the paragraph immediately preeding his desription of standard simultaneity [2]: We might, of ourse, ontent ourselves with time values determined by an observer stationed together with the wath at the origin of the o-ordinates, and o-ordinating the orresponding positions of the hands with light signals, given out by every event to be timed, and reahing him through empty spae. But this o-ordination has the disadvantage that it is not independent of the standpoint of the observer with the wath or lok, as we know from experiene. We arrive at a muh more pratial determination along the following line of thought. 2

3 Evidently, one reason Einstein favored standard simultaneity over bakward null one simultaneity was its property of being a muh more pratial method of synhronizing distant loks. In addition, Einstein objeted to synhronizing loks via the bakward null one method beause of its dependene on the loation of the observer, laiming that this onflits with experiene. We now address both of these assertions. With respet to standard simultaneity being more pratial, we note that the equations of a theory based on standard simultaneity are indeed less omplex than the equations of a theory based on bakward null one simultaneity. However, this advantage omes with a signifiant ost beause the only way to ahieve standard simultaneity is to insert into the theory the additional and unproven assumption that the one-way speed of light is equal to the experimentally measured two-way speed of light. This high ost is undersored by the observation that this assumption has not been experimentally onfirmed even though more than a entury has passed sine Einstein s seminal paper. All experimental efforts to measure the one-way speed of light independent of a synhronization sheme have failed. Furthermore, this assumption has led to a long-standing, and still unresolved, debate in the literature regarding the onventionality of simultaneity [5, 6, 7, 8, 9, 10, 11, 12]. Regarding Einstein s onern that the position dependene of bakward null one simultaneity is inonsistent with experiene, we note that this position dependene is no more troubling than the veloity dependene of standard simultaneity. Indeed, standard simultaneity and bakward null one simultaneity are omplementary ways of desribing nature. Whereas standard simultaneity is independent of an observer s loation, bakward null one simultaneity is independent of an observer s veloity [3]. Experiene does not require one hoie over the other, although a theory based on bakward null one simultaneity does not suffer from the disadvantage of introduing the unmeasurable one-way speed of light. In onventional relativity, the dependene of standard simultaneity on an observer s veloity does not prelude the equations of motion to be formulated in a veloity-independent manner. On the ontrary, relativity theory demands Lorentzinvariant equations in order to be onsistent with experiene. Similarly, the dependene of bakward null one simultaneity on an observer s position does not prelude the equations of motion to be formulated in a position-independent manner. On the ontrary, a theory based on bakward null one simultaneity demands equations that do not depend on an observer s loation in order to be onsistent with experiene. The 3

4 group of transformations that transform the position of the observer is the fundamental symmetry group of a theory based on bakward null one simultaneity just as the Lorentz group is the fundamental symmetry group of standard relativity. We argue that bakward null one simultaneity is loser to reality than standard simultaneity and other simultaneity relations beause it eliminates the unmeasurable one-way speed of light from the mathematial formalism. Indeed, the one-way speed of light aording to bakward null one simultaneity is a measured quantity. This is a signifiant advantage over Einstein s hoie of standard simultaneity sine all attempts to measure the one-way speed of light independent of a synhronization sheme have failed. Nevertheless, bakward null one simultaneity has reeived very little attention in the literature. Presumably, the omplexity of the mathematial formalism that results from the bakward null one simultaneity relation along with no obvious advantages of adopting this formalism, have disouraged many from investigating it further. Notable exeptions inlude Sarkar and Stahel [3] and Ben-Yami [4]. Sarkar and Stahel [3] demonstrated that bakward (forward) null one simultaneity satisfies Malament s riteria [8] for defining simultaneity one the assumption that any simultaneity relation must remain invariant under temporal refletions is removed. Ben-Yami [4] explored some of the advantages as well as the kinemati and dynami aspets of bakward null one simultaneity. In addition to eliminating the unmeasurable one-way speed of light of standard relativity, the bakward null one simultaneity framework helps illuminate the origin of quantum physis as well as the eletromagneti field. Consider an observer O at position 1 {x i O } and a oloated lok that reords t O. We assume that the observer O attributes events aording to bakward null one simultaneity, using the spherial oordinate set {t O, r, θ, φ} to desribe the null one struture of spae-time. Aording to bakward null one simultaneity, the two-way speed of light is and therefore the observer at O reords the time dt O = 2dL for two-way light propagation along any infinitesimal distane dl. The one-way speed of light aording to bakward null one simultaneity is no longer fored upon us as an assumption, being a measured quantity that depends on the angle, α, between the diretion of light propagation, ˆn, and ˆr. Speifially, for light propagating radially towards (away from) observer O the one-way speed of light is infinite (/2). For light propagating 1 Greek indies run from (0... 3) and lowerase Latin indies run from (1... 3). 4

5 transverse to the observer the speed of light is. To emphasize, these values are measured quantities and not assumptions of the theory, in ontrast to the one-way speed of light of standard simultaneity, whih an never be measured. Generally, the speed of light aording to observer O is: = 1 + ˆn ˆr = 1 + os α. (1) Whereas the Lorentz transformation follows from the invariane of the one-way speed of light for all observers in relative motion in onventional relativity, Equation (1) follows from the invariane of 2dL for all observers in spae that attribute events aording to bakward null one simultaneity. For infinitesimal light propagation at onstant radius (i.e., in the ˆθ- ˆφ plane), the null one struture aording to observer O is equivalent to the null one struture of Einstein s flat spae-time beause transverse light propagation aording to bakward null one simultaneity is equal to. However, for radial light propagation, the null one struture of O differs from the null one struture of the relativisti observer at rest in the same inertial frame. Indeed, in order to reonstrut the null one struture of relativity from observers that attribute events aording to bakward null one simultaneity, one must introdue an infinite number of observers that view light propagation in the ˆr diretion from a transverse vantage point. In other words, the flat spae null one struture of relativity for radial light propagation requires an observer O at eah position r that attributes events aording to bakward null one simultaneity and is situated with loal ˆr in the loal ˆθ- ˆφ plane. Therefore, we immediately see the orrespondene between relativity and a theory based on bakward null one simultaneity. The flat spae-time metri of relativity an only be onstruted by an infinite number of observers that attribute events aording to bakward null one simultaneity. In addition, for every olletion of infinite observers there is an infinite number of transformations that lead to another set of infinite observers due to the invariane of rotation in the ˆθ- ˆφ plane. Indeed, onsider the salar field ψ(x µ ) that defines the angle of eah bakward null one observer in eah loal ˆθ- ˆφ plane. One may ontemplate the transformation: ψ(x µ ) ψ (x µ ) = ψ(x µ ) + f(x µ ), (2) where f(x µ ) is an arbitrary salar field that leads to another equivalent set of observers. By reonstruting the null one struture of Einstein s inertial frame using observers that attribute events 5

6 aording to bakward null one simultaneity we are led to an infinite number of observers as well as a U(1) invariane embedded in flat spae-time. Naturally, we identify the U(1) invariane with the fundamental invariane group that, when gauged, produes the eletromagneti field. Charge results from introduing a set of observers that annot be ahieved by the simple transformation (2). Furthermore, we see that Einstein s inertial frame possesses an intrinsi non-loality due to the requirement that an infinite number of non-oloated observers is needed. Herein we see the onnetion between lassial and quantum physis. The observer assigned to a partiular Einstein inertial frame is indeed an infinite number of observers that attribute events aording to bakward null one simultaneity, and the problems enountered in desribing the quantum domain with lassial relativisti onepts are a result of our insistene on using a spae-time built from standard simultaneity. It is important to point out that aording to this interpretation, a mass, m 0, with de Broglie frequeny, ν 0 = m02 h, is equivalent to introduing a rotation of the observer in the loal ˆθ- ˆφ plane at a frequeny ν 0. Returning to Equation (1), we note that the angular dependene of the speed of light leads to observable onsequenes that are not predited by standard relativity. We onsider light propagating from a point A in spae to the observer O. In order to asribe the onventional values of wavelength and frequeny to light in the bakward null one simultaneity framework, we must reonstrut Minkowski spae-time along the propagation path using observers that attribute events aording to bakward null one simultaneity, as desribed above. Therefore, we onsider an infinite number of observers along the propagation path, eah with a line of sight perpendiular to the vetor AO. In other words, eah observer, O, along the propagation path must be situated with loal ˆr suh that ˆr is perpendiular to the vetor AO. When light travels from point A to O in a straight line, the wavelength at emission and wavelength at reeption are idential sine aording to this set of observers, the phase of light is: 2π (n t), (3) λ where n is the loal oordinate along the diretion of propagation, ˆn, and we have ignored an arbitrary phase onstant. However, if light leaves A and arrives at O but is defleted along the path, suh as via gravitational lensing, so that ˆn is no longer parallel with AO at the point of emission, then the observers introdued above do not view the propagation from a transverse perspetive at O, the point of reeption 6

7 (see Figure 1). Aording to the observers at the point of reeption, the speed of light follows from Equation (1), whih is ahieved by a transformation of time suh that: t t os α n + t 0, (4) where t 0 is an arbitrary onstant. Therefore, the phase of the eletromagneti wave aording to observer O is: 2π (n t), (5) λ where λ = λ 1+os α, = 1+os α, and we have disarded the arbitrary phase shift due to t 0. We see that transformation (4) leads to the orret transformation of the speed of light. Sine α = π 2 + δ, where δ is the defletion angle in free spae, we obtain: λ = = λ 1 sin δ, (6) 1 sin δ. Therefore, bakward null one simultaneity predits a wavelength shift that is not predited by standard relativity when light does not follow a straight path in free spae. It is important to point out that the frequeny of the wave remains onstant under transformation (4), suh that ν = ν. Assuming sin δ 1, the shift in wavelength is: λ λδ. (7) Aording to the bakward null one simultaneity framework, astrophysial wavelength shifts will be observed when light is defleted in free spae. This wavelength shift will appear to be anomalous to the relativisti frame that ignores the bakward null one simultaneity framework. The wavelength shift of Equation (5) is not predited by standard relativity beause standard relativity assumes the speed of light is in all diretion. Bakward null one simultaneity, on the other hand, avoids this assumption and therefore reognizes that the standard frame of relativity along the path of propagation is omposed of an infinite number of observers that attribute events aording to bakward null one simultaneity. This frame is defined by the diretion of light emission. A hange in the propagation diretion that is not due to mehanial means (e.g., mirror) hanges the angle of propagation of the light relative to this frame. This 7

8 manifests itself in a hange in wavelength at reeption, but not a hange in frequeny. The frequeny of light an be understood physially as a rotation of the observers that attribute events aording to bakward null one simultaneity around the axis of propagation. Evidently, this rotation of observers orresponds to the transport of energy, E = hν, in flat spae-time. It is reasonable to onlude that Equation (7) is responsible for a wide range of observed astrophysial wavelength shifts, suh as those measured on osmologial sales. Indeed, bakward null one simultaneity predits wavelength shifts from stationary soures without resorting to ad-ho mehanisms suh as an expanding universe, dark energy, or dark matter. All astrophysial observations are based on light, and therefore all derived quantities must aount for the wavelength shift of Equation (5). Figure 1: Geometry of light propagation. In onlusion, we ontend that our urrent formulation of relativity is a flawed, yet powerful, representation of nature that must be reexamined in order to develop a onsistent quantum theory of gravity and also to understand the relationship between the gravitational and eletromagneti fields. The fundamental gaps that exist between lassial and quantum physis are a diret result of Einstein s gaze down the road towards a theory based on bakward null one simultaneity and his deision to give standard simultaneity the better laim. Einstein s relativity employs standard simultaneity rather than bakward null one simultaneity for reasons of mathematial simpliity, but, in so doing it moves us further from reality by introduing the un- 8

9 measurable and hene unphysial one-way speed of light of traditional relativity. Therefore, one must return to the diverging roads and reasend the slope of nature by taking the first. Only by traveling Einstein s road not taken an our travels onverge. And that will make all the differene. Referenes [1] Robert Frost. The road not taken. In Louis Untermeyer, editor, The Road Not Taken: An Introdution to Robert Frost : a Seletion of Robert Frost s Poems with a Biographial Prefae and Running Commentary. Holt, [2] A. Einstein. On the eletrodynamis of moving bodies. Annalen der Physik, 17, [3] Sahotra Sarkar and John Stahel. Did Malament prove the non-onventionality of simultaneity in the Speial Theory of Relativity? 66(2): , June [4] Hanoh Ben-Yami. Apparent simultaneity. Marh [5] M. Reihenbah and J. Freund. The Philosophy of Spae and Time. Dover Publiations, New York, [6] H. Grunbaum. Philosophial Problems of Spae and Time, volume 12 of Boston Studies in the Philosophy of Siene. Dordreht, Boston, 2nd enlarged edition, [7] W. Salmon. The philosophial signifiane of the one-way speed of light. Noûs, 11: , [8] D. Malament. Causal theories of time and the onventionality of simultaniety. Noûs, 11: , [9] Y. Zhang. Speial Relativity and Its Experimental Foundations. World Sientifi, Singapore, [10] Ronald Anderson, E. Stedman, Geoffrey, and I. Vetharaniam. Conventionality of synhronisation, gauge dependene and test theories of relativity. Phys. Rept., 295:93 180, [11] H. Ohanian. The role of dynamis in the synhronization problem. Amerian Journal of Physis, 72: ,

10 [12] Allen Janis. Conventionality of simultaneity. In Edward N. Zalta, editor, The Stanford Enylopedia of Philosophy. Fall 2014 edition,