On Lorentz Transformation and Special Relativity: Critical Mathematical Analyses and Findings

Transcription

1 vixra.g physics On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings Radwan M. Kassir DAH (S & P, Mechanical Dept., Verdun St., PO Bo , Beirut , Lebanon. In this paper, the Lentz transfmation equations are closely eamined in connection with the constancy of the speed of light postulate of the special relativity. This study demonstrates that the speed of light postulate is implicitly manifested in the transfmation under the fm of space-to-time ratio invariance, which has the implication of collapsing the light sphere to a straight line, rendering the frames of reference igin-codinates undetermined with respect to each other. Yet, Lentz transfmation is shown to be readily constructible based on this conflicting finding. Consequently, the fmulated Lentz transfmation is deemed to generate mathematical contradictions, thus defying its tenability. A rationalization of the isolated contradictions is then established. An actual interpretation of the Lentz transfmation is presented, demonstrating the unreality of the space-time conversion property attributed to the transfmation. 1 Introduction The well-known Lentz transfmation, named after the Dutch physicist Hendrik Lentz, is a set of equations relating the space time codinates of two inertial reference frames in relative unifm motion with respect to each other, so that codinates can be transfmed from one reference frame to another. Length contraction time dilation are supposedly the principle outcome of the Lentz transfmation. Originally, Lentz developed the transfmation to eplain, with other physicists (Larm, Fitzgerald, Pointcaré, how the speed of light seemed to be independent of the reference frame, following the puzzling results of the famous Michelson-Mley eperiment [1]. These equations fmed later the basis of Einstein s special relativity. Einstein [2, 3] derived Lentz transfmation on the basis of two postulates: 1 the principle of relativity (i.e. the equations describing the laws of physics have the same fm in all proper frames of reference, 2 the principle of the constancy of the speed of light in all reference frames. Einstein they of special relativity has received much criticism [4 9]. Doubts on the bases of scientific, mathematical, philosophical contentions have been epressed. Criticism, on both academic non-academic levels, has been mainly motivated by the undinary physical phenomena of the time dilation length contraction of moving objects, emerging from the purely mathematical fmulation of the they, in addition to numerous paradoes combined with the inconsistency ambiguity in their resolutions [10]. In this paper, the Lentz transfmation, along with the special relativity speed of light constancy principle used in its derivation, is thoughly eamined in an attempt to reach rational conclusions regarding its ever questioned tenability. Pure mathematical analysis geometrical tools are used as the main arguments in achieving the objective of this study. 2 Lentz Transfmation Consider two inertial reference frames of reference, K(, y, z, t K (, y, z, t, in translational relative motion with parallel cresponding aes, let their igins be aligned along the overlapped - -aes. Let v be the relative motion velocity. The space time codinates of K K are then interrelated by the Lentz transfmation equations given in their present fm by Poincaré [11], subsequently by Einstein [2, 3] as follows: = γ( vt ( t = γ t v (1 = γ( + vt t = γ (t + v γ = y = y (2 z = z (3 1 1 v2 Equations (1 (2 result in the following relativistic velocity transfmation equations: u = u = u v 1 uv u + v 1 + u v (4 (5 Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings 1

2 vixra.g physics Where c is the speed of light propagation in empty space, u u are the velocity of a moving body in the -direction, when measured with respect to K K, respectively. It is to be noted that equation (4 requires that v be smaller than c. Also, equations (5 limit the values of u u to c (i.e. if u = c, then u is brought to c as well, vice versa. 3 Lentz Transfmation Analysis 3.1 Constancy of the Speed of Light Consider two inertial frames K(, y, z, t K (, y, z, t moving relative to each other with a unifm velocity v, suppose at an instant of time t = t = 0, the frame igins are coinciding. Let a light beam be emitted at this time from the point of the coinciding igins in an arbitrary direction. At time t in K, cresponding to time t in K, the position vect associated with the light beam will acquire the space-time codinates (, y, z, t (, y, z, t in K K, respectively. In line with the special relativity constancy of the speed of light principle [3], the light beam position vect codinates shall satisfy the following equations, referred to as the light sphere equations. 2 + y 2 + z 2 = t 2, (6 2 + y 2 + z 2 = t 2. (7 Subtracting equation (7 from equation (6, given that the y z codinates remain unaltered, leads to 2 2 = t 2 t 2. (8 Equation (8 ehibits only one solution, as it will be demonstrated below, readily obtained as 2 = t 2, (9 2 = t 2. (10 Indeed, Lentz transfmation equations (1 can lead to 2 = γ 2 ( 2 + v 2 t 2 2vt, (11 t 2 = γ 2 ( t 2 + v2 2 2vt. (12 Eliminating the term 2vt from equations (11 (12, yields 2 + v 2 t 2 2 γ 2 = c2 t 2 + v2 2 c2 t 2 γ 2. (13 Similarly, Lentz transfmation equations (2 bring about the following epression; 2 v 2 t γ 2 = c2 t 2 v2 2 + c2 t 2 γ 2. (14 Adding equations (13 (14 will lead to the following epression; 2 (1 + 1 γ 2 2 (1 + 1 γ 2 + v2 (t 2 t 2 = = t 2 (1 + 1 γ 2 c2 t 2 (1 + 1 γ 2 + v2 (2 2 ; which can be simplified to ( 2 2 (1 + 1 γ 2 v2 = c2 (t 2 t 2 (1 + 1 γ 2 v2 ; 2 2 = (t 2 t 2. (15 which is actually nothing but equation (8. Whereas, the subtraction of equation (14 from equation (13, results in ( (1 1 γ 2 v2 = c2 (t 2 + t 2 (1 1 γ 2 v2 ; = (t 2 + t 2. (16 Equations (15 (16 readily reduce to equations (9 (10, namely 2 = t 2, 2 = t 2. In other wds, equation (16 makes equations (9 (10 the only solution f the constancy of the speed of light equation (8. Another practical verification of equation (9 can be implemented through Fig. 1 depicting a graphical representation of Lentz transfmation equations (1, where K K are shown traveling along the overlapped time aes, t t. Noticing the similar triangles within the graph, we can write, yielding vt v = t, 2 = t 2. Similarly, equation (10 can be verified using Fig. 2, showing a graphical representation of Lentz transfmation equations (2, as follows; 2 Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings

3 vixra.g physics Fig. 1: Graphical representation of Lentz transfmation equations (1. yielding vt v = t, 2 = t 2. Consequently, the light sphere equations (6 (7 are collapsed to the line equations (9 (10. In fact, when equations (9 (10 are substituted into equations (6 (7, they result in the vanishing of y, z, y z, indicating that the constancy of the speed of light equations (6 (7 are preliminarily restricted to light beam propagation along the frame aes parallel to the direction of the relative motion. This can be reconfirmed by adding equations (6 (7, using equation (16. Now, dividing equation (9 by equation (10 yields ( 2 = ( ct ct 2, = ± ct ct. (17 Assuming, f the time being, that c > v (this assumption will turn out to be essential, then will always have the same sign (positive negative, whether the light beam was emitted at t = t = 0 in the positive negative direction, with respect to the coinciding K K igins. It follows that given that 0, Fig. 2: Graphical representation of Lentz transfmation equations (2. equation (17 becomes ct ct 0, = ct ct. (18 Hence, equation (18 combined with equations (9 (10, leads to c = t = t. (19 Equation (19 can also be readily obtained using Fig. 1, leading to Fig. 2; t = /γ t /γ = t, t = /γ t/γ = t, along with equation (8. In fact, using 2 t2 = 2 t. 2 in the epression resulting from dividing equation (8 by 2, leads to t 2 c2 1 = t 2 2 (t2 t 2, Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings 3

4 vixra.g physics which yields equation (10, equation (9 will follow when equation (10 is substituted into equation (8. Hence, equation (19 can be readily deduced. Consequently, it can be concluded that the constancy of the speed of light f the light beam propagation in the relatively moving reference frames can be epressed by equation ( Direct Inference It will be first shown that the constancy of the speed of light postulate is certainly unviable f relatively moving inertial reference frames, without a space-time distting transfmation. In fact, assuming the space-time is preserved (i.e. cannot be modified, the codinates (Fig. 3 would then be related by the following equation with respect to K; Fig. 4: -codinate with respect to K. = vt. (20 Substituting equation (23 in equation (24, we get =. (25 Using equations (22 (25, we can replace t with t, with in equations (23 (24, to obtain the following contradiction, =, 1 = 1, Fig. 3: -codinate with respect to K. Whereas, with respect to K, the same codinates (Fig. 4 would be related by the following equation = + vt. (21 Substituting equation (20 into equation (21, we get t = t. (22 Dividing both sides of equations (20 (21 by c, applying the speed of light constancy principle as determined above (c = /t = /t, the following epressions are obtained; indicating the set of equations (20, (21, (23 (24, resulting from the principle of the constancy of the speed of light, yield an impossible solution, generating an imaginary space-time. Thus the constancy of the light speed principle is unviable, at least in the case of no space-time distting transfmation. On the other h, although equations (20, (21, (23 (24, result in mathematical impossibility, they satisfy the constancy of the speed of light general criteria given by equation (8, implying that the speed of light constancy principle is merely a mathematical impossibility. In fact, equations (20 (23 lead to 2 = 2 + v 2 t 2 2vt, t 2 = t 2 + v2 2 2vt. Eliminating 2vt from the above two equations yields t = t v. (23 t = t + v. ( v 2 t 2 2 = t 2 + v2 2 t 2. (26 Similarly, equations (21 (24 can lead to 2 v 2 t = t 2 v2 2 + t 2. (27 4 Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings

5 vixra.g physics Adding equations (26 (27, rearranging simplifying the terms, returns equation (8: 2 2 = t 2 t 2. Indeed, the addition of equations (26 (27 results in the following epressions, 2( v 2 (t 2 t 2 = 2 (t 2 t 2 + v2 (2 2 ; ( 2 2 (2 v2 = (t 2 t 2 (2 v2 ; yielding the speed of light constancy principle equation, ( 2 2 = (t 2 t Lenz Transfmation Re-derivation Assuming that the principle of the speed of light constancy must result in a space-time distting transfmation, a length conversion by a fact of β along the direction of motion is hypothesized; the longitudinal length in one frame is scaled by a fact of β with respect to the other frame. This length conversion can therefe be epressed with respect to K K, using Figs. 3 4, respectively, as follows. = vt + β. (28 = vt + β. (29 Where β is a positive real number. Rearranging equations (28 (29, we can write = 1 ( vt, (30 β = 1 β ( + vt. (31 Dividing both sides of equations (30 (31 by c, applying the speed of light constancy principle equation, as demonstrated above through equations (6 to (19 restated here below; c = t = t, (32 the following epressions are obtained. t = 1 β t = 1 β ( t v, (33 (t + v. (34 Solving equations (30, (31, (33 (34 f β results in β = 1 v2 c, 2 1 β = 1 1 v2 = γ. (35 In fact, f = 0, equations (30 (34 yield = vt, t = βt, respectively, reducing equation (33 to βt = 1 ( t v2 t ; β therefe β = 1 v2 c. 2 Conversely, f = 0, equations (31 (33 yield = vt, t = βt, respectively, reducing equation (34 to βt = 1 (t v2 t ; β hence β = 1 v2 c. 2 We note that (35 is valid f c > v only, thus satisfying our assumption made above in connection with the set criteria of the speed of light constancy principle. It follows that, since β < 1, the hypothesized length conversion is a length contraction, as inferred from equations (28 (29. The obtained set of equations (30, (31, (33, (34, (35 are the Lentz transfmation, representing the spacetime transfmation resulting from the reduced constancy of the speed of light principle given by equation (32. It is noted that the velocity transfmation equations (5 can be readily derived from the invalid equations (20, (21, (23 (24 by dividing equation (20 by equation (23, equation (21 by equation (24 implying that the Lentz velocity transfmation equations are merely invalid velocity criteria of the speed of light constancy principle, independent of any space-time distting transfmation. 4 Lentz Transfmation Contradictions The fact that the constancy of the speed of light principle is manifested as c = /t = /t, as demonstrated above, etracted from the very Lentz transfmation, is sufficient to conclude the invalidity of the Lentz transfmation, eplicitly fully constructed in this paper from this fact, since the igin codinates of one frame are undetermined with respect to the other frame unless K K were combined Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings 5

6 vixra.g physics in one codinate system, their codinates became invariable, in which case the Lentz transfmation collapses to 1 = 1. In fact, f the igin of K(0, 0, 0, 0, the cresponding K codinates shall satisfy the relation t = t = 0 0, yielding the set of K codinates ( = 0 0, 0, 0, t = 0 0 with undetermined t. Hence, the constancy of the speed of light principle shall not principally be applicable at the reference frame igins, restricting the codinates from acquiring zero value. Consequently, Lentz transfmation, implicitly incpating equation (32 as demonstrated earlier, results in various conflicts unresolved paradoes. F instance, substituting equation (33 into equation (34, returns ( ( t = γ γ t v + v. (36 Equation (36 is simplified in the following steps. t = γ 2 t γ2 v t(γ 2 1 = v + γv, (γ 2 γ. Replacing = ct back in the above equation, we get c (γ2 1 = vt (γ 2 γ. (37 c From equation (30, we note that f = 0, = vt. Therefe, equation (37 reduces to yielding the contradiction, (γ 2 1 = γ 2, (γ 2 1 = γ 2, 0 = 1, which is the consequence of violating the restriction imposed by the light speed constancy principle on the codinates (in this case setting = 0, equivalent to = vt. Yet, this conflicting condition of setting the spatial codinate in the primed reference frame to zero under the speed of light invariance principle constitutes a vital strategy in the Lentz transfmation derivation, the interpretation of the time dilation, in the special relativity fmulation [2, 3]. Similar contradiction is obtained by substituting equation (34 into equation (33, replacing = ct back in the equation, using equation (31 f = 0 ( = vt. Furtherme, substituting equation (30 into equation (31, equation (31 into equation (30, yields And = γ ( γ( vt + vt, (γ 2 1 = γv(γt t. (38 = γ ( γ( + vt vt, (γ 2 1 = γv(t γt. (39 Dividing equation (38 by equation (39, we get γt t = t γt = t Using equation (32, we obtain (γ t t t ( t t γ γ = t t γ. From equation (34, we note that f = 0, t = γt. Therefe, the latter equation reduces to the following contradiction resulting from violating the codinate value restriction imposed by the speed of light invariance principle, namely = 0, equivalent to t = γt. γ = γt t γ, 1 = 0. It follows that, the Lentz transfmation arrived at under the principle of the constancy of the speed of light is deemed to be refuted. Consequently, the length contraction hypothesis iginally introduced as an ad hoc [3] to resolve the null result of the Michelson-Mley eperiment [1] (inconsistency between eperiment they with respect to a light ray fied-length round trip travel time in the earth travel direction compared to that in the respective transverse direction cannot be appropriately reconciled by the light velocity relativity principle space-time transfmation. The Lentz transfmation obtained contradictions are indeed epected, as it satisfies the constancy of the speed of light equation (8 that has been demonstrated in subsection to be mathematically impossible, since, as shown in 3.1.1, equation (8 can be derived from the unviable equations (20, (21, (23 ( Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings

7 vixra.g physics 5 Conflict Rationalization In the constancy of the speed of light principle equations [3] 2 + y 2 + z 2 = t 2, 2 + y 2 + z 2 = t 2, imposed as the governing aspect describing the space-time, (, y, z, t (, y, z, t represent the space-time codinates of an arbitrary light beam position vect in the reference frames K K, respectively. Therefe, f instance, assigning the entity vt to the -codinate of the reference frame K igin imposes a conflict with the light beam position vect -codinate, which is fced in this case to take the value of vt. Therefe, imposing the constancy of the speed of light equations on the space-time codinate systems prohibits the system codinates from taking other values than those associated with, describing the light beam position vect. Hence, the codinates of the igin of the moving frame are in conflict with the light beam position vect codinates. Therefe, the Lenz transfmation equations = γ( vt, = γ( + vt, return the equations = vt = vt f the igin of K ( = 0 K( = 0, respectively, which are in contradiction with the constancy of the speed of light equations in which represent the - -codinates of the light beam position vect. Indeed, this justifies the appearance of the identified contradictions upon using the Lentz transfmation equations under the particular condition of = 0 ( = 0, f which = vt ( = vt, with the constancy of the speed of light condition. 6 Apparent Space-time Transfmation Let s consider the classical codinate transfmation equation (20, hypothesize a general length conversion fact β from the perspective of the reference frame K, under the light speed constancy assumption: = vt + β, = 1 ( vt. (40 β Dividing equation (40 by c applying the constancy of the speed of light equation (32, we get t = 1 β ( t v. (41 Substituting = vt + β from equation (40 into equation (41, the following operations are perfmed to solve f t. t = 1 ( t v(vt + β, β then letting we get t β t = t β v2 t β v, (1 v2 γ = = t + v ; 1 1 v2 ( t 1 = t + v β γ 2 c, 2, ( t = βγ 2 t + v. (42 Substituting equation (42 into equation (40, rearranging, simplifying the terms, we get Indeed, = βγ 2 ( + vt. (43 = vt + β ; ( = vβγ 2 t + v + β ; = vβγ 2 t + β ( 1 + v2 γ 2 ; ( 1 = vβγ 2 t + β γ 2 γ + v2. 2 Since the latter equation terms between the brackets add to unity, it reduces to equation (43. F the particular case of β = 1/γ, equations (40 to (43 take the fm of the known Lentz transfmation equations. In addition, the relativistic velocity transfmation equations can be derived fm equations (41 to (44, irrespective of the value of β. It can be concluded from the transfmation resulting from the light velocity invariance principle, that f a length fact of β with respect to K, equations (40 (41 lead to the following equations f simultaneous ( t = 0 co-local ( = 0 events, respectively; = β, (44 Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings 7

8 vixra.g physics t = βt. (45 Whereas, f simultaneous ( t = 0 co-local ( = 0 events with respect to K, we have, respectively, from equations (43 (42, = 1 ( 1, (46 β γ 2 t = 1 β ( 1 t. (47 γ 2 F the case of preserved space-time where the length fact is β = 1, if we consider the events of a light ray being emitted returned, after being reflected, to the same point ( = 0 in the longitudinal direction in K, it can be easily shown that, in line with the constancy of the light speed, the light ray travel time in K would be ( 2L T = γ 2, (48 c where 2L is the round trip length. Accding to equation (46, the cresponding round trip length in K would be 2L = 2L γ 2, (49 the cresponding travel time in K becomes, accding to equation (47, T = T γ 2 = γ2 (2L/c γ 2 = 2L c. (50 Now, with the length fact of β being applied in K, the round trip length becomes β(2l, the light ray round trip travel time in K becomes, using equation (48, T = γ 2 ( β(2l c ( 2L = βγ 2 c, (51 whereas, accding to equation (47, the cresponding travel time in K, using equation (51, becomes, T = 1 ( T = 1 ( βγ 2 (2L/c = 2L β γ 2 β γ 2 c, (52 while, from equation (46 the cresponding round trip length becomes 2L = 1 ( β(2l = 2L β γ 2 γ. (53 2 It follows from equations (49, (50, (52 (53 that the transfmed longitudinal light ray round trip length travel time in K are independent of the introduced length conversion fact β in K, always converted to (2L/γ 2 (2L/c, respectively. It becomes then obvious that the transfmation (converting from K codinates to K codinates resulting from a length conversion fact with respect to K under the application of the contradicty light velocity invariance criteria (c = /t = /t, simply reverses the length fact to recover the iginal length in K, scales the recovered length down by a fact of 1/γ 2 so that the local time in K is obtained. This is indeed an amazing, tricky transfmation; when the introduced length conversion fact is 1/γ, thus changing the travel time in K from γ 2 (2L/c to γ(2l/c, the light velocity constancy resulting transfmation would reverse the length fact, returning the iginal time of γ 2 (2L/c in K, apply a new length fact of 1/γ 2 converting the travel time to 2L/c in K, with a net length fact of (1/γ 1 (1/γ 2 = 1/γ, giving the impression of a space-time distting transfmation, with a time dilation fact of γ a length contraction of 1/γ, although the actual length contraction fact in K is 1/γ 2! It follows that the length conversion fact of 1/γ is nothing but a particular fact resulting in [conflicting] symmetrical transfmation equations, when applying the restricted speed of light constancy principle on the classical spatial transfmation equation with a length conversion fact. Otherwise, any length conversion fact β introduced to the classical spatial codinate equation = vt + (changing it to = vt + β under the restricted assumption of the constancy of the speed of light, would result in inapplicable time space transfmation equations (40 to (43 invariantly satisfying the basic criteria of the light velocity assumption given by equation (8. Indeed, squaring both equations (40 (c equation (41, eliminating the similar terms from the resulting two equations, leads to 2 + v 2 t 2 2 β 2 = t 2 + v2 2 t 2 β 2. (54 Similar application of equations (42 (43 will result in 2 v 2 t β 2 γ = 4 c2 t 2 v2 2 + c2 t 2 β 2 γ. (55 4 It should be noted that equations (54 (55 reconfirm the equalities 2 = t 2, 2 = t 2. Adding equations (54 (55, simplifying rearranging the terms, leads to 2 2 = t 2 t β 2 γ 2 (c2 t 2 2 β 2 γ 2 ( t 2 2, (56 which reduces to the constancy of the speed of light equation (8: 2 2 = t 2 t Apparent Space-time Transfmation

9 vixra.g physics Finally, the above discussion, carried out from equation (20, from the perspective of K, can also be repeated based on equation (21, from the perspective of K, with identical results being obtained. 7 Conclusion Analysis of the Lentz transfmation revealed mathematical restrictions in terms of the deduced, simplified fm of the constancy of the speed of light equations residing in the transfmation. The Lentz transfmation, readily reconstructed using these basic, restricted light velocity invariance equations, resulted in mathematical contradictions. The principle of the constancy of the speed of light was thus demonstrated to be an unviable assumption, the ensuing Lentz transfmation was subject to refutation. Rationalization of the revealed contradictions was established. The actual interpretation of the Lentz transfmation demonstrated the unreal aspect of the space-time conversion attributed to the transfmation. References 1. A.A. Michelson E.H. Mley, On the Relative Motion of the Earth the Luminiferous Ether, Am. J. Sci. 34, ( A. Einstein, Zur elektrodynamik bewegter Körper, Annalen der Physik 322 (10, ( A. Einstein, On The Relativity Principle And The Conclusions Drawn From It, Jahrbuch der Radioaktivitat und Elektronik 4 (1907. English translations in Am. Jour. Phys. Vol. 45, NO. 6, June P. Beckmann, Einstein Plus Two. (Golem Press, Boulder, CO I. McCausl, The Persistent Problem of Special Relativity, Physics Essays 18 ( R. Hatch, Escape from Einstein (Kneat Kompany, Wilmington, CA, F. Selleri, in Open Questions in Relativistic Physics, F. Selleri, ed. (Apeiron, Montreal, H. Dingle, Science at the Crossroads (Martin Brian O keeffe, London, R. A. Monti, They of Relativity: A Critical Analysis, Physics Essays: 9 (2, ( A. Kelly, Challenging Modern Physics: Questioning Einstein s Relativity Theies (Brown Walker Press, Boca Raton, H. Poincaré, On the Dynamics of the Electron, Comptes Rendus 140, (1905. Radwan M. Kassir. On Lentz Transfmation Special Relativity: Critical Mathematical Analyses Findings 9