ABSTRACT

The stellar aberration finds explanation in classical time concept with which the principle of constancy of light speed in all inertial frames is incompatible. It is the Euclidean nature of space time that explains stellar aberration as well as principle of constancy of light speed in vacuum.

# Euclidean Nature of Space-Time : Experimental Proof

(PACS 03.30. + P, 04.20. Gz)

The aberration of star light, first reported by Bradley in 1727, is as follows. Let us assume that the rest frame S' of Fig. 1 is attached with astronomical coordinates fixed with the solar system, with respect to which the frame S attached with the earth is moving towards right with orbital velocity m at time instant t' = -1 as shown in Fig.l. Consider a starlight photon in S' at point P in the x' - y' plane with x' and y' coordinates as c sin q' and c cos q', respectively. This photon would be observed simultaneously in S and S' when O and O' coincide at t = t' = 0. Due to stellar aberration [1] the observer O sees the incoming direction of the starlight photon along line OP' at t = t' = 0 such that PP' = m and point P' coincides the point P at t' = -1. This coincidence of P and P' is possible only in classical time concept, t = t', and not in Einstein’s new time concept, t = b (t' + m x' / c2) where b = 1/ (1 - m2/c2)0.5. The problem lies in the fact that the stellar aberration as outlined above can be explained only in classical time concept with which the principle of constancy of light speed in S and S' is incompatible. It is this incompatibility that lead Einstein to establish new time concept [2]. Our endeavour is to find a time concept that explains stellar aberration as well as principle of constancy of light speed in S and S’.

In frame S this photon approaches the point O along the straight line OP which occupies the position O'P' in S' at t' = 0. But there is nothing worth moving length in Fig. 1 to substantiate the preceding statement. In objective reality there are only rest lengths OP and O'P' of S'. This means that the coinciding of O and O' at t = t' = 0 is accompanied by dislocation of initial point P of starlight photon to the point P' in S. In other words, the length OP at t' = -1 is a moving length which moves to position O'P' at t' = 0 in Fig. 1. But the principle of constancy of light speed in S and S' demands that the moving length OP must be Lorentz contracted along the direction of motion of S so that the rest length OP in S' reveals itself as a dilated value rather than a contracted value in S. Moreover, the dislocation of initial point P of starlight photon to point P' in S implies that the relativity of simultaneity has got meaning only when the spatial and temporal axes coincide each other in S and S'.

The angle q of tilt of the telescope to see the photon in the earth frame S can be obtained by considering the rest length OS of the frame S' as Lorentz contracted moving length of the earth frame S, and we get tan q = OS / PS = b (m + c sinq') / c cos q' where b = 1/ (1 - m2/c2)0.5. This is the relativistic formula for the aberration of starlight which was already deduced by Einstein in his first paper [3]. This clearly establishes that the rest length OS in S' does reveal itself as a dilated value in moving frame S in Fig. 1 rather than a contracted value. Further, in Fig. 1 the point P is the origin of temporal axis O'P in S' and of temporal axis OP in S for the starlight photon at P. Due to stellar aberration the physical existence of these temporal axes in S and S' can be experimentally proved in the form of apparent and true star positions observed simultaneously in S and S' when O and O' coincide at t = t' = 0 and observe the starlight photon previously at point P. This implies that (a) the distinction between space and time disappears completely. (b) there is no difference between time direction and space direction, (c) time should be measured in imaginary numbers rather than real ones. On the basis of these implications the conclusion that space time is Euclidean appears to be convincing and reasonable enough, since with imaginary time it has a Euclidean metric.

Moreover, the principle of relativity demands that the rest length in S must also reveal itself as a dilated value in S'. This is possible only when frame S must also get Lorentz contracted along with frame S' when one moves from frame S' to moving frame S in Fig. 1. This implies that the length OO' = bm of frame S in Fig. 1 gets Lorentz contracted as OO' = m when one goes from frame S' to S in Fig. 1. This reveals another aspect of Euclidean space time that when rest clock at O' registers t'= -1 in Fig. 1, the observer at O adjusts itself spatially as well as temporally when one moves from S' to S so that the rest clock at O in S registers t = -1.

References

[1] Albert B. Stewart, The Discovery of Stellar Aberration, Scientific American, p.100.(March, 1964)

[2] Harald Nordenson, Relativity, Time and Reality, p.84 (George Allen and Unwin Ltd., Ruskin House, Museum Street, London 1969).

[3] W. Pauli, Theory of Relativity, p.l7 (Pergamon Press, Oxford, 1958).

Rajan Dogra, Instructor, Wireless Operator Trade Industrial Training Institute, Chandigarh.

Mail: House No. 3291, Sector 27-D,
Chandigarh - 160 019, India
Phone : 0091-172-650148
E-mail: rajandogra@mailcity.com

APPARENT LACK OF SYMMETRY IN STELLAR

APPARENT LACK OF SYMMETRY IN STELLAR

ABERRATION AND EUCLIDEAN SPACE TIME

ABSTRACT :-

The effect of stellar aberration seems to be one of the simplest phenomena in astronomical observations. But there is a large literature about it betraying a problem of asymmetry between observer motion and source motion. This paper addresses the problem from the point of view of Euclidean space-time, arising from the proposition that stellar aberration (or Bradley aberration) gives rise to a Lorentz expansion.

( 1) INTRODUCTION:-

It is known that the earth completes a full circumference around the Sun every year. Consequently, since the Earth-Sun radius (Re) is well known, it is easy to determine the earth tangential velocity (Vt) required to complete the circumference in twelve months (T seconds). We have:

2 p (Re) =(Vt)T       [equation 4]

Equation (4) predicts that the average translational velocity V of the earth around the Sun is 29.79 Km/s. Of course, the earth velocity vector changes continuously in direction and completes a full cycle during a one year period while the earth circles the Sun.

On Figure A [26], an observer on Earth detects the photons emitted by a stationary star S, located in a direction perpendicular to the Earth velocity Vt. The star is located at such a large distance from the earth that the parallax caused by the orbit diameter around the Sun is completely negligible. Only the transverse velocity matters here. The stationary star S is emitting photons in all directions. The Earth and the telescope are moving upward at a velocity V . The telescope must make an angle q with respect to the real direction of the arriving photons in order to collect them at its focus.

In astronomy textbooks, the relative velocity between photons (at velocity c) and the Earth (Vt), explains why a telescope on Earth (see Fig. A) must be pointing at the angle q , with respect to the Earth-star direction, to be able to point at the star. Figure A shows that while the photons move in straight line toward the Earth, they will always remain in the axis of an inclined telescope, since it is moving sideways with the Earth. The angle q is equal to:

Tan q =(Vt)/c        [EQUATIO N 5]

Equation (5) gives q equals 20.5 arc-s. This is in perfect agreement with the value of aberration observed so many times since Bradley[11] in 1727. During the year, the observed direction of the stars makes one oscillation with an amplitude of 20.5 arc-s., as expected from the Earth motion around the Earth. The value of 20.5 arc-s. is called the constant of stellar aberration. The equation (5) ,in fact, gives classical formula for aberration in absolute time in which the waves (strictly the normals of the wave fronts) are not subjected to aberration to the great disappointment of Fresnel [1]. The relativity theory has solved this problem in terms of the relativity of simultaneity. Equal aberration for light photons and light waves requires this revision of simultaneity concept. Aberration is understood as aberration of the wave front normals. Such an aberration is the corollary of the relative simultaneity. According to Born [27], the relativistic stellar aberration formula calculation with a wave-front normal or phase through Lorentz transformation turned out to be identical to that with a particle through Einstein’s composition rule for velocities. Consequently, formulas are derived either by the first method in text books [28-33], or by the second method in the text books [34-36]. Today, the aberration is formally derived in text-books [6-8] from the relativistic transformation of the phase. But in these text-books [6-8] and [28-33], in the derivation of aberration from the relativistic transformation of the phase, the inverse Lorentz transformation equations are used as if one is going from moving frame to the rest frame. The preceding statement becomes quite clear later on in Section 3 (Conclusion–2) where on substituting inverse Lorentz transformation in place of Lorentz transformation [equation (11)] in right hand side of equation (10) and on comparing the coefficients of x, y, z and t in the resulting equation, the relativistic stellar aberration formula is derived. Aberration can also be derived from relativistic composition law of velocities which when applied in moving frame of source, using inverse Lorentz Transformations, predicts an aberration effect produced by the motion of the source in rest frame as discussed below.

Consider an inertial rest frame K with respect to which another inertial frame K’ is moving with velocity v along x-axis in positive (+X) direction. Further, consider a light photon emitted by a light source at rest in frame K’ such that the direction of emission lies in x-y plane and makes an angle q and q’ with +Y- axis in frame K and K’ respectively. Using inverse Lorentz transformations, the velocity components c cos q and c sin q of light photon along Y and X-axis respectively in rest frame K are expressed in terms of velocity components (c cos q’) and (c sin q’) in K’ as

c sin q = (c sin q’ + v)/(1+v sin q’/c)     ------(1)

c cos q = (c cos q’)/g(1+v sin q’/c) where g = 1/(1-v2/c2)0.5 ------(2)

Taking ratio of (1) and (2), we get

Tan q = g (c sin q’ + v)/(c cos q’)     ------(3)

The equation (3), derived using inverse Lorentz Transformation, predicts active aberration effect produced by the motion of source in rest frame K. This prediction is based on , relative motion between source and observer, which is claimed to be the cause of aberration even by Einstein [12] and Pauli [13]. But in the beginning of twentieth century, the observational evidence for the lack of longitudinal additions to the velocity of light by the radial motion of the source (Guthnick [14], desitter [15], Zurhellen[16]), later improved for extragalactic objects (Stromberg [17], Heckmann [18], Schmidt [19], Brecher [20], Barnet, Davis and Sanders [21]), convinced about the observational absence of active aberration. This absence was first tested by H. Seeliger [9] and Nyren [10] and also, recently, some authors [22-25] have shown this absence. This lack of observational confirmation of active aberration was interpreted as an error in the theory of relativity and as a hint that the latter had to be revised accordingly [2-5]. History forgot their [2-5] argument because they failed to provide correct explanation to the observed absence of active aberration. This revision of relativity theory is in the form of Euclidean Space –time in place of the Einstein’s four dimensional space time. Euclidean Space –time is such that both relativistic transformation of the phase and relativistic composition law of velocities make use of Lorentz transformations in deriving the relativistic aberration and doppler formulas. Further, the statement [1], that the aberration is a correction to be applied between observers in relative motion, is self-evident in the Euclidean Space-time

Moreover, the absence of active aberration leads to apparent lack of symmetry in aberration i.e. aberration is observed when observer is moving and not when source is moving. In order to find correct and rational explanation for the observed absence of active aberration produced by the motion of source, the mechanism of stellar aberration is reviewed in the next section.

(2) MECHANISM OF ABERRATION :

The aberration of star light is as follows. Let us assume that the rest frame S' of Fig. 1 is fixed by astronomical coordinates to the solar system, with respect to which the frame S attached to the Earth is moving towards right with orbital velocity m at time t' = -1 as shown in Fig.l. Consider a starlight photon in S' at point P in the x' - y' plane with x' and y' coordinates as c sin q' and c cos q', respectively. This photon would be observed simultaneously in S and S' when O and O' coincide at t = t' = 0. Due to stellar aberration [37] the observer O sees the incoming direction of the starlight photon along line O'P' at t = t' = 0 such that PP' = m and point P' coincides the point P at t' = -1. This coincidence of P and P' is possible only in classical time concept, t = t', and not in Einstein’s new time concept, t = b (t' + m x' / c2) where b = 1/ (1 - m2/c2)0.5. The problem lies in the fact that the stellar aberration as outlined above can be explained only in classical time concept with which the principle of constancy of light speed in S and S' is incompatible. It is this incompatibility that lead Einstein to establish new time concept [38]. Our endeavour is to find a time concept that explains stellar aberration as well as principle of constancy of light speed in S and S’.

In frame S this photon approaches the point O along the straight line OP which occupies the position O'P' in S' at t' = 0. But there is nothing worth moving length*(In [39] Einstein defined the concept of moving length as “By means of stationary clocks set up in the stationary system and synchronizing in accordance with Einstein’s definition of simultaneity, the observer ascertains at what points of stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring rod already employed which in this case is at rest, is also a length which may be designated the length of the rod. The length to be discovered by the above operation, we will call the length of the (moving) rod in stationary system.) in Fig. 1 to substantiate the preceding statement. In objective reality there are only rest lengths OP and O'P' of S'. This means that the coinciding of O and O' at t = t' = 0 is accompanied by dislocation of initial point P of starlight photon to the point P' in S. In other words, the length OP at t' = -1 is a moving length which moves to position O'P' at t' = 0 in Fig. 1. But the principle of constancy of light speed in S and S' demands that the moving length OP must be Lorentz contracted along the direction of motion of S so that the rest length OP in S' reveals itself as a dilated value rather than a contracted value in S. Moreover, the dislocation of initial point P of starlight photon to point P' in S implies that the relativity of simultaneity acquires meaning only when the spatial and temporal axes coincide each other in S and S'.

In Fig.1 the angle q of tilt of the telescope to see the photon in the earth frame S can be obtained by considering the rest length OS of the frame S' as Lorentz contracted moving length of the earth frame S, and we get tan q = OS / PS = b (m + c sin q') / c cos q' where b = 1/ (1 - m2/c2)0.5. This is the relativistic formula for the aberration of starlight which was already deduced by Einstein in his first paper [40].

The formula for the angle of aberration, instead of being tan q = m/c, then becomes sinq = m/c when q’ = 0. The relativistic formula agrees with the classical formula for quantities of first order in case of relative transverse velocity between star and earth. Further, Let assuming that Fig. 2 represents. Fig.1 at time instant t’ = +1 such that q’ = 0. Additionally, there is present at origin O of earth frame S a small horizontal plane mirror in x-z plane that reflects the incoming star light photon at

t’ = t = 0. In accordance with laws of reflection the starlight photon is reflected at angle sin
q = m / c along OP in earth frame S and at normal along O’P = c in frame S’ because of normal incidence in S’. After reflection the earth becomes moving source that has emitted reflected starlight photon at t = t’= 0. It is quite obvious that the direction OP of outgoing reflected starlight photon in earth frame S is apparent direction (as incoming starlight photon direction was apparent in S) and the direction O’P of outgoing reflected starlight photon in frame S’ is true direction. This implies that the earth frame S as moving source with transverse velocity emits the reflected star-light photon without any aberration. Consequently, the apparent lack of symmetry (described in section 1 of this paper) in stellar aberration, when the star is moving at transverse velocity, can be explained on the basis of above discussion. The fact, that Einstein’s theory fails to explain this lack of symmetry in aberration, implies that something is wrong with Einstein’s theory. Let us see what new conclusions can be drawn about relativity from the above discussion of stellar aberration that explains apparent lack of symmetry in aberration.

(3) CONCLUSION (1) :

The angle q of tilt of the telescope to see the photon in the earth frame S can be obtained by considering the rest length OS of the frame S' as Lorentz contracted moving length of the earth frame S, and we get tan q = OS / PS = b (m + c sinq ') / c cos q' where b = 1/ (1 - m2/c2)0.5. This clearly establishes that the rest length OS in S' does reveal itself as a dilated value in moving frame S in Fig. 1 rather than a contracted value. Further, in Fig. 1 the point P is the origin of temporal axis O'P in S' and of temporal axis OP in S for the starlight photon at P. Due to stellar aberration the physical existence of these temporal axes in S and S' can be experimentally proved in the form of apparent and true star positions observed simultaneously in S and S' when O and O' coincide at t = t' = 0 and observe the starlight photon previously at point P. This implies that (a) the distinction between space and time disappears completely. (b) there is no difference between time direction [41] and space direction, (c) time should be measured in imaginary numbers rather than real ones. On this basis the conclusion that space time is Euclidean appears to be convincing and reasonable enough, since with imaginary time it has a Euclidean metric.

Moreover, the principle of relativity demands that the rest length in S must also reveal itself as a dilated value in S'. This is possible only when frame S also becomes Lorentz contracted along with frame S' when one moves from frame S' to moving frame S in Fig. 1. This implies that the length OO' = bm of frame S in Fig. 1 becomes Lorentz contracted as OO' = m in going from frame S' to S in Fig. 1. This reveals another aspect of Euclidean space time that when rest clock at O' registers

t'=-1 in Fig. 1, the observer at O adjusts itself spatially as well as temporally on moving from S' to S so that the rest clock at O in S registers t = -1.

1. CONCLUSION (2)

In Euclidean space-time the relativistic transformation of the phase and relativistic composition law of velocities make use of Lorentz transformation in deriving the relativistic aberration and doppler formulas as discussed below.

In Fig. 1 the length OP of moving frame S in Euclidean-space time is calculated as

OP = [OS2 + PS2] 0.5

Substituting OS = b (m + c sin q’) and PS = c cos q’ in above equation, we get

OP = b (c + m sin q’)

This means that cosq in S is represented in Euclidean space-time as

cos q = (PS / OP) = (c cos q’)/ b (c + m sin q’)   ------(6)

Keeping in view the relativistic law of conservation of linear momentum, the linear momentum components along Y – axis of the starlight photon in frames S and S’ of Fig. 1 must be equal i.e.

(hw/2pc) cos q = (hw’/2pc) cos q’     ------(7)

Where h is planck’s constant and w, w’ are angular frequencies of starlight photon in S and S’ respectively. Rearranging terms in equation (7) and substituting equation (6) in (7), we get

w = bw’ (1+m cos q’/c)       ------(8)

The relation between wave-vectors k and k’ of starlight photon in frames S and S’ of Fig.1 can be found easily by substituting w = ck and w’=ck’ in equation (8) and we get

k = b k’ (1+m cos q’/c)      ------(9)

Consider a plane wave of frequency w’ and wave-vector k’ associated with the starlight photon in frame S’ of Fig.1. In moving frame S of Fig.1 due to linearity of Lorentz transformation this wave indeed remains a plane wave but will have, in general, a different frequency w and wave vector k but the phase f of the wave is an invariant quantity.

wt – k.r = f = w’ t’ – k’.r’      -------(10)

Where r and r’ are the radial motion vectors of starlight photon of Fig. 1 in frames S and S’ respectively.

If we substitute equations (8) and (9) in left hand side (wt – k.r) of equation (10) along with Lorentz transformation equations

x = b (x’ + mt’), y = y’, z = z’ and t = b (t’+mx’/c2)  -------(11)

we get right hand side (w’t’ – k’.r’) of equation (10). This verification of equation (10) leads to the result that the statement, for w and k in equation (8) and (9), that are derived on the basis of Euclidean space-time concept, represent angular frequency and wave-vector of starlight photon in moving frame S and are in accordance with Lorentz transformation equation. In fact, the equation (8) and (9) represent the relativistic doppler effect statements derived on the basis of Euclidean space-time concept. Moreover, the relativistic stellar aberration formula, derived in Section 3(1) of this paper using Euclidean space-time concept, also satisfies the Lorentz transformation equations (11).

Thus, the above discussion leads to the result that Euclidean space-time is such that both relativistic transformation of the phase and relativistic composition law of velocities make use of Lorentz transformation in deriving the relativistic aberration and doppler effect formulas.

(3) CONCLUSION (3)

It is stated in [1] that aberration is a correction to be applied between observers in relative motion. This statement is self-evident in Euclidean space-time concept as discussed below.

In Fig. 1, if the velocity m of frame S is allowed to take an arbitrary value between (-c) and (+c), the distance ( OO’ = m) is to be corrected for each observer at O in relative motion in order to find the apparent stellar position at angle q in Euclidean space-time. This obviously leads to the result [1] that aberration is a correction to be applied between observers in relative motion.

REFERENCES :

[1] D.-E. Liebscher, P.Brosche, Bonn, Aberration and relativity, Astron.Nachr.319 (1998).

[2] Hayn, A.N. (1920) : Didaktisches zur Aberration II, Astron.Nachr.212, 81-88 (5070).

[3] Tomaschek, R. (1925) : Uber die Aberration, Z.Physik 32, 397-402.

[4] Osten, H.(1925) : Aberration und Relativitat, Astron.Nachr.224, 66-67 (Nr.5356).

[5] Mohorovicic, St. (1928) : Optik bewegter Korper, in : E.Gehrke, Handbuch der physikalischen Optik 2, 917 ff., Leipzig, J.A. Barth.

[6] W. Pauli, Theory of Relativity, p.19 (Pergamon Press, Oxford, 1958)

[7] Robert Resnik, Introduction to Special Relativity, p.82 (Wiley Eastern Private Limited, New Delhi, First Edition 1962, Reprinted 1972).

[8] J.D. Jackson, Classical Electrodynamics, p.521 (Wiley Eastern Private Limited, New Delhi, Second Edition 1975, Reprinted 1989).

[9] Seeliger, H. 1884, Astron.Nachr. 109, 275-280 (2610).

[10] Nyren, M. (1888) : Zur Aberration der Fixsterne, Bull. Acad.Imp.St. Petersburg 32, 402-412.

[11] J. Bradley “Account of a New Discovered Motion of the Fixed Stars” Phil. Trans. 35p.637 (1728).

[12] Einstein, A. 1916, Uber spezielle und allgemeine Relativitatstheorie, Braunschweig, Vieweg.

[13] Pauli, W. 1921, Enzyklopadie der mathematischen Wissenschaften V/2, S.563.

[14] Guthnick, P.1913, Astron.Nachr. 195, 265-272 (4670).

[15] deSitter, W. 1913, Phys.Z. 14,429.

[16] Zurhellen, W.1914, Astron.Nachr.198, 1-10 (4729).

[17] Stromberg, G.1931, Publ.Astr.Soc.Pacific 43, 266.

[18] Heckmann, O. 1960, Ann. d' Astrophysique 23, 168-173.

[19] Schmidt, Th. 1964, Z.Astrophys. 60, 106-111.

[20] Brecher, K. 1977, Phys.Rev.Lett. 39, 1051-1054.

[21] Barnet, C., Davis, R., Sanders, W.L. 1985, Astrophys. J. 295, 24-27.

[22] H.E. Ives “Extrapolation from the Michelson-Morley Experiment” J.Opt. Soc. Am. 40, pp. 185-190 (1950).

[23] E. Eisner, “Aberration of Light from Binary Stars – a Paradox?” Am. J. Phys. 35, pp.817-819 (1967).

[24] T.E. Phipps “Relativity and Aberration,” Am. J. Phys., 57, pp., 549-550 (1989) also Phipps T.E., Jr., “Stellar Aberration from the standpoint of the Radiation Convection Hypothesis.” Phys. Essays 4, 368, (1991).

[25] H.C. Hayden, Stellar Aberration, “Galilean Electrodynamics, 4.pp. 89-92 (1993).

[26] Paul Marmet, Stellar Aberration and Einstein’s Relativity, Physics Essays, Vol. 9, No : 1p.96-99, 1976.

[27] Born, M. 1920, Die Relativitatstheorie Einsteins, Vieweg Verlag Braunschweig.

[28]  Bohm, D. 1965, The special theory of relativity, W.A. Benjamin.

[29] Joos, G. 1956, Lehrbuch der theoretischen Physik, Leipzig, Geest & Porting, 9.Aufl.

[30] Herlt, E., Salie, N. 1978, Spezielle Relativitatstheorie, Berlin, Akademie-Verlag.

[31] Sommerfeld, A. 1959, Vorlesungen, IV. Optik, 2. Aufl., Leipzing, Geest & Portig.

[32] Schneider, M. 1979, Himmelsmechanik, BI Mannheim.

[33a] Varicak 1910, Phys. Z.11, 93-96.

[33b] Varicak 1910, Phys. Z.11, 287-294.

[33c] Varicak 1910, Phys. Z.11, 586-587.

[34] Landau, L.D., Lifshic, E.M. 1962, Klassische Feldtheorie, Berlin, Akademie-Verlag 1964.

[35] Goenner, H. 1996, Einfuhrung in die spezielle and allgemeine Relativitatstheorie, Heidelberg, Spektrum AV.

[36] Rindler, W. 1986, Essential Relativity, Heidelberg, Springer.

[37] Albert B. Stewart, The Discovery of Stellar Aberration, Scientific American, p.100. (March, 1964).

[38] Harald Nordenson, Relativity, Time and Reality, p.84 (George Allen and Unwin Ltd., Ruskin House, Museum Street, London 1969).

[39] Albert Einstein, On the Electrodynamics of Moving Bodies, Ann. Physik, 17, 891 (1905).

[40] W. Pauli, Theory of Relativity, p.17 (Pergamon Press, Oxford, 1958).

[41] Stephen Hawking, A Brief History of Time, p.142, (Bantam Press Edition reprinted, 1996).

Rajan Dogra, Instructor, Wireless Operator Trade Industrial Training Institute, Chandigarh.

Mail: House No. 3291, Sector 27-D,
Chandigarh - 160 019, India
Phone : 0091-172-650148
E-mail: rajandogra@mailcity.com

Mathematical Contradiction of Special Relativity (By Eric J. Lange) and Euclidean Space -Time

Einstein based his special theory of relativity on two postulates:

1. The laws of physics are the same in all inertial systems (reference frames that move uniformly and without rotation). There are no preferred inertial systems. When a certain reference-frame moves with constant speed with respect to another, processes of nature will obey the same laws of physics in either reference-frame.

2. The speed of light in vacuum has the same constant value c in all inertial systems.

Let a light signal travels along X-axis with speed c in rest frame C such that  x = ct

The same light signal travels, according to Einstein, also with respect to reference-frame C' with a speed equal to c. With respect to C' the equation for the movement of the light signal becomes: x'= ct '

The movement of frame C' with respect to frame C is described by: a = v t

(For the three equations above and all still to derive, applies that C' coincides with C at t = 0 and t' = 0. The covered distances x and x' of the light signal are at that moment also equal to zero.)

Einstein had to make concessions in relation to absolute time which was generally accepted back then. Because distances x and x' (for t>0) were obviously not equal, t and t' had to be different too.

x = ct

x' = c t'

x' < x

c' = c

t' < t

The result of Einstein's principle of the constancy of the speed of light, was therefore that the course of time in C and C' had to be different. A clock would run slower in C' than in C.

The following derivation of the Lorentz-transformation according to the special theory of relativity, is probably the most frequently occurring version of the derivation of the transformation-equations that can be found in the literature1, 2. It is this version that is taught on universities around the world.

The light signal moves uniformly rectilinear with respect to both reference-frames. This means there exists a linear relation between coordinates x, y, z, t and x', y', z', t'. The most common form of this relation is as follows:

X' = a11 x + a12 y + a13 z + a14 t (1)

y' = a21 x + a22 y + a 23z + a 24t (2)

z' = a31 x + a32 y + a33 z + a 34t (3)

t' = a41 x + a 42 y + a 43 z + a 44 t (4)

Because C' moves linearly along the x-axis of C, it always applies that y=y' and z=z'. From this follows for the coefficients:

a21 = a 23 = a 24 = a31 = a32 = a34 = 0

a 22 = a33 = 1

Because of symmetry reasons t' can only depend on x and t. From this follows:

a42 = a 43 = 0

Then it is assumed that x'=0 in C' has to correspond with x = vt in C. It follows for the coefficients that:

a12 = a 13 = 0

a14 = -v a11

When the coefficients in equations (1), (2), (3) and (4) are substituted by there values we get:

x' = a11 (x - v t) (5)

y' = y (6)

z' = z (7)

t' = a 41 x + a 44 t (8)

The falsity in this derivation can be located in the assumption that x'=0 has to correspond with x = v t.

At t=t'=0, when x=0, x' will be equal to zero too. In a small time-interval greater than zero the speeds of C' and the light signal can be defined. As determined before, x is defined as the distance a light signal travels with speed c during a time interval t, and a the distance C' travels with speed v. With x = v t it has become unclear what movement we are talking about:

1. When it is meant to describe the propagation of light, it is not proper to use the variable v in this equation, since it equals the constant c (always). Substituting occurrences of v by c in this derivation shows clearly that this makes the transformation equations meaningless and useless.

2. It is more likely that v is intended to describe the movement of coordinate-system C', as was determined before. In the schematic figures and derivations I have seen, the speed v was never determined beforehand as being a divided by t, like I did in this article. v is usually left undetermined (although definitely designated to the speed of C') up to the point where the assumption is being made that x'=0 has to correspond with x = v t. I wanted to make clear from the start that the distance C' travels should not be described by x since this variable already describes the distance light travels. Therefore it can now be absolutely clear that x may not be used in this equation when it describes the movement of C'. Using the same variable x for different distances corrupts the derivation and makes it false. (When we would assume that C' and the light signal always coincide, so that a = x and the equation becomes justified, then v = c and the problem arises that their coinciding is contradictory to the second postulate.) Thus the equation x = v t is inconsistent with the assumptions on which the whole derivation is based, which falsifies the results as derivatives of the two postulates of the special theory of relativity.

The mistake which was being made in the assumption that led to equations (5), (6),

(7) and (8) do indeed show up in those equations. Specifically in equation (5) an error can be found, for in equation (5) a11 can't be solved (as is intended later on):

x' = 0

x - vt = 0

x' = a11 (x - vt)

a11 = 0/0

For the sake of completeness, the rest of the derivation will be shown in short. When a light signal is transmitted in arbitrary direction at t=t'=0, then the covered distance of the signal with respect to C can be determined with Pythagoras and satisfies therefore:

x2 + y2 + z 2 - c2 t 2 = 0 (9)

With respect to C' applies:

x2 + y'2 + z' 2 - c 2 t'2 = 0 (10)

Substitution of equations (5), (6), (7) and (8) in equation (10) gives:

X 2 (a 211 - c2 a 241) - 2 xt (a 211 v + a41 a44 c2) + y2 + z2

- t2 (a244 c2 - a211 v2) = 0 (11)

Equation of (9) and (11) leads to the next three equations for determining a11, a41 and a44:

a 211 - c2 a 241 = 1

a211 v + a41 a44 c2 = 0

a244 c2 - a 211 v2 = c2

a11 = a44 = 1 / (1-v2/c2) 0.5 (12)

a 41 = -v/c2 (1-v2/c2) 0.5 (13)

In deriving (12) and (13), the signs (- or +) are chosen so that when v becomes equal to zero (!), the equations (5), (6), (7) and (8) change into:

x' = x

y' = y

z' = z

t' = t

Substitution of equations (12) and (13) in (5), (6), (7) and (8) gives the Lorentz-transformation:

x' = (x - vt) b Where b = 1 / (1-v2/c2) 0.5 (14)

y' = y

z' = z

t' = (t - vx/c2) b (15)

(H.A.Lorentz derived equations (14) and (15) first (based on an ether theory), years before Einstein first published his special relativity theory.)

Summarizing, I think the mathematical error in the above derivation can be located where c is falsely substituted by v (or where x is wrongly equated to vt). The concluding transformation equations may appear alright but note that v entered those equations unjustified. Replacing this variable with c in (14) en (15) shows the falsity of this derivation convincingly.

The derivations of the Lorentz-transformation by Einstein himself (in his original article3 and much later in another and also "simplified" form in his book about relativity4 ) contain the same error as in the above derivation.

The falsity of these derivations does not prove of course that in principle a good derivation can't be made at all. A proof of this sort lies more on a philosophical en empirical plane I think. I consider, for instance, the postulate of the constant speed of light to be in contradiction with Galilean transformation and therefore false. The relativity of speeds in general makes one absolute speed impossible, whilst the postulate itself should make usage of Galilean transformation in the derivation of the relativistic transformation equations out the question. Yet, all of the derivations are based on Galilean transformation. Furthermore and foremost, I think it should be obvious to those who know the Kennedy-Thorndike experiment5 (an altered version of the famous Michelson-Morley experiment) that the relativistic transformation equations can never explain or describe the results thereof.

__________________

Erik J. Lange

__________________

1. Introduction to Special Relativity, R. Resnick, J. Wiley & sons Inc., 1968

2. Klassieke Mechanica en Relativiteitstheorie, J.J.J. Kokkedee, Uitg. T.U. Delft, 1992

3. A. Einstein, "Zur elektrodynamik bewegter Körper", Annalen der Physik. IV. Folge. 17, page 891, 1905.

4. Relativiteit, speciale en algemene theorie, A. Einstein, Het Spectrum, 1978

5. R.J. Kennedy and E.M.Thorndike, "Experimental establishment of the relativity of time", the Physical Review, vol. 42, second series, page 400, 1932.

# My explanation

The Mathematical error in the above derivation of Lorentz transformation equation is located where c is falsely substituted by v as viewed by Eric J. Lange. But in Euclidean space - time the path followed by light signal in moving frame C' is the moving length and therefore (x - vt) denotes the moving length that is transversed by the light signal with speed c. The initial point x' = 0 of this moving length has to be denoted by x = vt because the moving length itself is defined by the path transversed by the light signal with speed 'c'. Therefore, it is clear that there is no mathematical error in the derivation of Lorentz transformation equations in Euclidean space time when c is substituted by v (or x is equated to vt)

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