ISSN 0029-3865
CENTRO BRASILEIRO OE PESQOISAS FÍSICAS Notas de Física
CBPF-NF-046/86 ANALYSIS OF ABSOLUTE SPACE-TIME LORENTZ THEORIES
by
A.K.A. Maciel and J. Tiorano
RIO OE JANEIRO 1986 NOTAS DE FÍSICA c uaa pré-publicação de trabalho original em Física
NOTAS DE FlSICA is a preprint of original works un published in Physics
Pedidos de cópias desta publicação devem ser envia dos aos autores ou ã:
Requests for copies of these reports should be addressed to:
Centro Brasileiro de Pesquisas Físicas Xrea de Publicações Rua Dr. Xavier Sigaud, 150 - 49 andar 22.290 - Rio de Janeiro, RJ BRASIL ISSN 0029 - 3865
. CBPF-NF-046/86 ANALYSIS OF ABSOLUTE SPACE-TIME LORENTZ THEORIES by A.K.A. Maciel and J. Tiomno
Centro Brasileiro de Pesquisas Físicas-CBPF/CNPq Rua Dr. Xavier Sigaud, 150 22290 - Rio de Janeiro, RJ - Brasil CBPF-NF-046/86
ABSTRACT
Two particular forms of absolute space time theories are pre- sented together with a derivation of their predictions for meas- urements that are within present day detection limits.
Key-words: Absolute space-time; Ether theories; Special Relativity. CBPF-NF-046/86
1 INTRODUCTION
The problem of a clear experimental evidence in favour of Special Relativity (SR) versus absolute space-time theories is as old as SR itself and seems to have eluded physicists to this date . Claims of experimental evidence against SR have motivated its confrontation ' with versions of a rival theory generally called the Lorentz Ether Theory (LET), which should be better called Lorentz Absolute (space-time) Theory. It has been shown that, for experiments not involving di- rect measurement of time, but only of time (or frequency) diJf ferences at the same point, LET is undistinguishable from SR for Aon accelerated experiments in vacuum. We like to mention that although Lorentz proposed his absolute ether theory not in vacuum but in presence of dispersive media we are presently restricting ourselves to vacuum situations for the sake of simplicity. In a preliminary analysis of .past experiments designed to test the validity of SR, we concluded that SR a«d one specific form of LET, called strict LET (S-LET) , can be ex per imentally distinguished with presently existing technology but no experiment performed to date has had the necessary charac teristics to unquestionably favour one of these competing the ories. One of us (JD showed that another form of LET, extended LEI' s, however, ruled out by already existing experimental results The present paper serves the purpose of organizing prelim . lnary theoretical information that is scattered through re- ferences 1/3,4, presenting it in a complete and generalized form. CBPF-NF-046/86
-2-
Sections 2,3 and 4 are devoted respectively to the presen- tation of LET and of two forms of LET, strict LET and extended LET, and to the derivation of their equations of angular mo-1 tion for the free rotation of a solid body in Einstein's co- ordinates .
In Section 5 quantitative predictions of measurable quan- tities that would evidence departures from SR are derived in both models. Some results have already appeared in approximated forms, adapted to specific cases, in previous publications ' .
In Section 6, comments and conclusions on the theoretical aspects of the problem are presented.
2 ABSOLUTE SPACE-TIME THEORIES AND LET
A common feature of most absolute space-time theories is the existence of a preferred (absolute) frame of reference S experimentally distinguishable from all other inertial fra- mes S, those in uniform translation with respect to S . These absolute S-T theories impose the postulate :
(i) The round trip vacuum propagation of light is iso- tropic in S (with speed c = 1) and independent of the veloci ty of the source. Theories have been proposed with various characteristics, and here we state the basic features of the theory we shall refer to as LET. We present them in the form of four postula- tes. Thus, besides (i) LET imposes. CBPF -NF-046/86
-3-
(ii) Lenghts contract and time rates dilate in the usual
SR abiding way for uni$c*müj translating systems in SQ. Postulate (ii) has been experimentally tested to one part in 107 for the Fitzgerald-Lorentz longitudinal contraction relation to the transverse behaviour of solid bodies, and to one part in 10s for the Lorentz time dilation *'.A remaining indeterminacy of a possible uniform relative space dilation exists of ± 10~ . This makes postulate (ii) highly plausible. Postulates (i) and (ii) are coordinate independent in S and entirely consistent with SR (where S is any inertial fra me). However, the preparation of a reference frame takes many identical rods and clocks to be scattered through space in relative rest. This brings in the problem of synchronization of these distant clocks and is where LET and SR do part. Both LET and SR state that if all identical clocks and rods are brought together to make their phases coincide and lenghts to be standardized, then measurers can slowly scatter them away to define an inertial frame when at relative rest. This is known to be equivalent to Einstein's light signal synchronization method. If S and S are both prepared in this way and S is in uniform translation V = Vx with respect to S , then their mea surements of position and time will be related by the Lorentz transformations,
x « Y(X0 -vto) ; y « y0 (1) 2 -1/2 t « Y(t0 -Vx0) ; Y " d -V ) CBPF-NF-046/86
which define what we will refer to as "Einstein Coordinates" (x,t) and the t-synchronization, valid for both LET and SR. In these coordinates the propagation of light will be isotropic in LET in all frames S, even if SR is violated. SR and LET are thus far indistinguishable since S has not yet been singled out. It is the third postulate which does this job: (iii).LET admits, besides Einstein's synchronization (1), the existence of other synchronizations that are inequivalent to Einstein's. Within this class, we choose to single out that synchrony zation in which positions and times measured in S and S will relate by
X — *v / v •Vi- *t • V s v • 7 s 7
T = "f1^ ; T = t + V.x(t) . (2)
Equations (2) define what we will refer to as "Fitzgerald- Lorentz Coordinates" (X,T) and the T-synchronization, according to which, events that are simultaneous in one inertial frame, are so in all others. If this T-synchronization is achieved in some experiment in S, then the "absolute" t -synchronization in S is promptly obtained (and conversely), since V is neces- sarily a by-product of that experiment. Of course this is in- consistent with SR. The electromagnetic theory is known to be invariant under the transformations (1) which -preserve the Maxwell equations and the isotropy of light propagation in CBPF-NK-046/86 -5-
all frames, but not under (2). In Fitzgerald-Lorentz Coordina tes the propagation of light will be anisotropic in all frames other than S . o It is then clear that LET contains all of SR as a special case but is less restrictive in that it allows for an absolute time that is inequivalent to Einstein's time. We have not yet discussed how synchronization (2) could be achieved. This can only be done after we have tentatively decided in what domain of physics the absolute time would manifest itself. Inspired by past experiments we have decided to examine the rotation-translation of rigid bodies as a possible domain of Lorentz invariance violations. Postulates (i) to (iii) must then be complemented with the definition of specific mechanisms that are consistent with them and that lead to the violation of SR. This is the object of the next section where we consi_ der two possible ways of violating SR within LET theories.
3 STRICT. AND EXTENDED LET
Consider a freely rotating rigid body at rest in some iner- tial reference frame S which is in uniform translation V =V x with respect toS. Rigidity properties in SR are such that all points in this body have, in S, the same constant angular velocity u in Einstein coordinates. Such rotation is described
by
and leads to Einstein's t-synchronization. He now investigate two other forms of rigidity leading to the T-synchronization (in which case the underlying theory is called Strict LET or S-LET) and to the absolute or t -syn- o chronization (Extended LET ' or E-LET). It will be assumed, for simplicity that V is perpendicular to the rotation axis z. In general, V should be replaced by its component in the plane of rotation which defines the direction x, z = V.x x In our future formulae x has been chosen as the angular origin. 3.1 Strict LET The physics of free rotation according to S-LET is presented as a fourth postulate . (iv-S) The free rotation of a rigid" body as observed in a co-moving in ertial frame S is uiiform when described by the Fitzgerald-Lorentz coordinates. Thus, if a general point solidary to this body is described by the polar vector R(T) - (R, • (T) » •«>) • flT and leads to the T-synchronization. In Einstein coordinates, CBPF-NF-046/86 -7- 4(t) = 4(0) + fl(t +V.R(t)) (4) where T = t+tf.R(t) stems from eq. (2), n is a constant of the motion, and R(t) = Rcos4(t)x + Rsin4(t)y (5) Substitution of (5) into (4) with v "^ SIR gives ' 4(t) .= 4(0).. + fit + W cos (4(0) +Qt) + O(v2V2) • (6) Thus, according to S-LET, the free rotation of a disk as seen in S (Einstein .coordinates) is such that the angular velocity of its points will be oscillating harmonically around an aver- age value SI, each with a position dependent phase of its own. Time varying phase differences between different points will imply departures from a rigid like motion as we know it in' Einr*ein coordinates. More specifically, rigidity in S-LET means |R\ (T) - iL(T.) | «constant for all T for any two points 1 and 2, instead of the usual SR constancy of |r.(t) -r-(t)|. This is precisely the effect that we suggest11'*tk^ should be looked for in an experiment designed to test S-LET against SR in the dynamics of rotating bodies. 3.2 Extended LET The work of Kolen and Torr( 'has led Rodrigues and Tiomno CBPF-NF-046/86 -8- to consider one other model of absolute space-time physics, E-LET. Postulates (i) to (iii) are maintained as well as the assumption that violations of SR occur in the dynamics of ro- tating bodies. In parallel with S-LET,the physics of free ro- tation according to E-LET is presented as a fourth postulate : .:•>• (iv-E) The free rotation of a rigid body is uniform when observed in So and described by the absolute coordinates, Thus, if a general point solidary to this body is described in S3 by the polar vectoir R (t ) • (R ($ ) ,4» (t )), its an- o 00 0000 gular motion is given by Vo) and leads to the t -synchronization. We must translate this e guation into Einstein coordinates in the laboratory frame S. Remembering that angles in both frames differ by a Lorentz boost, tan 4>0(t0) = Y tan and using the transformation t = y(t + V.R(t)) we find (v «oR> +VvQ cos 2 2 - (V /4)sin 2 (<|>(0) + flot) + O(V*fvoV»,v*V ) (7) It is clear that S-LET and E-LET differ radically. Fitzgerald-Lorentz coordinates need not be considered in E-LET CBPF-NF-046/86 -9- as they have no privileged role. Departures from SR originate from the fact that it is an oblongue (Lorentz contracted) motion in S that has constant angular velocity, o In order to render future derivations and discussions more practical, we merge equations (6) (S-LET) and (7) (E-LET) into one single equation (8) of free rotational motion by means of the-binary parameter \ defined such that \ = 0(1) whenever S-LET (E-LET) is in operation. We also drop the indices (o) in E-LET constants since A will unambiguously indicate which theory, is being considered and therefore which constants should be used. Thus, *(t) = <|>(0) + (l+XV2/2)«t + Vv cos ($(0) - (AV2/4) sin 2(0(0) +fit) + O(V\vV3,v2V2) (8) 4 PARTIAL FITZGERALD-LORENTZ CONTRACTION EFFECTS We investigate next the consequences of yet another ten- tative mechanism of Lorentz invariance breaking which, being presumed independent in origin from S-LET and E-LET, might oc cur simultaneously with either. In the S description of a freely rotating body, sup- posedly at rest in S, we allow for the possibility that the Fitzgerald-Lorentz contraction along % may be slightly altered due to the rotation. For that purpose we introduce a modified boost parameter y = fW0) satisfying CBPF-NF-046/86 -10- lim y = Y » d -V2)"1/2 ft +0 o such that a full contraction is restored in the case of no ro- tation . As a model we take Y = d-d -3)V2)"1/2 where 6 = B (ÍÍ ) is a positively or negatively increasing func- tion of Í2 from 6(0) = 0 to any limit lower or equal to 1 in modulus at high ilQ. If such is the case, consider a solidly built non rotating circular disk in S of radius R. Und~i rotation, a peripheral point will have in S a position (polar) vector (R/Y) cos = (YR/Y) cos*(t)x + R sin + 0(V") . (9) CBPF-NF-046/86 -11- Equation (9) should hereafter replace eq. (5) and has an extra 8-term responsible for a dilation (8 >0) or contraction (8 < 0) of the Einstein coordinate x-components in case the mo- dified Fitzgerald Lorentz contraction between S and S takes place. This is however of no concern to the equations of angular motion of either S-LET or E-LET up to the orders of magnitude considered here. In fact, the added term in R(t) will contri- bute to 4>(t) in eqs. (6) and (7) only at the neglected vV3 scale. Its consequences will unfold clearly in the next section which is devoted to the derivation of experimentally detectable consequences of LET. We like to mention the work of Atkins ' who considers in- ertial. effects on the rotating body that prevent it fror. at- taining its instantaneous {Lorentz contracted) equilibrium con figuration. The consequences are of a similar nature to those originated from our "ad hoc" introduction of the parameter $. Knowing that Electromagnetism is Lorentz invariant if written in Eisnteln coordinates, we shall consider only experiments involving electromagnetic signals and detection devices. Thus our concern in writing the equations of motion in Einstein coordinates throughout, so that we are assured of the absence of coordinate effects in future confrontations between theory and experiment. SR violations such as those contained in eq. (8), if observed, necessarily result from a non Lorentz invariant dynamics of the rotating body in S. CBPF-NF-046/86 ' -12- 5 CONSEQUENCES OF LET We concentrate on two different methods of search for SR violations in the dynamics of rotating bodies. •5.1 Length measurements The first one deals with the detection of length (or time) variations SL(t) in the optical path between two points, say, two reflecting mirrors, solidary to. a rotating system. . The optical distance between points 1 and 2 is L(t) = |R2(t+6t) - R^t) | (10) where 6t(t) » L(t) is the light transit time from 1 to 2. Use of eqs. (8) and (9) here supposesR1,R2 and V coplanar, a view we adopt for simplicity. Should any of these lie outside the rotation plane, its modulus in the remaining formulae must be multiplied by the cosine of its latitude. All following calculations re- main true to the order considered. From eqs. (9) and (10), 2 2 L (t) •« Rj + R - 2R1R2cos[>2(t+5t) - !]^ cos -13- This equation is to be iterated with (8) up to terms of order V2, Í2R and Í2RV. with the convention that and *02 the LET result for L(t) can be written as L(t) =Lo+6L(t) = LQ < 1 +qfiLo sin$Q -q VÍÍX(t) sin<|)0 2 2 (XV /2)q sin i>0 cos(<|>0+ 2i2t) BV2X2(t)/(2L2) + 0(W2) \ • (11) wnere /L X(t) = R cosfit R cos q = R1R2 o ' l " 2 and - (RJ2 i T*2 ^T> •> —-. — J. \*'* (9) The term in fiL contributes to the Sagnac effect and o does not evolve with time. Our conventions imply that in the op- tical measurement of L(t)'or 5L(t) the light path starts off at site (1) being reflected back or absorbed at (2). 4>0 .is the angular separation (^"^oi* at rest *no rotation) be~ tween points (1) and (2) as measured in SQ (E-LET) or in S(S-LCT] As most experimental situations involve optical paths whose CBPF-NF-046/86 -14- ends are equidistant from the center of rotation, we re-write eq.- (11) for the special car,e R^ = R_ = R: 6L(t)/L = v cos ($ /2) - vV cos ($rt/2) sin { + J2 +C±) 0 0 O 2Qt) 2 2 2 •f ($V /2)sin (<(>0/2 +fit) +0(W ) (12) 5.2 Doppler shift measurements The second method of SR violation detection we discuss is based on Doppler shift technology with an emitter and an .ab- sorber as endpoints of the rotating optical path. Let us con si. der three inertial frames of reference S,S' ano S" which are, respectively, co-moving with the laboratory (S) and instantane ously co-moving with emitter (S1) and absorber IS"), all of them using Einstein coordinates. Then, according to either SR or LET in Einstein Coordinates, phase invariance of the Elec- tromagnetic radiation allows us t? write -Jc'.r») - w"(t" -Jc".r") In particular, for the emitter (e)at any instant we have u(t-k.R (t)) = u'(t' -k'.R e e CBPF-NF-046/86 -lb- Differentiation with respect to t gives -k.ve(t)) where v (t) = R (t), and since f =Ye(t-Ve(t).Re(t)) df/dt = Y.d-v" we find ÍÜ« = wv (l -k\v (t)) (13-a) where k is the unit vector in the direction of propagation as observed in S (Einstein Coordinates). Analogously, for the ab sorber (a) we may write w" = wYavI -Jc.va(t)) . (13-b) Now let v, v and v be the emitted light frequency (Einstein time) as measured respectively in S, S1 and S". Then, from eqs. (13) in either SR or LET we have (1 -i^.v (t))l = v [Y(Y (dl -k\v-Je.v (t))(t))l l . (14) e aaLa a aa J Contrary to a common belief, equation (14) does not neceis sarily imply, SR or LET. It is a consequence only of postulate CBPF-NF-046/66 -16- - . (i) (in section 2) and of the use of Einstein coordinates, being thus valid for any absolute Space-time theory. It is only when expressions (8) and (9) are used with V ^ 0 (V = 0) that LET (SR) will be imposed. In Doppler type experiments the observed quantity is the frequency shift 6v/v0 = (va-vo)/vo which, from eq. (14), is readily found to be 3 3 6v/vQ =ic. (ve(t) -va(t+6t)) +0(v ) - -L(t) +0(v ) (15) where v represents the magnitude of tangential velocities(< 10~6 typically) and 6t=L.is the emitter to absorber light transit time as before. Time differentiation of L(t) in eq. (11) gives the LET prediction 6v/v0 = V(vave/ftLo)sin *oi 2 2 - AV (v av e/ÍÍLjsin o * osin U o+ 2flt) O(v2V2) (16) where R and R replace R, and R, in the definitions of L arid e a . i i o X(t). In the equal radii case R • R = R, eq. (16) simplifies to * C ft CBPF-NF-046/86 -17- ív/v = v2Vsin* cosi* /2 +ílt) O 0 0 2 - ÀvV sin +2I2t) + O(v2V2) (17) 6 COMMENTS AND CONCLUSIONS Analysing the then current status of experimental verifica tions of Special Relativity, Mansouri and Sexl^5' argue that, contrary to the general feeling of a high degree of accuracy beLtfeen predictions and measurements, there is still room left for speculation on possible competing theories*1'3' . Indeed it is difficult to express this accuracy in specific numbers, unless rival theories are defined and compared with SR. This has been the subject of our previous sections where LET has been conpared with ar.d dif- > ferentiated from SR for experiments involving the rotation of rigid bodies. LET has already been the object of previous studies and many dilutes in the literature (see, for example, references 5. vs 10, 11 vs 7, 3 vs 7). One of the reasons for these confrontations is the imprecise definition of the specific LET in question. For instance, Moller's work of 1957 on the experimental distinction between SR and the pre-relátivistic lenght contraction and time dilation Newton-Fresnel ether theory is frequently analysed in connection with LET, many wrong con- clusions ensuing. Besides, several papers dealing with LET do CBPF-NF-046/86 Tia-. not state clearly what the postulates of the theory in question are. We have made it clear that LET is more like a. class of theories, rather than one specific theory, of which S-LET and E-LET are examples. Both have been clearly defined, and predic tions of measurable effects that may lead to their distinction from SR and from one another have been derived. Equations (11) and (16) are the central results from which experimentally measu table quantities of interest can be easily obtained. Prior to any experimental analysis, a serious objection to E-LET may be placed here on theoretical grounds. The V-indepen dent term in eq. (11) for 6L is such that Um 6L/L + 5L/L 1 =0 *•» ° °Jv.o This discontinuous character iis highly undesirable of a phy- sical theory, and constitutes strong theoretical evidence against E-LET in our view. CBPF-NF-046/86 -19- REFERENCES (1) A.K.A. Maciel and J. Tiorano, Phys. Rev. Lett. 55, 143, (1985) (2) S. Marinov, Czech J. Phys. 24_, 965 (1974) and Gen. Rel. Grav. 12, 57 (1980) (3) W.A. Rodrigues and J. Tiomno, Found. Phys. 15, 945 (1985) (4) J.. Tiomno, Proc. 4— Marcel Grossman Meeting on Gen. Rel., Rome (1985), North Holland Pub. Co. (in press). Note that in this reference a misprint in the second member of Eq. (1) omitted a term - v2/2 which was actually detected by Ives and Stilwell (5) R. Mansouri and R. Sexl, Gen. Rel. Grav. 8, 497 (1977), H.P. Robertson, Rev. Mod. Phys. 2_1, 378 (1949), J.G. Vargas, Found. Phys. 1±, 625 (1984). (6) R. Mansouri and R. Sexl, Gen. Rel. Grav. 8, 515 (1977) (7) D.G.Tbrr and P.Kolen, Found. Phys. ^2, 265 (1982) and Found. Phys. 12/ 401 (1982). (8) K.R. Atkins, Phys. Lett. 30A, 243 (1980) (9) ^. Sagnac, C.R. Acad. Sci. Paris, 157, 1410 (1913) (10) A.A. Tyapkin, Lett. Nuovo Cim. 7, 760, (1973) (11) S.J. Prokhovnik,.Found. Phys. 10, 197 (1980) (12) C. Moller, Suppl. Nuovo Cim. 6, 381 (1957)