LETTERE AL NUOVO CII$IENTO VOL. 9, N. 12 23 Matzo 1974

K i n e m a t i c s o f a ~ R i g i d ~) R o t o r .

T. E. PHIPPS jr.

Naval Ordnance Laborato'ry - White Oak (Silver Spring), Md.

(ricevuto il 28 Agosto 1973)

Several papers (1-4) have suggested t h a t the Minkowski-spaec metr ic of a ro ta t ing disk m a y be nons ta t ic as well as non-Euc l idean ; i.e. t ha t the metr ica l curva ture general ly associated (5) with a con t inuous ly accelerated reference system m a y not be res t r ic ted to the spat ia l dimensions. For example WEINSTEIS (4), by a radia l in tegra t ion of Thomas precessions assumed to occur locally wi th in such a disk, infers a lag t ime of the ma te r i a l a t rad ius r of av~'(r)/c 2 per ro ta t iona l per iod (to second order), where v{r) is the disk speed at rad ius r, re la t ive to the l abora to ry in which the disk center is a t rest , c is l ight speed, and a is a cons tan t e q u a l i n his theory to 1/6. Such a radia l dependence of angular velocity implies a progressive re t rograde curving of an in i t ia l ly s t raight radia l l ine marked on the disk surface, as observed in the l abora to ry dur ing rota t ion. The s tretching of this l ine (which m a y coincide wi th a metr ic s tandard) implies a nons ta t ic metr ic of the ro ta t ing body. F rom hypothes ized group proper t ies of angular veloci ty TAKENO (3) infers a s imilar resul t wi th a : �89 and ROSEN (1) and HILL (2) by different conside- ra t ions deduce a : �89 Most au thor i t ies (5) favor ~ : 0.

Latitud(~' exists for such conflicting theoret ical possibil i t ies because the (~ r igidi ty ~ of a metr ic s tandard , or of an y ex tended s t ructure such as a disk, has never found a consis tent (6) relat ivis t ic definit ion as a pure ly k inemat ic a t t r ibute . Rigidi ty has there- fore general ly been conceived (5) as a (nonexistent) physical proper ty . However, the consequent dismissal of r ig id i ty as an (~ impermiss ible ideal iza t ion ~ m a y be difficult to reconcile wi th cer ta in physical observat ions, e.g., the Moessbauer effect. Here the lat t ice <~ as a whole ~ recoils in appa ren t ly nonloeal ized rad ia t ion react ion, simul- t aneous ly in l abora tory t ime, hence wi th an appearance of (~ r igidi ty ~). Similar ly in Compton pho ton scat ter ing (7) from high-Z a toms thc mass of the a tom (~ as a whole ~> is effective, so t ha t the a tom seems to recoil r igidly in the Ncwtonian sense (of (( act ion a t a distance ~)). Such observat ions, be ing of an essent ia l ly k inemat ic na ture , create

(1) N. ROSEN: Phys. Rev., 71, 54 (1947). (2) E. HILL: Phys. Rev., 71, 318 (1947). (~) H. TAKFNO: Progr. Theor. Phys., 7, 367 (1952). (~) D. H. WEI,~STEIN: Nature, 232, 548 (1971). (t) l~. ARZELII~S: Relativistic Kinematics (New York, 1966). (e) G. CAVALLERI: Nttovo Cimento, S3A, 415 (1968). (7) For th i s example the wr i te r is i ndeb ted to Dr. 11. G. NEWBURGH.

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4 6 8 T . E . PHIPPS jr.

a difficulty for existing kinematic theory. In these circumstances it is prudent to gather all relevant observational evidence.

The experiment suggested by WEINSTEIN (4) has therefore been performed. A 1.35 cm diameter 17-4 PH stainless steel disk was spun continuously in air for approximately four months at about 6072 r.p.s. (r.m.s. speed) by means of a small air bearing air tur- bine. During rotation 20 ns laser flash photographs were taken of the disk surface, on which various radial lines had been scribed. Ruby lasers were used having peak powers of roughly 7 MW (dye Q-switch) and 100 ~ W {frustrated total in ternal reflection Q-switch). Figure 1 shows a photograph taken with the lower-powered laser when the

Fig. 1. - Disk stationary before experiment.

disk was at rest before the experiment. Figure 2, taken with the other laser, shows the disk in counterclockwise rotation at about 6116 r.p.s, near the end of the run. Despite surface contamination by airbone dirt, the radial lines are easily discerned in Fig. 2. A dashed curve has been superposed on this Figure to indicate the lagging curvature tha t would have been observed if an effect of magnitude ~ = �89 had been present. Analysis of the corresponding lines marked with arrows in these Figures yields an upper bound to the effect constant, 1~1~ 6-10-4. It can be inferred that there is a null effect of line curvature to order v2/c 2, and presumably to all orders in v/c. I t seems unlikely tha t this simple result depends on experimental details such as disk material or manner of producing the motion.

In addition to nonstatic metric hypotheses, these observations rule out certain optical effects otherwise conceivably characteristic of the aberration of light from a continuously accelerated source. Specifically, formal application to the k-vector of light propagation

K I N E M A T I C S O F A (( R I G I D >) R O T O R 469

F i g . 2. - D i s k i n m o t i o n n e a r e n d o f r u n .

(from any fixed point on the disk surface) of the same kinematics (s) responsible for the Thomas precession of spin vectors leads to a predicted slow turning of the k-vector as viewed in the laboratory. The resulting second-order aberration imparts to the disk surface on appearance of rim lag identical (and additive) to tha t associated with the kine- matic effects discussed above, with ~ = �89 The Thomas precession of co-ordinate axes is a purely logical consequence of the group structure of the Lorentz transformation. But the Thomas precession of physical vectors is an independent, additional physical hypothesis. Evidence weighing against the latter now includes

a) the present nonobservation of an optical analogue of the Thomas effect;

b) doubts raised by WmT~II~E (9) Concerning the validity of the Thomas effect for vectors other than electron spins;

c) the remark by FISKER (lo) tha t spectroscopic evidence for the Thomas preces- sion of electron spins is accounted for by the Dirae equation without geometrical (vector turning) hypotheses;

d) the fact that the Thomas precession of physical vectors is the only know single- particle effect that violates Newton's laws in the Newtonian limit (of slow rotation for a very long time), whereby it alone prevents Newtonian particle mechanics from being contained as a limiting case within relativistic mechanics;

(s) C. ~[OLLER: The Theory o/ Relativity ( O x f o r d , 1952) . (g) D . P . W H I T M m E : Nature, 239 , 207 (1972) . (lo) G. P . FISHER: Am. Journ. Phys., 40, 1772 (1972) .

470 T.E. PmPPS jr.

e) the fact tha t the Thomas precession of angular-momentum vectors violates l~ach's principle, according to which the fixed stars determine local inertial properties, presumably including angular momentum;

f) the circumstance that the energy associated with the Thomas precession is physically unique in arising from torque-free causes in flat space.

Present evidence, if otherwise unsupported, would be highly inconclusive, since it admits of alternative interpretations. However, in conjunction with the other points noted above it appears to justify further investigation. Consequently, additional experimentation is planned with the object of testing directly the geometrical aspect of the Thomas precession of electron spins.

Figure 2 exhibits a (~ rigid body ,> (to order v2/c 2) and thus provides further evidence that rigidity is not always a physically impermissible idealization. The fact tha t straight lines on the disk surface remain straight on hyperplanes of constant laboratory time is consistent with both the classical and the Born (xl) definition of rigidity. The latter, in conjunction with Einstein's kinematics, is known to contradict experience by pre- dicting the existence of too few degrees of freedom of the Born rigid body (12). This may reflect a parametric deficiency of the accepted kinematic theory. If we postulate 1) tha t the number of degrees of freedom of a body is a kinematic (relative motional) invariant, 2) tha t Newtonian rigid-body kinematics is contained as a limiting case within relativistic kinematics (this postulate implies defining a rigid body kinematically as one lacking in internal degrees of freedom), then the need to bring extra parameters into the formalism become logically unavoidable. Einstein's parameterization, which employs a universal homogeneous Lorentz transformation, involves transforming all particles of the structure by a collective 4-rotation about a shared space-time co-ordinate origin. The resulting coalescence of individual-particle origins in a single , preferred )> world point has an invariant descriptive significance (and an effect on worldline shapes) that appears physically questionable, since the only legitimate invariants of the theory are event-related. (Because no two particles of the body share an event, none should share any invariant for arbitrary motions.) This suggests tha t a) the logical difficulty in rigid-body kinematics results from over-reliance on the 6-parameter homogeneous Lorentz group, b) the needed extra parameters can be introduced by use of the full (inhomogeneous, 10-parameter) Lorentz group--seemingly the last unexploited source of parameters compatible with the known validity of Einstein's single-particle kinematics. This tentative suggestion is offered in the hope of stimulating a wider awareness of the existence and nature of the extended-structure kinematic problem.

WEINSTEIN (I~) has done a similar disk experiment with the same null result.

(11) l~I. BORN: Ann. Phys., 30, 8~0 (1909). (1~) W. PAULI: Theory o] Relativity (New York, 1958). (i~) D. H. WEINSTEIN: private communication.