Conformal transformations and maximal acceleration

IL NUOVO CIMENTO VOL. 104 B, N. 6 Dicembre 1989

Conformal Transformations and Maximal Acceleration (*).

W. R. WOOD, G. PAPINI and Y. Q. CAI

Department of Physics, University of Regina - Regina, Saskatchewan, $4S OA2, Canada

(ricevuto il 12 Luglio 1989)

S u m m a r y . - - The restricted conformal transformation to a uniformly accelerating frame is reviewed and a physical interpretation of the singularities of the transformation is given. By applying the transformation to an extended object, a limit on the four-acceleration is shown to be necessary in order that the tenets of special relativity not be violated. This result is consistent with other derivations of maximal acceleration, one of which is equivalent to a conformal transformation of the classical line element. The relationship between the restricted conformal transformation and the Rindler coordinates of a family of uniformly accelerating observers is also given.

PACS 03.65 - Quantum theory; quantum mechanics. PACS 02.20 - Group theory. PACS 11.30.Ly - Other internal and higher symmetries.

1 . - I n t r o d u c t i o n .

Conformal t ransformations have a relat ively long history and have been throughly analyzed by Fulton, Rohrlich and Wit ten (1). Of part icular in teres t here is the less often cited paper (2) in which these authors studied the special case of the t ransformation to a uniformly accelerated frame. The singularities of this transformation, identified in ref. (2), are here given a physical interpretat ion.

Real in teres t in uniform acceleration seems to have been t r iggered several years later by Rindler 's example (3) of a uniformly accelerating rod. Recently, the

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) T. FULTON, F. ROHRLICH and L. WITTEN: Rev. Mod. Phys., 34, 442 (1962). (2) T. FULTON, F. ROI-IRLICH and L. W!TTEN: Nuovo Cimento, 26, 652 (1962). (3) W. RINDLER: Am. J. Phys., 34, 1174 (1966).

653

654 W.R. WOOD, G. PAPINI and Y. Q. CAI

Rindler example has been used (4) to illustrate some consequences of the novel concept of maximal acceleration that was derived by considering an eight- dimensional line element which reverts to the Minkowski line element when h--* 0.

For the acceleration of a massive particle such as an electron to be bounded, a necessary condition is that the particle be treated as having a finite extension (5), i.e. there appears to be no reason why a pointlike particle cannot undergo an infinite acceleration. The purpose of this work is to show that the maximal acceleration of an extended particle follows naturally from the theory of conformal transformations. The formalism in ref. (2) will provide the basis for the uniform acceleration transformation discussed in sect. 2. The effect of this transformation on an extended object (sect. 3) will provide the link with maximal acceleration and the Rindler example (sect. 4). Possible extensions of this work are mentioned in the conclusions.

2. - The c o n f o r m a l t r a n s f o r m a t i o n to a u n i f o r m l y acce l era ted f r a m e .

A transformation can be envisaged as being active, where it is the points themselves that are mapped from one domain of an observer to another domain of the same observer, or passive, where it is the components of the points as observed in one frame that are transformed to components of the same points as observed in a different frame. The active conformal transformation (1) from the point x to the point ~ is characterized by the property that the transformed line element is scaled:

(1) d~2(u = z(s) dz2(x),

where z(x) is an arbitrary positive differentiable function of x. Due to the definition

(2) dz2(x) = - g~(x) dx ~ dx y ,

it follows that the metric transforms as

~-2~ ~u _ ~(x)g~(x) (3) g,~(u ~x ~ ~x ~ �9

(4) E. R. CAIANIELLO, A. FEOLI, M. GASPERINI and G. SCARPETTA: Quantum corrections to the spacetime metric from geometric phase space quantization, preprint (1988). (~) E. R. CAIANIELLO: Lett. Nuovo Cimento, 32, 65 (1981); E. R. CAIANIELLO, S. DE FILIPPO, G. MARMO and G. VILASI: Lett. Nuovo Cimento, 34, 112 (1982); E. R. CAIANIELLO: Lett. Nuovo Cimento, 41, 370 (1984).

CONFORMAL TRANSFORMATIONS AND MAXIMAL ACCELERATION 655

The conformal transformations of the metric,

1 (4) gC,(x) = z (x ) g,,,(x) and g C ( x ) -- - ~ g~' (x) ,

form a group Cg with the set of all manifolds isomorphic under Cg constituting a conformal space. The group of all coordinate transformations together with C~ form a larger group C, of which the former are proper subgroups. The space in which equations are invariant under C is called a Weyl space. The subgroup of C defined by those transformations that transform flat space (R = 0) into flat space (R ~ -- 0) is called the restricted conformal group Co. This restricted group is a 15- parameter Lie group: four parameters for space-time translations; six for homogeneous Lorentz transformations; one for dilatations; and four for acceleration transformations, as defined below. The restricted p r o p e r conformal group C~ consists of the transformations of the restricted group with the additional condition that the direction of time not be reversed.

Under a passive or coordinate transformation S-~ S' with x ~ ~ x ' ' , the metric and the line element transform according to

~x ~ ~x ,~ (5) g'~(x') - ~ x '~ ~ x '~ g ~ ( x )

and

(6) dr'2(x ') = dr2(x).

A c o n f o r m a l coordinate transformation that scales the line element as in eq. (1) is defined as that coordinate transformation which gives the same coordinates of the actively transformed point u in S' as x has in S. Under this particular frame- dependent transformation (~),

(7) g~,~(x') = ~(x) ~x~ ~xz ~ x ',~ 8 x ''~ g ~ ( x ) -- cr(x)g'~(x')

and

(8) c ! t u r ,~ dz~ = - g~.,(x ) d x d x = ~(x) d~ 2.

The conformal coordinate transformations of interest in this work are those based on the restricted proper group C~ from Minkowski space g~,~(x)= rj,~ to Minkowski space g C,~(x') = ~ ..... with r,~v = ( - 1, 1, 1, 1). The particular coordinate transformation that will be considered is the ,,acceleration transformation, effected by the inversion-translation-inversion:

x~ _ a , x 2 ( 9 ) x ' ' -

l - 2x'~a~ + a 2 x 2 '

656 w . R . WOOD, G. PAPINI and Y. Q. CAI

where x 2 = x , x , , a2=a.~a~ and a ~" is a constant four-vector. Under this transformation the metric becomes

(10) g'~(x') = ~2(x) ~ ,

where

(11) ~(x) = 1 - 2a'x~ + x2a 2 .

The corresponding conformal coordinate transformation to flat space, wherein ~,S' is a frame in a Weyl space, uniformly accelerating with respect to an inertial f rame, (interpretation B, in ref. (2)), is obtained by choosing

(12) z(x) -- ~-2(x).

The inverse coordinate transformation from S' to S is given by

(13) x ' - X " + a,~x '2

1 + 2a~x,~ + a2x '2"

Due to the nonlinear nature of transformations (9) and (13), in contradistinction to Lorentz transformations, the transformation of objects of finite size is nontrivial. Also, unlike the case for two inertial frames, the scale factor must be determined if a measurement in one frame is to be compared with a measurement in the other frame.

Under the coordinate transformation (9), fur ther restricted to motion along the z-axis, the frame S' experience a uniform acceleration

(14) a . ~ = ( O , O , o , l = c o n s t )

with respect to the frame S. An observer at rest at the origin in the S' frame will determine the coordinates of a particle at rest in the S frame with coordinates x ,~ = (T, 0, 0, z) and rest mass m0 to be

a ( 1 - ~ ) - z (15) ~, = Z x' = y' = 0, z' ~' ~ '

where

(16)

As in ref. (2), the transformation from (z, z) to (r', z') is singular along the lines

(17) r = + (z - ~).


The physical significance of this can be seen as follows. Consider the family of world-lines parallel to the r-axis in the (z, .-)-plane defined by

where s is an arbi t rary nonzero parameter . The world lines (18) intersect the singular lines (17) at .-= +(as/2). This family of world-lines, represent ing particles at res t in the S-frame, are mapped into a family of hyperbolae in the (z', .-')-plane defined by

(19) z' + a ~ s2

with asymptotes of slopes + 1 and vertices at z" = - ~ and z' -- ~ ( 2 / s - 1). The observer determines the velocity of the particle to be

(20) v' - dz__~' _ + r' _ + sr' d.-' ' '

where

1 s z ' (21) ~ ' ( s ) - - ~- s - 1 .

~ /1 - V '2

In terms of the unprimed coordinates, eq. (20) becomes

(22) v' - +_ ~xSv

~ 2 s 2 / 4 + _2, ,

which has a maximum of I v'l = 1 at the intersection points ~= +(~s/2). Consequently, the lines in eq. (17) represent events in the (z, ~)-plane for which the observer in S' determines that the magnitude of the massive particle's three- velocity is c, a situation contrary to the tenets of special relativity. This provides a physical interpretat ion of the singularities noted in ref. (2). Since a particle's world-line cannot cross the singular lines, it is natural to ask which of the disconnected regions may be considered as being physically meaningful under this transformation.

The portions of the world lines (18) lying in the left and right wedges given by ~2< (~s/2)2 with z X ~ are mapped into the hyperbolae lying in the regions for which =,2< (z '+ ~)2 with vertices at z ' < > - a . Points in these left and right wedges are t ransformed by eqs. (15) and (16) such that r' and .- have the same sign, i . e . these regions correspond to proper transformations. In contrast, the branches of the hyperbolae (19) lying in the regions _,2 > (z' + ~)2 with vertices at z ' = - ~ , which are (discontinuously) mapped from the segments of the lines in

658 w . R . WOOD, G. PAPINI and Y. Q. CAI

eq. (18) with ~2 > (as/2)2 correspond to improper transformations since the signs of ~' and z are opposite to one another. As a result, the regions defined by ~2>~ (as/2)2 will be considered inaccessible to the transformation (15) and (16); only the transformation of those portions of world-lines in the (z, r)-plane within the regions ~2< (as/2)2 will be deemed physically meaningful.

Before concluding this discussion on the acceleration transformation, one last observation needs to be made. As a--)0, i.e. as the magnitude of the uniform acceleration approaches infinity, the hyperbolae in the (z', z')-plane collapse to the lines ~' = + z' for which the observed velocity of the particle is again the unphysical value of the speed of light. Clearly, while the acceleration can approach infinity, physical principles prohibit a truly infinite acceleration. In the following sections, a limiting value for the acceleration will be considered.

3. - T h e a c c e l e r a t i o n t r a n s f o r m a t i o n f o r e x t e n d e d o b j e c t s .

In order that the transformation of an extended object be considered physically meaningful, one requires that the object maintain a rigid shape under the transformation, i.e. that the proper length of each of its infinitesimal elements dl be preserved (6). To see if such is the case for transformation (9), consider a rod of finite length at rest in the S frame and lying along the z-axis. Treat the rod as being composed of a series of points, the world-lines of which are parametrized by s, with the condition that no point of the rod coincide with the singularity at s = 0. The observer at rest in the S' frame will observe a uniformly accelerating rigid rod only if each of the infinitesimal observed elements of the dl' are contracted by the appropriate instantaneous Lorentz factor.

By treating the description of the motion within the conformal coordinate transformation interpretation, the observer in S' must take the scale factor used in the conformal transformation to Minkowski space into account; at a fixed time ~', the infinitesimal length that the observer should compare is dl'--dz' /~' . In order to determine the ratio dl/dl', the inverse transformation (13) must be used to determine ~'(3z/3z')lT,. From eq. (13),

~(~' - 1) - z' (23) z

where, using eq. (19),

(24) ~, = 2 (z' + a). a8

(~) W. RINDLER: Introduction to Special Relativity (Oxford University Press, London, 1982).

C O N F O R M A L T R A N S F O R M A T I O N S A N D M A X I M A L A C C E L E R A T I O N 659

Taking the partial derivative of eq. (23) with respect to z', one finds

~ ,3z 1 ( 2r'2~ (25) " ~z' :,= ~-- ~' + ~---~]

and, using eqs. (19) and (24), this becomes

:, az = sz___~' + s - 1. (26) ~ ~z ~,

Hence, by eq. (21),

dl (27) ~-~ = ~/(s)

so that, for each value of the parameter s along the rod the observer in S' determines, by explicitly compensating for ~' in his measurement procedure, that each infinitesimal element of the rod is in fact contracted by the correct Lorentz factor. Hence, the acceleration transformation (9) can be applied to extended objects in a physically meaningful way. This is of relevance in the present work because fundamental particles, e.g., electrons, will be treated as having a finite extension.

Over the years there have been numerous attempts, based on different approaches and for varying reasons, to treat particles not as mathematical points but rather as objects with some finite extension. While many prominent researchers have developed theories for extended particles (e.g., Lorentz(7), Dirac(S), etc.) a consensus to what a correct description might be has not yet been realized.

A detailed model of the structure of extended particles will not be given here. It will simply be assumed that, on the basis of the uncertainty principle, a ,~quantum, particle cannot be treated as a mathematical point in an operational sense (9.10), but must have some finite extension )~. A quantum particle would not be represented on a space-time diagram by a world-line, but rather by a world ,,tube, of radius ~/2, with the axis of the tube coinciding with the world-line of the corresponding classical pointlike particle. By treating quantum (i. e. physical) particles as having a definite extension, the maximal value of the acceleration is subsequently restricted.

(7) H.A. LORENTZ: The Theory of Electrons, 2nd ed. (Dover, New York, N.Y., 1952). (s) p. A. M. DIRAC: Proc. R. Soc. London, Sez. A, 268, 57 (1962). (9) E. PRUGOVECKI: Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht, 1984). (10) M. TOLLER: Nuovo Cimento B, 40, 27, 1977; 102, 261 (1988).

45 - II Nuovo Cimento B.

660 W . R . WOOD, G. PAPINI and Y. Q. CAI

4. - Maximal accelerat ion of quantum particles.

Consider a quantum particle at rest at the origin in the S frame, such that the axis of its world tube is coincident with the r-axis. By requiring that the singular point z = a does not lie within the world tube, a maximal limit of the uniform four-acceleration (14) results at ami n = ~/2:

(28) lalmax - 1 _ 2 . ~min

As the direction of the z-axis is arbitrary (more generally one would consider a plane orthogonal to the r-axis), the coincidence of the world tube and the r-axis corresponds to an upper limit on the acceleration that is independent of the orientation of the frame.

Based on a systematic analysis of transformations under the Poincar6 group, Toller (10) has suggested that, due to certain ~,geometric limitations~>, a maximal acceleration of the form (28) should exist. By setting ~ equal to the Compton wavelength of the particle, Caianiello's maximal acceleration (5)

moC (29) lalm~x = 2

h

is recovered(*). Prugove~ki's theory of relativistic quantum mechanics (9) has been shown (11) to support eq. (29) as well. The element common to each of these derivations of a maximal acceleration is the requirement that quantum particles have some finite extension. This relation between a minimal extension and a maximal acceleration is also in keeping with Misner, Thorne and Wheeler's discussion (12) on the constraints of the size of an accelerated frame.

The uniform acceleration defined by eq. (19) can be recast into the commonly used Rindler coordinates (1~) as follows. The s = 1 world-line

(30) z,2 _ r,2 = a2

is used as the generic equation for a family of observers by allowing the proper

(*) Caianiello's result actually differs from that given above by a factor of c 2. This difference arises because Caianiello calculates the maximum of the three-acceleration d 2 xi/dt 2. (11) W. R. WOOD, G. PAPINI and Y. Q. CAI: Maximal acceleration and Prugove~ki's stochastic quantum mechanics, in Proceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics (World Scientific, Singapore, in press). (12) C. W. MISNER, K. S. THORNE and J. A. WHEELER: Gravitation (W. H. Freeman and Company, New York, N.Y., 1973). (13) N. D. BIRRELL and P. C. W. DAVIES: Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982).

C O N F O R M A L T R A N S F O R M A T I O N S A N D M A X I M A L A C C E L E R A T I O N 661

acceleration a to vary from observer to observer. The Rindler coordinates (r,, ~) which cover the right wedge z' > 17.'1 are given by

(31) -:' =a-lexp[a~]sinhav, z' =a- lexp|a~]coshav,

where a = const > 0, - ~ < 7, ~ < ~ and ~ = a -1 exp [a~]. The lines of constant comprise a family of hyperbolae representing uniformly accelerating observers while lines of constant Rindler time r, are rays emanating from the origin. The existence of a maximal acceleration causes the event horizons at the null rays r, -- + ~ corresponding to an infinite acceleration hyperbola to be shifted to the maximal acceleration hyperbola

2 ( 3 2 ) z '2 - 7 . ' 2 - - 0:mi n .

This shift may be visualized more clearly by generalizing the (1 + 1)-dimensional Rindler space to a (2 + 1)-dimensional cylindrical Rindler space (see, e.g. (14)). The constant r~ rays are replaced by cones and the family of constant 5 hyperbolae become a family of single-sheeted hyperboloids, the left and right wedges no longer being causally disconnected. A uniformly accelerating pointlike particle would have an event horizon given by the asymptotes of its motion, i.e. by the null cones. In order that no region of an extended particle violate the null cone horizon, the axis of its world tube must be bounded by the maximal acceleration hyperboloid for which the circle of vertices corresponding to points where the particle is instantaneously at rest has a radius equal to the radius of the particle.

The consequences of a maximal acceleration in Rindler space have been considered by Caianiello et al. (4). In Caianiello's theory, maximal acceleration results when the incorporation of quantum mechanics leads to the replacement of the classical line element ds 2= dx~dx~ by the generalized line element

(33) h 2 o~ o ~_ ,

dT. 2 = dx ~ dx,~ + ~ dx , dx,~ ~ 0 moc

where ~ = dx~/ds. A quantum particle existing in a four-dimensional physical space-time that is embedded in eight-dimensional phase space has a line element of the form

(34) m02 C 2

which can be interpreted as a conformal transformation of the classical line

(14) I. H. REDMOUNT and S. TAKAGI: Phys. Rev. D, 37, 1443 (1988).

662 W . R . WOOD, G. PAPINI and Y. Q. CAI

element(15). Caianiello has suggested that the significance of this particular conformal transformation is that it provides a mechanism by which particles of different species, e .g . , leptons and quarks, experience different geometries, which may account for phenomena such as quark confinement. For the case of uniform acceleration, the Rindler line element is generalized from

(35) d s ~ = d X ~ _ X 2 d T 2 ,

where X = a - l e x p [ a $ ] and T = a r l , to

(36) dr 2=dX 2 - ( X 2 h 2 ~dT 2

which yields the shift in the horizon discussed above. In the Rindler example, this shift is significant as it leads to a discrete energy spectrum for uniformly accelerating quantum particles as well as a physical singularity at the horizon.

5 . - C o n c l u s i o n s .

The singularities of the restricted conformal transformation have been shown to represent the unphysical situation where a massive particle is observed to be moving with the speed of light. This, together with the requirement that only proper transformations be allowed, has provided the basis with which to consider the transformation of extended objects.

Based on the premise that a physical particle satisfying the basic principles of quantum mechanics cannot be treated as a mathematical point, a limit on the uniform acceleration parameters a~ of the restricted proper conformal group has been established. This maximal uniform acceleration follows from the requirement that no point in the interior region of the quantum particle world tube be permitted to contravene the tenets of special relativity. The role that the extended nature of quantum particles plays in the Rindler example, as well as the conformal nature of Caianiello's generalized line element, have been discussed.

There are several aspects of the present work that require further investigation. The more general case in which the acceleration of quantum particles results from transformations of the extended group C should be considered. The consequences of a maximal acceleration for the case that the quantum particle is charged, e .g . , for the radiation rate, should also be studied. A detailed model of the structure of extended particles on which to base such

(15) E. R. CAIANIELLO, M. GASPERINI, E. PREDAZZI and G. SCARPETTA: P h y s . Le t t .

A, 132, 82 (1988).


studies is needed as well. With regard to such a model, it is general ly accepted (16) that the gravitational and electromagnetic fields alone are not sufficient to support a field-induced s t ructure of an extended e lementary particle. In light of this, the Weyl-Dirac theory developed by Papini (17) and his co-workers, which contains a scalar field as well, may p~ovide an excellent f ramework in which to extend this work.

We are indebted to Dr. H. E. Brandt for discussing with us his work on the maximum possible proper acceleration relat ive to the vacuum (18). Unlike the results of ref. (5,10,11) and the present paper, his maximal acceleration is particle- independent.

This research was supported by the Natural Sciences and Engineer ing Research Council of Canada.

(16) F. I. COOPERSTOCK and N. ROSEN: Int. J. Theor. Phys., 28, 423 (1989). (17) D. GREGORASH and G. PAFINI: Nuovo Cimento B, 63, 487 (1981); G. PAPINI: Gravitation and electromagnetism, covariant theories a la Dirac, in Proceedings of the 20th Orbis Scientiae Conference dedicated to P. A. M. D1RAC'S 80th year, High Energy Physics, edited by B. KURSUNOGLU, S. L. MINTZ and A. PERLMUTTER (Plenum Publishing Corp., New York, N.Y., 1985); G. PAPINI: Conformally invariant gravitation and electromagnetism, in Procedings of the Sir Arthur Eddington Centenary Symposium, Vol. I, Relativistic Astrophysics and Cosmology, edited by V. DE SABBATA and T. M. KARADE (World Scientific, Singapore, 1984); G. PAPINI: The Weyl-Dirac equation from conformally invariant gravitation and electromagnetism, in Proceedings of the 4th Marcel Grossman Meeting on General Relativity, University of Rome, 17-21 June (North-Holland, Amsterdam, 1986); G. PAPINI: Quantum mechanics and the geometry of spacetime, in Proceedings of the 5th Marcel Grossman Meeting on General Relativity, Perth, Australia, 8-13 August (World Scientific, Singapore, 1988). (xs) g. E. BRANDT: Lett. Nuovo Cimento, 38, 522 (1983); 39, 192 (1984); Found. Phys. Lett., 2, 39 (1989).

�9 R I A S S U N T O (*)

Si rivede la trasformazione conforme ristretta rispetto ad una struttura con accelerazione uniforme e si da un'interpretazione fisica delle singolarita della trasformazione. Applican- do la trasformazione ad un oggetto esteso si mostra che un limite sulla quadriaccelerazione

necessario per non violare i principi della relativita speciale. Questo risultato ~ coerente con altre derivazioni di accelerazione massima, una delle quali ~ equivalente a una trasformazione conforme dell'elemento di linea classico. Si da anche una relazione tra la trasformazione conforme ristretta e le coordinate di Rindler di una famiglia di osservatori in accelerazione uniforrne.

(*) Traduzione a cura della Redazione.

Pe31oMe He llO./IyqeHo.

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Conformal transformations and maximal acceleration