- Email: [email protected]
The hypothesis of locality in relativistic physics
Volume 145, number 4
PHYSICS LETTERS A
9 April 1990
THE HYPOTHESIS OF LOCALITY IN RELATIVISTIC PHYSICS Bahram MASHHOON Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, MO 65211, USA Received 29 December 1989; accepted for publication 13 February 1990 Communicatedby J.P. Vigier
The standard extension of any Lorentz-invariant theory to arbitrary frames of reference as well as gravitational fields is based on the assumption that an observer is momentarily equivalent to a comoving inertial observer. This hypothesis of locality is indispensable to the interpretation ofthe results of experiments in physics. The origin, significance, and limitations ofthe hypothesis of locality are discussed.
1. Introduction This paper is about a basic question in the theory of measurement: Which law ofphysics specifies what accelerated observers measure? It has been proposed that the hypothesis of locality i.e., the presumed equivalence of an accelerated observer with a momentarily comoving inertial observer underlies the standard relativistic formalism by relating the measurements of an accelerated observer to those of an inertial observer. This fundamental assumption therefore replaces the customary hypotheses concerning classical measuring devices in accelerated motion. In particular, a physical basis can be provided for the discussion of the clock hypothesis. To see how this comes about, one must begin with classical mechanics. In Newtonian mechanics, one imagines that a complex system is composed of material particles which are represented by mathematical points. The state of a system of particles is characterized at each instant by the positions and velocities of the particles involved. The forces, whether true or fictitious, only affect accelerations at any moment; therefore, the instantaneous state of the system is unperturbed if the forces are all removed. Thus, the system is momentarily equivalent to an inertial systern. The dynamical evolution of a system of classical point particles can be thought of as passage through an infinite sequence of inertial states. This is ultimately a consequence of~.eprinciple of inertia —
—
and the assumption of existence of point particles in Newtonian mechanics. The state of an accelerated particle is at each instant the same as the state of a comoving inertial partide. Generalizing this result of Newtonian mechanics, the hypothesis of locality posits the local equivalence of an accelerated observer with a comoving inertial observer. As expected, the basic significance of this assumption first appears in the interpretation of electromagnetic phenomena in moving frames of reference. That is, in modern physics the hypothesis of locality first acquires the status of an independent assumption in the theory of relativity. The presumption that an observer along an arbitrary path is equivalent to an infinite sequence of instantaneously comoving inertial observers is reminiscent of Zeno’s third argument about motion as related by Aristotle. In Zeno’s argument, a flying arrow is at each instant locally equivalent to an arrow at rest. However, a comoving inertial observer is not equivalent to an inertial observer at rest; these observers are related instead by the principle of relativity. The hypothesis of locality involves the local immateriality of acceleration and thus goes one step beyond the local immateriality of velocity that Underlies Zeno’s argument. The local immateriality of acceleration means, in terms of realistic measurements, that the influence of inertial effects can be neglected over the length and
0375-960l/90/$ 03.50 © Elsevier Science Publishers By. (North-Holland)
147
Volume 145, number 4
PHYSICS LETTERS A
time scales characteristic of elementary local observations. This may be expressed in terms of characteristic length and time scales that can be constructed locally from the speed of light in vacuum (c) and the instantaneous acceleration of the observer. For an ideal inertial observer, these acceleration scales are infinitely long. The accelerated observer may be considered approximately inertial if the acceleration scales are sufficiently long cornpared to the intrinsic scales of the phenomena under observation. If physical phenomena could be described in terms of poin dike coincidences as in Newtonian mechanics, then the hypothesis of locality would be exactly valid since a point mass is, by definition, devoid of any intrinsic scale. Otherwise, this hypothesis must be regarded as an approximation. For instance, a classical charged particle of mass m and charge q in accelerated motion cannot be considered locally inertial since the particle radiates electromagnetic energy and its motion is thus affected by radiation reaction [1,2]. The accelerated motion of such a particle involves an intrinsic time scale of order q2/mc3. This is the time that light would take to cross the classical radius ofthe charged particle. The existence of such intrinsic scales is cxpected to lead to violations of the hypothesis of locality, The deviations from the hypothesis of locality would appear, in general, as novel effects that go beyond the standard theory. Why have such features not been discovered experimentally? The answer lies in the circumstance that the relevant acceleration scales are usually too long to allow such effects to become easily observable; for example, the intrinsic acceleration lengths for observers at rest with respect to the earth are longer than an astronomical unit. The assumption that the translational motion of a system can be considered locally uniform was already contained in the fundamental researches of Maxwell [31 and Lorentz [1]. It was given mathematical expression in the standard extension of the theory of relativity to accelerated frames and grayitational fields by Einstein and Grossmann in 1913 (see, e.g., ref. [4] for a concise history of the relativistic theory of gravitation). The study of the behavior of accelerated systems in the context of Mmkowski spacetime was initiated by Born [5]. The intrinsic geometry of a rotating disk has since been 148
9 April 1990
the subject of much discussion (see, e.g., ref. [61). A survey of the extensive literature on this subject is beyond the scope of this paper. The general viewpoint developed in this work has been discussed in a number of recent papers (cf. ref. [7] and the references cited therein). Here attention is focused on the intrinsic significance as well as limitations of the hypothesis of locality. Proper acceleration scales are discussed in section 2. To bring out the essential urnitation that is imposed on measurements by the hypothesis of locality, the problem of length measurement is discussed for uniformly rotating observers in section 3. A critical discussion of the results is given in section 4.
2. Proper acceleration scales
The basic distinction between an accelerated observer in Minkowski spacetime and a momentarily comoving inertial observer is the existence of acceleration scales associated with the noninertial observer. These scales are intrinsic measures of the rate of variation of the local reference frame of the observer along the accelerated path. To be sure, the local orthonormal frame is identical, except perhaps up to a spatial rotation, to the one carried by the instantaneously comoving inertial observer as a consequence of the hypothesis of locality. To describe the acceleration scales mathematically, let A~a)(t) be an orthonormal frame carried along an accelerated path C given by x°=x°(t),where i is the proper time, Greek indices run from 0 to 3, and Latin indices run from 1 to 3. In this paper conventions are used whereby c= 1 unless specified otherwise, the Minkowski metric tensor ~p has signature + 2, and the alternating tensors are defined such that ~ 123 = and ~ol23= 1. The frame consists of the vector tangent to the path ‘~o~ = dx°/dr and i= 1, 2, 3. which constitute a spatial frame defined along the path. Thus ~
=
,
(1)
where Ø~p(r) is an antisymmetric tensor with respect to the local tetrad frame. The acceleration scales can be constructed as rotationally invariant combinations of spacetime scalars Ø~and their proper-time
Volume 145, number 4
PHYSICS LETTERS A
derivatives. Such combinations provide, in principle, an infinite number of acceleration scales; however, for the sake of simplicity only the three dominant scales associated with the tensor Ø~will be considered here. It is useful to decompose this antisymmetric tensor into two spatial vectors A and Q in analogy with electric and magnetic fields, respectively; i.e., Øap~~ (A, Q). The three rotationally invariant combinations are thus AA, A~Q,and QQ. To obtain a physical characterization of this decomposition, let X~j)~ i= 1, 2, 3, be a spatial frame along C that is nonrotating, i.e., the local variation of each leg of the triad along C is purely tangential. This implies that
9 April 1990
that is subject to a given acceleration at time ‘r. The proper acceleration scales are thus frame-independent and can be constructed from two invariants given by I (5) —~
I
—
2
~
~
tv/i
~
2
—
~
2
afi
where Ø~= ap~Ø°i1s the dual tensor. These invariants can be written as I~= —A2+Q2 and 12= —AQ in complete correspondence with the electromagnetic analogy. For an observer with purely translational acceleration (i.e., an accelerated observer that carries a nonrotating spatial frame), the proper acceleration length is given by I~ 1/2 = c2 /A (t). Similarly, for an observer with purely rotational adceleration i.e., an observer at rest or in uniform translational motion that refers observations to a spatial frame rotating with frequency Q(t) the proper acceleration length is given by II~I 1~’2= —
—
(2) di where a °( ‘r) = d).~/dt is the acceleration vector, a 0M~O)=0. The spatial frame of the observer can be obtained from the nonrotating frame by a general rotation characterized by an orthogonal matrix M(t), 20(~) M 1j~~~J ~ U) ‘(3 ‘ The frequency of the rotation may be denoted by a pseudo-vector (relative to the local frame of the observer) Q(t) defined by the relation —
dM T~ I
(4)
fIklQ/MkJ.
It then follows that A (r), A
=
—
—
c/Q(t). An example of a mixed situation involving an observer at rest in a rotating system will be discussed in the text section. It will be shown there that 2y2 and its frame rotates alational uniformly acceleration rotating observer of rw at radius r has a transat a rate given by Q= y2w, where y is the Lorentz factor and w is the frequency of rotation of the system per unit time of the static inertial observer. It follows that I~ =y2w2 and 12=0. Thus, the rotating observer has acceleration length given by proper c/yw, where yw isa proper the frequency of rotation per unit time
a
9k~~), is the translational acceleration vector as referred to the spatial frame and Q(r) is the instantaneous rotational frequency of the spatial frame. The six components of these quantities indicate the noninertial character of the observer; therefore, the dominant acceleration (time and length) scales may be 2,constructed from AQ, and Q2. theThe speed of light together with A acceleration scales constructed above depend, in general, on the instantaneous speed of the observer. It would be interesting to develop proper acceleration scales which would depend purely on the acceleration of the observer. That is, at each point along the path the local frame may be subjected to a Lorentz transformation under which Ø~would transform as a Lorentz tensor. Physically, the parameters of the Lorentz transformation correspond to the freedom of choosing the initial velocity and the mitial orientation of the spatial frame for an observer
of the observer. The agreement of this outcome with the results of previous investigations of wave phenomena in rotating systems [8,9] indicates the significance of proper acceleration scales in physical applications. The acceleration scales are all infinite for an inertial observer. It isacceleration possible thatscales in special circumstances the proper become infinite for an accelerated observer as well; however, other finite acceleration scales would still be available. One must also consider the possibility that an acceleration scale might vanish at a certain instant. This would correspond to a singularity along the path [7]. At each instant t, the local tetrad frame is common to both the accelerated observer and the momentarily comoving inertial observer (up to a spatial rotation); these observers refer their respective measurements to the local frame. In a Lorentz-in149
Volume 145, number 4
PHYSICS LETTERS A
variant theory, the characteristics of accelerated ohservers could be based purely on measurements performed by inertial observers. The results of measurements performed by accelerated observers can then be related to those of inertial observers via the hypothesis of locality. This will be discussed in the next section for the measurement of spatial separation by observers at rest in a rotating system such as the earth. It would appear at the outset that no intrinsic limitation should be associated with the application of the hypothesis of locality either spatially (since Euclidean geometry applies at each instant t) or temporally (since an arbitrary time interval can be thought of as the integration of infinitesimal intervals at each of which locality holds). Nevertheless, the the mainhypothesis purpose of the following section is to show that spatial distance is unambiguously defined for rotating observers only when it is extremely small compared to the proper acceleration length.
9 April 1990
frame the coordinates are X°= (t, X) with Xt = c1n 2~a)is the natural orthonormal tetrad 0A~I), hereobserver, n°is a unit spacelike vector frame ofwthe orthogonal to the worldline at ‘r, and a is the proper distance away from the worldline along the direction defined by n°. If the spatial frame is nonrotating along the worldline, the geodesic frame reduces to a Fermi frame. To investigate the consistency of the conceptual framework provided by the hypothesis of locality, it proves useful to focus attention on a particular case, namely, the measurement of distance in a rotating frame. To this end, let P 1 and P2 be point particles at rest on a circle of radius angular separation 2)-plane of ranwith inertial frame f with zl in the (x’,x°x= (t, x). At some time 1<0, these parcoordinates tides are set in rotational motion along the circle in exactly the same way so that the relative angular separation of P 1 and P2 with respect to observers at rest in f remains the same. Imagine that at (=0 the particles at P1 and P2 reach a uniform rate of rotation w that is maintained for t ~ 0 as represented in fig. 1. According to static observers in the arc length from P~to P2 is a constant equal to I=rA. However, it would appear from a simple application of the hypothesis of locality that according to comoving observers the arc length should be larger than I by the Lorentz factor To see this, imagine a large number of particles along the arc from P1 and P2 that share their pattern of motion. At each instant I after the motion has begun, let f’ (t) denote collectively the ~,
3. Measurement of distance in a rotating frame
It is assumed in the Lorentz-invariant theory of relativity that spacetime measurements are performed by inertial observers using ideal rods and clocks. The hypothesis of locality permits the extension of these procedures to actual (i.e., accelerated) observers. Measurement devices for which this is possible are generally referred to as “standard”. Thus, a standard clock measures proper time along its worldline and along standard measure proper lengths spatialmeasuring directions rods orthogonal to the observer’s worldline. The spatial determinations
~‘.
2 x
P 2p
made by ideal inertial observers are assumed to be in accordance with the rules of Euclidean geometry. The same rules must be adopted for accelerated observers as well due to the instantaneous equivalence of all comoving observers. Thus, imagine a coordinate system set up along the path of the accelerated observer with the origin of spatial coordinates represented by the worldline of the observer and the spatial distance away from the origin at proper time i determined proper at length along a straight line normal tobythethe woridline t. Such a coordinate system will be referred to as a geodesicframe. In this 150
r
~ wt
Fig. I. The uniform rotation of point particles on a circle of radius r as viewed from the inertial frame fi. The coordinate axes I this plane, the observer co2)-plane. are chosen such that P1 is on the xIn-axis at t=0 and the motion takes place moving withinP the (x’, x 1 refers its measurements at each instant to spatial axes that point in the radial and tangential directions.
Volume 145, number 4
PHYSICS LETTERS A
inertial frames that are comoving instantaneously with the particles on the arc from P~to P2. The application of Lorentz—FitzGerald contraction to each infinitesimal portion of the arc leads to the conclusion that for comoving observers the arc length is in general larger than 1. In fact, for t ~ 0 the arc length accordinggiven to comoving in i~’is constant by I’ =yl,inertial where observers y (~[ _p2)_i/2 anda .8= rw/c. It is important to note that 1’ for J3-+ 1. The collection of instantaneously comoving inertial frames 5’ may be replaced by a single continuously comoving geodesic frame centered along the worldline of any of the particles on the arc from p, to P 2. On the basis of the hypothesis of locality, it is equally valid to calculate the length ofthe arcfrom p, to P2 in such a frame. For the sake of definiteness, the calculation will be performed in the geodesic frame ~ along the worldline of P To construct this coordinate system, it is necessary to identify the natural tetrad frame corresponding to a uniformly rotating observer at P1 for t?~0.This frame is given by
9 April 1990
Thomas precession and may be interpreted as being ultimately due to the difference between proper time ‘rand inertial time t. The transformation between the inertial coordinates x°in 5 and the natural coordinates X°in ~ is given by 2) (10) X°=y~(x°—flyX Xi =xi cos(ywX°)+x2sin(ywX°)—r, (11) ,
—~
~.
=
y( 1, —.8 sin ~‘, fi cos ç~,0), (0, dos ~, sin ç~,0)
=
y(/l,
=
‘
—
sin
qi,
cos qi, 0)
,
A~3)= (0,0,0, 1)
X2=y~[ —x1 sin(ywX°)+x2cos(ywX°)], X3=x3.
(12) (13)
In the inertial frame, the coordinates of any point P on the arc from P1 to P2 are given by P: (t, rcos r sin 1, 0), where the difference between i and ~‘ = wt is a fixed angle ö that ranges from 0 to A. In the corresponding coordinates are P: (—/3V+t/y, U, V, 0), where U+r=rcos(ô+w/JyV) (14) ~,
,
yV=rsin(ö+wfJyV)
.
(15)
(6) (7)
the point P is and on an with It 2)-plane follows from eqs. (14) (15)ellipse that in thesemi(Xi, major axis rand eccentricity/i as in fig. 2. Once ö is X
(8)
specified, eq. (15) has a unique solution for V with 0 ~ $2 ~ 1. It is important to note that the configuration of P 1 and P2 in the geodesic system ~ is constant in time. The arc length from P1 and P2 in ~ is given by
(9)
where qi=wt and the spatial axes correspond to directions that are radial, tangential and normal with respect to the orbital plane (cf. fig. 1). It follows from the treatmentofof 2 that acceleration the only nonzero components thesection translational vector and the rotational frequency vector are A 1= 2w, respectively. Thus, Ii=y2w2 and r12=0, and so Q3=y that the proper acceleration length is given
2O)~2dO,
(16)
L=rJ (1~fl2cos
—
by c/yw and the proper acceleration time is the proper period of rotation since ço=ywr, where r=t/ y is the proper time along the path ofP 1 for t>~0. Thus the motion of P is 2, characterized a centripetal where yw isbythe frequency acof celeration of r(yw) rotation of P 1 per unit proper time ‘r. Furthermore, the spatial frame (partially depicted in fig. 1) carried along by the observer at P1 rotates with respect to the nonrotating frame at the rate of yw per unit time t. It follows that the nonrotating frame rotates with respect to the inertial observers in .1 at a rate of (1 —y)w per unit time t, which corresponds to the
~~2Lx2
Fig. 2. The configuration of point particles P1 and P2 as viewed from the geodesic frame ~ centered on P1. The coordinate axes are based on the radial and tangential directions at P1 (cf. fig. 1). The ellipse with eccentricity $ can be thought of as a circle of radius r that is Lorentz contracted along the direction of motion of P1.
151
Volume 145, number 4
PHYSICS LETTERS A
where
u(fl) is the eccentric anomaly given by sin u A. It is interesting to note that for fl—f I the ellipse collapses to a line; therefore, in this limit L approaches afinite value, i.e., L—r( 1 cos u1) with u _fl2
—
u1 —sin u1 =4. L(/3, 4) is a monotonically increasing function of 4 when /3 is kept fixed. For a fixed 4, however. L does not always increase with/i. Consider, for instance, 4=it so that L is equal to 1= itr for /3=0 whereas L—~2rfor $—~l. The discrepancy between L and 1’ implies a restriction on the applicability of the hypothesis of locality; i.e., this hypothesis is valid only to the extent that the two approaches described above yield the sameseparation result. This occurs Pfor /30 only when the spatial between 1 and P2 vanishes. To see this, leti<< 1 and consider the expansion of L(fl, 4) in powers ofzL it can be shown that = I ~$2y1244+.... (17) —
Forafixedfl, it isclearthatL/l’~l as4~0. Fora fixed 4, however, eq. (17) becomes meaningless for /3—~l.It follows that in physical experiments the hypothesis of locality is approximately valid for /3<< 1. This means that r/ (c/yw) = y/3<< 1, so that the radius of rotation must be negligible in comparison to the proper acceleration length c/yw under the ohservational arrangement. The conditions I 4z r ‘K c/ yw are necessary for the approximate validity of the hypothesis of locality in the case under consideration; therefore, the dimensions of the sector must be negligible compared to the proper acceleration length. these conditions are satisfied for the Michelson— Morley experiment. More generally, the range of applicability of the hypothesis of locality is limited by r<< c/yw. It follows from these considerations that the hypothesis of locality is truly local, i.e., it is valid only for I—~0.The issue of spatial distance is left unresolved, however. That is, it remains to predict theoretically the separation between P1 and P2 in a general case according to rotating observers. If the nature of matter, radiation and their interaction could be based entirely on classical (i.e., nonquantum) laws of physics, then the pointwise embodiment of the hypothesis of locality in the classical laws would lead to a complete picture of physics in a rotating system. Once it is specified how the spatial separation is ac152
9 April 1990
tually measured, there would then be no difficulty of principle in predicting the outcome of measurement [101. However, a satisfactory extension of quantum mechanics to accelerated systems is not available at present.
4. Discussion Considerations based on the hypothesis of locality have led to a preferred geodesic reference frame for an accelerated observer. For instance, a uniformly rotating observer can refer physical events to a geo2= desic frame ~ given by the metric ds tlopdxOdxfl=G/21,dXOdXP, where eqs. (lO)—(13) imply that2[1—w2(r+X’)2—w2y2(X2)2] , (18) G00= —y G 0, = (QXX)1, (19) ~ (20) This geodesic frame reduces to the standard rotating coordinate system for r=0, i.e., when the observer is at the origin of spatial coordinates in 5. The rotating observer is characterized by a proper acceleration length c/yw. It is important to note that this length vanishes for fl—fl. An analysis of the determination of distance by rotating observers has revealed that an unambiguous interpretation of the hypothesis of locality requires that the relevant distance be negligible compared to the proper acceleration length. It can be demonstrated from the results of ref. [7 1 that the standard prescription for distance determination involving clocks and light signals does not overcome this limitation. More generally, the hypothesis of locality is approximately valid whenever the intrinsic length and time scales of phenomena under observation are negligibly small compared to the appropriate acceleration scales. The laws of classical mechanics ensure the validity of the hypothesis of locality for a Newtonian point particle. In modern physics the idea of wave—particle duality, which originated from Planck’s quantum hypothesis, has been a guiding principle. This standpoint naturally leads to the question of validity of the hypothesis of locality for classical wave phenomena. That is, any basic violation of the hypothesis of locality is expected to be due to the wave character-
Volume 145, number 4
PHYSICS LETTERS A
istics of a particle [8,9,11]. Moreover, such a violation should disappear for an inertial observer. Therefore, any deviation from the hypothesis of locality is expected to be in the form of a ratio of the intrinsic scale of the phenomenon under observation to the corresponding (proper) acceleration scale of the observer. Such effects could not have been detected at the level of sensitivity of experiments performed thus far (see, e.g., refs. [12] and [131). It appears that any fundamental extension of the hypothesis of locality must be based on the principle of complementarity [14].
9 April 1990
[2]P.A.M. Dirac, Proc. R. Soc. A 167 (1938) 148. [3] J.C. Maxwell, Nature 21(1880) 314. [4] E.T. Whittaker, History London, of the theories ofV. aether and electricity, Vol. IIA(Nelson, 1951) ch. [5] M. Born, Ann. Phys. (Leipzig) 30 (1909) 1. [6] J. Stachel, in: General relativity and gravitation, Vol. 1, ed. A. Held (Plenum, New York, 1980) p. 1. [7] B. Mashhoon, Phys. Lett. A 143 (1990) 176. [8] 13961(1988)2639. (1989) 103. [9] B. B. Mashhoon, Mashhoon, Phys. Phys. Lett. Rev. A Lett. [10] H.A. Lorentz, Nature 106 (1921) 793. [11] B. Mashhoon, Phys. Lett. A 122 (1987) 67. [12] J. Bailey ci al., Phys. Lett. B 55 (1975) 420. [13] U. Bonse and T. Wroblewski, Phys. Rev. Lett. 51 (1983) 1401. [14] B. Mashhoon, Phys. Lett. A 126 (1988) 393.
References [1] H.A. Lorentz, The theory of electrons (Dover, New York, 1952).
153
PHYSICS LETTERS A
9 April 1990
THE HYPOTHESIS OF LOCALITY IN RELATIVISTIC PHYSICS Bahram MASHHOON Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, MO 65211, USA Received 29 December 1989; accepted for publication 13 February 1990 Communicatedby J.P. Vigier
The standard extension of any Lorentz-invariant theory to arbitrary frames of reference as well as gravitational fields is based on the assumption that an observer is momentarily equivalent to a comoving inertial observer. This hypothesis of locality is indispensable to the interpretation ofthe results of experiments in physics. The origin, significance, and limitations ofthe hypothesis of locality are discussed.
1. Introduction This paper is about a basic question in the theory of measurement: Which law ofphysics specifies what accelerated observers measure? It has been proposed that the hypothesis of locality i.e., the presumed equivalence of an accelerated observer with a momentarily comoving inertial observer underlies the standard relativistic formalism by relating the measurements of an accelerated observer to those of an inertial observer. This fundamental assumption therefore replaces the customary hypotheses concerning classical measuring devices in accelerated motion. In particular, a physical basis can be provided for the discussion of the clock hypothesis. To see how this comes about, one must begin with classical mechanics. In Newtonian mechanics, one imagines that a complex system is composed of material particles which are represented by mathematical points. The state of a system of particles is characterized at each instant by the positions and velocities of the particles involved. The forces, whether true or fictitious, only affect accelerations at any moment; therefore, the instantaneous state of the system is unperturbed if the forces are all removed. Thus, the system is momentarily equivalent to an inertial systern. The dynamical evolution of a system of classical point particles can be thought of as passage through an infinite sequence of inertial states. This is ultimately a consequence of~.eprinciple of inertia —
—
and the assumption of existence of point particles in Newtonian mechanics. The state of an accelerated particle is at each instant the same as the state of a comoving inertial partide. Generalizing this result of Newtonian mechanics, the hypothesis of locality posits the local equivalence of an accelerated observer with a comoving inertial observer. As expected, the basic significance of this assumption first appears in the interpretation of electromagnetic phenomena in moving frames of reference. That is, in modern physics the hypothesis of locality first acquires the status of an independent assumption in the theory of relativity. The presumption that an observer along an arbitrary path is equivalent to an infinite sequence of instantaneously comoving inertial observers is reminiscent of Zeno’s third argument about motion as related by Aristotle. In Zeno’s argument, a flying arrow is at each instant locally equivalent to an arrow at rest. However, a comoving inertial observer is not equivalent to an inertial observer at rest; these observers are related instead by the principle of relativity. The hypothesis of locality involves the local immateriality of acceleration and thus goes one step beyond the local immateriality of velocity that Underlies Zeno’s argument. The local immateriality of acceleration means, in terms of realistic measurements, that the influence of inertial effects can be neglected over the length and
0375-960l/90/$ 03.50 © Elsevier Science Publishers By. (North-Holland)
147
Volume 145, number 4
PHYSICS LETTERS A
time scales characteristic of elementary local observations. This may be expressed in terms of characteristic length and time scales that can be constructed locally from the speed of light in vacuum (c) and the instantaneous acceleration of the observer. For an ideal inertial observer, these acceleration scales are infinitely long. The accelerated observer may be considered approximately inertial if the acceleration scales are sufficiently long cornpared to the intrinsic scales of the phenomena under observation. If physical phenomena could be described in terms of poin dike coincidences as in Newtonian mechanics, then the hypothesis of locality would be exactly valid since a point mass is, by definition, devoid of any intrinsic scale. Otherwise, this hypothesis must be regarded as an approximation. For instance, a classical charged particle of mass m and charge q in accelerated motion cannot be considered locally inertial since the particle radiates electromagnetic energy and its motion is thus affected by radiation reaction [1,2]. The accelerated motion of such a particle involves an intrinsic time scale of order q2/mc3. This is the time that light would take to cross the classical radius ofthe charged particle. The existence of such intrinsic scales is cxpected to lead to violations of the hypothesis of locality, The deviations from the hypothesis of locality would appear, in general, as novel effects that go beyond the standard theory. Why have such features not been discovered experimentally? The answer lies in the circumstance that the relevant acceleration scales are usually too long to allow such effects to become easily observable; for example, the intrinsic acceleration lengths for observers at rest with respect to the earth are longer than an astronomical unit. The assumption that the translational motion of a system can be considered locally uniform was already contained in the fundamental researches of Maxwell [31 and Lorentz [1]. It was given mathematical expression in the standard extension of the theory of relativity to accelerated frames and grayitational fields by Einstein and Grossmann in 1913 (see, e.g., ref. [4] for a concise history of the relativistic theory of gravitation). The study of the behavior of accelerated systems in the context of Mmkowski spacetime was initiated by Born [5]. The intrinsic geometry of a rotating disk has since been 148
9 April 1990
the subject of much discussion (see, e.g., ref. [61). A survey of the extensive literature on this subject is beyond the scope of this paper. The general viewpoint developed in this work has been discussed in a number of recent papers (cf. ref. [7] and the references cited therein). Here attention is focused on the intrinsic significance as well as limitations of the hypothesis of locality. Proper acceleration scales are discussed in section 2. To bring out the essential urnitation that is imposed on measurements by the hypothesis of locality, the problem of length measurement is discussed for uniformly rotating observers in section 3. A critical discussion of the results is given in section 4.
2. Proper acceleration scales
The basic distinction between an accelerated observer in Minkowski spacetime and a momentarily comoving inertial observer is the existence of acceleration scales associated with the noninertial observer. These scales are intrinsic measures of the rate of variation of the local reference frame of the observer along the accelerated path. To be sure, the local orthonormal frame is identical, except perhaps up to a spatial rotation, to the one carried by the instantaneously comoving inertial observer as a consequence of the hypothesis of locality. To describe the acceleration scales mathematically, let A~a)(t) be an orthonormal frame carried along an accelerated path C given by x°=x°(t),where i is the proper time, Greek indices run from 0 to 3, and Latin indices run from 1 to 3. In this paper conventions are used whereby c= 1 unless specified otherwise, the Minkowski metric tensor ~p has signature + 2, and the alternating tensors are defined such that ~ 123 = and ~ol23= 1. The frame consists of the vector tangent to the path ‘~o~ = dx°/dr and i= 1, 2, 3. which constitute a spatial frame defined along the path. Thus ~
=
,
(1)
where Ø~p(r) is an antisymmetric tensor with respect to the local tetrad frame. The acceleration scales can be constructed as rotationally invariant combinations of spacetime scalars Ø~and their proper-time
Volume 145, number 4
PHYSICS LETTERS A
derivatives. Such combinations provide, in principle, an infinite number of acceleration scales; however, for the sake of simplicity only the three dominant scales associated with the tensor Ø~will be considered here. It is useful to decompose this antisymmetric tensor into two spatial vectors A and Q in analogy with electric and magnetic fields, respectively; i.e., Øap~~ (A, Q). The three rotationally invariant combinations are thus AA, A~Q,and QQ. To obtain a physical characterization of this decomposition, let X~j)~ i= 1, 2, 3, be a spatial frame along C that is nonrotating, i.e., the local variation of each leg of the triad along C is purely tangential. This implies that
9 April 1990
that is subject to a given acceleration at time ‘r. The proper acceleration scales are thus frame-independent and can be constructed from two invariants given by I (5) —~
I
—
2
~
~
tv/i
~
2
—
~
2
afi
where Ø~= ap~Ø°i1s the dual tensor. These invariants can be written as I~= —A2+Q2 and 12= —AQ in complete correspondence with the electromagnetic analogy. For an observer with purely translational acceleration (i.e., an accelerated observer that carries a nonrotating spatial frame), the proper acceleration length is given by I~ 1/2 = c2 /A (t). Similarly, for an observer with purely rotational adceleration i.e., an observer at rest or in uniform translational motion that refers observations to a spatial frame rotating with frequency Q(t) the proper acceleration length is given by II~I 1~’2= —
—
(2) di where a °( ‘r) = d).~/dt is the acceleration vector, a 0M~O)=0. The spatial frame of the observer can be obtained from the nonrotating frame by a general rotation characterized by an orthogonal matrix M(t), 20(~) M 1j~~~J ~ U) ‘(3 ‘ The frequency of the rotation may be denoted by a pseudo-vector (relative to the local frame of the observer) Q(t) defined by the relation —
dM T~ I
(4)
fIklQ/MkJ.
It then follows that A (r), A
=
—
—
c/Q(t). An example of a mixed situation involving an observer at rest in a rotating system will be discussed in the text section. It will be shown there that 2y2 and its frame rotates alational uniformly acceleration rotating observer of rw at radius r has a transat a rate given by Q= y2w, where y is the Lorentz factor and w is the frequency of rotation of the system per unit time of the static inertial observer. It follows that I~ =y2w2 and 12=0. Thus, the rotating observer has acceleration length given by proper c/yw, where yw isa proper the frequency of rotation per unit time
a
9k~~), is the translational acceleration vector as referred to the spatial frame and Q(r) is the instantaneous rotational frequency of the spatial frame. The six components of these quantities indicate the noninertial character of the observer; therefore, the dominant acceleration (time and length) scales may be 2,constructed from AQ, and Q2. theThe speed of light together with A acceleration scales constructed above depend, in general, on the instantaneous speed of the observer. It would be interesting to develop proper acceleration scales which would depend purely on the acceleration of the observer. That is, at each point along the path the local frame may be subjected to a Lorentz transformation under which Ø~would transform as a Lorentz tensor. Physically, the parameters of the Lorentz transformation correspond to the freedom of choosing the initial velocity and the mitial orientation of the spatial frame for an observer
of the observer. The agreement of this outcome with the results of previous investigations of wave phenomena in rotating systems [8,9] indicates the significance of proper acceleration scales in physical applications. The acceleration scales are all infinite for an inertial observer. It isacceleration possible thatscales in special circumstances the proper become infinite for an accelerated observer as well; however, other finite acceleration scales would still be available. One must also consider the possibility that an acceleration scale might vanish at a certain instant. This would correspond to a singularity along the path [7]. At each instant t, the local tetrad frame is common to both the accelerated observer and the momentarily comoving inertial observer (up to a spatial rotation); these observers refer their respective measurements to the local frame. In a Lorentz-in149
Volume 145, number 4
PHYSICS LETTERS A
variant theory, the characteristics of accelerated ohservers could be based purely on measurements performed by inertial observers. The results of measurements performed by accelerated observers can then be related to those of inertial observers via the hypothesis of locality. This will be discussed in the next section for the measurement of spatial separation by observers at rest in a rotating system such as the earth. It would appear at the outset that no intrinsic limitation should be associated with the application of the hypothesis of locality either spatially (since Euclidean geometry applies at each instant t) or temporally (since an arbitrary time interval can be thought of as the integration of infinitesimal intervals at each of which locality holds). Nevertheless, the the mainhypothesis purpose of the following section is to show that spatial distance is unambiguously defined for rotating observers only when it is extremely small compared to the proper acceleration length.
9 April 1990
frame the coordinates are X°= (t, X) with Xt = c1n 2~a)is the natural orthonormal tetrad 0A~I), hereobserver, n°is a unit spacelike vector frame ofwthe orthogonal to the worldline at ‘r, and a is the proper distance away from the worldline along the direction defined by n°. If the spatial frame is nonrotating along the worldline, the geodesic frame reduces to a Fermi frame. To investigate the consistency of the conceptual framework provided by the hypothesis of locality, it proves useful to focus attention on a particular case, namely, the measurement of distance in a rotating frame. To this end, let P 1 and P2 be point particles at rest on a circle of radius angular separation 2)-plane of ranwith inertial frame f with zl in the (x’,x°x= (t, x). At some time 1<0, these parcoordinates tides are set in rotational motion along the circle in exactly the same way so that the relative angular separation of P 1 and P2 with respect to observers at rest in f remains the same. Imagine that at (=0 the particles at P1 and P2 reach a uniform rate of rotation w that is maintained for t ~ 0 as represented in fig. 1. According to static observers in the arc length from P~to P2 is a constant equal to I=rA. However, it would appear from a simple application of the hypothesis of locality that according to comoving observers the arc length should be larger than I by the Lorentz factor To see this, imagine a large number of particles along the arc from P1 and P2 that share their pattern of motion. At each instant I after the motion has begun, let f’ (t) denote collectively the ~,
3. Measurement of distance in a rotating frame
It is assumed in the Lorentz-invariant theory of relativity that spacetime measurements are performed by inertial observers using ideal rods and clocks. The hypothesis of locality permits the extension of these procedures to actual (i.e., accelerated) observers. Measurement devices for which this is possible are generally referred to as “standard”. Thus, a standard clock measures proper time along its worldline and along standard measure proper lengths spatialmeasuring directions rods orthogonal to the observer’s worldline. The spatial determinations
~‘.
2 x
P 2p
made by ideal inertial observers are assumed to be in accordance with the rules of Euclidean geometry. The same rules must be adopted for accelerated observers as well due to the instantaneous equivalence of all comoving observers. Thus, imagine a coordinate system set up along the path of the accelerated observer with the origin of spatial coordinates represented by the worldline of the observer and the spatial distance away from the origin at proper time i determined proper at length along a straight line normal tobythethe woridline t. Such a coordinate system will be referred to as a geodesicframe. In this 150
r
~ wt
Fig. I. The uniform rotation of point particles on a circle of radius r as viewed from the inertial frame fi. The coordinate axes I this plane, the observer co2)-plane. are chosen such that P1 is on the xIn-axis at t=0 and the motion takes place moving withinP the (x’, x 1 refers its measurements at each instant to spatial axes that point in the radial and tangential directions.
Volume 145, number 4
PHYSICS LETTERS A
inertial frames that are comoving instantaneously with the particles on the arc from P~to P2. The application of Lorentz—FitzGerald contraction to each infinitesimal portion of the arc leads to the conclusion that for comoving observers the arc length is in general larger than 1. In fact, for t ~ 0 the arc length accordinggiven to comoving in i~’is constant by I’ =yl,inertial where observers y (~[ _p2)_i/2 anda .8= rw/c. It is important to note that 1’ for J3-+ 1. The collection of instantaneously comoving inertial frames 5’ may be replaced by a single continuously comoving geodesic frame centered along the worldline of any of the particles on the arc from p, to P 2. On the basis of the hypothesis of locality, it is equally valid to calculate the length ofthe arcfrom p, to P2 in such a frame. For the sake of definiteness, the calculation will be performed in the geodesic frame ~ along the worldline of P To construct this coordinate system, it is necessary to identify the natural tetrad frame corresponding to a uniformly rotating observer at P1 for t?~0.This frame is given by
9 April 1990
Thomas precession and may be interpreted as being ultimately due to the difference between proper time ‘rand inertial time t. The transformation between the inertial coordinates x°in 5 and the natural coordinates X°in ~ is given by 2) (10) X°=y~(x°—flyX Xi =xi cos(ywX°)+x2sin(ywX°)—r, (11) ,
—~
~.
=
y( 1, —.8 sin ~‘, fi cos ç~,0), (0, dos ~, sin ç~,0)
=
y(/l,
=
‘
—
sin
qi,
cos qi, 0)
,
A~3)= (0,0,0, 1)
X2=y~[ —x1 sin(ywX°)+x2cos(ywX°)], X3=x3.
(12) (13)
In the inertial frame, the coordinates of any point P on the arc from P1 to P2 are given by P: (t, rcos r sin 1, 0), where the difference between i and ~‘ = wt is a fixed angle ö that ranges from 0 to A. In the corresponding coordinates are P: (—/3V+t/y, U, V, 0), where U+r=rcos(ô+w/JyV) (14) ~,
,
yV=rsin(ö+wfJyV)
.
(15)
(6) (7)
the point P is and on an with It 2)-plane follows from eqs. (14) (15)ellipse that in thesemi(Xi, major axis rand eccentricity/i as in fig. 2. Once ö is X
(8)
specified, eq. (15) has a unique solution for V with 0 ~ $2 ~ 1. It is important to note that the configuration of P 1 and P2 in the geodesic system ~ is constant in time. The arc length from P1 and P2 in ~ is given by
(9)
where qi=wt and the spatial axes correspond to directions that are radial, tangential and normal with respect to the orbital plane (cf. fig. 1). It follows from the treatmentofof 2 that acceleration the only nonzero components thesection translational vector and the rotational frequency vector are A 1= 2w, respectively. Thus, Ii=y2w2 and r12=0, and so Q3=y that the proper acceleration length is given
2O)~2dO,
(16)
L=rJ (1~fl2cos
—
by c/yw and the proper acceleration time is the proper period of rotation since ço=ywr, where r=t/ y is the proper time along the path ofP 1 for t>~0. Thus the motion of P is 2, characterized a centripetal where yw isbythe frequency acof celeration of r(yw) rotation of P 1 per unit proper time ‘r. Furthermore, the spatial frame (partially depicted in fig. 1) carried along by the observer at P1 rotates with respect to the nonrotating frame at the rate of yw per unit time t. It follows that the nonrotating frame rotates with respect to the inertial observers in .1 at a rate of (1 —y)w per unit time t, which corresponds to the
~~2Lx2
Fig. 2. The configuration of point particles P1 and P2 as viewed from the geodesic frame ~ centered on P1. The coordinate axes are based on the radial and tangential directions at P1 (cf. fig. 1). The ellipse with eccentricity $ can be thought of as a circle of radius r that is Lorentz contracted along the direction of motion of P1.
151
Volume 145, number 4
PHYSICS LETTERS A
where
u(fl) is the eccentric anomaly given by sin u A. It is interesting to note that for fl—f I the ellipse collapses to a line; therefore, in this limit L approaches afinite value, i.e., L—r( 1 cos u1) with u _fl2
—
u1 —sin u1 =4. L(/3, 4) is a monotonically increasing function of 4 when /3 is kept fixed. For a fixed 4, however. L does not always increase with/i. Consider, for instance, 4=it so that L is equal to 1= itr for /3=0 whereas L—~2rfor $—~l. The discrepancy between L and 1’ implies a restriction on the applicability of the hypothesis of locality; i.e., this hypothesis is valid only to the extent that the two approaches described above yield the sameseparation result. This occurs Pfor /30 only when the spatial between 1 and P2 vanishes. To see this, leti<< 1 and consider the expansion of L(fl, 4) in powers ofzL it can be shown that = I ~$2y1244+.... (17) —
Forafixedfl, it isclearthatL/l’~l as4~0. Fora fixed 4, however, eq. (17) becomes meaningless for /3—~l.It follows that in physical experiments the hypothesis of locality is approximately valid for /3<< 1. This means that r/ (c/yw) = y/3<< 1, so that the radius of rotation must be negligible in comparison to the proper acceleration length c/yw under the ohservational arrangement. The conditions I 4z r ‘K c/ yw are necessary for the approximate validity of the hypothesis of locality in the case under consideration; therefore, the dimensions of the sector must be negligible compared to the proper acceleration length. these conditions are satisfied for the Michelson— Morley experiment. More generally, the range of applicability of the hypothesis of locality is limited by r<< c/yw. It follows from these considerations that the hypothesis of locality is truly local, i.e., it is valid only for I—~0.The issue of spatial distance is left unresolved, however. That is, it remains to predict theoretically the separation between P1 and P2 in a general case according to rotating observers. If the nature of matter, radiation and their interaction could be based entirely on classical (i.e., nonquantum) laws of physics, then the pointwise embodiment of the hypothesis of locality in the classical laws would lead to a complete picture of physics in a rotating system. Once it is specified how the spatial separation is ac152
9 April 1990
tually measured, there would then be no difficulty of principle in predicting the outcome of measurement [101. However, a satisfactory extension of quantum mechanics to accelerated systems is not available at present.
4. Discussion Considerations based on the hypothesis of locality have led to a preferred geodesic reference frame for an accelerated observer. For instance, a uniformly rotating observer can refer physical events to a geo2= desic frame ~ given by the metric ds tlopdxOdxfl=G/21,dXOdXP, where eqs. (lO)—(13) imply that2[1—w2(r+X’)2—w2y2(X2)2] , (18) G00= —y G 0, = (QXX)1, (19) ~ (20) This geodesic frame reduces to the standard rotating coordinate system for r=0, i.e., when the observer is at the origin of spatial coordinates in 5. The rotating observer is characterized by a proper acceleration length c/yw. It is important to note that this length vanishes for fl—fl. An analysis of the determination of distance by rotating observers has revealed that an unambiguous interpretation of the hypothesis of locality requires that the relevant distance be negligible compared to the proper acceleration length. It can be demonstrated from the results of ref. [7 1 that the standard prescription for distance determination involving clocks and light signals does not overcome this limitation. More generally, the hypothesis of locality is approximately valid whenever the intrinsic length and time scales of phenomena under observation are negligibly small compared to the appropriate acceleration scales. The laws of classical mechanics ensure the validity of the hypothesis of locality for a Newtonian point particle. In modern physics the idea of wave—particle duality, which originated from Planck’s quantum hypothesis, has been a guiding principle. This standpoint naturally leads to the question of validity of the hypothesis of locality for classical wave phenomena. That is, any basic violation of the hypothesis of locality is expected to be due to the wave character-
Volume 145, number 4
PHYSICS LETTERS A
istics of a particle [8,9,11]. Moreover, such a violation should disappear for an inertial observer. Therefore, any deviation from the hypothesis of locality is expected to be in the form of a ratio of the intrinsic scale of the phenomenon under observation to the corresponding (proper) acceleration scale of the observer. Such effects could not have been detected at the level of sensitivity of experiments performed thus far (see, e.g., refs. [12] and [131). It appears that any fundamental extension of the hypothesis of locality must be based on the principle of complementarity [14].
9 April 1990
[2]P.A.M. Dirac, Proc. R. Soc. A 167 (1938) 148. [3] J.C. Maxwell, Nature 21(1880) 314. [4] E.T. Whittaker, History London, of the theories ofV. aether and electricity, Vol. IIA(Nelson, 1951) ch. [5] M. Born, Ann. Phys. (Leipzig) 30 (1909) 1. [6] J. Stachel, in: General relativity and gravitation, Vol. 1, ed. A. Held (Plenum, New York, 1980) p. 1. [7] B. Mashhoon, Phys. Lett. A 143 (1990) 176. [8] 13961(1988)2639. (1989) 103. [9] B. B. Mashhoon, Mashhoon, Phys. Phys. Lett. Rev. A Lett. [10] H.A. Lorentz, Nature 106 (1921) 793. [11] B. Mashhoon, Phys. Lett. A 122 (1987) 67. [12] J. Bailey ci al., Phys. Lett. B 55 (1975) 420. [13] U. Bonse and T. Wroblewski, Phys. Rev. Lett. 51 (1983) 1401. [14] B. Mashhoon, Phys. Lett. A 126 (1988) 393.
References [1] H.A. Lorentz, The theory of electrons (Dover, New York, 1952).
153