Acceleration-induced nonlocal electrodynamics in Minkowski spacetime

19 June 2000

Physics Letters A 271 Ž2000. 8–15 www.elsevier.nlrlocaterpla

Acceleration-induced nonlocal electrodynamics in Minkowski spacetime Uwe Muench, Friedrich W. Hehl 1, Bahram Mashhoon) Department of Physics & Astronomy, UniÕersity of Missouri, Columbia, MO 65211, USA Received 4 April 2000; accepted 25 April 2000 Communicated by P.R. Holland

Abstract We discuss two nonlocal models of electrodynamics in which the nonlocality is induced by the acceleration of the observer. Such an observer actually measures an electromagnetic field that exhibits persistent memory effects. We compare Mashhoon’s model with a new ansatz developed here in the framework of charge & flux electrodynamics with a constitutive law involving the Levi-Civita connection as seen from the observer’s local frame and conclude that they are in partial agreement only for the case of constant acceleration. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction As pointed out by Einstein w1x 2 , in special relativity theory it is assumed that the rate of a fundamental Ž‘ideal’. clock depends on its instantaneous speed and is not affected by its instantaneous acceleration. This is usually called the ‘clock hypothesis’; see w2–4x for more recent discussions of this assumption. The decay of elementary particles obeys this hypothesis very well as shown by Eisele w5x for the weak decay of the muon. If we study electrodynamics, for instance, in an accelerated reference frame Žsee w6x., then we have to

)

Corresponding author. E-mail address: [email protected] ŽB. Mashhoon.. 1 Permanent address: Inst. Theor. Physics, Univ. of Cologne, 50923 Cologne, Germany. 2 The hypothesis regarding accelerated rods and clocks is mentioned in a footnote on p. 60.

presuppose corresponding hypotheses for the measurement of the electric and magnetic fields, the electric charge, etc. In this way, we arrive at the hypothesis of locality that has been extensively investigated w7–11x. Replacing the curved worldline of the accelerated observer by its instantaneous tangent vector is reasonable if the intrinsic spacetime scales of the phenomena under consideration are negligibly small compared to the characteristic acceleration scales that determine the curvature of the worldline; otherwise, the past worldline of the observer must be taken into account. This would then result in a nonlocal electrodynamics for accelerated systems. Nonlocal constitutive relations have been studied in the phenomenological electrodynamics of continuous media for a long time w12,13x. In basic field theories, form-factor nonlocality has been the subject of extensive investigations. The main problem with such field theoretical approaches has been that they defy quantization. A review of nonlocal quantum

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 3 1 6 - 9

U. Muench et al.r Physics Letters A 271 (2000) 8–15

field theories and their insurmountable difficulties has been given by Marnelius w14x. The present work is concerned with a benign form of nonlocality that is induced by the acceleration of the observer. The hypothesis of locality refers directly to acceleration; therefore, one can develop an alternative approach in which the acceleration enters as the decisive quantity. This type of nonlocality, if it refers to time, would involve persistent memory effects. Materials with memory have been extensively studied. However, we are interested in the ‘material’ vacuum – and in this context our Letter is devoted to a comparison of two models involving accelerationinduced nonlocality.

9

ˆ refer to Žanholonomic. frame indices, from 0ˆ to 3, and we choose the signature Žq,y,y,y .. The hypothesis of locality implies that the field as measured by the observer is the projection of Fi j upon the frame of the instantaneously comoving inertial observer, i.e. Fab Ž t . s Fi j e i a e j b .

Ž 1.

On the other hand, measuring the properties of the radiation field would necessitate finite intervals of time and space that would then involve the curvature of the worldline. The most general linear relationship between the measurements of the accelerated observer and the class of comoving inertial observers consistent with causality is

2. Mashhoon’s model t

The observational basis of the special theory of relativity generally involves measuring devices that are accelerated; for instance, static laboratory devices on the Earth participate in its proper rotation. The standard extension of Lorentz invariance to accelerated observers in Minkowski spacetime is based on the hypothesis of locality, namely, the assumption that an accelerated observer is locally equivalent to a momentarily comoving inertial observer. The worldline of an accelerated observer in Minkowski spacetime is curved and this curvature depends on the observer’s translational and rotational acceleration scales. The hypothesis of locality is thus reasonable if the curvature of the worldline could be ignored, i.e. if the phenomena under consideration have intrinsic scales that are negligible as compared to the acceleration scales of the observer. The accelerated observer passes through a continuous infinity of hypothetical comoving inertial observers along its worldline; therefore, to go beyond the hypothesis of locality, it appears natural to relate the measurements of an accelerated observer to the class of instantaneous comoving inertial observers. Consider, for instance, an electromagnetic radiation field Fi j in an inertial frame and an accelerated observer carrying an orthonormal tetrad frame e i a Žt . along its worldline. Here t is its proper time, the Latin indices i, j, k, . . . , which run from 0 to 3, refer to spacetime coordinates Žholonomic indices., while the Greek indices a , b , g , . . . , which run

Fab Ž t . s Fa b Ž t . q

Ht K

gd ab

Ž t ,t X . Fgd Ž t X . dt X ,

0

Ž 2. where Fab is the field actually measured, t 0 is the instant at which the acceleration begins and the kernel K is expected to depend on the acceleration of the observer. A nonlocal theory of accelerated observers has been developed w9,10x based on the assumptions that Ži. K is a convolution-type kernel, i.e. it depends only on t y t X , and Žii. the radiation field never stands completely still with respect to an accelerated observer. The latter is a generalization of a consequence of Lorentz invariance for inertial observers to all observers. In the space of continuous functions, the Volterra integral Eq. Ž2. provides a unique relationship between Fab and Fa b . It is possible to express Ž2. as t

Fab Ž t . s Fa b Ž t . q

Ht R

gd ab

Ž t ,t X . Fgd Ž t X . dt X ,

0

Ž 3. where R is the resolvent kernel and if K is a convolution-type kernel as we have assumed in Ži., then so is R, i.e. R s RŽt y t X .. Assumption Žii. then implies that RŽt . s

d LŽ t q t 0 . dt

Ly1 Ž t 0 . ,

Ž 4.

U. Muench et al.r Physics Letters A 271 (2000) 8–15

10

where R and L are 6 = 6 matrices and L is defined by Ž1. expressed as Fˆ s L F in the six-vector notation. Here Fˆ denotes the field as referred to the anholonomic frame. This nonlocal theory, which is consistent with all observational data available at present, has been described in detail elsewhere w10x. It proves interesting to provide a concrete example of the nonlocal relationship Ž2.. Imagine an observer that moves uniformly in the inertial frame along the y-axis with speed c b for t - t 0 and for t G t 0 rotates with uniform angular speed V about the z-axis on a circle of radius r, b s r Vrc, in the Ž x, y .-plane, see Fig. 1. In this case, e i 0ˆ s g Ž 1,y b sin w , b cos w ,0 . , e i 1ˆ s Ž 0,cos w ,sin w ,0 . , e i 2ˆ s g Ž b ,y sin w ,cos w ,0 . , e i 3ˆ s Ž 0,0,0,1 . ,

Ž 5.

in Ž ct, x, y, z . coordinates with w s V Ž t y t 0 . s g V Žt y t 0 .. Here w is the azimuthal angle in the Ž x, y .-plane and g is the Lorentz factor. Using sixvector notation,

Ž Fab .

™ EBˆˆ

,

Ž Fa b .

™ EB

,

Ž 6.

one can show that with respect to the tetrad frame Ž5. E s Eˆ q

t

Ht

B s Bˆ q

v = Eˆ Ž t X . y

0

t

Ht

0

a c

a

= Bˆ Ž t X . dt X ,

Ž 7.

= Eˆ Ž t X . q v = Bˆ Ž t X . dt X ,

Ž 8.

c

where a is the constant centripetal acceleration of the observer and v is its constant angular velocity. These quantities can be expressed with respect to the triad e i A as a s Žyc bg 2 V , 0, 0. and v s Ž0, 0, g 2 V .. For an arbitrary accelerated observer, we expect that the relations analogous to Ž7. and Ž8. would be much more complicated.

Fig. 1. The path of an observer in space moving with constant angular velocity around the z-axis for t )t 0 .

Imagine now a general congruence of accelerated observers such that relations similar to Ž2. and Ž3. hold for each member of the congruence. The requirement that the electromagnetic field Fi j Žor Fa b . satisfy Maxwell’s equations would then imply, via Ž3., that the field Fab would satisfy certain complicated integro-differential equations, which could then be regarded as the nonlocal Maxwell equations for Fab . Instead of this system, we give here a different, but analogous, acceleration-induced nonlocal electrodynamics and study some of its main properties.

3. Charge & flux electrodynamics with a new nonlocal ansatz The electrodynamics of charged particles and flux lines, see w15,16x and the references cited therein, involves the electromagnetic field strength Fab – that is defined via the Lorentz force law and is directly related to the conservation law of magnetic flux – as well as the electromagnetic excitation H ab that is directly related to the electric charge conservation. The corresponding Maxwell equations are metric-free and in Ricci calculus in arbitary frames read Žcf. w17,18x.

E w a Fbg x y Cw a b d Fg x d s 0 , Eb H ab y 12 Cbg a H gb y 12 Cbg

Ž 9. b

H ag s J a .

Ž 10 .

a

Here J is the electric current and the C’s are the components of the object of anholonomicity: Cab g [ 2 e i a e j b E w i e j x g s yCba g .

Ž 11 .

U. Muench et al.r Physics Letters A 271 (2000) 8–15

Ordinarily for vacuum, we would have the constitutive equation H ab s y g g a m g bn Fmn .

'

Ž 12 .

However, this reformulation of electrodynamics allows for much more general constitutive relations between H ab and Fab . In particular, it is possible to develop a nonlocal ansatz based on a generalization of Eq. Ž12. along the lines suggested by Obukhov and Hehl w15x H ab Ž t , j . s y g g a m g bn

'

In the following, we explore the consequences of the new ansatz Eq. Ž14. for a general accelerated observer in Minkowski spacetime.

4. The new ansatz and the accelerating and rotating observer It has been shown in w19,20x, and the references cited therein, that the orthonormal frame ea of an arbitrary observer with local 3-acceleration a and local 3-angular velocity v reads

= Kmn rs Ž t ,t X , j . Frs Ž t X , j . dt X ,

H

Ž 13 . where the kernel K corresponds to the response of the medium and j A , A s 1,2,3, are the Lagrange coordinates of the medium. As an alternative to Mashhoon’s model but along the same line of thought, see equation Eq. Ž2., one can develop an acceleration-induced nonlocal constitutive relation in vacuum via equation Eq. Ž13. by using the ansatz,

e 0ˆ s 1q

H

Ž t . s 'y g g a m g bn t

Ht

X

r

X

qG 0 n Ž t y t . Fm r Ž t . dt

X

,

Ž 14 .

Gabg [ ggd Ga b s s yGagb .

1 2

Ž yCa bg q Cbga y Cga b .

c

/E

B

,

Ž 16 .

where the barred coordinates are the standard normal coordinates adapted to the worldline of the accelerated observer. The coframe q a can be computed by inversion. We find a

P x dx 0 s Ndx 0 ,

/

c2

q A s dx A q

where the integral is over the worldline of an accelerated observer in Minkowski spacetime as before. Here the response of the ‘medium’ is simply given by the Levi-Civita connection of the accelerated observer in vacuum and the local constitutive relation Eq. Ž12. is recovered for inertial observers. We recall that in an orthonormal frame the connection is equivalent to the anholonomicity, see w17x: d

Px

B

=x

e A s EA ,

Fmn Ž t .

0

ž

E0 y

c2

ž

G 0 m r Ž t y t X . Frn Ž t X .

yc

v

1 a

q 0ˆ s 1 q ab

11

v

ž

c

A

=x

/

dx 0 s dx A q N A dx 0 . Ž 17 .

In the Ž1 q 3.-decomposition of spacetime, N and N A are known as lapse function and shift Õector, respectively. The frame and the coframe are orthonormal. The metric reads as follows: ds 2 s ha b q a m q b s

ž

1q

c

2

Px

v y2

ž

c

v

2

a

=x

/ ž / y

A

c

2

=x

/

Ž dx 0 .

2

dx 0 dx A y dA B dx A dx B ,

Ž 18 . Ž 15 .

If we invert Ž15., we find that Cabg s y2 G w a b xg .

where Ž v = x . A s e A s e i A a i.

B C

v B x C , a s a A e A , and a A

U. Muench et al.r Physics Letters A 271 (2000) 8–15

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Starting with the coframe, we can read off the connection coefficients Žfor vanishing torsion. by using Cartan’s first structure equation dq a s yGb a n q b with Gb a s Gi b a dx i. By construction, the connection projected in spacelike directions vanishes, since we have spatial Cartesian laboratory coordinates. Thus we are left with the following nonvanishing connection coefficients:

G 00ˆ A s yG 0 A0ˆ s

aA c2

c

.

a Arc 2 1 q a P xrc 2

e A B C v Crc 1 q a P xrc 2

,

.

Ž 20 .

t

Ht Ž G ˆ

00

C

FC D

0

qG 0 D C F0ˆ C . dt X

Ž 21 .

or

y

v Žtyt X . =EŽt X .

c

t

= B Ž t X . dt X .

q

Ž 22 .

Ht

Ž 23 .

v Žtyt X . =BŽt X .

0

aŽ t y t X . c

= E Ž t X . dt X ,

Ž 24 .

respectively. Clearly, for constant a and v our nonlocal relations Ž22. and Ž24. are the same as Eq. Ž7. and Eq. Ž8. provided we identify H with F , i.e. we postulate that the field actually measured by the accelerated observer is the excitation H . This agreement does not extend to the case of nonuniform acceleration, however, as will be demonstrated in the next section.

5. Nonuniform acceleration To show that the new ansatz Eq. Ž14. is different from Mashhoon’s ansatz Eq. Ž2. for the case of nonuniform acceleration even when we identify H with F , we proceed via contradiction. That is, let us assume that Fab s Ha b and hence from Eq. Ž22. and Eq. Ž24.

K Žt . s

0

aŽ t y t X .

ˆ

qG 0 E 0 FD 0ˆ q G 0 E C FDC . dt X

HsBq

ˆ ˆˆ H 0 B Ž t . s h 00h B D F0ˆ D Ž t . y c

t

F0ˆ E q G 0 D C FC E

0

Ž 19 .

In general, of course, the translational acceleration a and the angular velocity v are functions of time. Let us return to Ž14.. If we study the electric sector of the theory, we find, because of Ž19.,

Ht

0ˆ 0D

C

The first index in G is holonomic, whereas the second and third indices are anholonomic. If we transform the first index, by means of the frame coefficients e i a , into an anholonomic one, then we find the totally anholonomic connection coefficients as follows:

DsEq

t

Ht Ž G

or

v

G 0ˆ A B s yG 0ˆ B A s

H A B s h A Dh B E FD E y c

,

G 0 A B s yG 0 B A s e A B C

G 00 ˆ ˆ A s yG 0ˆ A0ˆ s

Similarly, for the magnetic sector, the corresponding relations read

Kv Ka

yK a , Kv

Ž 25 .

where K v s v Žt . P I and K a s aŽt . P Irc. Here IA , Ž IA .B C s ye A B C , is a 3 = 3 matrix that is propor-

U. Muench et al.r Physics Letters A 271 (2000) 8–15

13

tional to the operator of infinitesimal rotations about the e A-axis. We must now prove that in general RŽt . given by Eq. Ž4. cannot be the resolvent kernel corresponding to K Žt . given by Eq. Ž25.. To this end, consider an observer that is accelerated at t 0 s 0 and note that for kernels of Faltung type in equations Eq. Ž2. and Eq. Ž3. we can write F s Ž I q K . Fˆ and

ˆ Ž I qR. F , Fs

Ž 26 .

Fig. 2. The acceleration of an observer that is uniformly accelerated only during a finite interval from t s 0 to t s a .

respectively, where f Ž s . is the Laplace transform of f Žt . defined by `

f Ž s. [

H0

ys t

f Žt . e

dt

Ž 27 .

It is now possible to work out Eq. Ž28. explicitly and conclude that for X Ž s . [a Ž t . sinhQ ,

and I is the unit 6 = 6 matrix. Hence, the relation between K and R may be expressed as

Ž I q K . Ž I qR. sI .

Ž 28 .

Z Ž s . [a Ž t . ,

Y s ZŽ1 q X . .

Y Ž s. s

ZŽ s. 1 yZ 2 Ž s.

V , U

.

Ž 33 .

On the other hand, we have `

U yV

Ž 32 .

These relations imply that

ZŽ s. s R Ž t . s aŽ t .

Ž 31 .

we must have X s YZ ,

Imagine now an observer that is at rest on the z-axis for y` - t - 0 and undergoes linear acceleration along the z-axis at t s 0 such that aŽt . s g ) 0 for 0 F t - a and aŽt . s 0 for t G a Žsee Fig. 2.. That is, the acceleration is turned off at t s a and thereafter the observer moves with uniform speed ctanhŽ g arc . along the z-axis to infinity. Thus in Eq. Ž25., K v s 0 and K a s aŽt . I3rc. On the other hand, one can show that Eq. Ž4. can be expressed in this case as

Y Ž s . [a Ž t . coshQ ,

ys t

H0 aŽ t . e

dt s

g s

Ž 1 y ey a s .

Ž 34 .

Ž 29 . and

where U s J3 sinhQ , V s I3 coshQ , and Ž J3 . A B s dA B y dA3 d B 3 . Here we have set c s 1 and t

Q Žt . s

H0 aŽ t . dt s

½

gt, g a,

0Ft-a , tGa .

Y Ž s. s

1

`

H aŽ t . Ž e 2 0

Q

q ey Q . eyst dt

g 1 y ey Ž syg . a

Ž 30 .

s

1 y ey Ž sqg . a q

2

syg

sqg

.

Ž 35 .

U. Muench et al.r Physics Letters A 271 (2000) 8–15

14

cality is induced by the acceleration of the observer in a similar way as in Mashhoon’s model. An explicit example of nonuniform acceleration has been used to show that the two nonlocal prescriptions discussed here are in general different.

Acknowledgements

Fig. 3. Plot of the functions Y Ž s . and W Ž s . [ ZŽ s .rw1y Z 2 Ž s .x for a g s 2.

™

We consider only the region s ) g in which X Ž s . and Y Ž s . remain finite for a `. Comparing Ž35. with gs Ž 1 y eya s .

Z 1yZ2

s

s 2 y g 2 Ž 1 y ey a s .

2

,

One of the authors ŽF.W.H.. would like to thank Bahram Mashhoon and the Department of Physics & Astronomy of the University of Missouri-Columbia for the invitation for a two-month stay and for the hospitality extended to him. This stay has been supported by the VW-Foundation of Hannover, Germany. We are grateful to Yuri Obukhov ŽMoscow. for highly interesting and critical discussions on nonlocal electrodynamics. Thanks are also due to Guillermo Rubilar ŽCologne. for helpful correspondence.

Ž 36 .

™

we find that, contrary to Ž33., they do not agree except in the a ` limit Žsee Fig. 3.. Therefore, we conclude that the two models are different if one considers arbitrary accelerations.

6. Discussion If one rewrites Mashhoon’s nonlocal electrodynamics in the framework of charge & flux electrodynamics in vacuum by substituting the generalization of equation Eq. Ž3. for a congruence of accelerated observers in equations Eq. Ž9. –Eq. Ž12., one finds a rather complicated implicit nonlocal constitutive law. The Maxwell equations expressed in terms of the excitations Ž D, H . and field strengths Ž E, B . remain the same, a fact which is significant since otherwise the conservation laws of electric charge and magnetic flux would be violated. In this Letter, we have developed an alternative nonlocal constitutive ansatz within the framework of charge & flux electrodynamics such that the nonlo-

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