Foundations of Physics, Vol. 33, No. 9, September 2003 (© 2003)

The Gravitational Field of a Circulating Light Beam
Ronald L. Mallett 1
Received April 27, 2003
Exact solutions of the Einstein field equations are found for the exterior andinterior gravitational field of an infinitely long circulating cylinder of light. The
exterior metric is shown to contain closed timelike lines.
KEY WORDS: general relativity and gravitation; exact solutions; closedtimelike lines; lasers.

1. INTRODUCTION
A number of interesting post-Newtonian phenomena are known to occur
for rotating distributions of matter in Einstein’s general theory of relativity.
Inertial frame dragging, for example, is a consequence of the weak gravitational field of a slowly rotating massive sphere. (1) In addition, exact solutions of the Einstein field equations indicate the presence of closed timelikelines for rotating Kerr black holes, (2) van Stockum rotating dust cylinders, (3, 4) and the rotating universe of Godel. (5)Recently, the author (6) solved the linearized Einstein field equations toobtain the gravitational field produced by the electromagnetic radiation ofa unidirectional ring laser. It was shown that a massive spinning neutral
particle at the center of the ring laser exhibited inertial frame dragging.
Photonic crystal wave guides with the capability of very sharp lightbending (short radius of curvature) have been demonstrated. (7) Directing alaser beam along a helical path to form a solenoid of light in a photonic
crystal (see Fig. 1) would enhance the strength of the gravitational field
generated by a circulating light beam. A generalized model of the system

1

Department of Physics, 2152 Hillside Road, University of Connecticut, Storrs, Connecticut
06269; e-mail: RLMallett@aol.com
1307
0015-9018/03/0900-1307/0 © 2003 Plenum Publishing Corporation1308

MallettFig. 1. Solenoidal laser beam throughwave guide photonic crystals.

just described would be a long circulating cylinder of light. Such a model isthe subject of the present paper.
In this paper exact solutions of the Einstein field equations are found
for the exterior and interior gravitational field produced by an infinitely
long circulating cylinder of light. The exterior solution is shown to containclosed timelike lines.

2. GRAVITATIONAL FIELD OF CIRCULATING LIGHT
We seek a solution of the Einstein field equations given by (c=G=1)
Gmn =8pTmn ,(1)with the energy momentum tensor of axially circulating radiation represented byTmn =Egm gn ,(2)where E is the radiation energy density with x m=(x 0, x 1, x 2, x 3 )=(t, r, z, j)
and gm a null vector such thatgm g m=0,(3)gm =(g0 , 0, 0, g3 ).

(4)withThe Gravitational Field of a Circulating Light Beam1309

Consider a canonical form of an axisymmetric line element (8, 9)
ds 2=f dt 2 − 2w dt dj − l dj 2 − e m(dr 2+dz 2),(5)with metric components
g00 =f,g03 =−w,

g33 =−l,g11 =g22 =−e m,(6)g 00=D −2 l,

g 03=−D −2 w,

g 33=−D −2f,

g 11=g 22=−e −m,(7)where D is defined by
D 2 — fl+w 2.

(8)Since Eq. (1) can be written as
Gmn =R mn − 12 gmn R=8pTmn ,(9)then Eqs. (1)–(3) and Eq. (9) imply
R mn =8pTmn ,

(10)

R=0.

(11)

Using Eqs. (6)–(8) to compute the Ricci components R 00 , R 03 , and R 33
for f=f(r), l=l(r), and w=w(r) gives
R 00 =12 e −mD[(D −1fr )r +D −3fF],(12)

R 03 =− 12 e −mD[(D −1 wr )r +D −3wF],(13)

R 33 =− 12 e −mD[(D −1 lr )r +D −3lF],(14)

where F is defined by
F — fr lr +w 2r ,(15)with fr — “f/“r, wr =“w/“r,... . Equations (7) and (11), with R 11 =R 22=0, imply
D −2(lR 00 − 2wR 03 − fR 33 )=0.

(16)

Equations (12)–(16) yieldD rr =0.

(17)

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Choosing the solution of Eq. (17) such that D=r simplifies the field
equations and allows the identification of r as the canonical cylindricalcoordinate. Equation (8) now has the constraint
D 2=fl+w 2=r 2.

(18)

Using Eqs. (3), (4), and (6), we have
g 0=tg 3,(19)

t=f −1(w+r).

(20)

where

Now for Tmn ] 0, using Eqs. (2), (6), (10), (20) with gm =gml g l, we findthe following combinations
R 00 +t −1R 03 =0,(21)R 00 − t −2R 33 =0,(22)−1(23)

R 03 +t R 33 =0.

Substituting Eqs. (12)–(14) in Eqs. (21)–(23), using Eqs. (18) and (20)yields, respectively,t(r −1fr )r − (r −1wr )r +r −2F=0,
2−1−1−2(24)t (r fr )r +(r lr )r +2r tF=0,(25)t(r −1wr )r +(r −1lr )r +r −2tF=0.

(26)

Equation (20) implies that
ft=w+r.

(27)

Substituting Eq. (27) in Eq. (24) for t constant gives
F=fr lr +w 2r =1.

(28)

Using Eq. (28) in Eq. (26) yields(r −1 Wr )r +r −2=0,(29)W — w+tl.

(30)

whereThe Gravitational Field of a Circulating Light Beam1311

A solution of Eq. (29) is given by
W=r.

(31)

For points outside the light cylinder Tmn =0, so that Eqs. (10) and (13)together with Eqs. (18) and (28) result in a differential equation for w givenbyr 2 wrr − rwr +w=0.

(32)

Equation (32) has as a solution
w=lr ln(r/a),(33)

where a is a positive constant and l is a constant proportional to theenergy density per unit length. Substituting Eq. (33) in Eq. (27), setting theconstant t=a, gives
f=r/a+l(r/a) ln(r/a).

(34)

The function l(r) can be obtained from the constraint Eq. (18) in the form
l=(r 2 − w 2)/f.

(35)

Substituting Eqs. (33) and (34) in Eq. (35) gives
l=ra − lra ln(r/a).

(36)

It is seen that Eqs. (33) and (36) are consistent with Eqs. (30) and (31) fort=a. It is also straightforward to show that Eqs. (33), (34), and (36)satisfy fr lr +w 2r =1 and are thus consistent with Eq. (28).
The Einstein field equations for R 11 =0 and R 22 =0 with F=1 giveR 11 =− 12 mrr +12 r −1mr +12 r −2=0,
12

12−1

R 22 =− mrr − r mr =0.

(37)
(38)

Equations (37) and (38) yield the result−1e m=(r/a) 2 .

(39)

Thus, the metric functions f, w, l, and e m in Eqs. (33), (34), (36), and(39) give the solution of the Einstein field equations for the exterior of thecirculating light cylinder.

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We now turn to a discussion of the interior solution. Consider them=0, n=3 component of Eq. (10)R 03 =8pT03 ,(40)T03 =Eg0 g3 .

(41)

whereFor g m=(1, 0, 0, t −1 ), using Eqs. (13), (18), (28), and (41), with t=a andgm =gml g l in Eq. (40), we find(r −1wr )r +r −3w=16pEe mr/a.

(42)

We proceed by determining the Green’s function which satisfies the equation(r −1Gr )r +r −3G=−d(r − rŒ).

(43)

The solution of Eq. (42) is then given by16p
F G(r, rŒ) Ee mrŒ drŒ,w(r)=a(44)with the Green’s functionln rŒ,3 −rrŒ−rrŒ ln r,G(r, rŒ)=r < rŒ,r > rŒ.

(45)

It can be verified by direct substitution that Eq. (44), integrated from r=0
to the boundary r0 of the cylinder, with Eq. (45) is a solution of Eq. (42).
Equation (44) with Eq. (45), matched at the boundary with Eq. (33), allows
a determination of the relation between the parameters l and a and E
and r0 . Using Eqs. (27) and (35) with Eq. (44) yields
r 16p
f= + 2 F G(r, rŒ) Ee mrŒ drŒ,a a(46)l=ra − 16p F G(r, rŒ) Ee mrŒ drŒ.

(47)

It can be shown that Eqs. (46) and (47) with Eq. (45) satisfy, respectively,
the Einstein field equations R 00 =8pT00 and R 33 =8pT33 . The interior functions f, w, and l in Eqs. (44), (46), and (47), with R 11 =0 and R 22 =0, alsosatisfy the equation R=r −2(lR 00 − 2wR 03 − fR 33 )=0. The function e m isagain given by Eq. (39).

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An interesting consequence of the exterior metric is obtained by considering the line element in Eq. (5) for fixed t, r, and z. Equation (5) thenreduces to
ds 2=−l dj 2.

(48)

Equation (36) implies that for l ln(r/a) > 1 then l < 0 so that the curves
given by Eq. (48) under these conditions are closed and timelike. Note that
under a coordinate transformation tŒ=t − aj then Eq. (5) with Eqs. (33),(34), and (36) takes the form
ds 2=f dt −2 − 2wŒ dtŒ dj − e m(dr 2+dz 2 ),(49)where wŒ=r with tŒ a periodic coordinate.

3. CONCLUSIONIt has long been known (3, 4) that the van Stockum solution for theexterior metric of an infinitely long rotating dust cylinder contains closed
timelike lines. The present paper has shown closed timelike curves also
occur for an infinitely long circulating cylinder of light. This model also
shares some of the same limitations as the van Stockum solution in that themetric is not asymptotically flat. Bonnor, (4) however, has emphasized thatcertain aspects of an infinitely long rotating dust cylinder may be sharedby a long finite one. This may also apply to a long but finite circulating
cylinder of light.

ACKNOWLEDGMENTS
The author thanks C. Roychoudhuri for a number of useful discussions and for the illustration that appears in Fig. l. He also wishes tothank A. Guth for a number of critical remarks, as well as, R. Bartram,
M. Silverman, W. Smith, W. Stwalley, K. Hakmiller, and participants ofthe third biennial conference of IARD for helpful comments.

REFERENCES
1. I. Ciufolini and J. A. Wheeler, Gravitation and Inertia (Princeton University Press, Princeton,
1995), Chap. 6.
2. B. Carter, Phys. Rev. 174, 1559 (1968).

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3. W. J. van Stockum, Proc. Roy. Soc. Edinburgh 57, 135 (1937).
4. W. B. Bonnor, J. Phys. A: Math. Gen. 13, 2121 (1980).
5. K. Gödel, Rev. Modern Phys. 21, 447 (1949).
6. R. L. Mallett, Phys. Lett. A 269, 214 (2000).
7. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow ofLight (Princeton University Press, Princeton, 1995).
8. T. Lewis, Proc. Roy. Soc. London Ser. A 136, 176 (1932).
9. M. Carmeli, Classical Fields: General Relativity Gauge Theory (Wiley, New York, 1982),
p. 365.