16May 1994 PHYSICS
Physics Letters A 188 (1994) 125-129
Topological constraints on Maxwell fields in Robertson-Walker space-times W. Oliveira Departamentode Fisica, UniversidadeFederal de Juiz de Fora, 36036-330 Juiz de Fora-MG,Brazil
‘, A.F.F. Teixeira
Centro Brasileirode PesquisasFisicas, 22290-180 Rio de Janeiro-M, Brazil Received 5 October 1993; revised manuscript received 8 March 1994; accepted for publication 14 March 1994 Communicated by J.P. Vigier
Abstract Two spatially homogeneous solutions of Maxwell’s equations with a source in the elliptic Robertson-Walker (RW) spacetime geometry are found. It is shown that although both solutions can be accommodated in the RW space-time manifolds whose sections t=const are three-spheres S3, only one of them is admissible when the sections are quatemionic manifolds Q’, making explicit the existence of topological constraints on Maxwell fields in Robertson-Walker space-times.
In most theories of gravitation of some generality and scope a model of the physical world commences with a four-dimensional space-time (representing the physical events) endowed with a locally Lorentzian metric. Representing the physical phenomena in this space-time we have fields satisfying appropriate differential equations (the physical laws). Finally, fields and geometry are coupled according to the corresponding gravitational theory one is dealing with. The space-time geometries arising as solutions of the gravitational field equations predetermine to some extent the laws which are supposed to govern the behaviour of the physical fields. The line element corresponding to a given geometry just tells us how to compute the separations beI E-mail: [email protected]
tween space-time events whose coordinates differ by infinitesimal quantities. However, it does not tell us whether two events are really physically far apart. Actually, even when the metric tensor gPy is given, the topological features of the space-time manifold are generally left unsettled, and for a further retinement to this picture of the physical world we ought to take into account possible underlying topologies. Since the topological properties are perhaps more fundamental than the differentiable structure on which tensor analysis is based, it is important to know which physical results concerning a space-time geometry depend upon or somehow involve the topological features of the underlying manifold. In other words, in order to be confident about the physical results one derives it is often necessary to have some degree of control over the large-scale structure of the space-time geometry. As electromagnetism is a long range interaction the
0375-9601/94/gO7.00 0 1994 Elscvier Science B.V. All rights reserved SSDZO375-9601(94)00212-S
W. Oliveira et al. /Physics Letters A 188 (1994) 125-129
electromagnetic field seems to be particularly suitable for revealing far away conditions imposed by global constraints [ 11. Regardless of the theory of gravitation one may be concerned with, the combination of Weyl’s postulate (which ensures the existence of a global cosmic time) and the cosmological principle of homogeneity and local isotropy determine the geometry of space-time as given by the Robertson-Walker line element
are endowed with the same elliptic (K= 1) Robertson-walker space-time metric ( 1.1). They are different in that the sections t=const are two distinct compact, globally homogeneous and locally isotropic manifolds, namely S3 and Q3. The elliptic Robertson-Walker geometry is usually given in the form ( 1.1) . Nevertheless we shall use cylindrical coordinates, which prove to be more suitable to work with the symmetries of the Maxwell fields we shall discuss. A straightforward but rather combersome calculation shows that the transformation
where ~=const, t is the cosmic time (t=const defines a world-wide simultaneity), and (r, 0, #) are the spatial coordinates. The above assumptions place no restriction on A as a function of time, nor do they impose any relation between A ( t ) and K.Actually the behaviour of the scalar factor A(t) as well as the possible values of the curvature parameter K (0, f 1) have to be eventually determined by dynamical considerations imposed by the particular theory of gravitation one is dealing with. In the present Letter we discuss two solutions of Maxwell’s equations in the elliptic (K= 1) Robertson-walker space-time geometry ( 1.1) with the section t = const endowed with two different compact topologies, namely the three-sphere S3 and the clockwise quatemionic topology Q3 [ 2-41. We show that the elliptic geometry ( 1.1) can accommodate both solutions of Maxwell’s equations if its section t = const is the simply connected compact manifold S3. However, if its three-space is endowed with the multiply connected compact quatemionic topology Q’, it can accommodate only one of the two solutions of Maxwell’s equations. In other words, this latter topology excludes a solution of Maxwell’s differential equations. Most of the calculations for this article were made by using the suite of algebraic computing programs CLASS1 [ 5,6] written in the symbolic manipulation system SHEEP [ 6,7 1.
sin 8 ,
r -co!38 I_ +2
brings the line element ( 1.1) into the form d.r2=dt2-A2(t)(dp2+sin2pd$2+cos2pdC2), (2.3) where (t, p, @, C) may be thought of as cylindrical coordinates #‘. We shall now study two solutions of Maxwell’s equations in the elliptic Robertson-Walker background ( 2.3 ) . To this end, we define at each point of our space-time manifolds a set of pseudo-orthonormal real one-forms aA (A =O, 1,2, 3), such that the element ( 2.3 ) can be put into the form ds2 = fj.&JY%B )
where rjAB=diag( 1, - 1, - 1, - 1). The basis of one-forms mA= EA,dxfi is defined up to Lorentz transformations. A set of ok’ particularly adequate for our purpose is given by wO=dt, w’=A(t)
(2.5) [cos([email protected]
+i sin2psin w2=A(t)[
+f sin2pcos(C-~)(d~+dC)l, 09=A(t)[cos2pdC-sin’pd#] .
2. Maxwell’s equations
X1Notice that for t = const and when p-0 the line element of the
The four-dimensional space-time manifolds, arenas for the Maxwell fields we shall be concerned with,
three-space reduces to that of the flat space in cylindrical coordinates.
W. Oliveira et al. /Physics Letters A 188 (1994) 125-129
It should be mentioned at this point that similarly to the ordinary Euclidean three-geometry, in the elliptic three-geometry one can have parallelism of finitely separated vectors regardless of the path joining them. This peculiar teleparallelism is known as Clifford parallelism. Actually there are two kinds of Clifford parallelisms, namely the clockwise and the anticlockwise Clifford parallelisms. At a point B the vector clockwise (anticlockwise) Clifford parallel to a vector at another point A, is obtained by Christoffel (parallel) transport along the geodesic joining A and B, followed by clockwise (anticlockwise) rotation, around the geodesic line, proportional to the distance between the two points (for more details see, e.g., p. 108 of Ref. [ 81). For a given A(t) the fields of the one-forms w’, w2 and w3 have been constructed, respectively, through clockwise Clifford transport of the Euclidean one-forms d_x=cos $ dp, dy = sin @dp and dz= dc, defined at the origin of the cylindrical coordinates (p=c=O). It is worth mentioning that the above clockwise and anticlockwise Clifford transports correspond, respectively, to the right and left translations recently discussed by King [ 9 ] in connection with two concepts of homogeneity in the three-sphere. In a local frame (&} Maxwell’s equations take the fOl-lll
FABiB= -JA , F[AB;c,
(2.10) z(t) =ylA2 ,
and & = -H(t) Plg=H(t) E23 =
time coordinate t. It is worth mentioning that another pseudo-orthonormal frame exists H2,relative to which the field FATihas only one nonvanishing component, i.e. Err= - H( t ) . For the sake of uniformity, though, we shall use just the frame defined by Eqs. (2.5)-(2.8) to study both fields. Before proceeding to the discussion of the electromagnetic fields, it seems important to stress that according to King [ 9 ] there are two types of spatial homogeneity for tensor fields in the three-sphere S3, namely left and right homogeneity. This is so because in S3 one can globally define both left and right invariant vector fields. Using King’s definitions of homogeneity the above Maxwell field FAB is right homogeneous, whereas f;;ABis left homogeneous. We notice that as far as the clockwise quaternionic manifold Q’ is concerned we can define only one type of homogeneity for tensor fields, namely right homogeneity, since in this manifold only the right invariant vector fields can be globally defined. This fact is in agreement with the exclusion of the left homogeneous Maxwell field Fa by the topology of Q’. Using now the package ELDYNF of the suite of algebraic computing programs CLASS1 we referred to in the Introduction [ $61, one can easily show that for the electromagnetic field FABgiven by (2.11) the Maxwellequations (2.9), (2.10) forJC=-2A-‘(0, X, Y, 2) are satisfied as long as
where FABis the Maxwell tensor, J* is the current fourvector and the bracket denotes antisymmetrization. To reveal our major goal in this work we shall study two spatially homogeneous magnetic fields FAB and p’. in the elliptic RW background (2.3 ) . In the frame (2.5H2.8) they are given by
cos 2p , sin2pcos([email protected]
) , sin2psin(c-#) ,
(2.12) (2.13) (2.14)
where H, X, Y and 2 are arbitrary functions of the
where cr, /3 and y are arbitrary real constants #3.As a matter of fact, in this case Maxwell’s equations reduce to three separate differential equations for X(t) , Y(t) andZ(t),allofthemoftheform (2.16) below, hence the solutions (2.15 ) . Similarly, for the electromagnetic field F&, given by (2.12H2.14) one can show using ELDYNF that for *2 This basis of one-forms can be obtained throughthe anticlockwise Clifford transport of the Euclidean one-forms defined at the origin of the coordinates @=C=O) and is given by cSO=dt, (3’=A(t)[cos([email protected]
)(dC-d))], 6% A(t)[sin([email protected]
)@-t sin2pcos(C+#)(dC-de)], cY=A(t)x [[email protected]
]. *’ The constant Kin the computer algebrapackage ELDYNF has been taken equal to - 2.
W. Oliveira et al. /Physics Letters A 188 (1994) 125-129
sin 2p sin(C-+), cos2p]
Maxwell’s equations ( 2.9 ) , ( 2.10 ) reduce to a single ordinary differential equation, namely ii+2H $0,
which can be easily integrated to give H=H,,A-2,
where Ho is an arbitrary real constant. To close this Section we mention that similarly to the fields of one-forms @A (A= 1, 2, 3) the electromagnetic fields FABand pABwere figured out through the use of two congruences of paratactics (clockwise and anticlockwise, respectively), which are equidistant geodetic lines in the elliptic geometry obtained through the use of Clifford parallelisms, starting from a given geodesic. Our basic idea was to introduce spatially homogeneous Maxwell fields along the paratactics,see,e.g.,Refs. [8,10-121.
3. Topological constraints For a specific scalar factor A (t), the elliptic Robertson-walker geometry ( 1.1) determines the spacetime only locally - it does not contain global information about the space-time topology. When we endow the t=const sections of ( 1.1) with the topologies of either S3 or Q3, the two space-time manifolds become substantially different. The effective understanding of global properties of space-time manifolds quite often depends to a large extent on visualization and intuition. Sometimes it is difficult to perceive global arguments unless one already has a good qualitative idea of the features of the particular manifold one is concerned with. Although the unit three-sphere S’ is relatively well known to physicists [ 13,141, this is not so for the clockwise quatemionic manifold Q3. An account of the manifold Q3 is beyond the scope of this Letter and we refer the reader to Refs. [ 2-41. However, for the sake of completeness we mention that one can obtain Q3 by separating from a unit three-sphere S3 a solid curved cube of height ix, and identifying each
face with the opposite after a one-quarter clockwise turn . It should be stressed that with the above definition for the quatemionic manifold Q3, the field of oneforms aA defined by Eqs. (2.5)-(2.8) is compatible with the topological identifications. When the t = const section of the space-time manifold with ( 1.1) is S3, and since the three-sphere is the universal covering manifold of the elliptic geometry, there are no topological identifications. Thus, both solutions of Maxwell’s equations we have studied in the previous Section can be accommodated in the space-time manifold whose three-space is S3. Moreover, each magnetic field, (2.11) with (2.15 ) and (2.12)-(2.14) with (2.17), defines a congruence of paratactics (clockwise and anticlockwise) which entirely fill the manifold S3. On the other hand, when the t=const section of the space-time manifold is Q3 one has to consider the constraints imposed by the topological identifications, and it is clear in this case that one can have solutions of Maxwell’s equations consistent and inconsistent with these identifications. Here, although the congruence of clockwise paratactics defined by the electromagnetic field (2.11) with (2.15) is compatible with the identifications imposed by the topology of Q3, the Maxwell field (2.12)-(2.14) with (2.17) defines a congruence of anticlockwise paratactics, which does not couple with the topological identifications imposed by the clockwise quatemionic topology Q’. Thus, for example, in cylindrical coordinates, the events defined by P= (t, in, 0, 0) and Q= (t, arc, n, 0) are the same; however, a straightforward calculation shows that the only nonvanishing component of & at P and Q, namely Fi3, is H(t) at P, whereas at Q it is -H(t), making clear that this solution of Maxwell’s equations is excluded by the topological constraints imposed by the quatemionic topology Q’. It should be noticed that an anticlockwise quaternionic manifold Q’ exists in which the Maxwell field PAiAB can be globally defined whereas the (clockwise constructed) field FAB cannot. It is worth remarking that in the integration of the various partial differential equations corresponding to the physical laws a number of arbitrary functions of the coordinates (or constants) appear, and are chosen to match the fields and the geometry to the
W. Oliveira et al. /Physics Letters A 188 (1994) 125-129
topology of the underlying space-time manifold. In dealing with fields produced by bounded sources in noncompact simply connected manifolds as, for example, the Newtonian space-time manifold, these arbitrary functions or constants of integration are usually chosen so that the physical fields vanish at spatial infinity. However, as far as compact and/or multiply connected space-time manifolds are concerned the choice of satisfactory boundary conditions usually is not as simple as that, and one is often confronted with somehow unexpected results. Thus, for example, it is easy to show that the electrostatic field associated to a point charge q in a three-sphere S3 (a compact manifold) implies the existence of an opposite charge -q in the position antipodal to that ofq.
Acknowledgement Special thanks are due to Graham Hall and Reza Tavakol for fruitful discussions, and to Francis Wright for useful comments. We also thank an anonymous referee for drawing our attention to Ring’s paper and for careful suggestions for improvement of our manuscript. One of us (W.O. ) is grateful to Departamento de Fisica - PontifIcia Universidade Catolica do Rio de Janeiro, where part of this work was carried out. He also thanks CNPq and FAPERJ for fellowships.
[ 1 ] J.V. Narlikar and T.R. Seshadri, Astrophys. J. 288 ( 1985) 43. [ 21 J.R. Weeks, in Pure and applied mathematics, Vol. 96. The shape of space (Dekker, New York, 1985). [ 3 ] W.P. Thurston, The geometry and topology of 3-manifolds, Princeton University report, unpublished.  P. Scott, Bull. London Math. Sot. 15 (1983) 401. [ 5 ] J.E. Aman, A Manual for CLASSI: classification programs for geometries in general relativity, University of Stockholm, Institute of Theoretical Physics technical report (1987), distributed with the CLASS1 source.  M.A.H. MacCallum and J.E.F. Skea, SHEEP: A computer algebra system for general relativity, in: Algebraic computing in general relativity: Lectures notes from the first Brazilian school on computer algebra, Vol. 2, eds. M.J. Recoucas and W.L. Roque (Oxford Univ. Press, Oxford, 1993). [ 71 I. Frick, The computer algebra system SHEEP, what it can and cannot do in general relativity, University of Stockholm, Institute of Theoretical Physics technical report ( 1977). [ 8 ] D.M.Y. Sommerville, The elements of non-Euclidean geometry (Dover, New York, 1958).  D.H. Ring, Phys. Rev. D 44 (1991) 2356. [lo] J.L.C. Costa, I. Wolk and A.F.F. Teixeira, Phys. Rev. D 29 (1984) 2402. [ll]A.F.F.Teixeira,Phys.Rev.D31 (1985) 2132. [ 121 M.J. Reboucas and A.F.F. Teixeira, J. Math. Phys. 32 (1991) 1861. [ 131 C.W. Misner, KS. Thome and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [ 141 R.A. d’Invemo, Introducing Einstein’s relativity (Oxford Univ. Press, Oxford, 1992).