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Electromagnetic field generated by cosmic vorticity
Volume 85A, number 8,9
PHYSICS LETFERS
19 October 1981
ELECTROMAGNETIC FIELD GENERATED BY COSMIC VORTICITY Nikos A. BATAKIS Department of Physics, University of loannina, Greece Received 9 July 1981
In a Bianchi type II cosmological model with dust the requirement of non-zero vorticity seems to make mandatory through the field equations the inclusion of a non-vanishing electromagnetic field in the energy content of the model.
In the framework of quantum field theory, anomalous magnetic moments of neutral elementaiy partides are always connected with a non-vanishing spin, It is interesting to consider whether general relativity allows for an analogous classical situation in rotating spacetimes. Although it seems impossible to define an intrinsic magnetic moment of a spacetime (because there would be no “external” em field with which such a magnetic moment would interact) one could rather start by looking at cosmological models with an intrinsic em field. Such models have of course been investigated [1,2], but in all cases the em fields are more or less independent additives in the energy content and not related with any rotational motion of the model. Consider however the following case. It is often asserted that a Bianchi type II cosmological model with perfect fluid cannot sustain vorticity [3]. One way to understand why this happens is to realize that in some of the Einstein field equations there appear extra terms for which the requirement that they vanish leads to the result of zero vorticity. But one could prevent the vanishing of these terms by providing additional contributions to the energy—momentum tensor. It will turn out that in our case such an additional contribution can be provided adequately by an em field whose existence is thus made “mandatoiy” by the presence of the non-vanishing vorticity. One could of course reverse the above argument and state, that the presence of the em field has allowed the existence of vorticity. The latter behaviour is not unknown in cosmology and has been demonstrated in similar contexts [4], namely in situations in which a Bianchi type 0 031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
B model with em field exibits features which would normally occur in a type B’ > B model. For standard results and notation see ref. [1]. The spacetime here is a locally rotationally symmetric (LRS) manifold and the metric is diag(— 1, 1, 1, 1) in the orthonormal frame {~}with w0 = dt, w1 = aa1, w2 = ba2, w~ be3 a, b are functions of the cosmic time t and {a’} is a basis of invariant oneforms pertinent to a Bianchi type II group of motions with.da1 = a2 A a3, da2 = da3 = 0. The pressureless perfect-fluid content has energy density (81T)~pand its four-velocity u is tilted with respect to the hypersurfaces of homogeneity: u = —u°w0+ u1w1. One can now determine the conservation laws 2
~
—
pab —M, u1a = N
(1) (2)
U
where M, N are constants and (with * the duality operator on forms) the vorticity one-form 1
A
*~UAdU=w(u~ —uw) with magnitude =
2 3 There is however one problem with this model as deyeloped thus far. The 01 component of the field equations forces on us pu°u1= 0 which gives u1 = 0 and consequently the standard result of zero vorticity. But this may well not be the case. One may rather consider that vorticity makes effectively necessary the presence of some additional physical entity, whose contribution
409
Volume 85A, number 8,9
PHYSICS LETTERS
19 October 1981
to the field equations will excactly counterbalance the pu0u1 term in the 01 equation. It turns out that the electromagnetic field f (with df = d * f = 0) with components
all the quantities described earlier. (For a comparison with known excact solutions see ref. [2] and references therein.) It should be clear that our result is certainlymodel
f~ ~
dependent. However it is not contradicted by the existing data on cosmic vorticity and primordial em fields. Unfortunately the present estimates of these
=
f~ eL/~hk, =
of magnitude e 1, 2 = —h3 = —A1(ab)
e 3
a = A4V1e2f,
(5)
quantities are quite vague and do not allow any real quantitative testing. The electric and magnetic fields do not have to be of equal magnitude, as we chose for simplicityFurther, in (4), but should case be orthogonal. theythey should bothinbeany orthogonal to the axis of LRS. Also orthogonal to each other (as well as to the em field) are the four-velocity and vorticity vectors. The approximation lu’ k~1 improves as t -÷ In the opposite directions, it breaks down
b A2vef,
(6)
before see that, thenotice singularity that weat must t = 0 have can be a ~reached N = 2A2M—1 [5]. To
=
=
A2(ab)~,
(4)
and e1 = = 0 (A1, A2 are constants), does precisely that. The field equations can now be1 I~ solved. sim1 i.e.,For small plicity in [5]. the solution we Aassume 1u (with the divorticity Then, with a constant mension of a length) we find:
where
f:
~M
which is well valid as long as the energy density of the em field is negligible compared to that of matter. How-
f v~dt2
(7)
and v := ab2 is a solution of (a dot is d/dt): 2uif + 4~2 — 1.6Mth +Mu + 12M2t2
=
0.
In particular the 01 field equation (with A2 + A~)gives A2(ab)_2 = ~pu0u1, or 2A2 = MN. Eq. (9) together with eqs. (1)—(4) gives
(8) :=
A~ (9) (10)
where H~:= u0b2a1 can be thought of (with proper choice of constants) as the Hubble radius and h2:= + = e~ + e~.Eq. (10) states that the ratio of the em energy density to the rate of rotation is equal to the ratio of the matter energy density to the Bubble constant. There is an interesting special case of our solution. Notice first that v = ~Mt2 satisfies eq. (8). Then, eq. (7) givesf~ lnA~tand eqs. (5), (6) reduce to a = ~A3/2M1t1/2, b = ~A3I~Mt~/~ (11) Eqs. (11) can be used to obtain simple expressions for .
410
~.
ever the first increases faster as t -÷ 0 so it will eventually dominate. In the limit of zero vorticity our model describes a LRS Bianchi type II spacetime with its dust content moving orthogonal to the hypersurfaces of homogeneity. It is on this spacetime that vorticity and the em field have been added as a perturbation but in the inseparable manner described by eq. (10), as dictated by the field equations. It should be clear however that this result does not depend directly on the approximation used; eq. (10) is the entire content of the 01 field equation. References [1] A.K. Raychaudhuri, Theoretical cosmology (Clarendon, Oxford, 1979). [2] D. Kramer, H. Stephani, M. MacCallum and E. Herlt, Excact solutions of Einstein’s field equations WEB Deutscher Verlag der Wissenschaften, Berlin, 1980) Ch. 12. [3] C.B. Collins and S.W. Hawking, Mon. Not. R. Astron.
Soc. 162 (1973) 307.
L41 A.J. Fennelly, Phys. Rev. D21 (1980) 2107. [51 N.A. Batakis, Phys. Rev. D23 (1981) 1681.
PHYSICS LETFERS
19 October 1981
ELECTROMAGNETIC FIELD GENERATED BY COSMIC VORTICITY Nikos A. BATAKIS Department of Physics, University of loannina, Greece Received 9 July 1981
In a Bianchi type II cosmological model with dust the requirement of non-zero vorticity seems to make mandatory through the field equations the inclusion of a non-vanishing electromagnetic field in the energy content of the model.
In the framework of quantum field theory, anomalous magnetic moments of neutral elementaiy partides are always connected with a non-vanishing spin, It is interesting to consider whether general relativity allows for an analogous classical situation in rotating spacetimes. Although it seems impossible to define an intrinsic magnetic moment of a spacetime (because there would be no “external” em field with which such a magnetic moment would interact) one could rather start by looking at cosmological models with an intrinsic em field. Such models have of course been investigated [1,2], but in all cases the em fields are more or less independent additives in the energy content and not related with any rotational motion of the model. Consider however the following case. It is often asserted that a Bianchi type II cosmological model with perfect fluid cannot sustain vorticity [3]. One way to understand why this happens is to realize that in some of the Einstein field equations there appear extra terms for which the requirement that they vanish leads to the result of zero vorticity. But one could prevent the vanishing of these terms by providing additional contributions to the energy—momentum tensor. It will turn out that in our case such an additional contribution can be provided adequately by an em field whose existence is thus made “mandatoiy” by the presence of the non-vanishing vorticity. One could of course reverse the above argument and state, that the presence of the em field has allowed the existence of vorticity. The latter behaviour is not unknown in cosmology and has been demonstrated in similar contexts [4], namely in situations in which a Bianchi type 0 031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
B model with em field exibits features which would normally occur in a type B’ > B model. For standard results and notation see ref. [1]. The spacetime here is a locally rotationally symmetric (LRS) manifold and the metric is diag(— 1, 1, 1, 1) in the orthonormal frame {~}with w0 = dt, w1 = aa1, w2 = ba2, w~ be3 a, b are functions of the cosmic time t and {a’} is a basis of invariant oneforms pertinent to a Bianchi type II group of motions with.da1 = a2 A a3, da2 = da3 = 0. The pressureless perfect-fluid content has energy density (81T)~pand its four-velocity u is tilted with respect to the hypersurfaces of homogeneity: u = —u°w0+ u1w1. One can now determine the conservation laws 2
~
—
pab —M, u1a = N
(1) (2)
U
where M, N are constants and (with * the duality operator on forms) the vorticity one-form 1
A
*~UAdU=w(u~ —uw) with magnitude =
2 3 There is however one problem with this model as deyeloped thus far. The 01 component of the field equations forces on us pu°u1= 0 which gives u1 = 0 and consequently the standard result of zero vorticity. But this may well not be the case. One may rather consider that vorticity makes effectively necessary the presence of some additional physical entity, whose contribution
409
Volume 85A, number 8,9
PHYSICS LETTERS
19 October 1981
to the field equations will excactly counterbalance the pu0u1 term in the 01 equation. It turns out that the electromagnetic field f (with df = d * f = 0) with components
all the quantities described earlier. (For a comparison with known excact solutions see ref. [2] and references therein.) It should be clear that our result is certainlymodel
f~ ~
dependent. However it is not contradicted by the existing data on cosmic vorticity and primordial em fields. Unfortunately the present estimates of these
=
f~ eL/~hk, =
of magnitude e 1, 2 = —h3 = —A1(ab)
e 3
a = A4V1e2f,
(5)
quantities are quite vague and do not allow any real quantitative testing. The electric and magnetic fields do not have to be of equal magnitude, as we chose for simplicityFurther, in (4), but should case be orthogonal. theythey should bothinbeany orthogonal to the axis of LRS. Also orthogonal to each other (as well as to the em field) are the four-velocity and vorticity vectors. The approximation lu’ k~1 improves as t -÷ In the opposite directions, it breaks down
b A2vef,
(6)
before see that, thenotice singularity that weat must t = 0 have can be a ~reached N = 2A2M—1 [5]. To
=
=
A2(ab)~,
(4)
and e1 = = 0 (A1, A2 are constants), does precisely that. The field equations can now be1 I~ solved. sim1 i.e.,For small plicity in [5]. the solution we Aassume 1u (with the divorticity Then, with a constant mension of a length) we find:
where
f:
~M
which is well valid as long as the energy density of the em field is negligible compared to that of matter. How-
f v~dt2
(7)
and v := ab2 is a solution of (a dot is d/dt): 2uif + 4~2 — 1.6Mth +Mu + 12M2t2
=
0.
In particular the 01 field equation (with A2 + A~)gives A2(ab)_2 = ~pu0u1, or 2A2 = MN. Eq. (9) together with eqs. (1)—(4) gives
(8) :=
A~ (9) (10)
where H~:= u0b2a1 can be thought of (with proper choice of constants) as the Hubble radius and h2:= + = e~ + e~.Eq. (10) states that the ratio of the em energy density to the rate of rotation is equal to the ratio of the matter energy density to the Bubble constant. There is an interesting special case of our solution. Notice first that v = ~Mt2 satisfies eq. (8). Then, eq. (7) givesf~ lnA~tand eqs. (5), (6) reduce to a = ~A3/2M1t1/2, b = ~A3I~Mt~/~ (11) Eqs. (11) can be used to obtain simple expressions for .
410
~.
ever the first increases faster as t -÷ 0 so it will eventually dominate. In the limit of zero vorticity our model describes a LRS Bianchi type II spacetime with its dust content moving orthogonal to the hypersurfaces of homogeneity. It is on this spacetime that vorticity and the em field have been added as a perturbation but in the inseparable manner described by eq. (10), as dictated by the field equations. It should be clear however that this result does not depend directly on the approximation used; eq. (10) is the entire content of the 01 field equation. References [1] A.K. Raychaudhuri, Theoretical cosmology (Clarendon, Oxford, 1979). [2] D. Kramer, H. Stephani, M. MacCallum and E. Herlt, Excact solutions of Einstein’s field equations WEB Deutscher Verlag der Wissenschaften, Berlin, 1980) Ch. 12. [3] C.B. Collins and S.W. Hawking, Mon. Not. R. Astron.
Soc. 162 (1973) 307.
L41 A.J. Fennelly, Phys. Rev. D21 (1980) 2107. [51 N.A. Batakis, Phys. Rev. D23 (1981) 1681.