Frame dragging and super energy

Transcription

1 Frame dragging and super energy arxiv: v1 [gr-qc] 5 Jul 007 L. Herrera 1, J. Carot 3, and A. Di Prisco 1, 1 Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela. 3 Departament de Física, Universitat Illes Balears, E-071 Palma de Mallorca, Spain February 6, 008 Abstract We show that the vorticity appearing in stationary vacuum spacetimes is always related to the existence of a flow of super energy on the plane orthogonal to the vorticity vector. This result, toghether with the previously established link between vorticity and super energy in radiative Bondi Sachs) spacetimes strength further the case for this latter quantity as the cause of frame dragging. 1 Introduction The appearance of vorticity in the congruence of world lines of observers in the gravitational field of a massive rotating ball Lense Thirring effect), led laherrera@cantv.net.ve jcarot@uib.es adiprisc@fisica.ciens.ucv.ve 1

2 Shiff [1] to propose the use of gyroscopes to measure such an effect. Since then this idea has been developed extensively see [, 3, 4] and references cited therein). However, the appearance of vorticity is not always related at least explicitly) to rotating sources. Indeed, this is the case of the field of a magnetic dipole plus an electric charge [5], and the case of gravitational radiation [6] [9]. In the former case the vorticity is accounted for by the existence of a flow of electromagnetic energy on the plane orthogonal to the vorticity vector. As a matter of fact it appears that in all stationary electrovacuum solutions [10], at least part of the vorticity has that origin. In the case of gravitational radiation Bondi Sachs) we have recently shown [11] that the appearing vorticity is related to the existence of a flow of super energy on the plane orthogonal to the vorticity vector. Here we prove that such a link between a flow of super energy and vorticity, is also present in stationary vacuum spacetimes. In order to motivate our study, we shall start by considering simple particular cases, before dealing with the general stationary situation. Thus, we shall first present the Lense Thirring case Kerr up to the first order in m/r and a/r), then we will treat the Kerr case, and finally we establish the aforementioned link for the general stationary vacuum spacetime. Doing so we provide an universal mechanism i.e. one which applies to all known situations) for the occurrence of frame dragging. At the same time this result brings out the relevance of the Bel Robinson tensor in the study of self gravitating systems. The Lense Thirring metric As is well known the Lense Thirring metric [1] ds = 1 m r ) dt m r ) dr + r dθ + r sin θ dφ ) + 4J sin θ dφdt 1) r

3 describes the gravitational field outside a spinning sphere of constant density, and is valid up to first order in m/r and J/r, with m and J denoting the mass and the angular momentum respectively. It is also well known that, up to that order, it is the Kerr metric, with the identification ma = J ) where a is the Kerr parameter [13] Now, the congruence of the world line of observers at rest in the frame of 1) is u α 1 =, 0, 0, 0, 3) 1 m r and the vorticity vector, defined as usual by yields, up to order a/r and m/r ω α = 1 ηαηιλ u η u ι,λ, 4) ω r ma cosθ = r 3 5) ω θ ma sin θ = r 4 6) or, for the absolute value of the vector ω α Ω = ω α ω α ) 1/ = ma r cos θ 7) At the equator θ = π) Ω = ma 8) r 3 which is a very well known result. The leading term of the super Poynting gravitational vector at the equator, calculated from 1), is see next section), implying that P φ = 0 a = 0 ω α = 0. P Φ 9 m a 1 9) r r r 5 We shall now prove our conjecture in the exact case Kerr metric). 3

4 3 The Kerr metric In Boyer Linquist coordinates, the Kerr metric takes the form ) ds mr 4mar sin = 1 + dt ) θ dtdφ r + a cos θ r + a cos θ r + a cos ) θ + dr + r + a cos θ ) dθ r mr + a + r sin θ + a sin θ + mra sin 4 ) θ dφ 10) r + a cos θ The congruence of world line of observers at rest in 10) is defined by the vector field u α 1 =, 0, 0, 0 11) 1 mr r +a cos θ There are two non vanishing components of the vorticity vector for such a congruence, which are ω r = mra cosθ r mr + a ) r + a cos θ ) r mr + a cos θ ) 1 1) and ω θ = ma sin θ r a cos θ ) r + a cos θ ) r mr + a cos θ ) 1 13) which of course coincide with 5) and 6) up to first order in m/r and a/r. Next, the super Poynting vector based on the Bel Robinson [14] tensor, as defined in Maartens and Basset [15], is P α = η αβγδ E β ρ Hγρ u δ, 14) where E µν and H µν, are the electric and magnetic parts of Weyl tensor, respectively, formed from Weyl tensor C αβγδ and its dual C αβγδ by contraction with the four velocity vector, given by E αβ = C αγβδ u γ u δ 15) 4

5 H αβ = C αγβδ u γ u δ = 1 η αγǫδc ǫδ βρ uγ u ρ, 16) Then, a direct calculation of P µ using the package GR Tensor running on Maple yields, for the Kerr space-time: P µ = P t, P φ, 0, 0 ) 17) where P t = 18m 3 ra sin θ r mr + a sin θ + a ) r + a cos θ ) 4 r mr + a cos θ ) r mr + a cos θ r + a cos θ ) 1/ 18) and P φ = 9m a r mr + a a cos θ ) r mr + a cos θ r + a cos θ r + a cos θ ) 4 r mr + a cos θ ) 1 ) 1/ 19) From the above it can be seen that P φ = 0 a = 0 ω α = 0. In other words: there is always an azimuthal flow of super energy, as long as a 0, and viceversa, the vanishing of such a flow, implies a = 0. Also observe that the leading term in power series of m/r and a/r in 17) is of order m/r). Let us now consider the general stationary and axisymmetric vacuum case. 4 The general case The line element for a general stationary and axisymmetric vacuum spacetime may be written as [16] ds = fdt + fωdtdφ + f 1 e γ dρ + dz ) + f 1 ρ fω )dφ 0) where x 0 = t; x 1 = ρ; x = z and x 3 = φ and metric functions depend only on ρ and z which must satisfy the vacuum field equations: γ ρ = 1 [ ρ f 4ρf ρ ) f z f 4 ωρ z)] ω 5 1)

6 γ z = 1 ρf [ ρ f ρ f z f 4 ω ρ ω z ] f ρρ = f zz f ρ ρ f3 ) ω ρ ρ + ωz 1 ) + f f ρ + fz ) 3) ω ρρ = ω zz + ω ρ ρ f f ρω ρ + f z ω z ) 4) The four velocity vector for an observer at rest in the frame of 0) is u α = f 1/, 0, 0, 0) 5) The super-poynting vector can now be calculated for the general class of spacetimes represented by the above metric 0), i.e.: without making any assumption about the matter content of the spacetime), and one gets using again GR Tensor): P µ = P t, P φ, 0, 0) with P t = ωp φ, hence P µ = 0, ρ f P φ, 0, 0) 6) Thus, the relevant quantity is P φ which is given by again in the general case, i.e.: without taking into account the field equations): P φ = f 3/ e 4γ ρ 5 {A11} 7) Substituting now the vacuum field equations 1-4) in the above expression one gets: P φ = 1 3 f 3/ e 4γ ρ 5 {A1} 8) where A11 and A1 are given in the Appendix. 6

7 Now, it has been shown in [10] that for the general metric 0) the following relations hold H αβ = 0 ω α = 0 ω = 0. 9) and of course we know that H αβ = 0 P µ = 0. 30) So, what we want to show here is that P µ = 0 implies necessarily that ω = 0 the solution becomes then static, not just stationary, and therefore the so-called dragging of inertial frames effect disappears as the vorticity vanishes. In other words we want to establish the relation P µ = 0 H αβ = 0 ω α = 0 ω = 0. 31) Now, as the above equation is far too complicated to be treated in full, we shall start instead by analyzying what happens in the neighborhood of the symmetry axis ρ = 0 and far away from any matter source along it, that is, for z. In so doing, the following two assumptions will be made: 1. The spacetime is regular at the axis.. The spacetime is asymptotically flat in spacelike directions. 4.1 The super-poynting vector in a neighborhood of the axis The geometry of axisymmetric spacetimes in the vicinity of the axis was studied in [18] see also [19] and[16]). It then follows that g tφ must tend to zero when ρ 0 at least as ρ, g φφ tends to zero as ρ, and g tt, g ρρ and g zz cannot vanish on the symmetry axis. All these imply but are not equivalent to) the so-called elementary flatness condition on the axis, namely: Let us then put X a X a 4X 1 X g φφ 3) ωρ, z) = ρ +k Az)+Oρ 3 ), fρ, z) = mz)+ρnz)+ρ Sz)+Oρ 3 ) 33) 7

8 where k 0 is a constant and mz) 0 necessarily; further, the elementary flatness condition mentioned above implies that γ 0 as ρ tends to zero, and expanding P φ in a power series around ρ = 0 one gets in the case k = 0): P φ 8Am n ρ 8m ρ { 4A 3 nm 4 + 3m n z A z + m na z,z + 4mA z nm z 4mAnS mam z,z n + 6mAn z m z + 3Am z n An 3} + m { 16 A 5 m A A z,z + 1 A z A ) m A 3 m z,z + 56 A z A m z 3 A 3 S ) m A 3 n + 4 A 3 m z ) m SA z,z + 4 A z S z + 8 m z,z A z,z )m 3 + [ 10 n A z,z + 76 A z n z 8 An z,z ) n +4 An z + 0 A z m z,z m z 16 AS 16 Am z,z S A z,z m z + 3 A z Sm z + 48 AS z m z ] m + [ 1 AS 0 Am z,z + 8 A z m z )n +88 An z nm z A z m z Am z S 4 Am z,z m z ] m +7 Am z Am z n An 4} + Oρ) 34) Setting P φ = 0 all over the spacetime implies that the above must also vanish in a neighborhood of ρ = 0, which in turn implies that the coefficients of ρ, ρ 1 and ρ 0 must be zero. Since m 0, it must be either A = 0 which is what we aim at showing), or else n = 0. Let us assume that n = 0, the coefficients of the terms in ρ and ρ 1 then vanish identically, whereas the last term is much reduced. Further, our second requirement above asymptotic flatness), implies that far away from the source and in a small neighborhood of the axis, mz) = 1, m z, m zz = 0, and also Sz) = 0 for otherwise f 1 at infinity in spatial directions), therefore one is left with: AA zz + 3A z 4A4 = 0 35) which must hold for z large and in a neighborhood of the axis. Further, its solution must be bounded, since otherwise ω would increase without limit thus violating again the condition of asymptotic flatness. 8

9 This is an autonomous equation, a first integral of which can be readily found to be: A z = ± 1 1 A4A 7 A 7 + C) 36) C being a constant of integration. It is then immediate to show using numerical simulations that the function A diverges for large values of z, and therefore it must vanish, which is what we wanted to show. So far, we have only analyzed the case k = 0, however it is a simple matter to check that field equation 4) together with 33) rule out all values of k > 0. In order to complete our proof we have to show that the result above the vanishing of vorticity, implied by the vanishing of super Poynting, within an infinite cylinder around the axis of symmetry), can be analytically extended to the whole spacetime. In other words we have to prove that it is not possible to smoothly match an static axially symmetric, asymptotically flat vacuum spacetime to a stationary non-static) axially symmetric vacuum spacetime which is also asymptotically flat, across an infinite cylinder say Σ) around the axis of symmetry. Such a matching is not possible [0], but it can also be checked at once from the continuity of the first fundamental forms on the cylinder. Indeed, this last requirement implies that ω should vanish also at the outer part of Σ, thereby indicating that the static condition can be analytically extended. 5 Conclusions We have seen that Bonnor s original idea to associate the frame dragging in some electrovac solutions with the existence of a flow of electromagnetic energy as described by the Poynting vector), can be successfully extended to the same effect in vacuum stationary spacetimes, by replacing the flow of electromagnetic energy by a flow of super energy, as described by the super Poynting vector defined from the Bel Robinson tensor. Due to the lack of a covariant definition of gravitational energy, super energy appears to be the best candidate for playing such a role. On the other hand, the fact that it is also associated to frame dragging in radiative spacetimes, reinforces further our conjecture, that it is responsible for such an effect in 9

10 any general relativistic scenario. Before concluding, the following remark is in order: All along this work, the vorticity is calculated for a congruence with a distinct physical meaning, namely the congruence of worldlines of observers at rest with respect to the source, i.e. observers at rest in the frame of 1), 10) and 0), respectively.this is particularly clear in the case of the Lense Thirring metric 1). Appendix A11 = [ ρω ρ ω ρ + ω z )γ ρ ρω z ω ρ + ω z )γ z ω z ω ρρρ +ω z ω zzρ + 4ω z ω ρ ω ρz ρ ω ρ ω zzρ ω 3 ρ + ω ρ ω ρρρ ] f 4 +3ρω ρ + ω z )ω zf z + ω ρ f ρ )f 3 ρ ργ z ω ρz + γ z ω ρρ +ργ ρω ρ + γ z ω z + γ ρ ω ρ + ργ ρ ω zz ργ ρ ω ρρ )f [ 4ρ 3 f z ω z + f ρ ω ρ )γ ρ + ρ ρf zz ω ρ ρf ρz ω z f z ω z +4f ρ ω ρ ρf z ω ρz ρf ρρ ω ρ ρf ρ ω ρρ + ρf ρ ω zz )γ ρ +4ρ 3 f z ω z + f ρ ω ρ )γ z + ρ 4f ρ ω z + ρf z ω ρρ ρf ρz ω ρ +ρf ρρ ω z ρf ρ ω ρz ρf zz ω z ρf z ω zz + ω ρ f z )γ z +4ρ 3 f ρz ω ρz ρ 3 f zz ω ρρ ρ 3 f ρρ ω zz + ρ f zz ω ρ ρ f ρz ω z ρ f ρρ ω ρ + ρ 3 f zz ω zz + ρ 3 f ρρ ω ρρ ] f 6ρ 3 f ρ + f z )ω ργ ρ 6ρ 3 f ρ + f z )ω zγ z + 3ρ 3 f ρρ f ρ ω ρ + f zz f z ω z + f ρz f z ω ρ f ρρ f z ω z + f ρz f ρ ω z f zz f ρ ω ρ ) 37) A1 = ω ρ 7ωz 4 6ωρω z + ωρ)f [ ρω ρ f ρ ωz 4 + ω4 ρ + ω ρ ω z ) ρf zω z ωρ 4 + ω z ω ρ + ω4 z )] f 8 + [ 4ρω zz ωρ + 3ωz) + 4ω ρ ρω z ω ρz + ωz) ] f 7 + [ 4ρω z 8ωz f z + ρωz f ρz 3ρω ρ ω z f zz 3ρωρ f ρz ρf ρ ω z ω zz 5ωρ f z ρf ρ ω ρ ω ρz ω z ω ρ f ρ ρf z ω ρ ω zz + ρf z ω z ω ρz ) 10

11 +4ρω ρ ρω ρ f z ω ρz + ρω ρ f ρ ω zz + ωρ f ρ + ρωρ f zz) ] f 6 + [ 6ρ ωρf 3 z + fρ) ρ f z ω ρ ω z ω ρ f ρ + 5ω z f z ) +ρ f ρ ωz ω zf z ω ρ f ρ ) ] f 5 + [ 8ρ f ρz ω z f ρ ω zz ) 16ρ 3 f ρz ω ρz + f zz ω zz ) + 10ρ 3 f z ω z f ρ ω ρ f z ω z + f ρ ω ρ ) ρ 3 f z f ρ f z ωρ 3 + fρω z) 3 + ρ 3 fρω 3 ρ 3 ω ρ ωz) +ρ 3 fz 3 ω3 z ω zωρ )] f 4 + [ 4ρ 3 f ρz ω ρ f z + ω z f ρ ) +4ρ f ρ f ρ ω ρ 4f z ω z ) + 4ρ 3 3fρω zz + f zz f ρ ω ρ ω ρz f ρ f z +fz ω zz 10f zz ω z f z ) ] f 3 + [ 4ρ 4 f ρz fρ ω z fz ω z + ω ρ f ρ f z ) +4ρ 4 f zz fz ω ρ fρ ω ρ + ω z f ρ f z ) + 4ρ 4 ω ρz fz 3 + 3f zfρ ) +4ρ 4 ω zz fρ 3 + 3f ρfz ) + 4ρ3 4fρ f zω z 3fz f ρω ρ fρ 3 ω ρ + 3fz 3 ω z) ] f + [ ρ 4 ω ρ 14fz f ρ 7f4 z + 5f4 ρ ) +4ρ 4 ω z fρ 3 f z + 5fz 3 f ρ) ] f ρ 5 ω z fz 3 f ρ + f5 z + f zfρ 4 ) ρ 5 ω ρ fρ 3 f z + f5 ρ + f rhofz 4 ) 38) Acknowledgements One of us JC) gratefully acknowledges financial support from the Spanish Ministerio de Educación y Ciencia through the grant FPA LH wishes to thank Universitat Illes Balears for financial support and hospitality. ADP also acknowledges hospitality of the Physics Department of the Universitat Illes Balears. LH and ADP acknowledge financial support from the CDCH at Universidad Central de Venezuela under grant PI References [1] L I Shiff, Phys. Rev. Lett ). [] W Rindler Phys. Lett. A ) 11

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