Hawking-Ellis classification of stress-energy: test-fields versus back-reaction
aa r X i v : . [ g r- q c ] F e b Hawking–Ellis classification of stress-energy:test-fields versus back-reaction
Prado Mart´ın-Moruno ID and Matt Visser ID Departamento de F´ısica Te´orica and IPARCOS, Universidad Complutense de Madrid,E-28040 Madrid, Spain School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand
E-mail: [email protected] , [email protected] Abstract:
We consider the Hawking–Ellis (Segr´e–Pleba´nski) classification of stress-energy tensors,both in the test-field limit, and in the presence of back-reaction governed by the usualEinstein equations. For test fields it is not too difficult to get a type IV stress-energyvia quantum vacuum polarization effects. (For example, consider the Unruh quantumvacuum state for a massless scalar field in the Schwarzschild background.) However, inthe presence of back-reaction driven by the ordinary Einstein equations the situationis often much more constrained. For instance: (1) in any static spacetime the stress-energy is always type I in the domain of outer communication, and on any horizonthat might be present; (2) in any stationary axisymmetric spacetime the stress-energyis always type I on any horizon that might be present; (3) on any Killing horizonthat is extendable to a bifurcation 2-surface the stress-energy is always type I; (4)in any stationary axisymmetric spacetime the stress-energy is always type I on theaxis of symmetry; (5) some of the homogeneous Bianchi cosmologies are guaranteedto be Hawking–Ellis type I (for example, all the Bianchi type I cosmologies, all theFLRW cosmologies, and all the “single mode” Bianchi cosmologies). That is, in verymany physically interesting situations once one includes back-reaction the more unusualstress-energy types are automatically excluded.
Date:
Saturday 27 February 2021; L A TEX-ed March 1, 2021
Keywords: stress-energy; Hawking–Ellis classification; Segr´e–Pleba´nski classification;test-fields; back-reaction. ontents
B.1 Test field stress-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 24B.2 Dilaton gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References 27 – 1 –
Introduction
The energy conditions are inequalities on the stress-energy tensor that are requestedto be satisfied by orthodox material content [1]. Although they are not derived fromfundamental physics [2], they can be useful to extract general consequences. In par-ticular, in the framework of a theory of gravity, they allow us to restrict attention tothe physically reasonable spacetimes and their characteristics [3]. Nowadays one of themore conservative energy conditions, the strong energy condition (SEC), is known to beviolated macroscopically during cosmic evolution [4–7]. Furthermore it is well-knownthat the weakest energy condition, the null energy condition (NEC), can be violated bythe renormalized stress-energy tensor when considering quantum effects [8]. However,it seems that if semi-classical physics could allow unrestricted violations of the nullenergy condition, one should appeal to additional physics to forbid potential problemsof the spacetime, such as causal paradoxes.Historically, almost all numerical and semi-analytic computations of the renormalizedstress-energy tensor have been preformed in the test-field approximation; ie , picking afixed background and ignoring back-reaction. See for example references [3] and [9–13]. Working in the test field approximation it is relatively straightforward to seethat all the classical energy conditions can be violated, and more subtly that onecan encounter the more outr´e of the stress-energy tensors occurring in the Hawking–Ellis (Segr´e–Pleba´nski) classification [14]. In particular, as discussed below, one canencounter type IV stress-energy tensors [8, 15, 16], which necessarily violate all theenergy conditions [1].In counterpoint, recently Hideki Meada realised that once one includes back-reactionand looks at self-consistent solutions to the Einstein equations the situation simplifiestremendously [17]. Specifically, Hideki Meada showed that in any static spacetimesolving the Einstein equations, then in those regions where the t coordinate is actuallytimelike, the stress-energy is always of type I according to the Hawking–Ellis (Segr´e–Pleba´nski) classification. That is: The stress-energy tensor is certainly of type I inthe so-called “domain of outer communication”, but one has to be careful below anyhorizon that might be present. Therefore, in the region of interest, for static spacetimesthe stress-energy tensor (of either classical or semi-classical fields) takes the simplestform possible; and, therefore, there is no reason a priori why one should think that theweakest energy conditions would necessarily have to be violated.– 2 –erein we shall generalize this result in various ways: • In any static spacetime the stress-energy is always type I in the domain of outercommunication, and on any horizon that might be present. • In any stationary spacetime the stress-energy is always type I on any horizon thatmight be present. • The stress-energy is always type I on any bifurcate Killing horizon. • In any stationary axisymmetric spacetime the stress-energy is always type I onthe axis of symmetry. • Some of the homogeneous Bianchi cosmologies are guaranteed to be Hawking–Ellis type I (for example, all the Bianchi type I cosmologies, all the FLRW cos-mologies, and all the “single-mode” Bianchi cosmologies). Other Bianchi cos-mologies require a case-by-case analysis.That is, in very many physically interesting situations the more outr´e stress-energytypes are automatically excluded. While we shall largely focus on (3+1) dimensions,some brief comments on (2+1) and (1+1) dimensions are relegated to the appendix.Finally, note that many proposed modifications or extensions of Einstein gravity haveequations of motion that can be rearranged into the form G ab = 8 π [ T effective ] ab . Forexample, consider Maeda’s f ( g ab , R abcd ) gravity [17], the more specific f ( R ) theoriesof gravity [18], Rastall gravity [19–22], Horndeski theories [23, 24], and the like [25].Then at a minimum our results imply that [ T effective ] ab is Hawking–Ellis type I underthe stated conditions. (And to say anything more about T ab itself, we would need tospecify the precise extended/modified gravity model in question.) The use of test fields (and even test particles) in general relativity has a very longand distinguished history. Even the seemingly utterly innocuous statement that fallingparticles follow spacetime geodesics is ultimately limited to the test-particle regime [26].In a quantum field theoretic context, Hawking’s original derivation of Hawking radiationis intrinsically a test-field calculation [27, 28].
The classical energy conditions of general relativity are used to keep some of the more outr´e aspects of gravitational physics somewhat under control. While the classical– 3 –nergy conditions are not truly fundamental physics [2], they are nevertheless veryuseful to work with [1–8, 29–33].Cosmologically, to make the “accelerating universe” compatible with general relativityone has to violate the strong energy condition (SEC) on cosmological scales [4–6]. Thisis not surprising once one notes that SEC is the requirement of gravity always beingattractive in general relativity, which is obviously not the case during inflation and thecurrent cosmic epoch. Its mathematical expression is V a (cid:18) T ab − T δ ab (cid:19) V b ≥ , (2.1)for any timelike vector V a .On the other hand, we emphasise that to make the existence of Hawking radiationcompatible with the black hole area increase theorem, one must violate the null energycondition (NEC) through quantum effects. The NEC is the weakest of the point-wiseenergy conditions. It consists of demanding k a T ab k b ≥ , (2.2)for any null vector k a . It is a limiting case of the weak energy condition (WEC),which states that the energy density measured by any observer has to be non-negative( V a T ab V b ≥ The Hawking–Ellis (Segr´e–Pleba´nski) classification of stress-energy tensors is composedof four different types [14]. Each type represents a different eigenvector structure.Those are: type I, which has 1 timelike and 3 spacelike eigenvectors; type II, with 1double null eigenvector and 2 spacelike eigenvectors; type III, which has 1 triple nulleigenvector and 1 spacelike eigenvector; and type IV, which has no causal eigenvector.Using Lorentz transformations, all these types can (in an orthonormal basis) be put– 4 –nto the form (see, for example, reference [1]) T ˆ a ˆ b = ρ f f f p σ f σ p
00 0 0 p . (2.3)Here f = f = σ = 0 for type I; f = 0, f = σ = 0, and ρ + p = 2 f for type II; f = 0, f = 0 and σ = 0 for type III; and f = 0, f = σ = 0, and p = − ρ for type IV.Investigating which of these Hawking–Ellis types a given stress-energy tensor belongsto can be somewhat tricky for the renormalized stress energy tensors obtained bynumerical methods, at least partly due to delicate numerical artefacts, and the potentialfor delicate numerical (almost) cancelations.Working in spherical symmetry for now let us identify the orthonormal components ofthe stress-energy as T ˆ a ˆ b = ρ f f p k p ⊥
00 0 0 p ⊥ . (2.4)So, type III is already discarded in spherically symmetric situations. The Lorentzinvariant eigenvalues, defined by det( T ˆ a ˆ b − λg ˆ a ˆ b ) = 0 , (2.5)are easily calculated to be λ ∈ (cid:26)
12 ( p k − ρ ) ± q ( ρ + p k ) − f ; p ⊥ , p ⊥ (cid:27) . (2.6)The sign of the quantity Γ = ( ρ + p k ) − f (2.7)is thus key to controlling the reality and degeneracy of the eigenvalues, which in turncontrols the Hawking–Ellis type of the stress-energy tensor. For a type I stress-energy tensor there is always some observer who sees that thereare no net fluxes, so such a stress-energy tensor can be fully diagonalized via Lorentz– 5 –ransformations. Most classical fields that can be found in nature have a type I stress-energy tensor, and for type I stress-energy tensors one can always find an observer“moving with the field”. Since in this situation f = 0, we have Γ > p and p . (One also has the degenerate case inwhich ρ = − p k , that still has four eigenvectors.)On the other hand, the renormalized stress-energy tensors of quantum vacuum statesfor which the flux term can be made to vanish by Lorentz transformation are alsotype I. Since there is no net flux in either the Hartle–Hawking [9] or Boulware [10]quantum vacuum states, the relevant stress-energy is automatically diagonal, and so ofHawking–Ellis type I. For type II stress-energy tensor we cannot find any timelike observer for which the fluxterm vanishes. If ρ + p k = ± f , then Γ = 0 and the Lorentz invariant eigenvalues ofthe stress-energy are λ ∈ (cid:26)
12 ( p k − ρ ) ,
12 ( p k − ρ ); p ⊥ , p ⊥ (cid:27) . (2.8)This implies the stress-energy would be Hawking–Ellis type II. Classical radiation andother zero-mass fields have type II stress-energy tensors.Regarding the test-field renormalized stress-energy tensors, unfortunately this situa-tion, even if it were to arise in theory, would in practice be unstable to numericalroundoff error [36]. (Any roundoff error in the stress-energy components would gener-ically lift the degeneracy in the eigenvalues — unless one has a symmetry principlekeeping the equality of the eigenvalues exact.) In short, numerically computed renor-malized stress-energy tensors are not a useful diagnostic for exploring Hawking–Ellistype II stress-energy tensors. For this type of stress-energy tensor one cannot find any timelike observer that measuresno fluxes and no stresses. Therefore, spherical symmetry is not compatible with atype III stress-energy tensor. In classical physics there are no natural matter fieldsthat lead to this type of tensors, although artificial models have been constructed [37].– 6 –or the renormalized stress-energy tensors, to open the door for type III stress-energytensors, we should go beyond spherically symmetric quantum vacuum states; for ex-ample, by considering a Kerr background. In this case, one has a renormalized stress-energy tensor (for a minimally coupled massless scalar field) for the Kerr geometry [38]that can in principle be of type III. It should be noted, however, that even if not explic-itly forbidden by symmetries, type III is (like type II) unstable to numerical roundofferror [36]. Numerically computed renormalized stress-energy tensors are not a usefuldiagnostic for exploring Hawking–Ellis type III stress-energy tensors.
In this case Γ <
0, so there are no causal eigenvectors. There are 2 real spacelike and2 complex eigenvectors, therefore, no causal (timelike or null) eigenvectors. Type IVstress-energy tensors can be understood as a complex extension of type I, since in orderto get an observer measuring zero fluxes, she/he would need be able to measure complexenergy densities and pressures. Therefore, there is no physical timelike observer whomeasures no fluxes.There are no known classical matter fields that lead to a type IV stress-energy tensor.Regarding renormalized stress-energy tensors of quantum vacuum state, as alreadymentioned, all cases without an intrinsic flux are automatically type I. So, this leavesus with only the Unruh [12] quantum vacuum state to investigate — and a priori therelevant stress-energy could be any one of types I, II, or IV.Up to the best of our knowledge, the first investigation about a type IV stress-energytensor was made by Roman in 1986 [15]. Establishing an analogy with a model de-scribing a singularity-free collapsing star, he argued that the renormalized vacuumexpectation value of spherically symmetric evaporating black hole has to be type IVin a neighborhood of the apparent horizon (see section II of reference [15]). Unawareof that result, in reference [8] we proved that the stress-energy tensor of the Unruhvacuum is type IV far from the horizon. However, we found a spurious pole in Γ due toround-off errors in the numerical calculations. Later on, in reference [16], Abdolrahimi,Page, and Tzounis showed that the stress-energy tensor for the massless conformalscalar in the Unruh state is Type IV everywhere outside the horizon.
In short, in the test-field framework Hawking–Ellis types I and IV are quite common.In contrast, while Hawking–Ellis types II and III are not absolutely forbidden in thetest-field framework, they are numerically and perturbatively unstable.– 7 –
Back-reaction via the Einstein equations
Once one introduces back-reaction by imposing the Einstein equations T ab = 18 πG N G ab , (3.1)the stress-energy becomes very tightly constrained in terms of the spacetime geometry.Purely geometrical considerations will then fully control the energy conditions and theHawking–Ellis classification. Let us seek to understand what it means geometrically to have a type I stress-energytensor. Supposing that we have a spacelike slicing of spacetime, we can adopt the ADMformalism and write d s = − N d t + 2 N i d t d x i + g ij d x i d x j . (3.2)Here N is the lapse function, N i is the shift vector, and g ij is the intrinsic 3-geometry.The normal to the spacelike slices is the timelike co-vector n a = ( ∇ t ) a / |∇ t | , the pro-jection operator onto the 3-hypersurfaces is h ab = g ab + n a n b , (3.3)and the extrinsic 3-geometry curvature is K ij = h ai h bj n a ; b , (3.4)where ; denotes (3+1) dimensional covariant derivative. One key result of the ADMformalism is that n a R ab h bc can be evaluated in terms of intrinsic derivatives of theextrinsic curvature K ij . From the Gauss–Coddazzi equations, one has R ni = K ij : j − g ij K : j , (3.5)where we have denoted R ni = n a R ab h ib and now : denotes covariant derivative in the3-space of the spacelike slices. As one has G ab n a h bc = (cid:18) R ab − Rg ab (cid:19) n a h bc = R ab n a h bc − Rg ab n a ( g bc + n b n c ) = R ab n a h bc , (3.6)– 8 –t follows that G ni = K ij : j − g ij K : j . (3.7)Certainly G ni = 0 is a sufficient condition to guarantee the stress-energy is type I. Allsuch geometries have a stress-energy tensor of the form R ti = 0 = G ti , so that T ab = " ρ T ij . (3.8)Using ordinary spatial 3-rotations, this is now manifestly Hawking–Ellis type I.In reference [17] Hideki Maeda uses hypersurface orthogonality to get the spatial slicing,and then adds the Killing condition to enforce K ij →
0. However, it is clear fromthe discussion above that it is more than sufficient to demand the somewhat weakercondition K ij : j − g ij K : j = 0 . (3.9)That is, one might reasonably expect to have many different classes of spacetimes thatsatisfy this sufficient condition. We shall explore various possibilities below. This particular case is trivial. For a static spacetime there exists a spacelike hypersur-face (3-surface) orthogonal to the timelike Killing vector. If one additionally requiresspherical symmetry, then that 3-surface retains this symmetry. Then, one can alwayschoose a coordinate system where the metric is diagonal and time independent. In thatspecific coordinate system the Einstein tensor (and so the stress-energy tensor) is alsodiagonal — so the stress-energy is automatically Hawking–Ellis type I. The physicallymore interesting question is what happens when one relaxes these stringent conditions.
The existence of a hypersurface orthogonal Killing vector implies that, in the regionwhere that Killing vector is timelike, one can choose coordinates x a = ( t, x i ) to write g ab = " − N g ij , (3.10)with ˙ N = 0 and ˙ g ij = 0. That is, the (3+1) metric is block diagonalizable into(1 × ⊕ (3 ×
3) blocks, where the 3-metric g ij has Euclidean signature, and the– 9 –omponents of the metric in this decomposition are time independent. A quick versionof the argument leading to this conclusion is given in reference [39, page 119]. Inthese coordinates the timelike Killing vector is K a = (1 , , , a with, by definition, g ab K a K b = − N . The Killing covector is K a = ( − N ; 0 , , a = − N ∇ a t .(Notice block diagonalizability of the metric is a choice . For instance, in static sphericalsymmetry the metric is block-diagonal in curvature coordinates, conformal coordinates,isotropic coordinates, and proper distance coordinates; but the metric is not block-diagonal in Painleve–Gullstrand and Eddington–Finklestein coordinates.)In this situation block diagonalizability of the metric implies block diagonalizability ofthe Ricci tensor. To most easily see this one appeals to the Gauss–Coddazzi equationsas argued in the previous section. In the coordinate system set up above, the unittimelike vector normal to the spacelike hypersurfaces is n a = N K a and, therefore, theextrinsic curvature K ij of the constant- t K ij = 12 N ˙ g ij = 0 . (3.11)Specifically this means, through equation (3.5), R ti = 0, so that R ab = " R tt R ij . (3.12)Consequently we also have G ab = " G tt G ij . (3.13)Applying the Einstein equations, the stress-energy tensor satisfies T ab = " T tt T ij . (3.14)The remaining 3 × T ij can be diagonalized using ordinary 3-space rotations,so the stress-energy is automatically type I, (in the region where the t coordinate istimelike). Because T ab has a timelike eigenvector types Hawking–Ellis II, III, and IVare explicitly excluded. (Similarly, in the domain of outer communication of any staticspacetime, where the Killing vector is timelike, all of the polynomial curvature invari-ants are guaranteed to be poitive semidefinite [40].) Note that block diagonalizability and time independence are both necessary for this particular proof.– 10 –n the other hand, below any horizon that might be present the t coordinate is space-like, and so the 4-metric (while it is still block diagonalizable) is then of the form g ab = " + N g ij . (3.15)Here g ij is now a (2+1)-dimensional Lorentzian signature metric. The Ricci tensor andEinstein tensor are still block diagonalize, but now the Killing vector K a is spacelike,and so the stress-energy has at least one spacelike eigenvector — and so it cannot bededuced that the stress-energy is type I, in fact below the horizon it could in principlebe any one of the Hawking–Ellis types I to IV. Let us now investigate what happens to the stress-energy on any horizon that maybe present. The key point is that on any horizon that might be present there is anenhanced symmetry and the stress-energy takes the restricted form[ T ˆ a ˆ b ] H = ρ H − ρ H
00 0 [ T ˆ i ˆ j ] H , (3.16)where the hats denote that the tensor is expressed in an orthonormal basis. This isenough to guarantee the stress-energy is Hawking–Ellis type I. In the previous section we have seen that the stress-energy tensor of static spacetimes istype I in the domain of outer communication, that is where the hypersurface orthogonalKilling vector is timelike. Now we will see what happens on the horizon, where thehypersurface orthogonal Killing vector is null. To see that the claimed result holds inspherical symmetry is essentially trivial [41]. Note that in this situation the metric canbe written on the formd s = − exp { +2Φ( r ) } (cid:18) − m ( r ) r (cid:19) d t + d r − m ( r ) /r + r dΩ . (3.17)– 11 – quick computation yields ρ + p k = Φ ′ ( r )( r − m ( r ))4 πr , (3.18)which vanishes on the horizon at r = 2 m ( r ).To see that this still works for arbitrary static spacetimes is considerably trickier [42].(The current argument is similar but not quite identical to that in reference [42].) Firstuse the static condition to write d s = − N ( x i )d t + g ij d x i d x j . Next, within the spatial3-slices, use the level-sets of N ( x i ) to define one of the coordinates z , the remaining 2coordinates being x µ = ( x, y ). That is our coordinates are x a = ( t, x i ) = ( t, z, x µ ) =( t, z, x, y ) and without loss of generalityd s = − f ( z ) d t + d z f ( z ) + 2 N µ ( x k ) d z d x µ + g µν ( x k ) d x µ d x ν . (3.19)(Note that this is just a redefinition of z to get the desired form for g zz .) We still havecoordinate freedom in the x µ = ( x, y ) 2-surfaces which we can use to set N µ →
0. Thatis, since the metric is of the form (1 × ⊕ (3 × × s = − f ( z ) d t + d z f ( z ) + g µν ( x, y, z ) d x µ d x ν . (3.20)Now, let us go to an orthonormal non-coordinated basis { e ˆ a } . Since g ab = η ˆ a ˆ b e ˆ aa e ˆ bb , wehave T ˆ a ˆ b = e ˆ aa e ˆ bb T ab . (3.21)A brief but slightly tedious computation now yields T ˆ a ˆ b = ρ − ρ + O ( f ) O ( √ f ) O ( √ f )0 O ( √ f ) O (1) O (1)0 O ( √ f ) O (1) O (1) , (3.22)where p ˆ z ˆ z contains two terms, one O (1) and other O ( f ). Now ρ can be written as ρ = ρ H + O ( f ) and, as we have stated, p ˆ z ˆ z = − ρ + O ( f ), so that p ˆ z ˆ z = − ρ H + O ( f ).On the horizon, where f ( z ) →
0, this has the claimed restricted form (3.16). That is:[ T ˆ a ˆ b ] H = ρ H − ρ H
00 0 [ T ˆ i ˆ j ] H . (3.23)– 12 –bviously, as in previous discussions, the remaining 2 × T ˆ i ˆ j ] H can easily bediagonalized. (A more brutally explicit calculation is carried out in reference [42].) Inboth sub-cases the on-horizon symmetry implies that the on horizon stress-energy isHawking–Ellis type I in any static spacetime. To see that this enhanced on-horizon symmetry also works for stationary spacetimeswith axial symmetry is very much trickier [43]. (The current argument is similar butnot quite identical to that in reference [43].) First, note that stationary axial symmetryis enough to block diagonalize both metric and stress-energy: g ab = ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ ; T ab = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . (3.24)To isolate the near-horizon behaviour we set up an ADM-like decomposition g ab = − f ( r, θ ) + g φφ ( r, θ ) ω ( r, θ ) g φφ ( r, θ ) ω ( r, θ ) 0 0 g φφ ( r, θ ) ω ( r, θ ) g φφ ( r, θ ) 0 00 0 h ( r,θ ) f ( r,θ )
00 0 0 g θθ ( r, θ ) , (3.25)expressed in the coordinate system { t, φ, r, θ } . This is carefully constructed to beLorentzian signature regardless of the sign of f ( r, θ ). This metric has co-tetrad e ˆ aa = p f ( r, θ ) 0 0 0 p g φφ ( r, θ ) ω ( r, θ ) p g φφ ( r, θ ) 0 00 0 q h ( r,θ ) f ( r,θ )
00 0 0 p g θθ ( r, θ ) . (3.26)with corresponding tetrad e ˆ aa = √ f ( r,θ ) − ω ( r,θ ) √ f ( r,θ ) 1 √ g φφ ( r,θ ) q f ( r,θ ) h ( r,θ )
00 0 0 √ g θθ ( r,θ ) . (3.27)– 13 –t the horizon, that is when g rr ( r, θ ) → ∞ , implying f ( r, θ ) →
0, we must enforcea constant surface gravity and a constant angular-velocity. To accomodate this it issufficient to demand: f ( r, θ ) = K × ( r − r H ) + O ([ r − r H ] ); h ( r, θ ) = H + O ( r − r H ); ω ( r, θ ) = Ω + O ( r − r H ) . (3.28)Here K and H are related to the surface gravity, while Ω is minus the angular velocityof the horizon. Once set up in this manner, a brief but slightly more tedious computation (
Maple or Mathematica ) now yields T ˆ a ˆ b = ρ O [( r − r H ) / ] 0 0 O [( r − r H ) / ] O (1) 0 00 0 − ρ + O ( r − r H ) O [( r − r H ) / ]0 0 O [( r − r H ) / ] O (1) . (3.29)On the horizon, where r → r H , this has the claimed restricted form (3.16). Specifically,in the chosen coordinate system the on-horizon stress-energy is actuallly diagonal[ T ˆ a ˆ b ] H = ρ H p ˆ φ ˆ φ ] H − ρ H
00 0 0 [ p ˆ θ ˆ θ ] H . (3.30)(A more brutally explicit calculation is carried out in reference [43].) This stress-energytensor is now manifestly of Hawking–Ellis type I. A bifurcate Killing horizon is composed of two Killing horizons which intersect in thebifurcation 2 − surface. To see that this enhanced on-horizon symmetry also works forany arbitrary bifurcate Killing horizon, and even more, to any Killing horizon which canbe extended to a bifurcate Killing horizon, is rather subtle [43]. The benefit of working Any attempt at making K , H , or Ω depend on θ will result in (naked) curvature singularities atthe would-be horizon. Indeed any deviation from the near-horizon behaviour specified in (3.28) willresult in (naked) curvature singularities at the would-be horizon r = r H . – 14 –ith bifurcate Killing horizons is that one does not need to require axisymmetry, thedrawback is that the posited existence of the bifurcation 2-surface limits one to eternalblack holes, or at the very least, something that is extendable to an eternal black hole.(One might also note that if one steps outside of the framework of general relativity,say into the context of the “analogue spacetimes” [44–47], then there are situationswhere non-Killing horizons can arise [48].) Be that as it may, let us now see what wecan do with bifurcate Killing horizons.First, we consider an arbitrary section of a Killing horizon, by definition the vectorspace perpendicular to this 2-surface can be spanned by 2 null vectors, one of which K a can be taken to be the Killing vector that is null on the Killing horizon, and theother of which N a can be taken to be normalized as K a N a = −
1. To complete the basiswe take 2 orthonormal spacelike vectors m a and n a tangent to the 2-surface. Then g ab = ( K a N b + N a K b ) + ( m a m b + n a n b ) . (3.31)Because we are dealing with a Killing horizon we can without loss of generality chooseour basis to be invariant under the Killing flow: L K K a = 0; L K N a = 0; L K m a = 0; L K n a = 0 . (3.32)For the stress-energy tensor (indeed for any symmetric tensor) G ab = { G KK K a K b + G KN ( K a N b + N a K b ) + G NN N a N b } + G Km ( K a m b + m a K b ) + G Kn ( K a n b + n a K b )+ G Nm ( N a m b + m a N b ) + G Nn ( N a n b + n a N b )+ { G mm m a m b + G mn ( m a n b + n a m b ) + G nn n a n b } . (3.33)In view of the fact that we are dealing with a Killing horizon L K G ab = 0, so all of thecoefficients above are constant along the integral curves of the Killing vector K a .Now on any section of the Killing horizon it is a standard result that the Killing vectoris a null eigenvector G ab K b = λK a . (3.34)See for instance reference [39, page 333, equation (12.5.22)]. What Wald precisely says is that on any Killing horizon R ab K a K b = 0. (And the proof is ratherroundabout.) But since K a is a null vector on the Killing horizon, this implies R ab K b ∝ K a on theKilling horizon. This in turn implies G ab K b = λK a on the Killing horizon. Note λ does not have tobe nonzero; in fact in Schwarzschild spacetime it is trivially zero. – 15 –ut G ab K b = − G KN K a − G NN N a − G Nm m a − G Nn n a . (3.35)Consequently G NN = G Nm = G Nn = 0, and on any arbitrary section S of the Killinghorizon [ G ab ] S = { G KK K a K b + G KN ( K a N b + N a K b ) } + G Km ( K a m b + m a K b ) + G Kn ( K a n b + n a K b )+ { G mm m a m b + G mn ( m a n b + n a m b ) + G nn n a n b } (3.36)Now at the bifurcation 2-surface, N a being perpendicular to the 2-surface, must beparallel to the “other” Killing vector that is null on the second sheet of the bifurcatehorizon. Thus, on the bifurcation 2-surface we must also have G ab N b ∝ N a . (3.37)But, explicitly calculating G ab N b this now implies G KK = G Km = G Kn = 0, so on thebifurcation 2-surface B we have[ G ab ] B = G KN ( K a N b + N a K b ) + { G mm m a m b + G mn ( m a n b + n a m b ) + G nn n a n b } . (3.38)Once this has been established on the bifurcation 2-surface, the Killing symmetry ofeach individual sheet lets us extend this result to the full Killing horizon.Thus the on-horizon stress-energy is indeed block diagonal and indeed we can write[ G ab ] H = ρ H − ρ H
00 0 [ G mn ] H (3.39)This is manifestly Lorentz diagonalizable, and so manifestly of Hawking–Ellis type I. On any Killing horizon that may be present, be it static, stationary, or bifurcate, thestress-energy tensor is automatically Hawking–Ellis type I.– 16 – .5 On-axis behaviour in stationary (3+1) spacetimes
Let us consider a (3+1) stationarity axisymmetric spacetime. We now adopt “quasi-cylindrical” coordinates x a = ( t, φ, r, z ). Then g ab ( r, z ) = g tt g tφ g tφ g φφ g rr g rz g rz g zz (3.40)We can always without loss of generality define the r coordinate by setting g φφ = r ,so that little circles around the axis of rotation have circumference C = 2 πr . We canalso without loss of generality choose to set g rz = 0. Then g ab ( r, z ) = g tt g tφ g tφ r g rr
00 0 0 g zz (3.41)The four remaining free functions g tt ( r, z ), g tr ( r, z ), g rr ( r, z ), g zz ( r, z ), all need to beregular on the rotation axis.Now we adopt an ADM-like parameterization in terms of “lapse” and “shift”: g ab ( r, z ) = − N ( r, z ) + r ω ( r, z ) r ω ( r, z ) 0 0 r ω ( r, z ) r g rr ( r, z ) 00 0 0 g zz ( r, z ) . (3.42)Then det( g ) = − N ( r, z ) r g rr ( r, z ) g zz ( r, z ) . (3.43)This is enough to tell us that (everywhere in the spacetime) the non-zero elements of G ab are constrained by G ab = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . (3.44)This is already enough (everywhere in the spacetime) to preclude a stress-energy tensorof type III. – 17 –ow (since g rr and g zz are both positive, set g rr = h and g zz = k ) we can define aco-tetrad by e ˆ aa ( r, z ) = N ( r, z ) 0 0 0 rω ( r, z ) r h ( r, z ) 00 0 0 k ( r, z ) , (3.45)and a corresponding tetrad by e ˆ aa ( r, z ) = N ( r,z ) − ω ( r,z ) N ( r,z ) r h ( r,z )
00 0 0 k ( r,z ) . (3.46)Then in the corresponding orthonormal basis we have G ˆ a ˆ b ( r, z ) = e ˆ aa ( r, z ) e ˆ bb ( r, z ) G ab ( r, z ) = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . (3.47)What can we say on-axis? (That is, r = 0.) To avoid a conical singularity at r = 0, weneed g rr ( r, z ) = 1 + o ( r ), so h ( r, z ) = 1 + o ( r ). Looking along the rotation axis at r = 0we can also without loss of generality set g zz ( r, z ) = 1 + o ( r ), so k ( r, z ) = 1 + o ( r ).(That is, z is chosen to be proper distance along the axis of rotation.) To be moreprecise, demanding regularity near the rotation axis forces us to set N ( r, z ) = N ( z ) + 12 N ( z ) r + ... ; ω ( r, z ) = ω ( z ) + 12 ω ( z ) r + ... ; (3.48) h ( r, z ) = 1 + 12 h ( z ) r + .... ; k ( r, z ) = 1 + 12 k ( z ) r + .... ; (3.49)A brief calculation ( Maple or Mathematica ) shows that at the axis of rotation ( r = 0)one has: ( G ˆ a ˆ b ) r =0 = ( e Aa e Bb G ab ) r =0 = ( G ˆ t ˆ t ) G ˆ φ ˆ φ )
00 0 ( G ˆ r ˆ r )
00 0 0 ( G ˆ z ˆ z ) . (3.50)– 18 –ere we explicitly have: ( G ˆ t ˆ t ) = h ( z ) − k ( z ); (3.51)( G ˆ φ ˆ φ ) = ( G ˆ r ˆ r ) = k ( z ) + N ( z ) N ( z ) + [ N ( z )] ,zz N ( z ) ; (3.52)( G ˆ z ˆ z ) = 2 N ( z ) N ( z ) − h ( z ); (3.53)with all other stress-energy components being zero. This is manifestly diagonal, and somanifestly Hawking–Ellis type I. Note that the use of the tetrad/co-tetrad formalismand orthonormal basis is essential to getting this to work cleanly.In short: The stress-energy is always Hawking–Ellis type I on the axis of rotation ofany axisymmetric stationary (3+1) spacetime. Instead of focusing on spacetimes with timelike Killing vectors, we shall now explorecosmological spacetimes with a (3+1) metric that is block diagonalizable into (1 × ⊕ (3 ×
3) blocks. We shall further assume that in this spacetime g = − s = − d t + g ij ( t, x k ) d x i d x j . (3.54)Here the 3-metric g ij has Euclidean signature and, for the moment, the functions g ij ( t, x k ) can depend on any or all coordinates. The unit timelike vector orthogonal tothe spatial slicing is n a = (1 , , , n a = ( − , , ,
0) and K ij = n i ; j = Γ ij = 12 ˙ g ij . (3.55)Condition (3.9), that is K ij : j − g ij K : j = 0 , (3.56)can now be analyzed in a simpler way under these more restrictive assumptions. Anysynchronous cosmology (3.54) satisfying (3.56) will have a stress-energy tensor of theform R ti = 0 = G ti . So, as we have already discussed, using ordinary spatial 3-rotationsit can be seen that T ab would then be manifestly of Hawking–Ellis type I.The simplest (trivial) case in which this condition is satisfied is a subclass of the staticspacetimes, that have already been analyzed above. In this case, we have˙ g ij = 0 = ⇒ K ij = 0 , (3.57)and the condition is clearly satisfied. – 19 – .6.1 Bianchi type I spacetimes The next simplest case that we can consider of a block diagonalizable metric of theform (3.54) with ˙ g ij = 0 is one for which the spatial slicing is proportional to a flatEuclidean 3-space: g ij ( t, x k ) → h ij ( t ). In this case g ij,k = 0 and we can write the metricas d s = − d t + h ij ( t ) d x i d x j . (3.58)This is simply the Bianchi I cosmology , the simplest of the Bianchi homogeneous space-times [49]. Since K ij = 12 ˙ h ij ( t ) , (3.59)is now position independent, we have K : j = 0. Then the sufficient condition for aguaranteed type I stress-energy tensor is just K ij : j = 0 . (3.60)Since the connection in the spatial 3-slices vanishes, this equation is clearly satis-fied. Hence all Bianchi I spacetimes are Hawking–Ellis type I. It is worth notingthat spatially-flat FLRW spacetimes belong to this class.
We now allow curvature for the spatial slicings, but impose self-similarity so that thespatial slices have the same shape but can differ in overall scale factor. That is, we set g ij ( t, x ) → a ( t ) h ij ( x ) so thatd s = − d t + a ( t ) h ij ( x k ) d x i d x j . (3.61)Here, for the moment, we allow the h ij ( x k ) to be generic functions of the spatial coor-dinates. Geometries of this type are still much more general than FLRW, allowing forboth spatial inhomogeneities and anisotropies. In this situation, we have K ij = a ˙ a h ij , and K = 3 ˙ aa . (3.62)Under these conditions the sufficient condition for a type I stress-energy tensor can bewritten as h ij : j = 0 . (3.63)But is trivially satisfied for any 3-metric h ij ( x k ), and, therefore the stress-energy tensoris Hawking-Ellis type I. – 20 – .6.3 Single-mode restriction of Bianchi types II to IX General Bianchi cosmologies (the spatially homogeneous cosmologies) can all be writtenin the form d s = − d t + h IJ ( t ) ω I ω J . (3.64)Here the ω I are the 1-forms dual to the invariant basis vectors used in setting up theBianchi classification of homogeneous 3-geometries. See reference [49] for an extensivediscussion. (Note especially table 6.1 on pages 110–113.) Since h IJ ( t ) is a symmetric3 × h IJ ( t ). Suppose now that we restrict attention to a singlemode by setting h IJ ( t ) → a ( t ) h IJ , where the h IJ are now constants. Then ds = − dt + a ( t ) (cid:8) h IJ ω I ω J (cid:9) = − dt + a ( t ) (cid:8) h ij ( x ) dx i dx j (cid:9) . (3.65)That is, single-mode Bianchi cosmologies of this form are automatically spatially self-similar, and the discussion above applies. Consequently the stress-energy tensor isHawking–Ellis type I.It is easy to note that all three Friedmann–Lemaˆıtre–Robertson–Walker spacetimes areof this form. As expected all three FLRW spacetimes (which are isotropic spatiallyself-similar sub-cases of Bianchi types I, V, and IX respectively) have a stress-energyof Hawking–Ellis type I.Unfortunately we can give no really general arguments for other more general multi-mode Bianchi cosmologies, and at best one has to resort to case-by-case analyses.
We have explicitly demonstrated that, when considering self-consistent solutions ofthe Einstein equations, the presence of symmetry often severely restricts the natureof the stress-energy tensor under the Hawking–Ellis (Segr´e–Pleba´nski) classification.The same considerations also apply to any proposed modifications or extensions ofEinstein gravity which have equations of motion that can be rearranged into the form G ab = 8 π [ T effective ] ab . Then at a minimum our results constrain the Hawking–Ellisclassification of [ T effective ] ab .Working in the test-field limit it is rather easy to find examples of Hawking–Ellis typesI and IV. (Hawking–Ellis types II and III are perturbatively unstable; either undernumerical round-off error or under generic physical perturbations.)– 21 –orking within the framework of self-consistent solutions of the Einstein equations,type IV is often excluded. Indeed in (3+1) dimensions the stress-energy tensor isguaranteed to be Hawking–Ellis type I in at least the following situations: • In the domain of outer communication of any static spacetime. • On any Killing horizon (static, stationary, bifurcate). • On the axis of rotation of any axisymmetric stationary spacetime. • In any Bianchi type I cosmology. • In any single-mode restriction of the Bianchi type II to type IX cosmologies.This list is not necessarily exhaustive, and we are actively seeking further examples ofthis or similar behaviour.It is worthwhile to note that if it could be proved that the stress-energy tensor of self-consistent geometries has to be type I in most physically relevant situations, then thestudy of some implications of semi-classical effects could be simplified. For example,one could study the properties of (possibly regular) black hole spacetimes taking intoaccount Hawking radiation. In the appendices below we also comment on circularsymmetry in (2+1) dimensions, and on dilaton gravity in (1+1) dimensions.
A Appendix: Circular symmetry in (2+1)-dimensions
For completeness, let us consider circular symmetry in (2+1)-dimensions. (Note that(2+1) dimensions is often surprisingly subtle. For instance there is a Birkhoff theoremfor rotating “stars” in (2+1) dimensions [50, 51], so it is well worth the effort to check(2+1)-dimensional physics explicitly.) Ordering the coordinates as ( t, φ, r ), we canwrite g ab ( r ) = g tt g tφ g tφ g φφ
00 0 g rr , (A.1)where the metric components only depend on r . Without loss of generality we definethe r coordinate by setting g φφ = r , then g ab ( r ) = g tt g tφ g tφ r
00 0 g rr . (A.2)– 22 –e now adopt an ADM-like decomposition, then: g ab = − N ( r ) + r ω ( r ) r ω ( r ) 0 r ω ( r ) r
00 0 h ( r ) ab . (A.3)Thence g = det( g ab ) = − N ( r ) r h ( r ) . (A.4)Now we define a co-triad by e Aa ( r ) = N ( r ) 0 0 rω ( r ) r
00 0 h ( r ) , (A.5)and the corresponding triad by e Aa ( r ) = N ( r ) − ω ( r ) N ( r ) 1 r
00 0 h ( r ) . (A.6)What can we say at the centre of rotation? (That is, r = 0.) To avoid a conicalsingularity at r = 0, at a minimum we need h ( r ) = 1 + o ( r ). To be more precise,demanding regularity near the centre of rotation forces us to set N ( r ) = N + 12 N r + ... ; ω ( r ) = ω + 12 ω r + ... ; (A.7) h ( r ) = 1 + 12 h r + .... (A.8)with N > N , ω , ω , and h aretypically though not necessarily non-zero.A brief calculation ( Maple or Mathematica ) shows that at the axis of rotation ( r = 0)one has: ( G AB ) r =0 = ( e Aa e Bb G ab ) r =0 = ( G ˆ t ˆ t ) G ˆ φ ˆ φ ) G ˆ r ˆ r ) . (A.9)– 23 –ere we explicitly have:( G ˆ t ˆ t ) = h ; ( G ˆ φ ˆ φ ) = ( G ˆ r ˆ r ) = N N ; (A.10)with all other stress-energy components being zero. This is manifestly diagonal, and somanifestly type I. Note that the use of the triad/co-triad formalism and orthonormalbasis is essential to getting this to work cleanly.In short: The stress-energy is always Hawking–Ellis type I at the centre of rotation ofany circularly symmetric stationary (2+1) spacetime. B Appendix: (1+1)-dimensions
For completeness, let us finally consider the situation in (1+1)-dimensions. (Note that(1+1) dimensions one can often obtain exact results [11], so it is well worth the effortto check (1+1)-dimensional physics explicitly.) Note that Hawking–Ellis type III isautomatically excluded in (1+1) dimensions, though all three of Hawking–Ellis types I,II, and IV are a priori possible. Furthermore in (1+1) any nonzero vector field ishypersurface orthogonal, so there is no distinction between static and stationary.
B.1 Test field stress-energy
Let us first consider a test-field computation: we consider a massless minimally coupledscalar field and follow the analysis of [11], which is in turn based on discussion in [52, 53].We take the orthonormal form of the stress-energy to be T ˆ a ˆ b = (cid:20) ρ ff p (cid:21) (B.1)Then the two Lorentz invariant eigenvalues are λ = 12 ( p − ρ ) ± p ( ρ + p ) − f . (B.2)In the Boulware of Hartle–Hawking vacuum states the flux is zero, so the stress-energyis automatically type I. So let us focus on the Unruh vacuum state. Specifically, forthe (1+1) dimensional version of Schwarzschildd s = − (1 − m/r )d t + d r − m/r , (B.3)– 24 –efining z = 2 m/r , the stress-energy in the Unruh vacuum is easily computed to be [11] ρ = p ∞ − z + 14 z − z ) ; p = p ∞ − z − z ) ; f = p ∞ − z ) . (B.4)Consequently Γ = ( ρ + p ) − f = p ∞ z (1 + z )(1 − z )(4 − z )1 − z . (B.5)This changes sign at z ∈ n √ , , √ o . The test-field stress-energy is Hawking–Ellistype I for z ∈ h √ , (cid:17) and z ≥ √ , and is type IV for z ∈ h , √ (cid:17) and z ∈ (cid:16) , √ (cid:17) . Sotest field stress-energies can easily be type IV in (1+1) dimensions. B.2 Dilaton gravity
To go beyond the test field limit one has to choose a specific theory of (1+1) gravity.The Einstein tensor is identically zero, so the usual Einstein equations are not relevant.The Ricci tensor identically satisfies R ab = R g ab and is automatically always type I.To get anything interesting one needs something more subtle such as a dilaton gravity.Let us choose for instance: S = Z d x √− g { ϕR − U ( ϕ ) } + S matter . (B.6)Then one has T ab = 2 ∇ a ∇ b ϕ − g ab ∇ ϕ − g ab U ( ϕ ) , (B.7)Thus, up to a trivial shift, the Lorentz invariant eigenvalues of the stress-energy aredetermined by the Lorentz invariant eigenvalues of the traceless tensorˆ T ab = 2 (cid:18) ∇ a ∇ b ϕ − g ab ∇ ϕ (cid:19) . (B.8)– 25 –ow in (1+1) dimensions any Killing vector must satisfy ∇ a K b ∝ ǫ ab . (B.9)Furthermore, if there is a Killing vector K a under which the scalar field is invariant( K a ∇ a ϕ = 0) then in (1+1) dimensions both K a and ǫ ab ∇ b ϕ are perpendicular to ∇ a ϕ ,so they must be parallel to each other: K a ∝ ǫ ab ∇ b ϕ. (B.10)But then K a ( ∇ a ∇ b ϕ ) = K a ( ∇ b ∇ a ϕ ) = ∇ b ( K a ∇ a ϕ ) − ( ∇ b K a ) ∇ a ϕ = − ( ∇ b K a ) ∇ a ϕ. (B.11)This implies K a ( ∇ a ∇ b ϕ ) ∝ ( ǫ ba ) ∇ a ϕ ∝ K b . (B.12)Consequently in (1+1) dimensions any Killing vector is an eigenvector of the dilatonstress-energy tensor. If the Killing vector is timelike, then we are done (the stress-energy is Hawking–Ellis type I). If the Killing vector is spacelike, then with one spacelikeeigenvector, the other must be timelike, then we are again done (the stress-energy isHawking–Ellis type I). The exceptional case is when the Killing vector is null, thenthe stress-energy is Hawking–Ellis type II. Therefore for any non-null Killing vector, in(1+1) dimensions the stress-energy tensor in dilaton gravity is Hawking–Ellis type I. Acknowledgments
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