Delta function singularities in the Weyl tensor at the brane Philip D. Mannheim Department of Physics, University of Connecticut, Storrs, CT 06269 Email address: [email protected] hep-th/0101047, January 8, 2001 Abstract 1 0 0 In a recent paper Shiromizu, Maeda and Sasaki derived the gravitational 2 equations of motion which would hold on a brane which is embedded in n a a higher dimensional bulk spacetime, showing that even when the Einstein J equations are imposed in the bulk, nonetheless the embedding leads to a 8 modification of the Einstein equations on the brane.In this comment on their 1 work we explicitly identify and evaluate a delta function singularity effect at v thebranewhich they donot appear to have discussedin their paper,an effect 7 4 which, while actually being of interest in and of itself, nonetheless turns out 0 not to modify their reported results. 1 0 1 Interest in the possible existence of extra dimensions has recently been given consid- 0 erable impetus through the work of Randall and Sundrum [1,2] who showed that if our / h 4-dimensional universe is a domain wall 3-brane embedded in a 5-dimensional bulk AdS t 5 - spacetime, the AdS bulk geometry would then lead to exponential suppression of the ge- p 5 e ometry away from the brane and thereby localize gravity to it. With gravitational signals h thus effectively being confined to our brane the size of such extra dimensions could poten- : v tially then be very large and yet nonetheless not lead to conflict with currently available i X gravitational information. However, even while such higher dimensions could thus nicely r a hide themselves from direct view, it turns out that through the very fact of there being an embedding into a higher dimensional space at all, the very presence of extra dimensions then has an indirect effect on gravitational measurements within our 4-dimensional world, leading (as nicely shown by Shiromizu, Maeda and Sasaki [3]) to a modification of the Ein- stein equations on the brane, with measurements on the brane potentially then being able to reveal the existence of higher dimensions and thus probe their possible existence.1 Thus given the potential significance and broad applicability of the work of Shiromizu, Maeda and Sasaki we have gone over their paper, to find that there is a delta function singularity 1Such embeddings can even lead to possiblemodifications of the equations of state of the fields on the brane, with the consistency of the embedding often being found [4] to lead to negative pressure brane fluids just like the ones currently being considered in cosmology [5], except that rather than being due to the explicit presence of fundamental 4-dimensional quintessence fields, the negative pressure is instead maintained by gravitational stresses coming from a higher dimensional bulk. 1 effect which they do not appear to have discussed, an effect which, while actually being of interest in and of itself, nonetheless turns out not to modify their reported results. In order to discuss the embedding of our universe into a 5-dimensional bulk space with metric g (A,B = 0,1,2,3,5) it is particularly convenient [3] to base the analysis on the AB purely geometric Gauss embedding formula (4)Rα = RA q αqB qC qD −Kα K +Kα K , (1) βγδ BCD A β γ δ γ βδ δ βγ which relates the 4-dimensional Riemann tensor (4)Rα (α,β = 0,1,2,3) of a general 4- βγδ dimensional surface to the Riemann tensor RA of a 5-dimensional bulk into which it is BCD embedded via a term quadratic in the extrinsic curvature K = qα qβ n of the 4-surface. µν µ ν β;α (Here q = g − n n ≡ q is the metric which is induced on the 4-surface by the AB AB A B µν embedding (viz. the one with which (4)Rα is calculated) and nA is the embedding normal.) βγδ On introducing the bulk Weyl tensor C = R −(g R −g R −g R + ABCD ABCD AC BD AD BC BC AD g R )/3 + RE (g g − g g )/12, contraction of indices in Eq. (1) immediately BD AC E AC BD AD BC allows us to relate the 4- and 5-dimensional Einstein tensors according to (4)G = 2G (qA qB +nAnBq )/3−GA q /6 µν AB µ ν µν A µν −KK +Kα K +(K2 −K Kαβ)q /2−E (2) µν µ αν αβ µν µν where E = CA n nCqB qD , (3) µν BCD A µ ν with the geometric content of Eq. (2) being first, that of the 35 components of C (viz. ABCD the 35 components of the 50 component R which are independent of G ) 10 of them ABCD AB can be determined once the induced metric on the 4-surface is known; and second, that since itslefthandsideonlycontains derivatives withrespect tothefourcoordinatesother thanthe one in the direction of the embedding normal nA, on its right hand side all derivative terms with respect to this fifth coordinate (labelled y below) must cancel each other identically.2 Dynamical implications of Eq. (2) follow on restricting the 5-dimensional metric to the form ds2 = q dxµdxν + dy2, imposing the y → −y Z Randall-Sundrum brane scenario µν 2 symmetry for a 4-surface 3-brane placed at y = 0 with normal nA = (0,0,0,0,1), taking the bulk Einstein equations to be of the form G = R −g RC /2 = −κ2[−Λ g +T δµδνδ(y)] (4) AB AB AB C 5 5 AB µν A B and imposing the 20 junction conditions K (y = 0+)−K (y = 0−) = −κ2(T −q Tα /3), q (y = 0+)−q (y = 0−) = 0 (5) µν µν 5 µν µν α µν µν which serve to determine the discontinuity in the extrinsic curvature at the brane [6] and enforce the continuity of the induced metric on it. With the brane symmetry requiring the 2For instance, for ds2 = f(y)(−dt2 +dx¯2)+dy2, nA = (0,0,0,0,1), term by term Eq. (2) yields (4)G0 = −f′′/f −f′2/f2+f′′/f +f′2/4f2−f′2/f2+f′2/4f2+2f′2/f2−f′2/2f2−0, i.e. 0 = 0. 0 2 induced metric coefficients to be functions of |y| = y[θ(y) − θ(−y)] (where |y|′ = θ(y) − θ(−y), |y|′2 = 1, |y|′′ = 2δ(y)), we see that the extrinsic curvature is an odd, discontinuous function of |y|, while the Einstein and Weyl tensors are even functions. Consequently, we can determine the extrinsic curvature from the Israel junction conditions, and since its contribution on the brane is quadratic in Eq. (2) and thus a net even function of |y| which is continuous across the brane, we can then evaluate its resulting contribution to Eq. (2) by averaging the values of the contributions of these quadratic terms on the two sides of the brane. Since the Einstein and Weyl tensors are even functions of |y| it is initially very tempting to apply this same prescription to evaluate their (generically of the form F¯ = F(y = 0+)/2 + F(y = 0−)/2) contributions on the brane as well, to thus lead in the T = −λq + τ case (we conveniently separate out the brane cosmological constant λ) µν µν µν to the relation given in [3], viz. (4)G = Λ q −8πG τ −κ4π −E¯ (6) µν 4 µν N µν 5 µν µν where G = λκ4/48π, Λ = κ2(Λ +κ2λ2/6)/2, (7) N 5 4 5 5 5 π = −τ τ α/4+τα τ /12+q τ ταβ/8−q (τα )2/24. (8) µν µα ν α µν µν αβ µν α As we thus see, in the event that the Einstein equations hold in some higher dimensional bulk spacetime (and even one such as a 10-dimensional one for instance), in general they will not in fact hold on any lower dimensional embedded brane as well, with the non-vanishing of the quantities π and E¯ signaling an explicit departure from the standard Einstein µν µν equations of motion on the lower dimensional brane. Before immediately identifying Eq. (6) as the explicit consequence of the embedding however, we note that as well as being even functions of |y|, the Einstein and Weyl tensors are also second derivative functions of the metric. Consequently they must also contain explicit (and equally even) discontinuous delta function terms in y as well at the brane (cf. d2f(|y|)/dy2 = d2f(|y|)/d|y|2+2(df(|y|)/d|y|)δ(y) for any function f(|y|)), terms which then contribute on the brane in addition to the continuous F¯ type averaging contributions. And indeed, the δ(y) contribution of the bulk Einstein tensor terms in Eq. (2) is readily determined from the Einstein equations, to yield a contribution 2G (qA qB +nAnBq )/3−GA q /6 = −2κ2[τ qα qβ −τα q /4]δ(y)/3 (9) AB µ ν µν A µν 5 αβ µ ν α µν on the brane. Now since (4)G is evaluated from the induced metric alone, and since q µν µν itself is continuous at the brane according to Eq. (5), (4)G cannot contain any second µν derivatives of |y|. Consequently, with Eq. (2) being a geometric identity, the discontinuous second order derivative terms in the Einstein tensor must be canceled identically by the discontinuous second order derivative terms in E (the K dependent terms are only first µν µν derivative functions of |y|). Thus as well as having an average contribution on the brane, E must contain a specific additional discontinuous δ(y) dependent term as well, viz. µν Edisc = −2κ2[τ qα qβ −τα q /4]δ(y)/3, (10) µν 5 αβ µ ν α µν 3 a term which reduces to Edisc = −2κ2(ρ +p )[U U +q /4]δ(y)/3 (11) µν 5 m m µ ν µν for a perfect fluid τ = (ρ + p )U U + p q . Now, as we had already noted above, µν m m µ ν m µν knowledge of the Einstein tensor on the brane provides us with information regarding the Weyl tensor. Equation (11) (a relation which is, as far as we know, new) thus emerges as the dynamical consequence of the absence of any discontinuity in (4)G or q .3 Now as far as µν µν the derivation of Eq. (6) is concerned, since the delta function terms in Eqs. (9) and (10) do cancel identically in Eq. (2), only the average values of the Einstein and Weyl tensors on the brane are ultimately needed to determine (4)G , with Eq. (6) as derived by the averaging µν procedure thus nicely remaining intact.4 As regards some possible practical applications of Eqs. (6) and (11), we note first that the non-vanishing of E¯ immediately entails a necessarily non-vanishing Weyl tensor in µν the bulk and a thus necessary (and possibly gravity non-localizing [4,7]) departure from the maximally 5-symmetric AdS bulk considered in [1,2]. Second, we note that even with 5 the vanishing of both E¯ and Edisc, an AdS bulk is still not necessarily secured, since µν µν 5 even if the entire 10-component E = CA n nCqB qD were to vanish, the 25 other µν BCD A µ ν components of the bulk Weyl tensor would still not be constrained. A specific case in point is the embedding of the ρ = 0, p = 0 (and thus ρ + p = 0) exterior Schwarzschild m m m m metric on a brane with cosmological constant λ and normal nA = (0,0,0,0,1) into a bulk with cosmological constant Λ = −κ2λ2/6, a situation for which there is an explicit exact 5 5 solution, one in which every single term in Eq. (6) vanishes, viz. [8] ds2 = e−2|y|[−(1−2MG/r)dt2 +dr2/(1−2MG/r)+r2dΩ]+dy2, (12) a solution in which the bulk Weyl tensor components C (=2MGe−2|y|/r3), C , C , 0101 0202 0303 C , C and C explicitly do not vanish away from the brane, with the bulk thus not 1212 1313 2323 being AdS in this particular case. 5 Now while we have identified Edisc as the discontinuous piece of E , it is important to µν µν note that since d2(Pa |y|n)/dy2 = Pa n(n−1)|y|n−2+2a δ(y), it is thus possible for Edisc n n 1 µν (∼ 2a δ(y)) to be non-vanishing even when E¯ (∼ 2a +6a |y|+...) itself does vanish, with 1 µν 2 3 theentire E thenbeing just a delta functionatthebrane.Infact wehave even foundacase µν in which this explicitly occurs, specifically the embedding into AdS of a static, p 6= −ρ 5 m m perfect fluid Roberston-Walker brane with non-zero spatial 3-curvature k, a model which 3The delta function singularity in Edisc arises explicitly because of the singular nature of the Z µν 2 symmetric Randall-Sundrum brane set up, with the Weyl tensor on the brane itself (viz. the one explicitly associated with the induced metric q ) not possessing any such singularity. µν 4Bearing in mind the different roles played by even and odd functions of |y|, we note in passing that since the 5-dimensional covariant conservation condition [TµνδAδBδ(y)] = 0 which follows µ ν ;B from Eq. (4) always involves products of even δ(y) functions with odd Christoffel symbol functions whenever Aor B is equalto 5, at thebranethis conservation condition then reduces tothe familiar Tµν = 0(i.e. toτµν = 0), acondition whichinvolves thefieldsandthemetriconthebranealone. ;ν ;ν 4 had been studied in [4,7]. In this specific case the solution to Eqs. (4) and (5) was found to be of the form5 ds2 = −dt2e2(y)/f(y)+f(y)[dr2/(1−kr2)+r2dΩ]+dy2 (13) where f(y) = αeν|y| +βe−ν|y| −2k/ν2, e(y) = αeν|y| −βe−ν|y|, ν = (−2κ2Λ /3)1/2, 5 5 3ν(β −α) = (α+β −2k/ν2)κ2(λ+ρ ), 6ν(α+β) = (α−β)κ2(ρ +3p −2λ), (14) 5 m 5 m m with the vanishing of the Weyl tensor in the bulk (as required if the bulk is to actually be the maximally 5-symmetric AdS despite the embedding into it of a much lower, only 5 maximally3-symmetric,Robertson-Walkerbrane)thenbeingenforced[4,7]bytheadditional requirement thatαβ−k2/ν4 = 0,with the fields then having to obey κ2(ρ +λ)(2ρ +3p − 5 m m m λ) = 6Λ . However, since for the metric of Eq. (13) 10 components of the bulk Weyl tensor 5 (the 6 C with µ 6= ν and the 4 C ) would not in fact vanish without this additional µνµν µ5µ5 requirement, and since these components are all kinematically proportional to C = e[2eff′′ −3ef′2 +2efk +3e′ff′ −2f2e′′]/4f3, (15) 0505 then even with the imposition of the αβ −k2/ν4 = 0 condition, we see that while these 10 Weyl tensor components (and thus also the averaged E¯ ) would then vanish everywhere µν else, they will still possess non-vanishing delta function singularities (C ≃ kδ(y)) at the 0505 brane.6 In this particular model then the non-vanishing of Edisc is not associated with the µν non-vanishing of the Weyl tensor anywhere else, and thus even while a non-vanishing E¯ µν would immediately signal a non AdS bulk, in and of itself a non-vanishing Edisc does not.7 5 µν The author would like to thank Drs. A. H. Guth and A. Nayeri for some very helpful discussions. The author would also like to thank Drs. R. L. Jaffe and A. H. Guth for the kind hospitality of the Center for Theoretical Physics at the Massachusetts Institute of Technology where part of this work was performed. This work has been supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818 and in part by grant #DE-FG02-92ER40716.00. 5In passing we note that this particular model was studied in [4,7] since it provides an explicit case where having a non-vanishing cosmological constant Λ in the bulk proved not sufficient (cf. 5 the presence of both converging and diverging exponentials) to localize the geometry to the brane. 6While the continuous piece of e′′/e−f′′/f is cancelled by the other terms in Eq. (15), its delta function term (= κ2(ρ +p )δ(y) = −4kδ(y)/[ν(α−β)]) is not, with the resulting geometry thus 5 m m having thesomewhatbizarre structureof beingconformal to flatboth in thebulkand on thebrane while having a Weyl tensor which nonetheless is not everywhere zero. 7With manipulation of Eq. (6) leading to [3] 6E¯µν +κ4(ρ +p )(UµUν +qµν)(ρ ) = 0, we ;ν 5 m m m ;ν note in passing that for a perfect fluid the only way that Edisc could not be zero even when E¯ µν µν does vanish is when p +ρ is non-zero and ρ is spatially homogeneous, this intriguingly being m m m none other than the situation which prevails in the standard Robertson-Walker cosmology. 5 REFERENCES [1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999). [2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999). [3] T. Shiromizu, K. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000). [4] P. D. Mannheim, Phys. Rev. D 63, 024018 (2001). [5] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. 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