Intersecting hypersurfaces in AdS and Lovelock gravity

Abstract

Colliding and intersecting hypersurfaces filled with matter (membranes) are studied in the Lovelock higher order curvature theory of gravity. Lovelock terms couple hypersurfaces of different dimensionalities, extending the range of possible intersection configurations. We restrict the study to constant curvature membranes in constant curvature AdS and dS background and consider their general intersections. This illustrates some key features which make the theory different to the Einstein gravity. Higher co-dimension membranes may lie at the intersection of co-dimension 1 hypersurfaces in Lovelock gravity; the hypersurfaces are located at the discontinuities of the first derivative of the metric, and they need not carry matter. The example of colliding membranes shows that general solutions can only be supported by (spacelike) matter at the collision surface, thus naturally conflicting with the dominant energy condition (DEC). The imposition of the DEC gives selection rules on the types of collision allowed. When the hypersurfaces don't carry matter, one gets a soliton-like configuration. Then, at the intersection one has a co-dimension 2 or higher membrane standing alone in AdS-vacuum spacetime \emph{without conical singularities.} Another result is that if the number of intersecting hypersurfaces goes to infinity the limiting spacetime is free of curvature singularities if the intersection is put at the boundary of each AdS bulk.

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arXiv:hep-th/0412273v5 11 Oct 2006
Intersecting hypersurfaces in AdS and Lovelock gravity
Elias Gravanis1∗and Steven Willison1,2 †
1Department of Physics, Kings College, Strand, London WC2R 2LS, U.K.
2Centro de Estudios Cient´ ıficos (CECS), Casilla 1469, Valdivia, Chile.
Abstract
Colliding and intersecting hypersurfaces filled with matter (membranes) are studied in the Lovelock higher
order curvature theory of gravity. Lovelock terms couple hypersurfaces of different dimensionalities, extending the
range of possible intersection configurations. We restrict the study to constant curvature membranes in constant
curvature AdS and dS background and consider their general intersections. This illustrates some key features which
make the theory different to the Einstein gravity. Higher co-dimension membranes may lie at the intersection of
co-dimension 1 hypersurfaces in Lovelock gravity; the hypersurfaces are located at the discontinuities of the first
derivative of the metric, and they need not carry matter.
The example of colliding membranes shows that general solutions can only be supported by (spacelike) matter
at the collision surface, thus naturally conflicting with the dominant energy condition (DEC). The imposition of
the DEC gives selection rules on the types of collision allowed.
When the hypersurfaces don’t carry matter, one gets a soliton-like configuration. Then, at the intersection one
has a co-dimension 2 or higher membrane standing alone in AdS-vacuum spacetime without conical singularities.
Another result is that if the number of intersecting hypersurfaces goes to infinity the limiting spacetime is free
of curvature singularities if the intersection is put at the boundary of each AdS bulk.
1Introduction
Lately, a strange idea has become popular in cosmology. It has been suggested [1] that we live on a (3+1)-dimensional
membrane, called a brane world, living in a higher dimensional space-time. Many general relativity models have
been invented to describe the gravitational behavior of such a brane-world. Although there is a clear conceptual link
with string theory, i.e. the extra dimensions and the existence of membranes with matter and gauge fields confined
to their world-sheets, it is also clear that this is a highly speculative idea.
∗E-mail: eliasgravanis@netscape.net
†E-mail: steve-at-cecs.cl
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This idea motivates a general study of hypersurfaces in d-dimensional curved spacetime. Co-dimension 1 hy-
persurfaces are understood as co-dimension 1 sub-manifolds which are the locus of the discontinuities of the first
derivative of the metric. To draw specific conclusions we need a theory of gravity, determining the metric of the
d-dimensional spacetime locally. Lovelock gravity is a natural choice in d dimensions in place of Einstein gravity in
four dimensions; it is the only theory (action functional) for the metric which gives second order field equations when
torsion is zero, that is, when the covariant derivative is given by the usual formula [2][3]. One can get a relation
between the discontinuity of the first derivative of the metric to the energy tensor of matter on the hypersurface.
Hypersurfaces of any co-dimensionality which (potentially) carry matter will be called membranes.
The more complicated, compared to Einstein’s theory, structure of derivatives in Lovelock gravity, makes possible
to have membranes of co-dimensionality higher than one, via intersections of co-dimension 1 hypersurfaces, without
any spacetime singularities. Put slightly differently, high co-dimension membranes can be embedded in spacetime
without causing conical or more pathological curvature singularities if they are embedded at the intersection of co-
dimension 1 hypersurfaces. In fact, in d dimensions there exist membranes of co-dimensionality up to the integer
part ofd−1
2, such that the metric is everywhere continuous, its first derivative has (bounded) discontinuities at the
hypersurfaces and spacetime is everywhere, and especially at the membranes, a manifold [4][5].
The higher dimensional gravity theory of Lovelock [3] is an interesting generalization of general relativity. In
d ≥ 5 the Einstein-Hilbert is not the most general Lagrangian that produces second order field equations and it was
extended by Lovelock to a more general theory with this property. The latter gives the theory familiar features, in
accordance with our experience from classical mechanics and field theory. It allows for a Hamiltonian formulation [6]
and the possibility of a well-posed initial value problem [7]. The Lagrangian which possesses this property was
found by Lovelock [3] and it is a linear combination of terms corresponding to the Euler densities in all lower even
dimensions [2].
L =
[(d−1)/2]
?
n=0
1
2nβnδµ1...µ2n
ν1...ν2nRν1ν2
µ1µ2· · · Rν2n−1ν2n
µ2n−1µ2n
√gddx.(1)
where [x] is the integer part x. The generalization of the Einstein tensor is the Lovelock tensor:
Hµ
ν= −
[(d−1)/2]
?
n=0
1
2n+1βnδµµ1...µ2n
νν1...ν2nRν1ν2
µ1µ2· · · Rν2n−1ν2n
µ2n−1µ2n
(2)
The delta is the generalized totally anti-symmetrized Kronecker delta. It is the determinant of a matrix with
elements δM
N,
δµ1...µp
ν1...νp= det









δµ1
ν1
δµ2
ν1
...δµp
ν1
δµ1
ν2
...
δµ2
ν2
...
...
...
δµp
ν2
...
δµ1
νp
δµ2
νp
...δµp
νp









= p!δµ1
[ν1···δµp
νp]. (3)
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The Lovelocktheories have been studied extensively. Higher dimensional black hole solutions have been found [8][9][10].
This has shed some interesting light on questions of black hole entropy. Some cosmological metrics have been stud-
ied [11].
The nmax= 2 Lovelock theory, which we call the Gauss-Bonnet theory, has a special physical significance. This is
because the n = 2 term is the only quadratic term which has a ghost free perturbation theory about flat space-time.
It has been conjectured that the Gauss-Bonnet term is the leading order, purely geometric, correction to the effective
action of an underlying unitary fundamental theory [12]. In particular, the Lovelock contributions, motivated by
string theory, have played a role in brane-world cosmology [13, 14].
It was Zumino [2] who formulated the theory in the way we prefer, as an elegant way to prove suggestions by
Zwiebach related to low energy string theory [12]. We use the vielbein formulation: Eais the vielbein frame, ωab
the spin connection and Ωabis the curvature two-form.
Ωab=1
2Rab
cdEc∧ Ed,
Rab
cd= Ea
µEb
νRµν
κλEκ
cEλ
d.
In this language, the Lovelock Lagrangian is:
L =
[(d−1)/2]
?
n=0
βnΩa1a2∧ ···Ωa2n−1a2n∧ ea1···a2n
(4)
where
ea1···ap=
1
(d − p)!ǫa1...adEap+1∧ ··· ∧ Ead
(5)
and we have defined the totally anti-symmetric tensor such that ǫ(1)...(d)= 1. The Latin letters from the beginning
of the alphabet are used for the local Lorentz indices (d-dimensional). Greek letters from the middle of the alphabet
are used for space-time co-ordinate indices (d-dimensional).
In the Lovelock theory, singular hypersurfaces of co-dimension 1 can be meaningfully defined in terms of distri-
butions [14, 15], due to the property of quasi-linearity in second derivatives [16]. Brane-worlds of co-dimension 1
have thus been the most well studied and understood. They can also be formulated by means of boundary terms in
the action. The correct boundary term is most elegantly derived by a dimensional continuation of the Gauss-Bonnet
theorem for a manifold with boundary [17]. This latter approach is the one we have adopted.
The possibility of colliding shells or branes of matter has been studied in the context of GR [18]. In Lovelock
gravity, there has been some study of intersecting brane-worlds, see e.g. [19] and more recently [20][21], but so far,
there has been no study of collisions in this context except our comments in our recent work [4, 5]. In that work,
we restricted the smoothness of the metric so that there were well defined ortho-normal vectors at the intersec-
tion/collision. The most striking fact, physically, about intersections or collisions is that they could carry their own
singular stress-energy tensor. This is a phenomenon that does not occur in the Einstein theory. That difference and
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related properties of the Gauss-Bonnet term was used in [19], to address the cosmological constant problem and for-
mulate higher co-dimension brane worlds via intersections, continuing on previous work in GR context, see e.g. [22].
Our aim here will be to discuss general properties of the intersections and collisions of hypersurfaces in Lovelock
gravity, mostly via the example of AdS and dS background, which may be useful also outside the brane-world context.
In the Einstein theory, singular matter can only be accommodated at an intersection of co-dimension 2 if there is a
conical singularity, with a deficit angle. Then, it is impossible to define two ortho-normal vectors normal to the inter-
section. Although Lovelock gravity with a conical singularity can be described in terms of distributions [23][24][25],
there is a certain ambiguity about the solutions- in general, we would not expect the thin brane to be the unique
limit of a thick brane solution [26, 27]. We shall not consider this kind of singularity in the present work.
There is though an interesting possibility. We consider intersections of hypersurfaces, non-null as well as null,
which carry zero energy tensor. At their intersections there appear higher co-dimension membranes. For non-null
co-dimension 1 hypersurfaces we have an intersection of soliton-like configurations, pure (cosmological constant-)
vacuum gravitational field self-supported and with a non-zero jump in the extrinsic curvature; for null hypersurfaces
we have intersection/collision of gravitational shock waves. In both cases one has at their intersections membranes
of co-dimension ≥ 2 surrounded by pure AdS background on a non-singular spacetime. This is a phenomenon not
possible in Einstein gravity [26].
Previous works on related problems in the brane-world were in the context of: Einstein gravity e.g. [22]; in
supergravity, where intersection rules for branes carrying form field charges were derived in refs. [28][29][30]; and in
various formulations when Gauss-Bonnet or higher Euler densities are included, e.g. [31][19][20][21][24]. An important
difference in this work and our previous ones [4][5] is that one may have a high co-dimension membrane without the
cost of making spacetime singular [31][24][21]. Our primary intention really is to point out properties of Lovelock
gravity which are interesting on their own, but our results may be useful to other endeavors.
In sections 2-4 we present the example of intersecting hypersurfaces in an anti-de Sitter background. In section 5
we discuss colliding hypersurfaces in de Sitter background and point out the spontaneous dominant energy condition
violation in collisions. In section 6, we discuss the dimensionalities of the intersection in relation to a 4-dimensional
universe. In section 7 we discuss higher co-dimension membranes using intersections of solitonic configurations and
shock waves.
1.1The intersection junction conditions
For our purposes, hyper-surfaces are (d−1)-dimensional surfaces which divide the space-time up into d-dimensional
bulk regions. We shall assume that they are space-time like (i.e. with space-like normal vector). If there is a non-zero
singular component to the stress-energy tensor with it’s support on the hyper-surface, we shall also call it a brane.
The mathematics of the intersections becomes simple if we consider a kind of minimal intersection, which involves
the minimum number of hypersurfaces needed to build the intersection of a given co-dimensionality (dimensionality of
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its normal space). Put differently, in such an intersection any bulk region has (a co-dimension 1) common boundary
with any other bulk region. Not without a reason we call them simplicial intersections: if abstractly we assign a
point to every bulk region in which the connection is continuous, then a co-dimension p intersection corresponds to a
p-dimensional simplex, that is, the p-dimensional polyhedron with the minimum number of vertices. This abstraction
turns into a practical method of calculating the Lagrangian densities integrated over the intersections [4, 5].
One of the simplifications related to the simplicial intersection, is that if we label the bulk regions with i (and
designate {i}) then the co-dimension p intersection can be labeled by an anti-symmetric symbol involving the labels
of the p + 1 bulk regions meeting there; the simplest example are the co-dimension 1 hyper-surfaces designated in
general as {i0i1} = −{i1i0}. So we introduce the following [4, 5]
Definition 1.1. (simplicial intersection) Let {i} be a bulk region. {i0...ip} is a simplicial intersection where
bulk regions i0,...,ip meet, if it is a (d − p)-dimensional submanifold. The connections in the bulk regions are
ω0,...,ωprespectively. {i0...ip} is a part of the boundary of the (p − 1)-intersection {i0...ip−1}. The orientation
is ∂{i0...ip−1} = +{i0...ip} + ···. Swapping any pair of indices reverses the orientation.
Note that intersections may be space-like, time-like or null (or vary between them).
There are junction conditions relating the singular stress-energy to the geometry [4, 5]. Let ωibe the connection
in region {i}. At a hypersurface {ij} there can be a discontinuity ωi?= ωj. The junction conditions at a p-intersection
are obtained from the intersection Lagrangian:
[(d−1)/2]
?
n=1
βnLn
(p), (6)
Ln
(p)(E,ω0,..,ωp) = Ap
?
s0...p
dpt(ω1− ω0)a1b1···(ωp− ω0)apbpΩ(t)ap+1bp+1...anbnea1...bn,
Ap= (−1)p(p−1)/2
n!
(n − p)!.
Ω(t) is the curvature of the interpolating connection ω(t):
ω(t) :=
p
?
i=0
tiωi
,Ω(t) = dω(t) + ω(t) ∧ ω(t)(7)
The Ω(t)ap+1...bnis short for the (n−p)-fold product: Ω(t)∧···∧Ω(t). The integral is over the p-dimensional simplex
p
?
The junction conditions come from explicit Euler variation w.r.t. to the vielbein: δEcL(p)= −2(T(d,d−p))b
where T(d,d−p)is the part of the singular stress-energy tensor with support on the intersection. The factor of −2 is
s0...p= {t ∈ Rp+1|
i=0
ti= 1, all ti≥ 0}.(8)
c˜ eb,
5