Arne Gödeke

Alan D Rendall

Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

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Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

Introduction

The solutions of the Einstein equations most commonly applied in cosmology are spatially homogeneous and include some that recollapse and some that expand for ever. In the former case the proper length of the timelike curves orthogonal to the hypersurfaces of homogeneity (the worldlines of comoving observers) is finite. Along a curve of this type the expansion of this congruence of curves is first positive, then zero and finally negative. This is what is meant by recollapse. In the latter case the expansion is always positive and the length of the curves orthogonal to the hypersurfaces of homogeneity is infinite towards the future. This is what is meant by saying that the solution expands for ever.

On physical grounds it is reasonable to expect that a forever expanding solution will be future geodesically complete, i.e. there should be no singularities in the future. A theorem which confirms this expectation is proved in [START_REF] Rendall | Global properties of locally spatially homogeneous cosmological models with matter[END_REF] (Theorem 2.1 of that paper). In order to explain this result let t be a Gaussian time coordinate based on one of the hypersurfaces of homogeneity and let trk be the mean curvature of these hypersurfaces. Then trk is a function of t. The 1 Confidential: not for distribution. Submitted to IOP Publishing for peer review 20 May 2010 expansion of the congruence of timelike geodesics mentioned above is - 1 3 trk. Define an expanding phase of the solution to be a time interval (t 1 , t 2 ) on which trk < 0. Here t 1 and t 2 may be finite or infinite. The theorem says that, under certain assumptions, if t 2 < ∞ the solution can be extended to a time interval (t 1 , t 3 ) with t 3 > t 2 . Moreover, if t 2 = ∞ then the spacetime is future geodesically complete. The theorem is formulated for spatially homogeneous solutions of the Einstein equations coupled to a matter model satisfying three assumptions. Two of these are inequalities on the energy-momentum tensor, the dominant energy and non-negative sum pressures conditions. The third is a continuation property for solutions of the Einstein-matter equations. The only thing about these conditions which is important in what follows is that they are all satisfied in the case of the Einstein vacuum equations. Hence the theorem applies in the vacuum case.

In a spatially homogeneous spacetime, by definition the isometry group acts transitively on the hypersurfaces of homogeneity. In four dimensions there is a well-known classification of the possible isometry groups. Either the group can be taken to have dimension three or a higher-dimensional group is necessary. In the second of these cases there is only one possibility, which is called Kantowski-Sachs symmetry. In the first case any connected three-dimensional Lie group can occur. For the purposes of what follows it may be assumed without loss of generality that the spacetime is simply connected. This is because passing to the universal cover of a given spacetime has no effect on the dynamics. With this assumption the hypersurfaces of homogeneity can be identified with the group itself and the induced metric on one of these hypersurfaces with a leftinvariant metric on that group. The classification of connected and simply connected Lie groups reduces to that of the corresponding Lie algebras. The three-dimensional Lie algebras were classified by Bianchi into types I to IX. It turns out that a maximally extended vacuum spacetime of Bianchi type I-VIII expanding at some time is forever expanding while one which is of type IX or has Kantowski-Sachs symmetry recollapses [START_REF] Lin | Proof of the closed universe recollapse conjecture[END_REF]. The cases where recollapse takes place are precisely those where there is a spatial metric in the class whose scalar curvature R is positive. That this is a necessary condition for recollapse is not hard to see. The Hamiltonian constraint on a hypersurface of homogeneity is (in the vacuum case)

R -k ab k ab + (trk) 2 = 0 (1)
where k ab is the second fundamental form. Recollapse means that there is a maximal hypersurface, i.e. one where trk = 0. There R is clearly non-negative. If it is zero then k ab = 0. Moreover it can be shown that if R = 0 the induced metric of that hypersurface is flat and that the solution arising from those data is a flat spacetime where trk vanishes at all times. The latter case cannot occur in the context of spatially homogeneous cosmological models which are expanding at some time. The aim of this paper is to investigate to what extent the results just stated for four-dimensional spatially homogeneous vacuum spacetimes extend to higher dimensions n + 1. In particular, sufficient conditions for future geodesic com-pleteness are derived. Only the case where the isometry group can be taken to be of dimension n is considered. This is the analogue in higher dimensions of Bianchi models. The analogue of Kantowski-Sachs models is not considered. The same argument as in four dimensions applies to show that a necessary condition for recollapse is the existence of a left-invariant metric of positive scalar curvature on the group of interest. This motivates us to restrict attention to Lie groups admitting no left-invariant metric of positive scalar curvature. In this case it has been shown that vacuum spacetimes with this type of symmetry exist for infinite Gaussian time in the future ( [START_REF] Rendall | Partial differential equations in general relativity[END_REF], Theorem 5.3). On the other hand it turns out that the arguments used to prove geodesic completeness in the four-dimensional case do not easily generalize to higher dimensions. In what follows partial results on this question are obtained.

For a general dimension there is no known analogue of the Bianchi classification of Lie algebras. In some of the lower dimensions there are results. A review of what is known in the case of four-dimensional Lie algebras can be found in [START_REF] Hervik | Multidimensional cosmology: spatially homogeneous models of dimension 4+1[END_REF]. The strategy in the following is to identify some classes of Lie groups for which geodesic completeness can be proved. The main case considered is that where there is an isometry group of the form G 3 × (R Z 2 ) n-3 for a three-dimensional group G 3 . The spatial manifold is given by the identity component of this group, G 3 ×R n-3 . The strategy for handling these groups is to use Kaluza-Klein reduction. The presence of the discrete symmetry defined by the group Z n-3 2 ensures that the result is the Einstein equations coupled to linear scalar fields in four dimensions. Without it a more complicated reduced system would be obtained. In the case n = 4, for instance, it would be the Einstein-Maxwell-dilaton system. Section 2 contains the basic notation and equations, including those for the Kaluza-Klein reduction. In the third section results on the Einstein-scalar field system in four dimensions are proved. These are used in treating higherdimensional vacuum spacetimes in later sections and are also of interest in their own right. A general criterion for geodesic completeness is stated and proved in Section 4. It is then combined with the results on the scalar field to obtain the main theorem, Theorem 5. Section 5 discusses geodesic completeness in some examples with other types of symmetry and illustrates how the results of Section 4 can be applied. Conclusions and an outlook are contained in Section 6.

This paper is based in part on the diploma thesis of the first author [START_REF] Gödeke | Long-term behavior of higher dimensional spatially homogeneous cosmological models[END_REF].

The basic equations

Let G be a connected and simply connected Lie group and θ a a basis of the space of left-invariant one-forms on G. Consider the metric

-dt 2 + g ab (t)θ a ⊗ θ b (2)
on the manifold [t 0 , ∞) × G for some constant t 0 . Let k ab be the components in the given basis of the second fundamental form of the level hypersurfaces of t.

By the definition of the second fundamental form

∂ t g ab = -2k ab . (3)
It will now be supposed that this metric satisfies the Einstein vacuum equations and these will be decomposed using the foliation by the level hypersurfaces of t. The Einstein constraint equations are

R -k ab k ab + (trk) 2 = 0, (4) ∇ a k a b = 0. ( 5
)
The evolution equations are given by

∂ t k a b = (trk)k a b + R a b . (6)
A real number λ is said to be an eigenvalue of k ab with respect to g ab if there is a non-zero vector v a which satisfies the equation k ab v b = λg ab v b . In n space dimensions there are n eigenvalues of k ab with respect to g ab , counting multiplicity. Call them λ a . The mean curvature trk = a λ a never vanishes in an expanding phase. Define the generalized Kasner exponents (GKE) by p a = λa trk . When the Lie group G is Abelian the p a are independent of time and the solution can be written explicitly as

-dt 2 + n a=1 t 2pa (dx a ) 2 (7)
where n a=1 p a = 1 and n a=1 p 2 a = 1. In the case n = 3 this is the Kasner solution.

These equations will now be specialized to the case that the Lie group is of the form G 3 ×R n-3 for a three-dimensional group G 3 and the left-invariant basis θ a is adapted to the product decomposition. Call the basis vectors tangent to the first factor θ i , i = 1, 2, 3. The basis vectors corresponding to the second factor commute and so can be chosen to be of the form dy I , I = 1, . . . , n -3. Suppose further that the action of the group extends to an action of G 3 × (R Z 2 ) n-3 which preserves the hypersurfaces of constant t and the product decomposition of the initial hypersurface. Then the generators of Z n-3 2 act by commuting reflections of R n-3 . Without loss of generality they may be assumed to be given by reflections in the coordinates y I . The invariance of the metric under these transformations implies that it can be written in the form

-dt 2 + g ij (t)θ i ⊗ θ j + i e 2φ I (t) (dy I ) 2 (8)
for scalar functions φ I . Let φ = I φ I and write g ij = e -φ gij . Let t be a Gaussian coordinate for the conformally rescaled metric. Then

d t dt = e φ 2 .
Rewriting the Einstein equations in terms of gij and φ I gives the equations

-kij kij + (tr k) 2 = ρ, (9)
∂ t ki j = (tr k) ki j + Ri j -ρδ i j , (10)
∂ 2 t φ I -(tr k)∂ tφ I = 0. ( 11
)
Here

ρ = 1 2 I (∂ tφ I ) 2 . ( 12
)
These equations are identical to the Einstein equations in four dimensions coupled to n -3 non-interacting massless scalar fields. For prescribed initial data it is possible to do a rotation in R n-3 so as to set all ∂ tφ I except ∂ tφ 1 to zero. By the equations of motion for the scalar fields all φ I except φ 1 will be constant during the evolution. Then only φ 1 makes a contribution to the energymomentum tensor. The essential dynamics is that of a single scalar field. The four-dimensional metric and the scalar field φ 1 define a five-dimensional metric.

The full spacetime is obtained by taking the product of the five-dimensional metric with a flat time-independent metric. The generalized Kasner exponents of the (n + 1)-dimensional metric are related to the four-dimensional quantities by

p a = λa + 1 2 e -φ 2 ∂ t φ tr k + 1 2 e -φ 2 ∂ t φ , (13)
p I = -e -φ 2 ∂ t φ I tr k + 1 2 e -φ 2 ∂ t φ . (14)
In a spatially homogeneous situation a scalar field is equivalent to an untilted stiff fluid. Thus to understand the dynamics of the scalar field case known results for a stiff fluid can be applied. The matter variables of the two models are related by ρ = 1 2 (∂ tφ) 2 . Note that e -φ 2 ∂ t = ∂ t. It was pointed out in the introduction that it is important whether a given Lie group G admits left-invariant metrics with positive scalar curvature. If G is a Lie group of any of the Bianchi types except IX it admits no metric of this kind. This is equivalent, for simply connected Lie groups, to the property that they are diffeomorphic to Euclidean space [START_REF] Bérard-Bergery | Sur la courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes[END_REF]. It follows that a Lie group with topology G × R n-3 is also homeomorphic to Euclidean space and thus also admits no left-invariant metrics of positive scalar curvature. Hence groups of this type fail the necessary condition for recollapse mentioned in the introduction. This is a sign that they are good candidates for the symmetry groups of future geodesically complete metrics.

Four-dimensional models with a scalar field

This section contains results on the late-time behaviour of solutions of the Einstein-scalar field system in four dimensions, some of which are deduced from results on untilted stiff fluids taken from the literature. When information on the dynamics of the energy density is available the asymptotics of φ can be obtained by integration. Much of the analysis is based on the Wainwright-Hsu formulation of the equations for Bianchi models [START_REF] Wainwright | A dynamical systems approach to Bianchi cosmologies: orthogonal models of class A[END_REF], which will now be recalled.

(Background information on this topic and the concepts of the theory of dynamical systems used in the present paper can be found in [START_REF]Dynamical systems in cosmology[END_REF] and [START_REF] Rendall | Partial differential equations in general relativity[END_REF].) Since the only case of interest in the following is that of a stiff fluid the equations are only written down for the case where the parameter γ in the equation of state p = (γ -1)ρ of the fluid is equal to two. The system of evolution equations is

N 1 = (q -4Σ + )N 1 , (15)
N 2 = (q + 2Σ + + 2 √ 3Σ -)N 2 , (16)
N 3 = (q + 2Σ + -2 √ 3Σ -)N 3 , (17)
Σ + = -(2 -q)Σ + -3S + , (18)
Σ -= -(2 -q)Σ --3S -, (19)
Ω = -2(2 -q)Ω (20)
where

q = 2(Ω + Σ 2 + + Σ 2 -), (21)
S + = 1 2 [(N 3 -N 2 ) 2 -N 1 (2N 1 -N 2 -N 3 )], (22)
S -= 1 2 (N 3 -N 2 )(N 1 -N 2 -N 3 ). ( 23
)
The prime denotes d dτ and τ is related to the Gaussian time t by dτ dt = -1 3 trk. There is also the equation which expresses the Hamiltonian constraint in these variables which is

Ω + Σ 2 + + Σ 2 -+ 3 4 (N 2 1 + N 2 2 + N 2 3 -2(N 1 N 2 + N 2 N 3 + N 1 N 3 )) = 1. ( 24
)
It will not be necessary to recall the definition of all the variables in the system here. What is important is that Σ + and Σ -are linear combinations of the generalized Kasner exponents which are linearly independent. This means that if Σ + and Σ -tend to finite limits as t → ∞ the same is true of p 1 , p 2 and p 3 . The density parameter Ω is equal to 3ρ (trk) 2 . The mean curvature can be recovered from a solution of the Wainwright-Hsu system using the equation

(trk) = -(1 + q)(trk). (25)
Theorem 1 Let (M, g) be a solution of the Einstein equations coupled to a scalar field φ in four dimensions with symmetry of a Bianchi type other than IX or VI -1 9 . If the solution is expanding at some time and the time interval of definition of the solution is maximal then the upper limit of the interval is infinity and the generalized Kasner exponents and the density parameter Ω converge to limits as t → ∞.

Proof Bianchi type I solutions are defined by the condition N 1 = N 2 = N 3 = 0. It follows from the Hamiltonian constraint that q = 2 and that Σ + , Σ -and Ω are constant. That the conclusion of the theorem holds for Bianchi type II solutions, which are defined by the conditions N 2 = N 3 = 0 and N 1 = 0, was shown in Lemma 7.1 of [START_REF] Ringström | The Bianchi IX attractor[END_REF]. Bianchi type VI 0 solutions are defined by the conditions N 1 = 0, N 2 > 0 and N 3 < 0. The evolution equation for Σ + reduces to

Σ + = -(2 -q)(Σ + + 1). ( 26
)
It follows from the Hamiltonian constraint that q < 2. This in turn implies that Σ + + 1 is strictly positive and that Σ + is strictly decreasing. Applying the monotonicity principle to Σ + shows that there can be no ω-limit point with q < 2. Since the solution stays in a compact region it follows that q → 2 as τ → ∞ and that N 1 and N 2 tend to zero. Since Σ + and Ω are monotone and bounded both of them tend to limits as τ → ∞. By the Hamiltonian constraint the same is true of Σ -. Bianchi type VII 0 solutions are defined by the conditions N 1 = 0, N 2 > 0 and N 3 > 0. Locally rotationally symmetric solutions of type VII 0 are defined by the additional conditions N 2 = N 3 and Σ -= 0. In this case S + , S 1 are zero and q = 2. Hence Ω and Σ + are constant. The proof for solutions of type VII 0 which are not locally rotationally symmetric results from a small modification of the proof of Proposition 5 of [START_REF] Ringström | The future asymptotics of Bianchi VIII vacuum solutions[END_REF], which treats the corresponding question in the vacuum case. First it can be shown by an argument similar to that used for type VI 0 case that Ω and Σ + are strictly monotone. This makes use of the equation

(N 2 -N 3 ) = (q + 2Σ + )(N 2 -N 3 ) + 2 √ 3Σ -(N 2 + N 3 ) ( 2 7 )
to treat the points where Σ + vanishes. Since Ω and Σ + are also bounded they must converge as t → ∞. It is not known a priori that the whole solution is bounded and so it may not have an ω-limit point. Suppose that it does have such a point. Using the monotonicity principle it can be shown that if τ n is a sequence of times along which the solution converges then either (Σ 2 -+ (N 2 -N 3 ) 2 )(τ n ) tends to zero or N 1 + N 2 does so. It can then be concluded as in [START_REF] Ringström | The future asymptotics of Bianchi VIII vacuum solutions[END_REF], using the monotone function Z -1 of that paper that Σ -→ 0 as τ → ∞. Note that Z -1 is monotone in the case with matter (see [START_REF] Ringström | The future asymptotics of Bianchi VIII vacuum solutions[END_REF], Lemma 10.1). It remains to treat the case in which the solution has no ω limit point. Then N 1 and N 2 must tend to infinity and it can be assumed without loss of generality that the limit of Ω + Σ 2 + is less than one. In particular this means that the limit of Σ + cannot be minus one. Hence, by (26) the quantity 2 -q must be in L 1 . It can, however, be shown by a slight modification of an argument of Ringström ([20], proof of Proposition 5) that 2 -q is not in L 1 . That argument uses two quantities called x and y and the necessary modification is to replace them by

x = Σ - (1 -Σ 2 + -Ω) 1 2 , ( 28
)
y = √ 3 2 N 2 -N 3 (1 -Σ 2 + -Ω) 1 2 . ( 29
)
Bianchi type VIII solutions are defined by the conditions N 1 < 0, N 2 > 0 and N 3 < 0. The result of the theorem in this case follows from Theorem 3.1 and Corollary 3.1 of [START_REF] Horwood | Latetime asymptotic dynamics of Bianchi VIII cosmologies[END_REF].

For models of Bianchi class B it follows from Proposition 5.3 of [START_REF] Hewitt | A dynamical systems approach to Bianchi cosmologies: orthogonal models of class B[END_REF] that the ω-limit set of any solution lies in a set Lk . The parameter k distinguishes the different Bianchi types and is equal to h -1 in types VI h and VII h . For each value of k, the set Lk consists of equilibrium solutions. It is an open curve L k together with its two endpoints. The ω-limit set of a solution is in fact a single point of Lk . This can be seen as follows. A generic point of the curve is transversely hyperbolic and so if it belongs to the ω-limit set no other point can. The remaining points form a discrete set and so by connectedness of the ω-limit set it cannot contain more than one of them.

The fact that Bianchi type IX is excluded in the hypotheses of this theorem is essential. For the metric product of a four-dimensional solution of Bianchi type IX with the real line fails to exist globally towards the future. On the other hand there is no reason to expect that type VI -1 9 has to be excluded. It appears nevertheless that up to now no proof is available in that case.

Some more precise information will now be given on the asymptotics of scalar field models. Let D be the disk in the space with coordinates (Σ + , Σ -, N 1 , N 2 , N 3 ) defined by the conditions that the N i are all zero and Σ 2 + + Σ 2 -< 1. Let L be the line defined by

N 1 = 0, N 2 = N 3 > 0, Σ + = -1 and Σ -= 0.
Theorem 2 Let (M, g) be a solution of the Einstein equations coupled to a scalar field in four dimensions with symmetry of Bianchi class A. If the Bianchi type is I, II or VI 0 then the variables in the Wainwright-Hsu system converge to limits as τ → ∞. If the Bianchi type is VIII the ω-limit set of each solution is empty. In type VII 0 both types of behaviour occur. The sets of points which occur as ω-limit points is the disk D in type I, the open subset of D defined by the inequality Σ + > 1 2 in type II, the point of D with (Σ + , Σ -) = (-1, 0) in type VI 0 and L in LRS type VII 0 . The limit of Ω is non-zero in types I and II and zero in types VI 0 and VIII.

Proof The Bianchi type I solutions of the Wainwright-Hsu system with a stiff fluid are time independent and are exactly the points of D. It follows from Theorem 1 that any solution of type II converges to a point of D as τ → ∞. It is a consequence of the evolution equation for N 1 that this point must satisfy Σ + ≥ 0. A more detailed analysis reveals that this inequality is strict. To see this note that all terms on the right hand side of the equations for Bianchi type II solutions have a common factor N 1 . Omitting this factor leads to a new system whose integral curves are the same as those of the original system in the region where N 1 = 0. The new system is

Ṅ1 = 2(1 -2Σ + ) - 3 2 N 2 1 , (30)
Σ+ = 3 1 - 1 2 Σ + N 1 , (31)
Σ-= - 3 2 N 1 Σ -. (32)
This system has a line of stationary solutions defined by the conditions N 1 = 0 and Σ + = 1 2 . Linearizing the system about one of these points reveals that it has the eigenvalues 0, ±3i. Because of the reduction theorem (cf. [START_REF] Rendall | Partial differential equations in general relativity[END_REF], section 5.6) the qualitative behaviour of the solutions can be determined by analysing the dynamics on a centre manifold. The system on the centre manifold is such that no solution can converge to the stationary point while staying in the region N 1 > 0 (cf. [START_REF] Perko | Differential equations and dynamical systems[END_REF], p. 144, Theorem 5) and this implies the desired result. Any point of D with Σ 0 > 1 2 occurs as the ω-limit point of a solution of type II since the linearization there has a negative eigenvalue. By Theorem 1 any solution of type VI 0 converges to a point of D. As a consequence of the evolution equations for N 2 and N 3 this can only be the point where Σ + = -1. In particular Ω → 0 and the solution looks asymptotically like a vacuum solution. Any locally rotationally symmetric solution of type VII 0 has Σ -= 0, N 2 = N 3 and a constant value of Σ + . As τ → ∞ the quantity N 2 tends to infinity. For type VII 0 solutions which are not LRS it has already been shown that Σ 2 + + Ω → 1 and that Σ -and N 2 -N 3 tend to zero. If the solution has no ω-limit point then Ω → 0 as τ → ∞ and Σ + → 1. For general type VIII it is shown in [START_REF] Horwood | Latetime asymptotic dynamics of Bianchi VIII cosmologies[END_REF] that Ω → 0 and N + → ∞.

A criterion for geodesic completeness

The aim of this section is to prove the following criterion for geodesic completeness. The general set-up is a spatially homogeneous spacetime defined by a one-parameter family g ab (t) of left-invariant metrics on a Lie group G of dimension n.

Theorem 3 Let (M, g) be a locally spatially homogeneous vacuum spacetime of dimension n + 1 < 10 whose symmetry is defined by a Lie group G which admits no left-invariant metrics of positive scalar curvature. If for some t 0 the generalized Kasner exponents satisfy

sup t≥t0 min a p a (t) -inf t≥t0 min a p a (t) ≤ 9 -n 4n (33)
then the spacetime is geodesically complete.

Proof Since the scalar curvature is non-positive it follows from the Hamiltonian constraint that k ab k ab ≤ (trk) 2 and hence that a p 2 a ≤ 1. The minimum of a p 2 a subject to the constraint a p a = 1 occurs when all p a are equal, as follows from the method of Lagrange multipliers. Hence

k ab k ab ≥ (trk) 2 n . ( 34
)
The minimum possible value of p a is -n-2 n , as has been proved in [START_REF] Rendall | Partial differential equations in general relativity[END_REF], p. 96. By symmetry this is also the minimum value of any other p a . It can be concluded that the maximum eigenvalue λ max satisfies

1 n trk ≤ λ max ≤ - n -2 n trk. ( 35
)
If the value of λ max is fixed the inequality (34) can be improved. Assume that λ max = λ 1 . Using Lagrange multipliers again it can be seen that the minimum of k ab k ab is attained when all λ a other than λ 1 are equal. Hence the lower bound for k ab k ab is got by setting

λ 2 = . . . = λ n = trk -λ max n -1 . ( 36
)
It follows that k ab k ab is bounded below by

λ 2 max + (trk -λ max ) 2 n -1 = (trk) 2 (np 2 min -2p min + 1)(n -1) -1 = (trk) 2 P (p min ). ( 37
)
The function P is positive and decreasing on -n-2 n , 1 n . Let p 0 be the supremum in t of p min (t). Then by definition p min (t) ≤ p 0 for all t. On the other hand the hypotheses of the theorem imply that p min (t) ≥ p 0 -∆ where ∆ = p 0inf t≥t0 min a p a (t). Assume that p 0 -∆ < 0. (Geodesic completeness in the case p 0 ≥ ∆ is treated in Theorem 4.) Now

∂ t (trk) = k ab k ab ≥ P (p 0 )(trk) 2 . ( 38
)
Integrating this in t gives

(trk)(t) ≥ -P (p 0 ) -1 C + t (39)
for a constant C depending on t 0 and trk(t 0 ). Let q a be the projection of the tangent vector to a causal geodesic onto the hypersurfaces of constant t. Then d dt (g ab q a q b ) = 2k ab q a q b ≤ 2(p 0 -∆)(trk)g ab q a q b . (

Replacing trk in this inequality using (39) and integrating gives

(g ab q a q b ) -1 2 ≥ C(1 + t) -λ (41)
where λ = ∆-p0 P (p0) . The affine parameter length of a causal geodesic up to time τ is given by

τ to ( + g ab q a q b ) -1 2 dt ( 42
)
where is one for timelike geodesics parametrized by arc length and zero for null geodesics. This integral diverges provided λ ≤ 1. It remains to show that this follows from the inequality ∆ ≤ 9-n 4n . Now ∆ -

p 0 P (p 0 ) ≤ 1 ( 4 3 )
is equivalent to

p 2 0 + n -3 n p 0 + 1 n - n -1 n ∆ ≥ 0. ( 44
)
The expression on the left hand side of this inequality has its minimum at p 0 = -n-3 2n . Its value there is

n -1 n 9 -n 4n -∆ . ( 45
)
This completes the proof of the theorem.

For simplicity this theorem has been stated for the vacuum case only. In fact the theorem, and its proof, generalize straightforwardly to the case where matter is present provided the matter satisfies the dominant and strong energy conditions and has reasonable evolution properties. The precise formulation of the last property is the matter continuation criterion (MCC) introduced in [START_REF] Rendall | Partial differential equations in general relativity[END_REF]. A variant of Theorem 3 is given by Theorem 4 Let (M, g) be a locally spatially homogeneous vacuum spacetime of dimension n + 1 whose symmetry is defined by a Lie group G which admits no left-invariant metrics of positive scalar curvature. If for some t 0 the generalized Kasner exponents satisfy p a ≥ -1 n then the spacetime is geodesically complete.

Proof The proof proceeds in a similar way to that of the previous theorem. This time the inequality ∂ t (trk)

≥ (trk) 2 n implies that (trk)(t) ≥ - n C + t (46)
and that (g ab q a q b ) -

1 2 ≥ C(1 + t) -λ . ( 47
)
This completes the proof.

This theorem also generalizes easily to the case with matter satisfying the DEC, SEC and MCC. In the case n = 3 the hypothesis on the p a is satisfied identically. Next a large class of Lie groups will be exhibited for which the assumptions of Theorem 3 are satisfied. They are obtained using products of three-dimensional Lie groups with Abelian groups of arbitrary dimensions.

Theorem 5 Let (M, g) be a spatially homogeneous solution of the vacuum Einstein equations in n+1 dimensions with 3 ≤ n < 9. Suppose that the group defining the homogeneity is of the form G 3 × (R Z 2 ) n-3 for some three-dimensional Lie group G whose Bianchi type is neither IX or VI - 1 9 . If the solution is maximally extended to the future then it is geodesically complete.

Proof Under the hypotheses of this theorem Kaluza-Klein reduction can be used to relate the generalized Kasner exponents of the higher dimensional vacuum model to the generalized Kasner exponents of the reduced Einstein-scalar field model in four dimensions. By Theorem 1 and the formulae ( 13) and ( 14) it follows that the generalized Kasner exponents of the (n + 1)-dimensional spacetime converge as t → ∞. This conclusion makes use of the fact that, since

Further examples

In the previous section geodesic completeness was proved for models based on Lie groups belonging to a certain class not admitting metrics of positive scalar curvature. To go further it is necessary to enter into the structure of more general Lie algebras. (A good general reference for the theory of Lie groups and Lie algebras is [START_REF] Varadarajan | Lie groups, Lie algebras and their representations[END_REF].) For this some definitions are necessary. If L is a Lie algebra denote by [L, L] the subalgebra generated by the elements of L of the form [x, y] for x and y in L. Define L (n) recursively by L (n+1) = [L (n) , L (n) ] and L (1) = L. If L (n) = {0} for some n then L is called solvable. The radical R of a Lie algebra L is an ideal of L which is solvable and which contains all other solvable ideals. A Levi subalgebra is a subalgebra which, as a vector space, is complementary to the radical. It is semisimple which means that it has no nontrivial solvable ideals. A derivation of L is a linear mapping D which satisfies D([x, y]) = [Dx, y] + [x, Dy] for all x and y. A particular type of derivations are the inner derivations which are of the form [x, ] for some element x of the algebra. Let L 1 and L 2 be two Lie algebras. Denote by DerL 2 the vector space of all derivations of L 2 . It is closed under commutators and in this way acquires the structure of a Lie algebra. Let φ be a Lie algebra homomorphism from L 1 to DerL 2 . A Lie algebra L can be defined as the direct sum of the underlying vector spaces of L 1 and L 2 with the Lie bracket defined by

[(x i , y 1 ), (x 2 , y 2 )] = ([x 1 , x 2 ], [y 1 , y 2 ] + φ(x 1 )(y 2 ) -φ(x 2 )(y 1 )). ( 48
)
The Lie algebra L is called the semidirect sum of L 1 and L 2 . The Levi-Malcev theorem says that any finite-dimensional Lie algebra can be written as a semidirect sum of its radical R with a semisimple Lie algebra. Consider now the case of four-dimensional Lie algebras. It is known that the only semisimple Lie algebras of dimension no greater than four are su(2) and sl(2, R). It follows that the only four-dimensional Lie algebras which are not solvable are semidirect sums of the real numbers with su(2) and sl(2, R). In fact the semidirect sum of the real numbers with a semisimple Lie algebra is isomorphic to a direct sum. To see this note that every derivation of a semisimple Lie algebra is inner. Hence the mapping φ defining the semidirect sum must be of the form φ(x) = φ S (x)[y 0 , ] for a linear map φ S from L 1 to R and some y 0 ∈ L 2 . It then follows that the linear map ψ from L to itself defined by

ψ(x, y) = (x, y + φ S (x)y 0 ) ( 4 9 )
is an isomorphism from the given Lie algebra to the direct sum. The group SU (2) × R admits left-invariant metrics of positive scalar curvature because SU (2) does. On the other hand it is well-known that the simply connected Lie group with Lie algebra sl(2, R) is diffeomorphic to R 3 . Hence the group corresponding to the Lie algebra sl(2, R)⊕R is diffeomorphic to R 4 . Any simply connected solvable Lie group is diffeomorphic to R n for some n and hence admits no metric of positive scalar curvature. Putting these facts together it can be concluded that the only connected and simply connected four-dimensional Lie group which admits a metric of positive scalar curvature is SU (2) × R. This fact was noted in the thesis of Hervik [START_REF] Hervik | Mathematical cosmology: Bianchi models, asymptotics and extra dimensions[END_REF], who refers to a paper of Patera et. al. [START_REF] Patera | Invariants of real low dimension Lie algebras[END_REF]. In the latter a list of Lie algebras of dimensions 4 and 5 is given which are not direct sums and they are all stated to be solvable. It may be noted in passing that on the basis of this the only types for which recollapse can occur in five space dimensions are su(2) ⊕ R 2 and su(2) ⊕ A 2,1 . A general criterion for the existence of left-invariant metrics of positive scalar curvature in any dimension has been given in [START_REF] Hervik | Negatively curved left-invariant metrics on Lie groups[END_REF]. The facts just mentioned suggest the following conjecture:

Conjecture Let G be a connected and simply connected four-dimensional Lie group which is not isomorphic to SU (2) × R. Then any spatially homogeneous solution of the vacuum Einstein equations with this symmetry group which is expanding at some time is future geodesically complete. Theorem 5 proves this conjecture for Lie algebras which are the direct sum of a three-dimensional Lie algebra and the real numbers and metrics which have an additional discrete symmetry. Another case where something is known is that where the four-dimensional algebra is a direct sum of two non-Abelian subalgebras [START_REF] Andersson | Eternal acceleration from M-theory[END_REF]. In the notation of [START_REF] Patera | Invariants of real low dimension Lie algebras[END_REF] and [START_REF] Hervik | Multidimensional cosmology: spatially homogeneous models of dimension 4+1[END_REF] this is the case A 2,1 ⊕ A 2,1 . The metrics covered by the theorem of [START_REF] Andersson | Eternal acceleration from M-theory[END_REF] are not the most general left-invariant metrics. They have a six-dimensional symmetry group. By Theorem 3.1 of [START_REF] Andersson | Eternal acceleration from M-theory[END_REF] the quantities P and Q converge to positive limits as t → ∞ and these are essentially the generalized Kasner exponents. Hence Theorem 4 above applies. Similar results are obtained in higher dimensions. These apply for example to Lie algebras of the form A 2,1 ⊕ R n for any n. The limits of the generalized Kasner exponents are non-negative. It is once again the case that only metrics with additional symmetry are included.

The Lie algebras which define Bianchi models of class A are those which are unimodular, i.e. those for which the trace of the structure constants is zero. A four-dimensional Lie algebra which is the direct sum of a three-dimensional algebra with the real numbers is unimodular if and only if the three-dimensional Lie algebra is of class A. The unimodular four-dimensional Lie algebras which are not of this form can be read off from Table 1 in [START_REF] Hervik | Multidimensional cosmology: spatially homogeneous models of dimension 4+1[END_REF]. In the notation used there they are A 4,1 , A -2 4,2 , A pq 4,5 for p + q = -1, A pq 4,6 for p + 2q = 0, A 4,8 and A 4,10 . Some explicit solutions are known. In the cases A -2 4,2 and A pq 4,5

explicit diagonal solutions are given in [START_REF] Demaret | Anisotropic Kaluza-Klein cosmologies[END_REF]. In both cases the generalized Kasner exponents converge to (1, 0, 0, 0) and thus these solutions are future geodesically complete by Theorem 4. In the case A 4,8 a solution which is diagonal and selfsimilar is given in [START_REF] Christodoulakis | Essential constants for spatially homogeneous Ricci-flat manifolds of dimension 4+1[END_REF]. Its generalized Kasner exponents are time-independent. They are equal to ( 3 4 , -1 4 , 1 4 , 1 4 ). Theorem 4 applies to this solution and it is interesting to note that this is a borderline case for that theorem. In the case A 4,10 an explicit solution is mentioned in [START_REF] Christodoulakis | Essential constants for spatially homogeneous Ricci-flat manifolds of dimension 4+1[END_REF] but the authors note that it has a higher dimensional symmetry group and can also be considered as corresponding to the Lie algebra which is the direct sum of the Bianchi type II Lie algebra with the real numbers. It admits a reflection symmetry of the type which makes it belong to the class covered by Theorem 5.

There is a way of fitting many of these Lie algebras in a common geometrical framework. In [START_REF] Rendall | Global properties of locally spatially homogeneous cosmological models with matter[END_REF] the notion of local U (1)×U (1) symmetry was introduced and it was shown how the Bianchi types I, II, VI 0 and VII 0 can be identified among spacetimes with two commuting spacelike local Killing vectors and a global topology which is a bundle over the circle with fibre T 2 . The different Bianchi types are distinguished by the algebraic structure of a certain 2 × 2 matrix with unit determinant. If both eigenvalues are real with unit modulus and the matrix is diagonalizable then type I is obtained. A matrix which is not diagonalizable gives rise to type II. When the eigenvalues are real and distinct type VI 0 results. Finally type VII 0 is obtained when the eigenvalues are imaginary. The same analysis can be repeated for one dimension higher, although it does not cover all relevant algebras. In this case local U (1) × U (1) × U (1) symmetry is the basic input and different cases are classified by the algebraic properties of a 3×3 matrix with unit determinant. If there is a real eigenvalue with unit modulus and the corresponding Jordan block is one by one then a case is obtained where the Lie algebra is the direct sum of one of the algebras seen in the three-dimensional case with the real numbers. The only other case where all eigenvalues have are real with unit modulus is that of a 3 × 3 Jordan block. This gives the algebra A 4,1 . When the eigenvalues are real and all distinct type A pq 4,5 results. In the case that there is pair of non-real eigenvlaues the Lie algebra which arises is A pq 4,6 . If there is a non-trivial Jordan block corresponding to a double eigenvalue then A -2 4,2 occurs 1 . This construction does not apply to types A 4,8 and A 4,10 since it can be shown straightforwardly that neither of these has a three-dimensional Abelian subalgebra. This point of view shows that in all the cases where the Lie algebra has a three-dimensional Abelian subalgebra the group can be made to act on the universal cover of a spacetime with a compact Cauchy surface by isometries of the pull-back of the spacetime metric. The topology of the Cauchy surface is that of a bundle over S 1 with fibre T 3 .

A convenient feature of vacuum Bianchi models of class A is that initial data whose components in a suitable basis are diagonal lead to solutions which are also diagonal in that basis. Call a basis of this kind a canonical basis. It is moreover the case that for any initial data of Bianchi class A there is a canonical basis in which it is diagonal. These facts together mean that in order to understand the dynamics of vacuum models of Bianchi class A it is enough to understand the dynamics of diagonal models. The fact that models which are initially diagonal remain diagonal can be proved using automorphisms of the Lie algebras involved. For convenience let the linear transformation which multiplies e i by (-1) ai , a i ∈ {0, 1}, be denoted by (a 1 a 2 a 3 ). In a canonical basis the transformations (110), ( 101) and (011) are automorphisms. In other words, they leave the structure constants invariant. Data which are diagonal in the given basis are invariant under these transformations. Because the structure constants are invariant the Einstein evolution equations are also invariant under these transformations. It follows from the unique determination of solutions by data that invariant data lead to invariant solutions. A solution which is invariant under all these transformations is diagonal. It would be nice if this argument could be generalized to the case of unimodular Lie algebras in higher dimensions. This is no problem if the Lie algebra is a direct sum of a three-dimensional Lie algebra with the real numbers. For in that case the automorphisms of the three-dimensional case can be trivially extended to get (in a hopefully obvious notation) the automorphisms (1100), ( 1010) and (0110). For the other Lie algebras this method has very limited success, at least if the bases in which the Lie algebras are presented in [START_REF] Hervik | Multidimensional cosmology: spatially homogeneous models of dimension 4+1[END_REF] are taken as candidates for a canonical basis. The only case in which there are enough automorphisms obtained by changing the signs of some of the basis vectors to make the argument go through is A pq 4,5 . For that Lie algebra any automorphism must fix e 4 but the signs of the other basis vectors can be changed independently. In other words it is possible to use the automorphisms (1000), (0100) and (0010). The Wainwright-Hsu system has been very useful for the study of Bianchi class A models. Its simplicity relies on the diagonalization property and so it may be that nothing of comparable power is available for general unimodular Lie algebras in any higher dimension.

The theme of the present paper is the late-time behaviour of cosmological models but the behaviour near the initial singularity is also of great interest. For the latter subject the issue of diagonalization plays a very important role. In spacetime dimension four it is believed that generic spatially homogeneous solutions of the Einstein vacuum equations show oscillatory (Mixmaster) behaviour near the initial singularity and this has been proved rigorously in some cases. (For Bianchi type IX see [START_REF] Ringström | The Bianchi IX attractor[END_REF].) In 4 + 1 dimensions it seems that this is still true but that restricting to the special case of diagonal models removes the generic oscillations [START_REF] Demaret | Are Kaluza-Klein models of the universe chaotic?[END_REF]. In particular, generic oscillations can occur for the group SU (2) × R due to the influence of the non-diagonal degrees of freedom. The situation is similar in dimensions up to 9 + 1 but beyond that the oscillations are no longer generic. For some rigorous results on this see [START_REF] Damour | Kasnerlike behaviour for subcritical Einstein-matter systems[END_REF].

Conclusions and outlook

In the preceding sections it has been proved that spatially homogeneous solutions of the Einstein equations in various dimensions with symmetries defined by a large class of Lie groups are geodesically complete in the future. The case considered in most detail is that of spacetime dimension five. Even there it is to be hoped that something much more general can be proved. A suggestion for a generalization of this kind is formulated as a conjecture in Section 5. Making a corresponding conjecture in general dimensions is dependent on a sufficently good understanding of the structure of general Lie algebras. A guide in formulating the conjecture was to restrict to those Lie groups which do not admit a metric of positive scalar curvature. Another direction of possible generalization is to consider spatially homogeneous spacetimes where the Lie group defining the symmetry is necessarily of a dimension higher than the space dimension. These generalize the Kantowski-Sachs models in spacetime dimension four. In five spacetime dimensions the list of possibilities is as follows [START_REF] Hervik | Multidimensional cosmology: spatially homogeneous models of dimension 4+1[END_REF]: the four-sphere S 4 with symmetry group SO(4), the complex projective plane CP 2 with an eight-dimensional symmetry group, S 2 × S 2 with a six-dimensional symmetry group, S 2 × R 2 and S 2 × H 2 with five-dimensional symmetry groups. All of these examples admit an invariant metric with positive scalar curvature and so are not promising candidates for allowing spacetimes which are future geodesically complete. The theorem of Bérard-Bergery [START_REF] Bérard-Bergery | Sur la courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes[END_REF] mentioned at the end of Section 2 extends to homogeneous spaces which are more general than Lie groups and may be helpful in investigating generalizations of Kantowski-Sachs models in higher dimensions.

In the previous sections the cosmological constant Λ has always been assumed to vanish. It is likely that the theorem of Wald [START_REF] Wald | Asymptotic behaviour of homogeneous cosmological models in the presence of a positive cosmological constant[END_REF] on the late-time asymptotics of spatially homogeneous cosmological models with Λ > 0 extends to arbitrary dimensions with the essential assumption (in the vacuum case) being the absence of metrics of positive scalar curvature. Combining this with the ideas in [START_REF] Lee | Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant[END_REF] should give geodesic completeness. These ideas have, however, not yet been worked out in detail.

There are very few results on future geodesic completeness for inhomogeneous spacetimes which do not rely on choosing initial data which are close to those of a known solution (small data results). This applies even in the a priori simpler case Λ > 0. In fact the only result of this kind for vacuum spacetimes is that of Ringström [START_REF] Ringström | On a wave map arising in general relativity[END_REF]. It could be that obtaining a better understanding of geodesic completeness for homogeneous models in higher dimensions would provide new ideas for understanding the inhomogeneous case in spacetime dimension four.

It is interesting to ask what happens when there is a metric of positive scalar curvature. Is there always recollapse in that case? It seems that very little is known about this in spacetime dimensions greater than four. There are also few results available for inhomogeneous solutions in four dimensions. Perhaps a better understanding of the homogeneous case in higher dimensions could help to change this. In particular, higher dimensions could be a good place to look for a counterexample.

The long-time behaviour of solutions of the vacuum Einstein equations in higher dimensions gives rise to a rich variety of problems concerning the dynamics of the gravitational field. In this paper we have obtained theorems on some of the simpler of these, while pointing out promising directions for future research. We have also emphasized that one motivation for studying these questions is the insights they may bring for the Einstein-matter equations in four dimensions.

2 ,

 2 the convergence of ∂tφ tr k is equivalent to that of Ω. Hence p 0 = 0 and the hypotheses of Theorem 3 are satisfied.

Remarks related to the material of this paragraph are contained in an unpublished manuscript of S. Hervik


Alice J Braga

M B Chb

Mrcp Amber

E R Young

Bsc Mb

Chb Frca

Preventing venous thrombosis in critically ill children: what is the right approach?

Keywords: venous thromboembolism, thromboprophylaxis, central venous catheters

Background: The incidence of venous thromboembolic (VTE) events in children has increased in recent years 1,2 yet there is currently no consensus as to what VTE prophylaxis, if any, should be applied to the paediatric population.

Objectives/Aims: Our aim was to audit current practice in paediatric VTE prophylaxis across England and Wales and to advocate simple measures for prevention. We illustrate the importance of the condition with a series of cases from the South West Paediatric Burns and Neurosurgical Services based in Bristol.

Methods: Every paediatric intensive care unit (PICU) and burns centre admitting children in England and Wales was invited to participate in a structured telephone questionnaire designed to find out how VTE in children were being prevented. We performed a literature review of specific risk factors and management of these factors.

Results: Only one of the 24 units surveyed had written guidelines specific for children. Four other units used modified adult guidelines in older children. In the remaining 19 units that had no written guidelines, decisions regarding prophylaxis were based on individual cases and consultant-led.

Conclusion:

There is no consensus in England and Wales as to which VTE prophylactic measures should be applied in patients of less than 18 years of age. The National Institute for Health and Clinical Excellence (NICE) guidelines apply to adults only. Given the rarity of VTE events in children it is unlikely that

(CVCs) and the survival of children with previously fatal conditions [START_REF] Levy | Deep venous thrombosis in children and adolescents[END_REF][START_REF] Chan | Venous thrombosis in children[END_REF][START_REF] Monagle | Outcome of pediatric thromboembolic disease: A report from the Canadian childhood thrombophilia registry[END_REF] . However, VTE is still comparatively rare in children with frequencies estimated to be one tenth to one hundredth of that in the adult population [START_REF] Levy | Deep venous thrombosis in children and adolescents[END_REF][START_REF] Chan | Venous thrombosis in children[END_REF] . Nevertheless, it is an important cause of potentially preventable mortality and morbidity in children.

The Canadian VTE Registry, which is the largest published, comprises prospectively collected data from 15 tertiary care centres. In a report from this registry analysing 405 consecutive cases of VTE in children aged one month to 18 years over a six year period, the mortality attributed to thromboses was 2.2% [START_REF] Monagle | Outcome of pediatric thromboembolic disease: A report from the Canadian childhood thrombophilia registry[END_REF] . Unlike adults, children often survive for sixty years or more following VTE, and post-thrombotic syndrome (PTS), a condition marked by limb swelling and pain aggravated by activity, may develop in 50% of children who have experienced VTE [START_REF] Manco-Johnson | Post-thrombotic syndrome (PTS) in children: validation of a new pediatric outcome instrument and results in a comprehensive cohort of children with extremity deep vein thrombosis (DVT)[END_REF] .

Current clinical guidelines issued by the National Institute for Health and Clinical Excellence (NICE) in January 2010 for reducing the risk of venous thromboembolism apply only to patients over the age of 18 years [START_REF]Venous thromboembolism : reducing the risk[END_REF] . In view of the potential mortality, morbidity, increased length of hospital stay and associated increased health costs of VTE in children, the case for similar guidelines for the paediatric population particularly in the critical care setting, (paediatric intensive care and high dependency services), is compelling.

However, given the rarity of paediatric VTE, the risk-to-benefit ratio for formal prophylaxis remains controversial. For more than 10 years there have been calls Given the large surface area of her burns, tinzaparin was used to minimize the number of injections. After a 72 day hospitalization she was discharged home on LMWH for 3 months. The patient made a good recovery.

Survey Results

Twenty-eight centres in England and Wales were contacted between 10 th August 2009 and 29 th September 2009 by one of the authors (AB). A senior member of staff was prepared to answer the structured questionnaire at 24 units (86%).

Only one of the 24 units surveyed had written guidelines designed specifically assessed daily in a ward round proforma in only one unit. TEDs were used in some larger children and adolescents in all but one unit (96%). No unit reported using Flowtron boots routinely.

Discussion

The case for thromboprophylaxis in children is less clear than in adults because of the relative rarity of paediatric VTE. Why children should have a lower incidence of DVT than adults is not clear but possible explanations include reduced capacity for thrombin generation, increased capacity of α 2- macroglobulin to inhibit thrombin and enhanced antithrombotic potential by vessel walls [START_REF] Chan | Venous thrombosis in children[END_REF] . VTE prophylaxis itself is associated with complications and cost, therefore it is important to ascertain which children might benefit.

The aetiology and risk factors associated with VTE in children are well known and have been described in international registries [START_REF] Monagle | Outcome of pediatric thromboembolic disease: A report from the Canadian childhood thrombophilia registry[END_REF] . More than ninety percent of children who develop VTE will have more than one predisposing factor [START_REF] Parasuraman | Venous Thromboembolism in Children[END_REF] . By far the most common risk factor is the presence of a CVC [START_REF] Bratton | A Prospective Study of Femoral Catheter-Related Thrombosis in Children Arch[END_REF] which suggests that vessel size may be important. In the upper venous system the risk of VTE is increased significantly with a CVC in the subclavian veins, suggesting that placement in the brachial or jugular veins may be preferable [START_REF] Male | Central venous line-related thrombosis in children: association with central venous line location and insertion technique[END_REF] . Young children with diabetic ketoacidosis have an increased incidence of clinical DVT associated with the placement of CVCs which may be related to dehydration [START_REF] Gutierrez | Femoral central venous catheter-associated deep venous thrombosis in children with diabetic ketoacidosis[END_REF] . . The literature therefore reflects that CVCs are a major risk factor for childhood VTE particularly in those children that are less than one year of age, those admitted following trauma or with underlying malignancy. It is therefore important that CVCs should be removed as soon as possible and a simple risk benefit analysis should be undertaken at least daily [START_REF] De Jonge | Central Venous Catheter Use in the Pediatric Patient: Mechanical and Infectious Complications[END_REF] .

Despite the documented increase in the incidence of paediatric VTE and calls for the effects of prophylactic measures to be studied, our survey confirms that in England and Wales only one PICU uses written guidelines devised specifically for children and there is wide variation in clinical practice. Awareness of the risk of VTE within units also appears to vary enormously, some units addressing such risk daily and others not. When non-pharmaceutical measures were taken, the age and patient groups deemed at risk varied between units.

In contrast to VTE prevention in adults, the options for non-pharmaceutical prophylactic measures for smaller children are limited: TEDs are only available for a calf circumference greater than 25cm and a toe to knee measurement patients HB CVCs were shown to reduce the incidence of CVC related thrombosis significantly [START_REF] Pierce | Heparin-bonded central lines reduce thrombotic and infective complications in critically ill children[END_REF] , although there seems to be no advantage in using HB CVCs in infants less than one year old [START_REF] Anton | Heparin-bonded central venous catheters do not reduce thrombosis in infants with congenital heart disease: a blinded randomized, controlled trial[END_REF] . The use of LMWH for primary prophylaxis in children must not be undertaken lightly. In a review of the literature since 1980, LMWHs were found to be associated with minor bleeding in 21% and major bleeding in 3% of patients receiving primary prophylaxis 17 . There is one previously published paediatric VTE guideline 18 .

We propose the following based on our survey, case review within our regional childrens services and a formal literature review; that all paediatric critical care areas formally assess risk of VTE daily, that CVCs are removed as soon as possible particularly in children less than one year of age and that non-pharmacological prophylactic measures are used where appropriate and feasible. LMWHs should be used only in high-risk patients and at the discretion of the consultants responsible.

In our unit we have implemented a risk assessment table similar to that used in our adult population according to the new adult NICE guidelines [START_REF]Venous thromboembolism : reducing the risk[END_REF] (Figure 1). Five of the six cases seen at our hospital occurred in association with a CVC and all six had a combination of other risk factors. In most instances the CVC could have been removed earlier, the patients identified as medium to high risk for the development of

  Preventing venous thrombosis in critically ill children: Whatis the right approach? will provide the answer. We therefore propose that simple empirical measures be formally implemented in critically ill children to reduce the risk of developing this important but under-recognised condition. venous thromboembolic (VTE) events in children has increased in recent years, perhaps because of increased use of central venous catheters

Case 2 :

 2 A 14 month old girl admitted with a 65% scald subsequently developed Staphylococcus aureus septicaemia and hypernatraemia. An ultrasound performed when leg swelling occurred after 12 days confirmed a left femoral DVT extending into the iliac vein. A CVC had been present in the vein for three days and had been removed four days prior to diagnosis. She was treated with heparin before completing a three-month course of enoxaparin. A persistent thrombocytosis (platelet count up to 1,299 x 10 9 /l) attributed to iron deficiency could have been a contributory factor to the development of the thrombus. Case 3: An 18 month old boy was admitted for elective craniotomy with anterior vault reconstruction and re-modelling for metopic synostosis. During seven hours of surgery he received a total of 4240mls of fluid and blood products.Swelling of the right leg was noted on the third post-operative day and ultrasound confirmed the presence of a thrombus in the right femoral vein, associated with the central venous catheter that had been inserted at time of surgery. A decision had been made to keep the line in situ after surgery to allow for daily blood sampling whilst avoiding the need for phlebotomy. He was treated with three3 months of enoxaparin. Case 4: An 18 month old boy was admitted for craniotomy and excision of an atypical teratoid/rhabdoid tumour. Right femoral arterial and venous lines were at the time of surgery, 600 mls of blood loss were replaced with 600 mls of blood products. Surgical duration was 6.5 hours. On the second post-operative day his right leg was found to be swollen; ultrasonic examination confirmed the presence of a thrombus in the right common femoral vein. Intravenous heparin infusion was commenced in view of his recent extensive surgery and the need for a further craniotomy. Following a second procedure he was commenced on LMWH for 3 months. He died 8 months after diagnosis from recurrent disease. Case 5: A 7 year old girl presented acutely with an extensive cerebral gliosarcoma. Two craniotomies a week apart lasted 11 and 7 hours. Three days after the second procedure the patient developed pain and swelling in the left leg. Ultrasound demonstrated an extensive left common femoral vein thrombus despite no cannulation of this vessel. A CVC was present in the right femoral vein. The significant risk of pulmonary embolism prompted treatment with unfractionated heparin despite relative risk of neurosurgical haemorrhage. After five days enoxaparin was substituted for heparin, whereupon she developed subdural and intraventricular haemorrhages and anticoagulation was stopped. The risk of inferior vena cava (IVC) rupture was considered too great for an IVC filter to be placed. She died six weeks after diagnosis after receiving palliative care. Case 6: A 7 year old boy presented after falling from a tractor and being crushed by the trailer. He had multiple injuries comprising: fractures of the basal skull, temporal bone, ribs and clavicle as well as bilateral pneumothoraces and bilateral pelvic fractures. He spent seven days sedated and ventilated. In the Stapholococcus aureus was isolated from blood cultures. He was treated with intravenous antibiotics. Nineteen days after presentation he underwent open pelvic fixation lasting three hours. A decision was made to avoid LMWH prophylaxis, although TED stockings and pneumatic compression boots were employed. On the sixth post-operative day pain and swelling in the left thigh prompted an ultrasound scan which confirmed the presence of thrombus in the proximal superficial and common femoral veins. LMWH was commenced and factor Xa levels were monitored as per the hospital protocol.

  for the prevention of VTE in children. Two units used adult guidelines in adolescent patients aged over 12 years based on those issued by NICE. A fourth unit admitting children with burns used adult guidelines for children 13 years and over and modified guidelines for children under 13 years. A fifth unit had hospital-wide guidelines for patients, including children, admitted with diabetic ketoacidosis (DKA), which included LMWH prophylaxis.In units with no written guidelines, individual case-based decisions regarding VTE prophylaxis were ad hoc and usually consultant-led. Risk of VTE was

  .5cm. Pneumatic compression boots come in only one size. However, both TEDs and pneumatic combression devices can be used in adolescents and larger younger children. Heparin bonded (HB) CVCs are currently under evaluation. In a prospective randomized controlled trial of 209

  for prospective randomised trials to define an effective strategy for preventing deep vein thrombosis (DVT) in children judged to be at risk 3 . To date such trials have not materialised.The aims of this paper were to audit the current practice in paediatric VTE prophylaxis in critical care units across England and Wales, to promote daily VTE risk review and propose simple preventative measures. We illustrate the importance of the condition with a series of patients who developed DVT over the course of 24 months at the South West Paediatric Burns and Neurosurgical Services based in Bristol.

	of medical or nursing staff was invited to participate in a structured
	questionnaire. The nature and purpose of the study were explained and
	reassurance that the data collected would be anonymised such that no
	healthcare professional or unit would be identifiable. If agreement to participate
	was obtained, a short structured interview was conducted to determine the
	following: whether a written protocol for VTE prophylaxis in children existed
	and was in use; whether risk of deep vein thrombosis was specifically assessed
	F o r each day; which, if any, groups of patients wore graduated compression F stockings (Thrombo-Embolus Deterrent stockings or TEDs) or pneumatic o r compression boots (Flowtron® DVT Prophylaxis Systems, Huntleigh Healthcare
	P P Ltd); and if no written protocol existed, which grade of doctor made individual
	e e
	e r e r
	R R
	e e
	v i e The hospital admits approximately 2900 paediatric surgical patients per year including v i e
	w approximately 350 admitted with burns, 500 admitted for neurosurgery, 1200 w
	with plastic surgical conditions and 500 with orthopaedic trauma. Roughly fifty
	percent of these patients remain in hospital for at least one night and
	approximately 180 CVCs are sited within this population per year.
	Every paediatric intensive care unit (PICU) listed on the 'Paediatric Intensive Care
	Audit Network' website 6 and every centre admitting critically ill children with
	burns in England and Wales was contacted by telephone. A single senior member

Methods

A review of all diagnosed cases of DVT occurring in children in the South West Paediatric Burns and Neurosurgical Services over the last two years was carried out with particular attention to risk factors present. decisions regarding thromboprophylaxis in patients of less than 18 years of age. If no one was able to respond to the survey a single follow-up phone call was made.

Case Studies:

Six cases of VTE were diagnosed in our own service between September 2007 and September 2009: that is, 6 out of 3070 inpatients (0.2%) and 5 of the 360 patients with CVCs (1.4%).

Case 1: A 13 month old girl with a 70% scald required four PICU admissions complicated by sepsis and hyponatremia. Facial swelling was noted one month after presentation. CT scanning demonstrated a thrombus in the right internal jugular (RIJ) vein and superior vena cava associated with a RIJ catheter that had been in place for five days. After receiving a heparin infusion for 6 days, she was given low molecular weight heparin (LMWH).

  [START_REF] Chan | Venous thrombosis in children[END_REF] . More than ninety percent of VTEs in neonates and greater than fifty percent of VTEs in older children are secondary to CVCs 7 . CVCs, however, are an essential component of the care of critically ill children in order to allow accurate fluid management, haemodialysis or inotropic use. Venous thrombosis, usually undetected clinically, has been shown by routine ultrasound examination to have an incidence as high as 44% of children with CVCs in place for greater than 48 hours[START_REF] Krafte-Jacobs | Catheter-related thrombosis in critically ill children: comparison of catheters with and without heparin bonding[END_REF] . Risk is highest during the first four days after insertion and decreases thereafter[START_REF] Beck | Incidence of deep vein thrombosis related to peripherally inserted central catheters in children and adolescents[END_REF] . Peripherally-inserted central catheters are also associated with the development of VTE although the incidence is lower (up to 9%) than when the catheter is inserted centrally 10 . Younger and smaller children are more likely to develop catheterrelated VTE

	F o r
	P
	e
	e r
	R
	e
	v i e
	w

  Other potential areas of development include further clinical trials into the use of heparin bonded CVCs, as well as the possible development and evaluation of TEDs and Flowtron boots for use in younger children. 17. Nowak-Göttl U, Bidlingmaier C, Krümpel A, Göttl L, and Kenet G. Pharmacokinaetic, efficacy, and safety of LMWHs in venous thrombosis in neonates, infants and children. Br J Pharmacol. 2008; 153(6): 1120-1127. 18. Jackson PC and Morgan JM. Perioperative thromboprophylaxis in children: Paediatric Thromboprophylaxis Flow Chart NB. Daily Consultant Review Decisions must be documented in the Medical Notes Child must have Thromboprophylaxis reviewed daily by Consultant and management plan recorded in notes.If child has a short term CVC present, the line should be removed within 24 hours unless a consultant documents that it should remain in situ.CVC in an infant is a very high risk.

	prophylactic measures considered. The reasons for failure to remove a
	CVC within 24hrs is documented and reviewed.
	F o r P e This paper aims to raise the profile of VTE prevention in children. To devise a protocol which would be applicable across all areas and ages is difficult but we encourage units to develop their own guidelines in collaboration with the specialties involved. F o r P e F o r P e F o r P e development of a guideline for management. Pediatric Anesthesia 2008; 18(6): 478-487 F o r P e present? No Yes Low Risk: • Early mobilization • Adequate hydration F o r P e e r R e v i e w Risk Factors At Risk:
	e r e r e r e r e r	Interventions may include:
	R e R e R e R e R e		• Adequate hydration • Early mobilization if possible • TEDs/ Flowtrons if available for size of child. • Burns >20% consider anticoagulation
	v i e v i e v i e v i e v e i	• THE DECISION TO ANTICOAGULATE MUST BE MADE BY A CONSULTANT.
		w w w w w
	VTE, and additional

Summary

VTE is a recognised cause of preventable mortality and morbidity in children and with the increasing use of CVCs the incidence of VTE is rising. Our case series highlights the problem in a typical cohort of critically ill children following neurosurgery, trauma and major burn injuries. The survey confirms that there is no consensus in the England and Wales as to which prophylactic measures against VTE should be applied in patients of less than 18 years of age. Given the rarity of VTE events in children it is unlikely that randomised controlled trials will clarify this. We therefore propose that a risk assessment tool and simple empirical measures are employed within critical care areas.

F o r P e e r R e v i e w

Paediatric Thromboprophylaxis Risk Assessment Chart