Salvatore Capozziello

Mariafelicia De Laurentis

Arturo Stabile

Axially symmetric solutions in f

Keywords: numbers: 98.80.-k, 95.35.+x, 95.35.+d, 04.50.+h

come

I. INTRODUCTION

The issue to extend General Relativity (GR) to alternative theories of gravity has recently become dramatically urgent due to the missing matter problem at all astrophysical scales and the accelerating behavior of cosmic fluid, detected by SuperNovae Ia used as standard candles. Up to now, no final answer on new particles has been given at fundamental level so Dark Energy and Dark Matter constitute a puzzle to be solved in order to achieve a self-consistent picture of the observed Universe. f (R)-gravity, where f (R) is a generic function of the Ricci scalar R, comes into the game as a straightforward extension of GR where further geometrical degrees of freedom are considered instead of searching for new material ingredients [1]. From an epistemological point of view, the action of gravity is not selected a priori, but it could be "reconstructed", in principle, by matching consistently the observations [2][3][4]. This approach can be adopted considering any function of the curvature invariants as R µν R µν , R R and so on.

From a genuine mathematical point of view, alternative theories of gravity pose the problem to recover or extend the well-established results of GR as the initial value problem [5], the stability of solutions and, in particular, the issue of finding out new solutions. As it is well known, beside cosmological solutions, spherically and axially symmetric solutions play a fundamental role in several astrophysical problems ranging from black holes to active galactic nuclei. Alternative gravities, to be consistent with results of GR, should comprise solutions like Schwarzschild and Kerr ones but present, in general, new solutions that could be physically interesting. Due to this reason, methods to find out exact and approximate solutions are particularly relevant in order to check if observations can be framed in Extended Theories of Gravity [6].

Recently, the interest in spherically symmetric solutions of f (R)-gravity is growing up. In [7], solutions in vacuum have been found considering relations among functions that define the spherical metric or imposing a costant Ricci curvatue scalar. The authors have reconstructed the form of some f (R)-models, discussing their physical relevance. In [8], the same authors have discussed static spherically symmetric solutions, in presence of perfect fluid matter, adopting the metric formalism. They have shown that a given matter distribution is not capable of globally determining the functional form of f (R). Others authors have discussed in details the spherical symmetry of f (R)-gravity considering also the relations with the weak field limit. Exact solutions are obtained for constant Ricci curvature scalar and for Ricci scalar depending on the radial coordinate. In particular, it can be considered how to obtain results consistent with GR assuming the well-known post-Newtonian and post-Minkowskian limits as consistency checks. [9].

In this paper, we want to seek for a general metod to find out axially symmetric solutions by performing a complex coordinate transformation on the spherical metrics. Since the discovery of the Kerr solution [10], many attempts have been made to find a physically reasonable interior matter distribution that may be considered as its source. For a review on these approaches see [11,12]. Though much progress has been made, results have been generally disappointing. As far as we know, nobody has obtained a physically satisfactory interior solution. This seems surprising given the success of matching internal spherically symmetric solutions to the Schwarzschild metric. The problem is that the loss of a degree of symmetry makes the derivation of analytic results much more difficult. Severe restrictions are placed on the interior metric by maintaining that it must be joined smoothly to the external axially symmetric metric. Further restrictions are placed on the interior solutions to ensure that they correspond to physical objects.

Furthermore since the axially symmetric metric has no radiation field associated with it, its source should be also non-radiating. This places even further constraints on the structure of the interior solution [START_REF] Stephani | Herlt Exact Solutions to Einstein's Field Equations[END_REF]. Given the strenuous nature of these limiting conditions, it is not surprising to learn that no satisfactory solution to the problem of finding Confidential: not for distribution. Submitted to IOP Publishing for peer review 11 May 2010 sources for the Kerr metric has been obtained. In general, the failure is due to internal structures whose physical properties are unknown. This shortcoming makes hard to find consistent boundary conditions.

Newman and Janis showed that it is possible to obtain an axially symmetric solution (like the Kerr metric) by making an elementary complex transformation on the Schwarzschild solution [15]. This same method has been used to obtain a new stationary and axially symmetric solution known as the Kerr-Newman metric [16]. The Kerr-Newman space-time is associated to the exterior geometry of a rotating massive and charged black-hole. For a review on the Newman-Janis method to obtain both the Kerr and Kerr-Newman metrics see [START_REF] Inverno | Introducing Einstein's Relativity[END_REF].

By means of very elegant mathematical arguments, Schiffer et al. [START_REF] Schiffer | [END_REF] have given a rigorous proof to show how the Kerr metric can be derived starting from a complex transformation on the Schwarzschild solution. We will not go into the details of this demonstration, but point out that the proof relies on two main assumptions. The first is that the metric belongs to the same algebraic class of the Kerr-Newman solution, namely the Kerr-Schild class [19]. The second assumption is that the metric corresponds to an empty solution of the Einstein field equations. In the case we are going to study, these assumptions are not considered and hence the proof in [START_REF] Schiffer | [END_REF] is not applicable. It is clear, by the generation of the Kerr-Newman metric, that all the components of the stress-energy tensor need to be non-zero for the Newman-Janis method to be successful. In fact, Gürses and Gürsey, in 1975 [20], showed that if a metric can be written in the Kerr-Schild form, then a complex transformation "is allowed in General Relativity." In this paper, we will show that such a transformation can be extended to f (R)-gravity.

The outline of this paper is as follows. In the Sec.II, we introduce the f (R)-gravity action, the field equations and give some general remarks on spherical symmetry. In Sec III, we give a summary of the Noether Simmetry Approach [6] and find some spherically symmetric exact solutions for f (R)-gravity. In Sec IV, we review the Newman-Janis method to obtain axially symmetric solutions starting from spherically symmetric ones. The resulting metric is written in terms of two arbitrary functions. A further suitable coordinate transformation allows to write the metric in the so called Boyer-Lindquist coordinates. Such a transformation makes the physical interpretation much clearer and reduces the amount of algebra required to calculate the metric properties. In Sec.V, the Newman-Janis method is applied to the spherically symmetric exact solution, previously derived by the Noether Symmetry, and an axially symmetric exact solution is obtained. This result shows that the Newman-Janis method works also in f (R)-gravity. A physical application of the result is discussed in Sec VI. Discussion ans concluding remarks are drawn in Sec. VII.

II. SPHERICAL SYMMETRY IN f (R)-GRAVITY

Let us consider an analytic function f (R) of the Ricci scalar R in four dimensions. The variational principle for this action is:

δ d 4 x √ -g f (R) + X L m = 0 (1)
where X = 8πG c 4 , L m is the standard matter Lagrangian and g is the determinant of the metric 1 . By varying with respect to the metric, we obtain the field equations2   

H µν = f (R)R µν -1 2 f (R)g µν -f (R) ;µν + g µν f (R) = X T µν H = g ρσ H ρσ = 3 f (R) + f (R)R -2f (R) = X T (2)
where

T µν = -2 √ -g δ( √ -gL m )
δg µν is the energy-momentum tensor of standard fluid matter and the second equation is the trace. The most general spherically symmetric solution can be written as follows :

1 We are adopting the convention Rµν = R ρ µρν for the Ricci tensor and R α βµν = Γ α βν,µ -..., for the Riemann tensor. Connections are Levi-Civita :

Γ µ αβ = 1 2 g µρ (g αρ,β + g βρ,α -g αβ,ρ ) . ds 2 = m 1 (t , r )dt 2 + m 2 (t , r )dr 2 + m 3 (t , r )dt dr + m 4 (t , r )dΩ , (3)
where m i are functions of the radius r and of the time t . dΩ is the solid angle. We can consider a coordinate transformation that maps the metric (3) in a new one where the off -diagonal term vanishes and m 4 (t , r ) = -r 2 , that is3 :

ds 2 = g tt (t, r)dt 2 -g rr (t, r)dr 2 -r 2 dΩ . (4)
This expression can be considered, without loss of generality, as the most general definition of a spherically symmetric metric compatible with a pseudo -Riemannian manifold without torsion. Actually, by inserting this metric into the field Eqs.

(2), one obtains :

   f (R)R µν -1 2 f (R)g µν + H µν = X T µν f (R)R -2f (R) + H = X T (5)
where the two quantities H µν and H read :

H µν = -f (R) R ,µν -Γ t µν R ,t -Γ r µν R ,r -g µν g tt ,t + g tt ln √ -g ,t R ,t + g rr ,r + g rr ln √ -g ,r R ,r + +g tt R ,tt + g rr R ,rr -f (R) R ,µ R ,ν -g µν g tt R ,t 2 + g rr R ,r 2 (6)
H = g στ H στ = 3f (R) g tt ,t + g tt ln √ -g ,t R ,t + g rr ,r + g rr ln √ -g ,r R ,r + g tt R ,tt + g rr R ,rr + +3f (R) g tt R ,t 2 + g rr R ,r 2 . ( 7
)
Our task is now to find out exact spherically symmetric solutions.

In the case of time-independent metric, i.e., g tt = a(r) and g rr = b(r), the Ricci scalar can be recast as a Bernoulli equation of index two with respect to the metric potential b(r) (see [9] for details):

b (r) + r 2 a (r) 2 -4a(r) 2 -2ra(r)[2a(r) + ra(r) ] ra(r)[4a(r) + ra (r)] b(r) + 2a(r) r 2 + r 2 R(r) 4a(r) + ra (r) b(r) 2 = 0 . ( 8
)
where R = R(r) is the Ricci scalar. A general solution of ( 8) is:

b(r) = exp[-dr h(r)] K + dr l(r) exp[-dr h(r)] , (9)
where K is an integration constant while h(r) and l(r) are two functions that, according to Eq.( 8), define the coefficients of the quadratic and the linear term with respect to b(r) [START_REF] Ince | Ordinary Differential Equations[END_REF]. We can fix l(r) = 0; this choice allows to find out solutions with a Ricci scalar scaling as -2 r 2 in term of the radial coordinate. On the other hand, it is not possible to have h(r) = 0 since, otherwise, we get imaginary solutions. A particular consideration deserves the limit r → ∞. In order to achieve a gravitational potential b(r) with the correct Minkowski limit, both h(r) and l(r) have to go to zero at infinity, provided that the quantity r 2 R(r) turns out to be constant: this result implies b (r) = 0, and, finally, also the metric potential b(r) has a correct Minkowski limit.

In general, if we ask for the asymptotic flatness of the metric as a feature of the theory, the Ricci scalar has to evolve to infinity as r -n with n 2. Formally, it has to be:

lim r→∞ r 2 R(r) = r -n , (10)
with n ∈ N. Any other behavior of the Ricci scalar could affect the requirement to achieve a correct asymptotic flatness.

The case of constant curvature is equivalent to GR with a cosmological constant and the solution is time independent. This result is well known (see, for example, [START_REF] Barrow | [END_REF]) but we report, for the sake of completeness, some considerations related with it in order to deal with more general cases where a radial dependence for the Ricci scalar is supposed. If the scalar curvature is constant (R = R 0 ), field Eqs. (5), being H µν = 0, reduce to:

   f 0 R µν -1 2 f 0 g µν = X T µν f 0 R 0 -2f 0 = X T (11)
where

f (R 0 ) = f 0 , f (R 0 ) = f 0 .
A general solution, when one considers a stress-energy tensor of perfect-fluid

T µν = (ρ + p)u µ u ν -pg µν , is ds 2 = 1 + k 1 r + qX ρ -λ 3 r 2 dt 2 - dr 2 1 + k1 r + qX ρ-λ 3 r 2 -r 2 dΩ . ( 12
)
when p = -ρ, λ = -f0 2f 0 and q -1 = f 0 . This result means that any f (R)-model, in the case of constant curvature, exhibits solutions with de Sitter-like behavior. This is one of the reasons why the dark energy issue can be addressed using these theories [1].

If f (R) is analytic, it is possible to write the series:

f (R) = Λ + Ψ 0 R + Ψ(R) , (13)
where Ψ 0 is a coupling constant, Λ plays the role of the cosmological constant and Ψ(R) is a generic analytic function of R satisfying the condition

lim R→0 R -2 Ψ(R) = Ψ 1 , (14)
where Ψ 1 is a constant. If we neglect the cosmological constant Λ and Ψ 0 is set to zero, we obtain a new class of theories which, in the limit R → 0, does not reproduce GR (from Eq.( 14), we have lim R→0 f (R) ∼ R 2 ). In such a case, analyzing the whole set of Eqs. (11), one can observe that both zero and constant = 0 curvature solutions are possible. In particular, if R = R 0 = 0 field equations are solved for any form of gravitational potential entering the spherically symmetric background, provided that the Bernoulli Eq. ( 8), relating these functions, is fulfilled for the particular case R(r) = 0. The solutions are thus defined by the relation

b(r) = exp[-dr h(r)] K + 4 dr a(r) exp[- R dr h(r)] r[a(r)+ra (r)] , (15)
being g tt (t, r) = b(r) from Eq.( 4). In [9], some examples of f (R)-models admitting solutions with constant = 0 or null scalar curvature are discussed.

III. THE NOETHER SYMMETRY APPROACH AND THE SPHERICAL SYMMETRY

Besides spherically symmetric solutions with constant curvature scalar, also solutions with the Ricci scalar depending on radial coordinate r are possible in f (R)-gravity [9]. Furthermore, spherically symmetric solutions can be achieved starting from a point-like f (R)-Lagrangian [6]. Such a Lagrangian can be obtained by imposing the spherical symmetry directly in the action (1). As a consequence, the infinite number of degrees of freedom of the original field theory will be reduced to a finite number. The technique is based on the choice of a suitable Lagrange multiplier defined by assuming the Ricci scalar, argument of the function f (R) in spherical symmetry.

Starting from the above considerations, a static spherically symmetric metric can be expressed as

ds 2 = A(r)dt 2 -B(r)dr 2 -M (r)dΩ , (16)
and then the point-like f (R) Lagrangian4 is

L = - A 1/2 f 2MB 1/2 M 2 - f A 1/2 B 1/2 A M - Mf A 1/2 B 1/2 A R - 2A 1/2 f B 1/2 R M -A 1/2 B 1/2 [(2 + MR)f -Mf] , (17)
which is canonical since only the configuration variables and their first order derivatives with respect to the radial coordinate r are present. Details of calculations are in [6]. Eq. ( 17) can be recast in a more compact form introducing the matrix representation :

L = q t T q + V (18)
where q = (A, B, M, R) and q = (A , B , M , R ) are the generalized positions and velocities associated to L. It is easy to check the complete analogy between the field equation approach and point-like Lagrangian approach [6].

In order to find out solutions for the Lagrangian (17), we can search for symmetries related to cyclic variables and then reduce dynamics. This approach allows, in principle, to select f (R)-gravity models compatible with spherical symmetry. As a general remark, the Noether Theorem states that conserved quantities are related to the existence of cyclic variables into dynamics [START_REF] Arnold | Mathematical Methods of Classical Mechanics[END_REF][START_REF] Marmo | Dynamical Systems. A Differential Geometric Approach to Symmetry and Reduction[END_REF][START_REF] Morandi | [END_REF].

It is worth noticing that the Hessian determinant of Eq. ( 17),

∂ 2 L ∂q i ∂q j
, is zero. This result clearly depends on the absence of the generalized velocity B into the point -like Lagrangian. As matter of fact, using a point-like Lagrangian approach implies that the metric variable B does not contributes to dynamics, but the equation of motion for B has to be considered as a further constraint equation. Then the Lagrangian ( 17) has three degrees of freedom and not four, as one should expect a priori. Now, since the equation of motion describing the evolution of the metric potential B does not depend on its derivative, it can be explicitly solved in term of B as a function of the other coordinates :

B = 2M 2 f A R + 2Mf A M + 4AM f M R + Af M 2 2AM [(2 + MR)f -Mf] . (19)
By inserting Eq.( 19) into the Lagrangian [START_REF] Inverno | Introducing Einstein's Relativity[END_REF], we obtain a non-vanishing Hessian matrix removing the singular dynamics. The new Lagrangian reads5

L * = L 1/2 (20)
with

L = q t Lq = [(2 + MR)f -fM] M [2M 2 f A R + 2MM (f A + 2Af R ) + Af M 2 ] .
If one assumes the spherical symmetry, the role of the affine parameter is played by the coordinate radius r. In this case, the configuration space is given by Q = {A, M, R} and the tangent space by T Q = {A, A , M, M , R, R }. On the other hand, according to the Noether Theorem, the existence of a symmetry for dynamics described by Lagrangian (17) implies a constant of motion. Let us apply the Lie derivative to the (17), we have6 :

L X L = α • ∇ q L + α • ∇ q L = q t α • ∇ q L + 2 ∇ q α t L q , (21)
that vanishes if the functions α satisfy the following system

α • ∇ q L + 2(∇ q α) t L = 0 -→ α i ∂ Lkm ∂q i + 2 ∂α i ∂q k Lim = 0 . (22)
Solving the system (22) means to find out the functions α i which assign the Noether vector [START_REF] Arnold | Mathematical Methods of Classical Mechanics[END_REF][START_REF] Marmo | Dynamical Systems. A Differential Geometric Approach to Symmetry and Reduction[END_REF]. However the system (22) implicitly depends on the form of f (R) and then, by solving it, we get also f (R)-models compatible with spherical symmetry. On the other hand, by choosing the f (R)-form, we can explicitly solve (22). As an example, one finds that the system ( 22) is satisfied if we choose

f (R) = f 0 R s , α = (α 1 , α 2 , α 3 ) = (3 -2s)kA, -kM, kR , (23)
with s a real number, k an integration constant and f 0 a dimensional coupling constant 7 . This means that, for any f (R) = R s , exists, at least, a Noether symmetry and a related constant of motion Σ 0 :

Σ 0 = α • ∇ q L = 2skM R 2s-3 [2s + (s -1)MR][(s -2)RA -(2s 2 -3s + 1)AR ] .
A physical interpretation of Σ 0 is possible if one gives an interpretation of this quantity in GR, that means for f (R) = R and s = 1. In other words, the above procedure has to be applied to the Lagrangian of GR. We obtain the solution

α GR = (-kA, kM ) . ( 24
)
The functions A and M give the Schwarzschild solution, and then the constant of motion acquires the standard form

Σ 0 = 2GM c 2 . ( 25
)
In other words, in the case of Einstein gravity, the Noether symmetry gives, as a conserved quantity, the Schwarzschild radius or the mass of the gravitating system. This result can be assumed as a consistency check.

In the general case, f (R) = R s , the Lagrangian [START_REF] Inverno | Introducing Einstein's Relativity[END_REF] becomes

L = sR 2s-3 [2s + (s -1)MR] M [2(s -1)M 2 A R + 2MRM A + 4(s -1)AM M R + ARM 2 ] ,
and the expression (19) for B is

B = s[2(s -1)M 2 A R + 2MRM A + 4(s -1)AM M R + ARM 2 ] 2AM R[2s + (s -1)MR] (26)
As it can be easily checked, GR is recovered for s = 1.

Using the constant of motion [START_REF] Barrow | [END_REF], we solve in term of A and obtain

A = R 2s 2 -3s+1 s-2 k 1 + Σ 0 R 4s 2 -9s+5 2-s dr 2ks(s -2)M [2s + (s -1)MR] (27)
for s = 2 and k 1 an integration constant. For s = 2, one finds

A = - Σ 0 12kr 2 (4 + r 2 R)RR . ( 28
)
These relations allow to find out general solutions for the field equations giving the dependence of the Ricci scalar on the radial coordinate r. For example, a solution is found for

s = 5/4 , M = r 2 , R = 5r -2 , ( 29
)
obtaining the spherically symmetric space-time

ds 2 = (α + βr)dt 2 - 1 2 βr α + βr dr 2 -r 2 dΩ , ( 30
)
where α is a combination of Σ 0 and k and β = k 1 . In principle, the same procedure can be worked out any time Noether symmetries are identified. Our task is now to show how, from a spherically symmetric solution, one can generate an axially symmetric solution adopting the Newman-Janis procedure that works in GR. In general, the approach is not immediately straightforward since, as soon as f (R) = R, we are dealing with fourth-order field equations which have, in principle, different existence theorems and boundary conditions. However, the existence of the Noether symmetry guarantees the consistency of the chosen f (R)-model with the field equations.

IV. AXIAL SYMMETRY DERIVED FROM SPHERICALLY SYMMETRIC SOLUTIONS

We want to show now how it is possible to obtain an axially symmetric solution starting from a spherically symmetric one adopting the method developed by Newman and Janis in GR. Such an algorithm can be applied to a static spherically symmetric metric considered as a"seed" metric. Let us recast the spherically symmetric metric (4) in the form

ds 2 = e 2φ(r) dt 2 -e 2λ(r) dr 2 -r 2 dΩ, (31)
with g tt (t, r) = e 2φ(r) and g rr (t, r) = e 2λ(r) . Such a form is suitable for the considerations below. Following Newman and Janis, Eq. ( 31) can be written in the so called Eddington-Finkelstein coordinates (u, r, θ, φ), i.e. the g rr component is eliminated by a change of coordinates and a cross term is introduced [START_REF] Ch | Gravitation[END_REF]. Specifically this is achieved by defining the time coordinate as dt = du + F (r)dr and setting F (r) = ±e λ(r)-φ(r) . Once such a transformation is performed, the metric (31) becomes

ds 2 = e 2φ(r) du 2 ± 2e λ(r)+φ(r) dudr -r 2 dΩ. (32)
The surface u = costant is a light cone starting from the origin r = 0. The metric tensor for the line element (32) in null-coordinates is

g µν =     0 ±e -λ(r)-φ(r) 0 0 ±e -λ(r)-φ(r) -e -2λ(r) 0 0 0 0 -1/r 2 0 0 0 0 -1/(r 2 sin 2 θ)     . ( 33
)
The matrix (33) can be written in terms of a null tetrad as

g µν = l µ n ν + l ν n µ -m µ mν -m ν mµ , (34)
where l µ , n µ , m µ and mµ are the vectors satisfying the conditions

l µ l µ = m µ m µ = n µ n µ = 0, l µ n µ = -m µ mµ = 1, l µ m µ = n µ m µ = 0 . ( 35
)
The bar indicates the complex conjugation. At any point in space, the tetrad can be chosen in the following manner: l µ is the outward null vector tangent to the cone, n µ is the inward null vector pointing toward the origin, and m µ and mµ are the vectors tangent to the two-dimensional sphere defined by constant r and u. For the spacetime (33), the tetrad null vectors can be

                   l µ = δ µ 1 n µ = -1 2 e -2λ(r) δ µ 1 + e -λ(r)-φ(r) δ µ 0 m µ = 1 √ 2r (δ µ 2 + i sin θ δ µ 3 ) mµ = 1 √ 2r (δ µ 2 -i sin θ δ µ 3 ) (36)
Now we need to extend the set of coordinates x µ = (u, r, θ, φ) replacing the real radial coordinate by a complex variable. Then the tetrad null vectors become 8

                   l µ = δ µ 1 n µ = -1 2 e -2λ(r,r) δ µ 1 + e -λ(r,r)-φ(r,r) δ µ 0 m µ = 1 √ 2r (δ µ 2 + i sin θ δ µ 3 ) mµ = 1 √ 2r (δ µ 2 -i sin θ δ µ 3 ) (37)
A new metric is obtained by making a complex coordinates transformation

x µ → xµ = x µ + iy µ (x σ ) , (38)
where y µ (x σ ) are analityc functions of the real coordinates x σ , and simultaneously let the null tetrad vectors Z µ a = (l µ , n µ , m µ , mµ ), with a = 1, 2, 3, 4, undergo the transformation

Z µ a → Zµ a (x σ , xσ ) = Z ρ a ∂ xµ ∂x ρ . (39)
Obviously, one has to recover the old tetrads and metric as soon as xσ = xσ . In summary, the effect of the "tilde transformation" (38) is to generate a new metric whose components are (real) functions of complex variables, that is

g µν → gµν : x × x → R (40) with Zµ a (x σ , xσ )| x=x = Z µ a (x σ ). (41)
For our aims, we can make the choice

xµ = x µ + ia(δ µ 1 -δ µ 0 ) cos θ →                      ũ = u + ia cos θ r = r -ia cos θ θ = θ φ = φ ( 42
)
where a is constant and the tetrad null vectors (37), if we choose r = r, become

                           lµ = δ µ 1 ñµ = -1 2 e -2λ(r,θ) δ µ 1 + e -λ(r,θ)-φ(r,θ) δ µ 0 mµ = 1 √ 2(r-ia cos θ) ia(δ µ 0 -δ µ 1 ) sin θ + δ µ 2 + i sin θ δ µ 3 mµ = 1 √ 2(r+ia cos θ) -ia(δ µ 0 -δ µ 1 ) sin θ + δ µ 2 -i sin θ δ µ 3 ( 43
)
8 It is worth noticing that a certain arbitrariness is present in the complexification process of the functions λ and φ. Obviously, we have to obtain the metric (33) as soon as r = r.

From the transformed null tetrad vectors, a new metric is recovered using (34). For the null tetrad vectors given by ( 43) and the transformation given by (42), the new metric, with coordinates xµ = (ũ, r, θ, φ),

is gµν =     -a 2 sin 2 θ Σ 2 e -λ(r,θ)-φ(r,θ) + a 2 sin 2 θ Σ 2 0 -a Σ 2
.

-e -2λ(r,θ) -a 2 sin 2 θ

Σ 2 0 a Σ 2 . . -1 Σ 2 0 . . . -1 Σ 2 sin 2 θ     (44)
where Σ = √ r2 + a 2 cos 2 θ. In the covariant form, the metric (44) is gµν =     e 2φ(r,θ) e λ(r,θ)+φ(r,θ) 0 ae φ(r,θ) [e λ(r,θ) -e φ(r,θ) ] sin 2 θ . 0 0 -ae φ(r,θ)+λ(r,θ) sin 2 θ .

.

-Σ 2 0 . . . -[Σ 2 + a 2 sin 2 θe φ(r,θ) (2e λ(r,θ) -e φ(r,θ) )] sin 2 θ     (45)
Since the metric is symmetric, the dots in the matrix are used to indicate g µν = g νµ . The form of this metric gives the general result of the Newman-Janis algorithm starting from any spherically symmetric "seed" metric. The metric given in Eq. ( 45) can be simplified by a further gauge transformation so that the only off-diagonal component is g φt . This procedure makes it easier to compare with the standard Boyer-Lindquist form of the Kerr metric [START_REF] Ch | Gravitation[END_REF] and to interpret physical properties such as the frame dragging. The coordinates ũ and φ can be redefined in such a way that the metric in the new coordinate system has the properties described above. More explicitly, if we define the coordinates in the following way dũ = dt + g(r)dr and dφ = dφ + h(r)dr

where

     g(r) = -e λ(r,θ) (Σ 2 +a 2 sin 2 θe λ(r,θ)+φ(r,θ) )
e φ(r,θ) (Σ 2 +a 2 sin 2 θe 2λ(r,θ) ) h(r) = -ae 2λ(r,θ) Σ 2 +a 2 sin 2 θe 2λ(r,θ) (47) after some algebraic manipulations, one finds that, in (t, r, θ, φ) coordinates system, the metric (45) becomes θ) [e λ(r,θ) -e φ(r,θ) ] sin 2 θ .

g µν =     e 2φ(r,θ) 0 0 ae φ(r,
-Σ 2 /(Σ 2 e -2λ(r,θ) + a 2 sin 2 θ) 0

0 . . -Σ 2 0 . . . -[Σ 2 + a 2 sin 2 θe φ(r,θ) (2e λ(r,θ) -e φ(r,θ) )] sin 2 θ     . (48)
This metric represents the complete family of metrics that may be obtained by performing the Newman-Janis algorithm on any static spherically symmetric "seed" metric, written in Boyer-Lindquist type coordinates. The validity of these transformations requires the condition Σ 2 + a 2 sin 2 θe 2λ(r,θ) = 0, where e 2λ(r,θ) > 0. Our task is now to show that such an approach can be used to derive axially symmetric solutions also in f (R)-gravity.

V. AXIALLY SYMMETRIC SOLUTIONS IN f (R)-GRAVITY: AN EXAMPLE

Starting from the above spherically symmetric solution [START_REF] Capozziello | [END_REF], the metric tensor, written in the Eddington-Finkelstein coordinates (u, r, θ, φ) of the form (33) is

g µν =      0 2 βr 0 0 . -2 -2α βr 0 0 . . -1/r 2 0 . . . -1/(r 2 sin 2 θ)      . (49)
In addition, since the Hamiltonian is independent of the affine parameter λ, one can directly use the coordinate time as integration parameter. The problem is so reduced to solve six equations of motion. Using the above definitions, it is easy to achieve the reduced Hamiltonian (now linear in the momenta)

H = -p 0 =   p i g 0i g 00 + p i g 0i g 00 2 - m 2 + p i p j g ij g 00 1/2   (57)
with the equations of motion

dx i dt = ∂H ∂p i , dp i dt = - ∂H ∂x i , (58)
that give the orbits. The method can be applied to the above solution (52) considering the following line element

ds 2 = r(α + βr) + a 2 β cos 2 θ Σ dt 2 + 2 a(-2αr -2βΣ 2 + √ 2βΣ 3/2 ) sin 2 θ 2Σ dtdφ + - βΣ 2 2αr + β(a 2 + r 2 + Σ 2 ) dr 2 -Σ 2 dθ 2 -Σ 2 - a 2 (αr + βΣ 2 - √ 2βΣ 3/2 ) sin 2 θ Σ sin 2 θdφ 2
by which the elements of the inverse metric can be easily obtained:

g tt = 4Σ 2 Σ 2 - a 2 sin 2 θ(rα- √ 2 √ βΣ 3/2 +βΣ 2 ) Σ a 2 sin 2 θ 2rα - √ 2 √ βΣ 3/2 + 2βΣ 2 2 + 4Σ (a 2 β cos 2 θ + r(rβ + α)) Σ 2 - a 2 sin 2 θ(rα- √ 2 √ βΣ 3/2 +βΣ 2 ) Σ g rr = - β a 2 + r 2 + Σ 2 + 2rα βΣ 2 g θθ = - 1 Σ 2 g tφ = 2aΣ -2rα + √ 2 √ βΣ 3/2 -2βΣ 2 a 2 sin 2 θ 2rα - √ 2 √ βΣ 3/2 + 2βΣ 2 2 + 4Σ [a 2 β cos 2 θ + r(rβ + α)] Σ 2 - a 2 sin 2 θ(rα- √ 2 √ βΣ 3/2 +βΣ 2 ) Σ g φφ = - 4Σ csc 2 θ a 2 β cos 2 θ + r(rβ + α) a 2 sin 2 θ 2rα - √ 2 √ βΣ 3/2 + 2βΣ 2 2 + 4Σ [a 2 β cos 2 θ + r(rβ + α)] Σ 2 - a 2 sin 2 θ(rα- √ 2 √ βΣ 3/2 +βΣ 2 ) Σ (59)
and the null ones

g tr = g tθ = g rθ = g rφ = g θφ = 0 . (60)
Let us consider the equatorial plane, i.e. θ = π 2 , θ = 0, and assume α = 1 and β = 2. The reduced Hamiltonian can be written as It is independent of φ (i.e. we are considering an azimuthally symmetric spacetime), and then the conjugate momentum p φ is an integral of motion. From Eqs. (58), one can derive the coupled equations for {r, θ, φ, p r , p θ } and integrate them numerically (the expressions are very cumbersome and will not be reported here). To this goal, we have to specify the initial value of the position-momentum vector in the phase space. A Runge-Kutta method can be used to solve the differential equations. In Fig 1, the relative trajectories are sketched.

H(r, θ, φ, p r , p θ , p φ ; t) = 2ap φ -2r 3 + r 2 -1 a 2 (-2(r -1)r 2 -1) + r 5 + 4a 2 p 2 φ -2r 3 + r 2 -1 2 -a 2 -2(r -1)r 2 -1 -r 5 a 2 r 2 (r(2r -3)(2r + 1) + 6) -2 + (2r + 1)r 4 × - p φ (2r + 1) a 2 (r 2 (r(2r -3)(2r + 1) + 6) -2) + (2r + 1)r 4 - p r a 2 + r 2 + r + p θ r 4 -p r + 1 1 2 . (61)

VII. DISCUSSION AND CONCLUDING REMARKS

We have shown that the Newman-Janis method, used to derive axially symmetric solutions in GR, works also in f (R)-gravity. In principle, it could be consistently applied any time a spherically symmetric solutions is derived. The method does not depend on the field equations but directly works on the solutions that, a posteriori, has to be checked to fulfill the field equations.

The key point of the method is to find out a suitable complex transformation which, from a physical viewpoint, corresponds to the fact that we are reducing the number of independent Killing vectors. From a mathematical viewpoint, it is useful since allows to overcome the problem of a direct search for axially symmetric solutions that, in f (R)-gravity, could be extremely cumbersome due to the fourth-order field equations. However, other generating techniques exist and all of them should be explored in order to completely extend solutions of GR to f (R)-gravity. They can be more general and solid than the Newman-Janis approach. A good source for references and basic features of generating techniques is reference [START_REF] Stephani | Herlt Exact Solutions to Einstein's Field Equations[END_REF]. In particular, the paper by Talbot [START_REF] Talbot | [END_REF], considering the Newman-Penrose approach to twisting degenerate metrics, provides some theoretical justification for the scope and limitations of adopting the "complex trick". As reported in Chap. 21 of [START_REF] Stephani | Herlt Exact Solutions to Einstein's Field Equations[END_REF], several techniques can be pursued to achieve axially symmetric solutions which can be particularly useful to deal with non-empty space-times ( in particular when perfect fluids are the sources of the field equations) and to deal, in general, with problems related to Einstein-Maxwell field equations. We have to stress that the utility of generating techniques is not simply to obtain a new metric, but a metric of a new spacetime with specific properties as the transformation properties of the energy-momentum tensor and Killing vectors. In its original application, the Newman-Janis procedure transforms an Einstein-Maxwell solution (Reissner-Nordstrom) into another Einstein-Maxwell solution (Kerr-Newman). As a particular case (setting the charge to zero) it is possible to achieve the transformation between two vacuum solutions (Schwarzschild and Kerr). Also in case of f (R)-gravity, new features emerge by adopting such a technique. In particular, it is worth studying how certain features of spherically simmetric metrics, derived in f (R)-gravity, result transformed in the new axially symmetric solutions. For example, considering the f (R) spherically symmetric solution studied here, the Ricci scalar evolves as r -2 and then the asymptotic flatness is recovered. Let us consider now the axially symmetric metric achieved by the Newman-Janis method. The parameter a = 0 indicates that the spherical symmetry (a = 0) is broken. Such a parameter can be immediately related to the presence of an axis of symmetry and then to the fact that a Killing vector, related to the angle θ, has been lost. To conclude, we can say that once the vacuum case is

Figure 1 :

 1 Figure 1: Relative motion of the test particle with m = 1.

It is possible to take into account also the Palatini approach in which the metric g and the connection Γ are considered independent variables (see for example[21]). Here we will consider the Levi-Civita connection and will use the metric approach. See[3,22] for a detailed comparison between the two pictures.

This condition allows to obtain the standard definition of the circumference with radius r.

Obviously, the above choices are recovered for A(r) = a(r), B(r) = b(r), and M (r) = r 2 . Here we deal with A, B, M as a set of coordinates in a configuration space.

Lowering the dimension of configuration space through the substitution(19) does not affect the dynamics since B is a non-evolving quantity. In fact, inserting Eq. (19) into the dynamical equations given by[START_REF] Inverno | Introducing Einstein's Relativity[END_REF], they coincide with those derived by[START_REF] Inverno | Introducing Einstein's Relativity[END_REF].

From now on, q indicates the vector {A, M, R}.

The dimensions are given by R 1-s in terms of the Ricci scalar. For the sake of simplicity, we will put f 0 = 1 in the forthcoming discussion.

discussed, more general spherical metrics can be transformed in new axially symmetric metrics adopting more general techniques [13]. These approaches will be examined and discussed in future works.

The complex tetrad null vectors (37) are now

By computing the complex coordinates transformation (42), the tetrad null vectors become

Now by performing the same procedure as in previous section, we derive an axially symmetric metric of the form (48) but starting from the spherically symmetric metric [START_REF] Capozziello | [END_REF], that is

.

-

It is worth noticing that the condition a = 0 immediately gives the metric [START_REF] Capozziello | [END_REF]. This is nothing else but an example: the method is general and can be extended to any spherically symmetric solution derived in f (R)-gravity.

VI. A PHYSICAL APPLICATION: GEODESICS AND ORBITS

Let us discuss now a physical application of the above result. We will take into account a freely falling particle moving in the space-time described by the metric (52). For our aims, we make explicit use of the Hamiltonian formalism. Given a metric g µν , the motion along the geodesics can is described by the Lagrangian

where the overdot stands for derivative with respect to an affine parameter λ used to parametrize the curve. The Hamiltonian description is achieved by considering the canonical momenta and the Hamiltonian function

that results H = 1 2 p µ p ν g µν . The advantage of the Hamiltonian formalism with respect to the Lagrangian one is that the resulting equations of motion do not contain any sign ambiguity coming from turning points in the orbits (see, for example, [START_REF] Chandrasekhar | The mathematical theory of black holes[END_REF]) . The Hamiltonian results explicitly independent of time and it is H = -1 2 m 2 , where the rest mass m is a constant (m = 0 for photons). The geodesic equations are

∂g αβ ∂x µ p α p β = g γβ Γ α µγ p α p β .

(56)


Kate ; Khair

Keywords: Haemophilia, genetics, inheritance, carriers, children, young people Haemophilia

'Just an unfortunate coincidence': children's understanding of haemophilia genetics and inheritance

Kate Khair, Faith Gibson, Liz Meerabeau

Introduction

Haemophilia A and B are X linked inherited disorders of coagulation, which are carried by females and affect males. Healthcare workers readily draw family trees of affected individuals and easily identify 'obligate' and 'at risk' females within the family. The focus of genetic counselling and support is weighed heavily in favour of these women [START_REF]Ross J Perspectives of haemophilia carriers[END_REF] with little thought being given to the current and future genetic consequences of the affected male proband.

There is limited research describing communication of genetic information within haemophilia families [START_REF] Soronson | Communication about carrier testing within hemophilia A families[END_REF] or about how parents share genetic information within families [3,4,]. Spitzer [START_REF] Spitzer | Children's knowledge of illness and treatment experiences in haemophilia[END_REF] wrote, in 1992, that children (aged 6-13 years) with haemophilia, knew 'very little about their disease and especially about the way a person contracted it'. Since then genes and genetics have become more widely discussed and debated, and at this haemophilia comprehensive care centre forms part of routine clinical review of all children, by the haemophilia team [START_REF] Ludlam | A framework for genetic service provision for haemophilia and other inherited bleeding disorders[END_REF][START_REF]The RCN Haemophilia Nurses Association. A Competency Framework for Nurses Caring for People with Haemophilia and Other Related Bleeding Disorders[END_REF].

Children and young people with haemophilia in the twenty-first century, are the first to grow up in the genetic age with the human genome project, genetic testing, genetic modification of food, recombinant factor VIII/IX technology and gene therapy are talked about in everyday parlance [START_REF] Rew | A systematic review of literature about the genetic testing of adolescents[END_REF]. Genetics and inheritance are taught in English schools as part of the National Curriculum for Science [9]. This is divided into four age appropriate 'key stages' with each 'key stage' being further divided into areas of scientific interest. Human genetics, reproduction and inheritance are taught in key Children and genetics 2 stages 2-4 as part of the 'life process and living things' theme. The content of this teaching is shown in Table 1.

What has yet to be described in any detail is where young people with haemophilia learn about their own genetic make-up and its consequences. This paper presents findings about genetic knowledge and understanding as part of a larger qualitative study exploring the lived experience of boys with severe haemophilia.

Methods

This was a single site, qualitative study of 30 boys aged 4-16 years with severe haemophilia A (n= 27) or B (n= 3) undertaken in 2009 (see Table 2). Prior to commencement of the study ethical approval was granted by the South East Research Ethics Committee. The boys were invited to participate in a larger study evaluating the lived experience of haemophilia, where individual or group interviews were undertaken to better understand haemophilia, its treatment and its limitations, from the child's perspective.

The interviews were semi-structured and tape recorded, thus allowing the researcher to give full attention to what was being said. The tape recordings were listened to on several occasions, were transcribed by a professional transcriber and then analysed and coded by hand. Narrative content was identified and coded into recurring 'themes' which were identified as important aspects of modern day life with haemophilia. These themes were: family and siblings, school, sport, career, haemophilia -bleeds and treatment, good and bad aspects of living with haemophilia and genetics and inheritance. All of the themes, with the exception of genetics and inheritance, fit into a core category around the daily life experience of having

Results

Twenty four boys of the 30, who were interviewed, offered a range of responses in terms of complexity and detail about genetic knowledge and/or understanding of inheritance of haemophilia. Those who did not offer a response were aged seven or younger.

Boys with haemophilia grow up in families affected by haemophilia and develop a gradual understanding of the implications of haemophilia for them and their family members. The youngest boys in this study are at an age where they are beginning to understand 'symptoms, treatments and physical differences from peers' [START_REF] Gallo | Disclosure of genetic information within families: how nurses can facilitate family communication[END_REF], and their parents are starting to reveal genetic information, such as 'you were born with it' as they perceive their child to be developmentally ready for it [START_REF] Gallo | Parents sharing information with their children abou genetic conditions[END_REF][START_REF] Metcalfe | Family communication between children and their parents about inherited genetic conditions: a metasynthesis of the research[END_REF]. It is therefore not surprising that the smallest boys in this study have a lack of 'real' knowledge about haemophilia genetics and inheritance as these are complex issues [START_REF] Skene | Patients rights or family responsibilities? Two approaches to genetic testing[END_REF][START_REF] Keenan | It's their blood not mine': Who's responsible for (not) telling relatives about genetic risk?[END_REF] probably beyond their developmental ability [START_REF] Spitzer | Children's knowledge of illness and treatment experiences in haemophilia[END_REF].

A rudimentary understanding of haemophilia genetics is demonstrated by some of the youngest boys through statements such as 'I just came out that way' (Jack aged 4*)

'girls can't get haemophilia can they?' (Nick aged 9) and 'she didn't get it because she is a girl' (Murray aged 5). The data from boys is presented in two sub themes:

• Genetics, their haemophilia gene and its mutation and The knowledge demonstrated by these boys, exceeds the background genetic knowledge that is expected for boys of this age [9]. Haemophilia specific genetic knowledge has thus been gained either from their families, their haemophilia care providers or from both. Most parents see themselves as having 'prime responsibility' in discussing genetics with their children yet struggle with 'what and how to tell' at different stages of children's lives [START_REF] Plumridge | The internet as an information source for parents yalking to children about genetic conditions[END_REF]. Women (mothers) are often seen as 'genetic housekeepers' and 'kinkeepers' (those who hold the family (or kin) together) [START_REF] Thomas | Attitudes towards and beliefs about genetic testing in the haemophilia community: a qualitative study[END_REF] for family members, often negotiating genetic testing and coordinating counselling for family members, it is imperative that these women have the correct information and support to correctly inform family members [START_REF]Ross J Perspectives of haemophilia carriers[END_REF].

F

Inheritance

Approximately half to two thirds of boys with haemophilia come from 'affected families' i.e. those where there has been haemophilia in previous generations.

Spontaneous genetic mutations, either in the mother or the child are responsible for causing haemophilia in the remainder of those affected. This prevalence is reflected in this study where 10 of the 30 boys were the first affected member in their family.

Despite being the first, or only, affected individual in their family the boys demonstrated an awareness of the impact of haemophilia on other family members:

their mothers, siblings and their own future children.

Edward, aged 9, the only affected member of his family with haemophilia described its inheritance as 'just a rare unfortunate coincidence', something that his parents confirm they have told him repeatedly, and that his children might not be affected as 'it might not run in the family'. Edward has an older unaffected brother, his mother has chosen not to be carrier tested and there are no other 'at risk' females in the family. If Edward only has sons then he is correct in thinking that haemophilia will not run in the family, however all of his daughters will be carriers and their children might be affected. This notion is better narrated by boys with previously affected family members:

'Well it runs in the family so my granddad had it and he had a girl, my mum, and so she's a carrier and she carried it to me' Oliver aged 14 Sixteen year old George, who knew that his maternal grandfather was not his biological relative, described his inheritance of haemophilia from a known affected individual as:

'My mums' birth father had it so that made her a carrier and I got it'

These four boys were able to describe complex genetic inheritance patterns both from knowledge of their own families and from learning about genetics and inheritance at school and at the haemophilia centre where frequent discussion about this occurs.

Miller [START_REF] Miller | Counselling about diagnosis and inheritance of genetic bleeding disorders: haemophilia A and B[END_REF] describes treatment centre education and counselling protocols and states how affected boys and men are often 'forgotten' in this, assuming that they know the obligatory carrier status of their daughters. Several of the boys in this study showed a good understanding of the impact of haemophilia on their own children and their children's children:

'It means when men go on to have kids the boys will be OK and the girls will be carriers ….. it's a never ending circle really' Mark aged 16

'The girls will have the gene and her children, if she had a boy or a girl there is a 50/50 chance that either of them would have it' David aged 13

'Well the boys will be fine but the girls will carry it and the girl's boys will be half haemophilia' Matt aged 13

'Our [sic] Sixteen-year old George (who had an affected Grandfather whom he never knew) describes how being an affected grandfather himself would benefit his grandchildren:

'It would be a big advantage because they would know …….. all the stuff I did when I was young -so it can only be good'

This 'untapped expertise' [START_REF] Petersen | The best experts: the narratives of those who have a genetic condition[END_REF] of the expert patient whose life story is more knowledgeable about the impact of the illness than the expertise of the haemophilia health care professionals, shows how people can be 'successfully ill' [START_REF] Frank | The Wounded storyteller: body, illness and ethics[END_REF]. This is enhanced in inherited congenital disease where there is no 'before and after' to compare life with and without the illness, and where older affected individuals can have a significant impact on younger people:

'I've got a friend who is about 42-ish and he's got haemophilia and we chat, I think I look to him for a friend because he is older and he has lived with it longer than me, then we have a discussion about it, and I find it nice because things that kept on going for him he has been through as well, and I might go through exactly the same so that means quite a lot. He is someone I can actually relate to on a level where they know where I am coming from I suppose'. Mark aged 16

The sense of belonging to an extended family or a unique 'haemophilia community' [START_REF] Thomas | Attitudes towards and beliefs about genetic testing in the haemophilia community: a qualitative study[END_REF] through shared genetics (or genetic mutations) makes living with haemophilia an unexceptional everyday reality [START_REF] Gregory | Communicating about haemophilia within the family: the importance of context and of experience[END_REF] bound within individual, family, and cultural values. Modern day, openly communicated day-to-day lived experience enables boys to communicate freely about genetic inheritance and haemophilia management within their families. This leads to an understanding of the potential carriership of their sisters, and a relationship that 'becomes stronger as siblings grow up' [START_REF] Thomas | Attitudes towards and beliefs about genetic testing in the haemophilia community: a qualitative study[END_REF]. Having this knowledge enables the next generation to confront their fate [START_REF] Miller | Counselling about diagnosis and inheritance of genetic bleeding disorders: haemophilia A and B[END_REF], laying down a foundation for their future communication within his family [START_REF] Metcalfe | Family communication between children and their parents about inherited genetic conditions: a metasynthesis of the research[END_REF].

'She is fine, she is normal, she has been tested' James 13 Family membership and relatedness is a construct of genes and genetics that cause disease as well as those that do not -for example those that cause haemophilia and those which lead to blue eyes or curly hair and a family resemblance. Genetically unaffected siblings may still be affected by having a sibling with, or who is a carrier of, haemophilia. These siblings experience haemophilia on a day-to-day basis and understand how it affects family life and relationships both today as well as in the past and future.

'

Discussion

This is the first paper that the authors can identify, which presents the genetic understanding and knowledge of children and young people who have haemophilia.

Studies of young people and adults with haemophilia have shown that 69% of patients aged 13-25yrs [START_REF] Lindvall | Complinace with treatment and understanding of own disease in patients with severe and moderate haemophilia[END_REF] and 96% of patients aged over 25 [START_REF] Lindvall | Knowledge of disease and adherence in adult patients with haemophilia[END_REF] were aware of heredity i.e.

a healthy son and carrier daughter. When asked to rank genetic knowledge about haemophilia 80% of respondents, patients (aged > 18) and parents, ranked knowledge as high or medium high [START_REF] Miller | Haemophilia A: patients knowledge level of treatment and sources of treatment-related information[END_REF]. These data are from studies about disease and treatment

and not specifically about genetic knowledge and understanding.

Within this study the role of the families in supporting and educating boys about haemophilia is apparent. Growing up with a brother with haemophilia leads to strong sibling relationships [START_REF] Thomas | Attitudes towards and beliefs about genetic testing in the haemophilia community: a qualitative study[END_REF], experiential knowledge [START_REF] Petersen | The best experts: the narratives of those who have a genetic condition[END_REF] The boys in this study will have all had science education at school using the National Curriculum for England (science) teaching programme [9]. This knowledge will be enhanced through parental discussion of genetic knowledge with children. This occurs in a developmental way from basic information for preschoolers, family relationships and genetics for school age children and understanding of the implications of genetic disease on their own reproduction for adolescents [START_REF] Gallo | Disclosure of genetic information within families: how nurses can facilitate family communication[END_REF]. The boys in this study demonstrated genetic and inheritance knowledge that exceeds that expected for their educational age (see Table 1) this demonstrates knowledge acquisition from parents, families and haemophilia care providers.

The oldest boys in this study were able to perform 'verbal drawing of family trees' [START_REF] Skirton | A legacy for the children -attitudes of older adults in the United Kingdom to genetic testing[END_REF] fully describing the genetic implications and inheritance patterns of haemophilia for them, their ancestors and their descendants including Georges biological (and unknown) grandfather: For some this will be following formal education about genetics and inheritance in science at school [9] however in day-to-day life communication about haemophilia within the family is in the domain of the everyday (as treatment is given etc) and judgement about it as a condition, where individuals can lead fulfilling and worthwhile lives. [START_REF] Soronson | Communication about carrier testing within hemophilia A families[END_REF][START_REF] Gregory | Communicating about haemophilia within the family: the importance of context and of experience[END_REF].

'
The narratives told by these children are not only about themselves but are also 'partly stories about others in their family' [START_REF] Petersen | The best experts: the narratives of those who have a genetic condition[END_REF] providing self identity which relates them to 'others': these are family members identified as relatives as well as those belonging to the haemophilia community as described by Mark when talking about his 42 year old friend with haemophilia.

Conclusion

Genetic knowledge, testing and technology have been major scientific breakthroughs in haemophilia care in the twentieth century, and will continue to improve as further developments occur in genomics and genetic therapeutic options. Comprehensive strategies are needed to ensure that those affected, either directly or indirectly by haemophilia, fully understand their own genetic risk. This paper is a first insight into genetic knowledge of young people affected by haemophilia and should be used as a evidence for future education and support of both affected boys and their (potential or actual) carrier sisters who may pass on the haemophilia gene to future generations.

  It's in part of our genes -our mums got part missing, but she's got two and why girls can't get haemophilia … but boys only have one so when part of ourgene is missing which means our blood doesn't clot.' Henry aged 9 on behalf of himself and Isaac aged 6.

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

boys will be perfect … but my ……if I get a girl she will be a carrier … she then has a boy and he has haemophilia' Jonathan aged 10

  things that mean a lot……… we have had discussions about it, ………… and obviously it's not his fault [that he doesn't have it] and he says he finds it hard sometimes I know that he's always stuck close to me cos of that'.Mark 16

	She had like a test to see if she was a carrier, like my mum, and she wasn't a
	carrier ……..that's good because is she has some boys when she is older that
	means that it won't have haemophilia' Tom 12
	'They [his mum and dad] had this special treatment -they had like IVF and
	then some special treatment which had never been done before and it worked
	out she's not a carrier …….. she can't ever get it again …….. it wont just
	randomly pop into the boy again ' Oliver 14
	F o r
	P
	e
	e r
	R
	e
	Unaffected siblings are recognised as not being part of the immediate 'haemophilia v i e
	family' but of being affected by it, and therefore being part of, the global haemophilia w
	community:
	[It's good he] 'didn't have it, ……. because he does not have haemophilia
	cause he is only little ……I don't want my little brother to have haemophilia
	so I am better off having it myself'. Nick aged 9
	'I think my brother, my older brother who hasn't got haemophilia it affected, I

Genetic testing of female siblings in families with X linked inherited disorders is an area of debate, but many children state that they are glad to have been tested and been able to grow up with knowledge about carriership

[START_REF]Ross J Perspectives of haemophilia carriers[END_REF][START_REF] Metcalfe | Family communication between children and their parents about inherited genetic conditions: a metasynthesis of the research[END_REF][START_REF] Thomas | Attitudes towards and beliefs about genetic testing in the haemophilia community: a qualitative study[END_REF][START_REF] Miller | Counselling about diagnosis and inheritance of genetic bleeding disorders: haemophilia A and B[END_REF][START_REF] Mcconkie-Rossell | Genetic risk communication: experiences of adolescent girls and young women from families with fragile X syndrome[END_REF]

. However in a paper about Duchenne Muscular Dystrophy (DMD) carrier testing and status was not routinely known by 'at risk' girls as parents felt a need to protect them from the knowledge that they were carriers

[START_REF] Plumridge | Family communication about genetic risk information: particular issues for Duchenne Muscular Dystrophy[END_REF]

. The girls in this study felt that they should know everything about the genetics of DMD even if that knowledge was upsetting at the time, as this gave a better 'family view' of inherited disease. think in a way he feels guilty that we have haemophilia and he doesn't and he goes out of his way to help us out. In fact I know he does because we are only a year apart me and my older brother, me and him have a good relationship so I know for a fact that if certain things he'll do for me which are just little

  genetic disease on their own, their siblings and their future children's health. This knowledge develops in an age appropriate way: where the youngest of the children knew only that girls do not get haemophilia ('she didn't get it because she is a girl'

	Murray aged 5), in contrast to older boys such as Henry aged 9 who showed a greater
	understanding of inheritance:
	'It's in part of our genes -our mums got part missing, but she's got two and
	why girls can't get haemophilia … but boys only have one so when part of our
	gene is missing which means our blood doesn't clot.'
	F o r
	P
	e
	e r
	R
	e
	v i e
	w

, and communication of family values about the condition which may support carriers and boys with haemophilia in future reproductive choices. The boys in this study understand the implications of

  'It will be good for my sisters' kids, my girls and my grandsons… 'cos they will have me to help them through it .. so better for them than for me who had no one who knew'

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	My mum's birth father had it so that made her a carrier and I got it'
	Simon (aged 15, who is the first in his family to be affected) understands impact of
	haemophilia on future children within his family:

Knowledge/Education about genetics/inheritance in National Curriculum in England and (in column 4) haemophilia genetic knowledge demonstrated by this cohort of boys

  from within the family whilst the remainder comes from haemophilia care providers whose responsibility it is to inform and correct misinformation. The study is limited in that it only includes children from a single site where knowledge of inheritance and genetics are seen as an important part of haemophilia care and are repeatedly addressed at review clinics. Haemophilia health care workers should provide tailored, age appropriate information to those with haemophilia as well as to those who do, or may, carry it addressing age appropriate issues such as reproductive choices, as they occur. 'it's from my mum … she carrier it' (David aged 13)• that some diseases are inherited • 'the girls will carry it and the girl's boys will be haemophilia' (Matt aged 13) • I was born with it … because my mum's a carrier' (James aged 13• genetic engineering is possible • 'they had this treatment ……..like IVF …. So she's not a carrier' (Oliver aged 14)

	Table 1
	National	Expected	National Curriculum teaching programme :	Quotes which demonstrate advanced genetic knowledge in
	Curriculum	age of	Scientific knowledge about genetics/inheritance	boys with haemophilia
	Key Stage	achievement
	2	11		• recognises the roles of drugs as medicine	• children aged 5 (data not shown)
				• are knowledgeable about human life cycles	• 'it might not run in the family' (Edward aged 9)
	3	14		• human reproduction with inherited	• 'she didn't get it -she's a girl (Murray aged 5)
				variations within species	• 'it's good he didn't have haemophilia' [about his little
					brother](Nick aged 9)
				• selective breeding can lead to new varieties	• 'I just came out that way' (Jack aged 4)
					• 'Just a rare unfortunate coincidence' (Edward aged 9)
					• 'I was born with it … because my mum's a carrier'
					(James aged 13)
	4	16	• composition and function of blood, • the mechanism of inheritance with recessive and dominant genes r F o	• 'part of the gene missing' (Henry aged 9) • 'you don't clot fully' (Matt aged 13) • 'mum's got part missing …but she's got two (Henry aged 9) •
				P
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				w

This paper demonstrates that the children and young people in this study have a good awareness of haemophilia genetics and inheritance. Some of this information comes *Pseudonyms have been used throughout this paper to protect the identity of the interviewees F o r P e e r R e v i e w F o r P e e r R e v i e w

Table 2 Pseudonym

 2

		Age (Years and Months)	Diagnosis	Previous Family History
	Jack	4 y 11m	Severe haemophilia A	Y (brother of Duncan)
	Murray	5 y 3m	Severe haemophilia A	N
	Isaac	6y	Severe haemophilia A	Y (brother of Henry)
	Jimmy	7y 10m	Mild haemophilia A with high titre inhibitor	Y
			(severe phenotype)
	Neil	8y 1m	Severe haemophilia A	Y
	Duncan	8y 9m	Severe haemophilia A	N
	Edward	9y 2m	Severe haemophilia A	N
	Nick	9y 2m	Severe haemophilia A	Y
	Robert	9y 8m	Severe haemophilia A with high titre inhibitor N
	Henry	9y 11m	Severe haemophilia A	N
	Will	10y 2m	Severe haemophilia A	Y (brother of Paul)
	Jonathan	10y 8m	Severe haemophilia A	Y (brother of David)
	Tom	12y 3m	Severe haemophilia A	Y
	Michael	12y 10m	Severe haemophilia A	N
	James	13y	Severe haemophilia A	Y
	Matt	13y	Severe haemophilia A	Y
	David	13y 3m	Severe haemophilia A	Y
	Peter	14y 5m	Severe haemophilia A	N
	Oliver	14y 5m	Severe haemophilia A	Y
	Paul	14 y 10m	Severe haemophilia A	Y
	Ian	15y 3m	Severe haemophilia A	Y (brother of Neil)
	Simon	15y 6 m	Severe haemophilia A	N
	George	16y 4m	Severe haemophilia A	Y
	Mark	16y 10m	Severe haemophilia A	Y
	Boys who discussed knowledge of genetic information.

Haemophilia