Graeme N Candlish

On the smoothness of the multi-BMPV black hole spacetime

We demonstrate that, in a multi-BMPV black hole spacetime, the event horizon is not smooth. We explicitly show that for a simpler configuration comprising a line of static extremal black holes and a single BMPV black hole, the metric at the horizon of the BMPV black hole is once, but not twice, continuously differentiable. We argue that this result is also valid when all the black holes are rotating. The Maxwell field strength is shown to be continuous, but not differentiable at the horizon.

Introduction

Einstein-Maxwell theory admits multi-centre extremal black hole solutions, thanks to a balance of gravitational and electrostatic forces. Such a solution in four dimensions has been known for a long time, and the event horizon has been shown to be analytic [START_REF] Hartle | Solutions of the Einstein-Maxwell equations with many black holes[END_REF]. In higher dimensions, however, such solutions do not generally have smooth event horizons [START_REF] Welch | On the smoothness of the horizons of multi-black hole solutions[END_REF][START_REF] Candlish | On the smoothness of static multi-black hole solutions of higher-dimensional Einstein-Maxwell theory[END_REF].

In [START_REF] Welch | On the smoothness of the horizons of multi-black hole solutions[END_REF] this lack of smoothness was shown for an axisymmetric set of static black holes by considering the behaviour of components of the Riemann tensor in a parallely propagated orthonormal frame along axial null geodesics. This result was extended in [START_REF] Candlish | On the smoothness of static multi-black hole solutions of higher-dimensional Einstein-Maxwell theory[END_REF] by removing the restriction to axial geodesics and allowing an arbitrary number of coaxial static black holes to be present. Rather than working with the Riemann tensor, the behaviour of a geometric invariant was investigated, as such invariants will be as differentiable as the metric itself. In that work it was shown that the event horizon is generally C 2 but not C 3 in five dimensions 1 , and C 1 but not C 2 in more than five dimensions. Furthermore, in the case of more than five dimensions, it was demonstrated that there is a parallely propagated curvature singularity at the horizon. By tuning the parameters of the solution it is possible to increase the smoothness in five dimensions, and in fact it is possible to ensure that at least one connected component of the horizon is analytic. This is not the case for six dimensions or more.

In five dimensions, the action for Einstein-Maxwell-Chern-Simons theory coincides with the bosonic part of N = 2 five dimensional supergravity. All supersymmetric solutions of this theory have been classified according to whether there exists a timelike or null Killing vector [START_REF] Gauntlett | All supersymmetric solutions of minimal supergravity in five dimensions[END_REF]. In the class of solutions corresponding to the timelike case, there are static black holes as discussed above, stationary rotating black holes [START_REF] Breckenridge | D-branes and spinning black holes[END_REF] (known as the BMPV solution), and black rings [START_REF] Elvang | A Supersymmetric black ring[END_REF].

As these solutions are supersymmetric (they satisfy a BPS bound), they are also extremal, and thus multi-black hole solutions may be constructed due to the balance of forces exactly as described above. A striking example of such a solution comprising a set of concentric black rings is given in [START_REF] Gauntlett | Concentric black rings[END_REF]. The general concentric ring solution is shown there to be analytic, although the spacetime may contain closed timelike curves. When the rings are coplanar or oriented on orthogonal two-planes the solution preserves both the U (1) rotational isometries of a single supersymmetric black ring and, given a certain restriction on the parameters, there are no closed timelike curves outside the horizon.

We generally expect to see a non-analytic metric at the event horizon when a rotational symmetry is broken. A straightforward argument for this is given in [START_REF] Horowitz | How Hairy Can a Black Ring Be?[END_REF], where a supersymmetric black ring of non-constant charge density is shown not to admit a smooth horizon. The presence of a horizon for a rotating black hole requires a limit → ∞ where is the coordinate along the direction of rotation. Any non-trivial periodic function of will therefore run through an infinite number of periods close to the horizon and so will not be continuous there. The behaviour of the coordinate is familiar from the Kerr solution, where the azimuthal angle of the Boyer-Lindquist coordinates diverges at the horizon. It is essentially this effect we will see in this paper: a rotating black hole metric that has a broken symmetry in one of the directions of rotation will exhibit a lack of smoothness at the horizon.

One motivation of this work was to investigate the effects of a spin-spin interaction between two (or more) BMPV black holes. The lack of smoothness of the multi-BMPV spacetime could conceivably have been dependent on the relative alignments of the rotations, with parallel rotation resulting in a higher degree of differentiability than for the non-parallel case. A special case of parallel spins is when only one black hole rotates. Thus in this paper we consider an axial configuration of black holes, with a single, central rotating BMPV black hole, while the other black holes are static. The single BMPV solution has two planes of equal rotation and isometry group R × SU (2) × U [START_REF] Hartle | Solutions of the Einstein-Maxwell equations with many black holes[END_REF]. The axial configuration we are considering preserves only two of the isometries of a single BMPV black hole: time translations and a single U (1) rotational symmetry. From the argument above we therefore expect lack of smoothness resulting from breaking the symmetry in a direction of rotation.

As we already know from [START_REF] Candlish | On the smoothness of static multi-black hole solutions of higher-dimensional Einstein-Maxwell theory[END_REF] that the five dimensional extremal static multi-black hole solution only admits a C 2 horizon, we might ask if our special case of a single rotating black hole amongst static black holes is at least as smooth. Somewhat surprisingly we find that this is not the case, and we will demonstrate that our configuration exhibits an event horizon that is C 1 but not C 2 . Thus we can conclude that a general multi-BMPV solution must not have a C 2 horizon.

To begin with then, in section 2 we briefly describe some important aspects of the single BMPV black hole solution, as well as the Gaussian null coordinates which are central to our work. In section 3 we give the metric for the multi-black hole spacetime we are considering. The main analysis and result is discussed in section 4. Finally we conclude and summarise the results.

Single BMPV Solution

The BMPV black hole [START_REF] Breckenridge | D-branes and spinning black holes[END_REF] is a stationary, asymptotically flat supersymmetric black hole solution of minimal five dimensional supergravity. A general stationary BPS solution to minimal five dimensional supergravity has the following metric [START_REF] Gauntlett | All supersymmetric solutions of minimal supergravity in five dimensions[END_REF]:

ds 2 = -H -2 (dt + ω) 2 + Hds 2 (M 4 ). (1)
For the BMPV solution the base space M 4 = R 4 . The function H and the one-form ω are defined on R 4 . We will use the following metric on the base space:

ds 2 (R 4 ) = dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 + cos θ 2 dψ 2 ), (2)
where 0 ≤ θ ≤ π/2 and 0 ≤ φ, ψ < 2π. H satisfies the four dimensional Laplace equation and is therefore harmonic.

Poles in H correspond to the locations of event horizons of black holes. Thus taking H to be

H = 1 + µ r 2 (3)
corresponds to the presence of a single asymptotically flat black hole with horizon located at r = 0. The parameter µ is proportional to the mass. As we have four spatial dimensions there are two independent planes of rotation. In the coordinates we are using, the two commuting U (1) rotational symmetries of a single axisymmetric five dimensional black hole have orbits whose tangent vectors are ∂ φ and ∂ ψ . For the BMPV black hole, regularity of the horizon demands equal rotations in both two-planes. Thus the four dimensional rotation group SO(4) ∼ = SU (2) × SU (2) is effectively restricted to one of the SU (2) factors. The one-form ω, which describes the rotation of the black hole, may be written in terms of the left-invariant one-forms on the 3-sphere. Without loss of generality, we choose

ω = J 2r 2 (sin 2 θdφ + cos 2 θdψ) ( 4
)
where J is a parameter proportional to the angular momentum in both two-planes of rotation. As investigated in [START_REF] Gibbons | Supersymmetric rotating black holes and causality violation[END_REF], the BMPV solution becomes a time machine, with closed timelike curves outside the horizon if the causality bound 4µ 3 > J 2 is violated. A further interesting property of this solution is that the angular velocity of the horizon vanishes, even though the asymptotic conserved quantity J = 0. This is a result of supersymmetry as demonstrated in [START_REF] Myers | Black holes of D=5 supergravity[END_REF], where it is shown that a negative fraction of the angular momentum resides behind the horizon. The single BMPV black hole metric is therefore given by

ds 2 = -1 + µ r 2 -2
(dt + J 2r 2 (sin 2 θdφ + cos 2 θdψ)) 2 + 1 + µ r 2 (dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 + cos θ 2 dψ 2 )). ( 5)

Gaussian null coordinates

To investigate the smoothness of the multi-BMPV solution we will put the metric into a Gaussian null system of coordinates. The method of construction of such a coordinate system is discussed in [START_REF] Reall | Higher dimensional black holes and supersymmetry[END_REF] and in detail in [START_REF] Friedrich | On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon[END_REF]. Here we present only a brief description. Consider a connected component of the future event horizon H + of a multi-black hole spacetime, which we will call H + 0 . Let us label the intersection of this component of the horizon with a spatial hypersurface as H 0 . The surface H 0 is topologically an S 3 , on which we introduce the coordinates x i . In the coordinates in (1) the time translation Killing vector is given by V = ∂/∂ t . As this is null on the horizon and the generator of a symmetry, this is tangent to the null geodesic generators of H + 0 . We now define v to be the parameter-distance from H 0 along the integral curves of V . This gives us a coordinate chart (v, x i ) on a neighbourhood of H 0 in H + 0 . We now extend this chart off the horizon along a null geodesic. Let U be the unique (past-directed) null vector field satisfying U • V = 1 and U • ∂/∂x i = 0 on H + 0 . We take γ(v, x i ) to be the null geodesic that begins at the point with coordinates (v, x i ) in H + 0 and whose tangent vector there is U . The affine parameter distance from H + 0 along this geodesic is given by λ. The Gaussian null coordinates on a neighbourhood of H 0 are therefore (v, λ, x i ), and the metric takes the general form

ds 2 = -H(λ, x) -2 dv 2 + 2dvdλ + 2λh i (λ, x)dvdx i + h ij (λ, x)dx i dx j . ( 6
)
As a warm up, we will perform this transformation to Gaussian null coordinates for the single BMPV black hole metric given in the previous section. For a null geodesic with affine parameter λ in this geometry we have

0 = -1 + µ r 2 -2 ṫ + J 2r 2 (sin 2 θ φ + cos 2 θ ψ) 2 + 1 + µ r 2 ( ṙ2 + r 2 ( θ2 + sin 2 θ φ2 + cos 2 θ ψ2 )) (7)
where the dot denotes differentiation with respect to λ. From time translation symmetry, and the choice of a past-directed geodesic, we have

-H -2 ( ṫ + ω) = E, (8)
where ω = J 2r 2 (sin 2 θ φ + cos 2 θ ψ).

We choose the parameterisation such that we can set the constant of the motion E = 1, so we have

ṫ + ω = -H 2 . ( 10
)
The rotational symmetries, generated by ∂ φ and ∂ ψ , give us

φ = 1 Hr 2 sin 2 θ J φ - J 2r 2 sin 2 θ ψ = 1 Hr 2 cos 2 θ J ψ - J 2r 2 cos 2 θ , (11)
where J φ and J ψ are the conserved angular momenta of the geodesic in the φ and ψ direction respectively. By definition of the Gaussian null coordinate system, we require the tangent vector to the geodesic to be orthogonal to the vector fields tangent to the S 3 at the horizon H 0 . Therefore we set

θ = J φ = J ψ = 0, (12)
where we have chosen a geodesic where θ is constant. The conditions on J φ and J ψ ensure that the Killing fields associated to the φ and ψ directions lie tangent to the horizon. Note that for the multi-black hole case we will no longer have a constant θ, although θ will still be chosen to vanish at the horizon. With these conditions on the geodesic we now have

φ = ψ = - J 2Hr 4 . ( 13
)
Substituting ( 10) and ( 13) into the null condition [START_REF] Elvang | A Supersymmetric black ring[END_REF], and setting θ = 0 we find

ṙ = H - J 2 4H 2 r 6 1/2 (14)
This can be expanded out for small r, integrated and the resulting series inverted to obtain

r(λ) = ∆ µ 1/2 λ 1/2 1 + J 2 + 2µ 3 4µ 2 ∆ λ + O(λ 2 ) ( 15
)
where the term in brackets is analytic in λ and we have defined the symbol

∆ ≡ (4µ 3 -J 2 ) 1/2 . ( 16
)
As stated earlier, to ensure the absence of closed timelike curves outside the horizon we must have ∆ > 0. As we want λ = 0 on the horizon, we have also imposed the initial condition r(λ)| λ=0 = 0. Clearly this is an outgoing null geodesic for increasing λ. Using ( 15) and ( 13), along with the initial conditions on our geodesic we can find

φ = - J 2∆ log λ + Φ + J(10µ 3 -J 2 ) 4∆ 2 µ 2 λ + O(λ 2 ) ψ = - J 2∆ log λ + Ψ + J(10µ 3 -J 2 ) 4∆ 2 µ 2 λ + O(λ 2 ) ( 17
)
with Ψ and Φ constants of integration. From these expressions we can immediately see the consequences of breaking symmetry in either the φ or ψ directions: at the horizon, λ = 0, the angles undergo an infinite shift. If the metric components involve periodic functions of φ or ψ then those functions will go through an infinite number of periods at the horizon and will not be well-behaved. We will see the consequences of this for the smoothness of the multi-centre metric at the horizon later.

We now have all we need to transform the metric to the Gaussian null coordinates. The Gaussian null coordinates on the S 3 , previously denoted x i , will now be given by Θ, Φ and Ψ. The coordinate Θ is the limiting θ value of our null geodesic at the horizon. In the present case we have chosen θ to be constant, so we have the trivial transformation θ = Θ. In the multi-black hole case θ will not be constant and the transformation will be nontrivial. The coordinates Φ and Ψ are the constants of integration in (17). We may use these as coordinates because they are finite and

∂/∂ φ = ∂/∂ Φ , ∂/∂ ψ = ∂/∂ Ψ . ( 18
)
The Gaussian null coordinate v is defined as

v ≡ t + (H 2 + ω)dλ. (19)
Using [START_REF] Friedrich | On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon[END_REF] to rewrite ω, and [START_REF] Chruściel | Nonsmoothness of event horizons of Robinson-Trautman black holes[END_REF] to transform all functions of r to functions of λ we find

dt = dv -H(λ) 2 - J 2 4H(λ)r(λ) 6 dλ. ( 20
)
The final coordinate in the Gaussian null coordinate system is λ, the affine parameter along our null geodesic, which becomes our new radial coordinate. Therefore the full set of coordinate transformations from the original form of the metric given in (5) to the Gaussian null form is given by ( 15), (20) and

dr = H(λ) - J 2 4H(λ) 2 r(λ) 6 1/2 dλ dφ = dΦ - J 2H(λ)r(λ) 4 dλ dψ = dΨ - J 2H(λ)r(λ) 4 dλ dθ = dΘ. (21)
Using these, we find that the metric in Gaussian null coordinates is

ds 2 = -H(λ) -2 dv 2 + 2dvdλ - J H(λ) 2 r(λ) 2 dv sin 2 θdΦ + cos 2 θdΨ + H(λ)r(λ) 2 dΘ 2 + H(λ)r(λ) 2 sin 2 Θ - J 2 sin 4 Θ 4H(λ) 2 r(λ) 4 dΦ 2 + H(λ)r(λ) 2 cos 2 Θ - J 2 cos 4 Θ 4H(λ) 2 r(λ) 4 dΨ 2 - J 2 sin 2 (2Θ) 8H(λ) 2 r(λ) 4 dΦdΨ. (22)
In these coordinates the metric is regular at the horizon λ = 0 where it takes the form

ds 2 | H0 = 2dvdλ + µdΘ 2 + µ - J 2 sin 2 Θ 4µ 2 sin 2 ΘdΦ 2 + µ - J 2 cos 2 Θ 4µ 2 cos 2 ΘdΨ 2 - J 2 cos 2 Θ sin 2 Θ 4µ 2 dΦdΨ, (23)
and the determinant is

g = 1 4 (4µ 3 -J 2 ) cos 2 Θ sin 2 Θ, (24)
which clearly satisfies g = 0 when the causality bound 4µ 3 > J 2 holds, and for Θ = 0, π/2 which we expect are just coordinate singularities. Expanding the metric components as a series in λ around λ = 0 we can find the form off the metric away from the horizon. This expansion is presented in the appendix for the leading order terms in each component. Obviously the single BMPV solution preserves both rotational U (1) symmetries (with associated Killing vectors ∂ Φ and ∂ Ψ in these coordinates). In addition to this, all the metric functions (i.e. H and ω) depend only on r 2 . From [START_REF] Chruściel | Nonsmoothness of event horizons of Robinson-Trautman black holes[END_REF] we see that r 2 is an analytic function of λ. This shows we have an analytic metric at the horizon.

Axial Multi-Black Hole Spacetime

We now give the metric for the configuration of black holes in which we are interested: a single BMPV black hole in a line of extremal static black holes. As argued in the introduction, the result of the smoothness analysis for this simpler special case will apply to a general multi-BMPV spacetime. The metric in this case is again given by

ds 2 = -H -2 (dt + ω) 2 + H(dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 + cos 2 θdψ 2 )) (25)
but now we wish to add additional static black holes to the spacetime. As they are static, the one-form ω will be the same as [START_REF] Kimura | Analyticity of Event Horizons of Five-Dimensional Multi-Black Holes with Non-Trivial Asymptotic Structure[END_REF],

ω = J 2r 2 (sin 2 θdφ + cos 2 θdψ), (26)
but we must add more monopole sources to the harmonic function H. Thus we write

H = 1 + µ 1 r 2 + N i=2 µ i r 2 + a 2 i -2a i r sin θ sin φ (27)
where µ i and a i denote, respectively, the mass parameter and position of the i-th black hole. Therefore the additional static black holes lie at the locations r = a i along the axis given by θ = π/2 and φ = π/2 or φ = 3π/2. We may think of this axis as lying entirely in one of the

R 2 factors of R 4 ∼ = R 2 × R 2 .
The polar angle in this two-plane is φ. We could, of course, have chosen a different axis, possibly one lying entirely in the other R 2 factor, i.e. the orthogonal two-plane with polar angle ψ. This would make no difference to the analysis. The important point is that adding an extra pole away from the origin breaks at least one of the two rotational U (1) isometries of the single BMPV black hole located at r = 0. Thinking of R 4 as R 2 × R 2 it is clear that it is impossible to add a pole away from the origin that does not break a rotational symmetry in either R 2 . Therefore we make an explicit choice of which symmetry to break in the above expression of H. We can expand the harmonic function H in terms of spherical harmonics in the following way:

H = µ 1 r 2 + ∞ n=0 h n r n Y n (sin θ sin φ) ( 28
)
where the harmonics Y n (sin θ sin φ) are given by the Gegenbauer polynomials

Y n (sin θ sin φ) = C 1 n (sin θ sin φ). ( 29
)
The coefficients are

h n = δ n,0 + N i=2 µ i a 2+n i . ( 30
)
For the single black hole case this of course reduces to h 0 = 1, h n = 0 for n ≥ 1. Given our choice of H it is clear that ∂ ψ is still a Killing vector for this metric.

To summarise then, we have added additional static black holes to the single BMPV black hole spacetime. These additional black holes all lie along a common axis, in order to preserve one of the two U (1) rotational isometries of the BMPV black hole.

Smoothness Analysis

We now face the task of examining the smoothness of the metric we have just constructed. We will restrict ourselves to determining the smoothness at the connected component of the spatial cross-section of the future event horizon that corresponds to the S 3 horizon of the BMPV black hole at r = 0 (or λ = 0 in the Gaussian null coordinates).

To do this we will consider the behaviour of the norm of the Killing vector field K = ∂/∂ ψ along a null geodesic that approaches the horizon. We will show that this function is not C 2 at the horizon and hence the horizon cannot be C 2 . We then construct Gaussian null coordinates to show that the metric is C 1 . Therefore we determine that the metric at the horizon is once continuously differentiable but not twice.

Geodesic equations

First we must solve the geodesic equations to determine a past-directed outgoing null geodesic. As before we have

ṫ + ω = -H 2 ( 31
)
where ω is given by ( 9), and again we have chosen the parameterisation such that the constant of the motion E = 1. The rotational symmetry in the ψ direction again gives

ψ = 1 Hr 2 cos 2 θ (J ψ - J 2r 2 cos 2 θ). ( 32
)
As we have broken the φ symmetry we no longer have the constant of the motion J φ , and we must solve the geodesic equation for φ explicitly.

The geodesic equation (after substituting for ṫ) for r is r -r θ2 -r φ2 sin 2 θ -r ψ2 cos 2 θ + H -1 ṙ( φ∂ φ H + θ∂ θ H)

+ H -1 ( ψf ψr + φf φr ) -∂ r H + 1 2 H -1 ṙ2 ∂ r H - 1 2 H -1 r 2 ∂ r H( θ2 + ψ2 cos 2 θ + φ2 sin 2 θ) = 0. ( 33
)
The geodesic equation for θ is

θ + 2r -1 ṙ θ + 1 2 ψ2 sin(2θ) - 1 2 φ2 sin(2θ) + H -1 θ( φ∂ φ H + ṙ∂ r H) + H -1 r -2 ( ψf ψθ + φf φθ ) -r -2 ∂ θ H + 1 2 H -1 θ2 ∂ θ H - 1 2 H -1 ∂ θ H(r -2 ṙ2 + ψ2 cos 2 θ + φ2 sin 2 θ) = 0. ( 34
)
The geodesic equation for φ is

φ + 2r -1 ṙ φ + 2 cot θ θ φ -r -2 csc 2 θ∂ φ H + 1 2 H -1 csc 2 θ∂ φ H sin 2 θ φ2 -cos 2 θ ψ2 -r -2 ṙ2 -θ -r -2 H -1 csc 2 θ ṙf φr + θf φθ + H -1 φ ṙ∂ r H + θ∂ θ H = 0. ( 35
)
We have used

f µν = ∂ µ ω ν -∂ ν ω µ ( 36
)
where ω ν are the components of the one-form ω (of which there are only ω φ and ω ψ ). The final equation we require is the null condition,

-H 2 + H( ṙ2 + r 2 θ2 + r 2 φ2 sin 2 θ + r 2 ψ2 cos 2 θ) = 0. ( 37
)
We therefore have to solve this set of coupled differential equations for the functions r(λ), θ(λ) and φ(λ). The functions t(λ) and ψ(λ) may then be obtained from (31) and (32) after substituting r(λ), θ(λ) and φ(λ). Finally the null condition must be imposed to ensure we have a null geodesic. In practice it was more straightforward to solve directly (32) and the other three geodesic equations (along with making sure the null condition is satisfied), rather than using (32) to eliminate ψ in the equations for r, θ and φ. We will solve these equations using an ansatz involving a power series in the affine parameter λ. In addition we subject the geodesic to the initial conditions required for constructing a Gaussian null coordinate system. These conditions were applied earlier for the single BMPV case, but we repeat them here:

r(λ)| λ=0 = 0, θ| H0 = 0, J ψ = 0. ( 38
)
As we have broken the φ symmetry we do not have a conserved angular momentum for the geodesic in that direction.

Arbitrarily close to the BMPV black hole, however, the spacetime is effectively that of the single BMPV solution discussed earlier and we can still use the J φ = 0 condition on the geodesic. These conditions ensure that λ = 0 coincides with the horizon and that the Gaussian null condition of U • ∂/∂ x i = 0 on the horizon is satisfied, where the S 3 coordinates x i are Θ, Φ and Ψ.

Solving the geodesic equations

Before we give the expressions for the power series ansatz, we will first give an idea of what form the ansatz must take. We know that to leading order in λ the geodesic equations must be solved by the expressions for r(λ), θ(λ), φ(λ) and ψ(λ) that we found for the single BMPV black hole. Therefore, we will use these expressions to write the forms of these functions in a general way, with the leading order behaviour as for the single BMPV case plus a subleading correction term:

φ = Φ + w 0 log λ + w 2 λ + δφ ψ = Ψ + y 0 log λ + y 2 λ + δψ r = a 1 λ 1/2 + a 3 λ 3/2 + δr θ = Θ + δθ. ( 39
)
So we consider the corrections δφ and δψ to be terms of some order higher than O(λ). The correction δr is of some order higher than O(λ 3/2 ) and the correction to θ is simply some non-constant order in λ. The constants w 0 , y 0 , w 2 , y 2 , a 1 and a 3 are precisely the coefficients of the relevant powers of λ in ( 15) and (17):

w 0 = y 0 = - J 2∆ w 2 = y 2 = J(10µ 3 1 -J 2 ) 4∆ 2 µ 2 1 a 1 = ∆ µ 1 1/2 a 3 = (J 2 + 2µ 3 1 ) 4µ 5/2 1 √ ∆ , ( 40
)
where we have written the BMPV mass parameter as µ 1 rather than µ, in keeping with the form of H given in (27) for the multi-black hole metric. We substitute these expressions into the geodesic equation for θ, and expand for small λ, keeping only the terms of lowest order in λ. We rewrite the trigonometric functions of φ such as sin(φ) using normal trigonometric identities and then we perform a Taylor expansion on any terms that do not involve log λ. After all this we find that, at lowest order in the expansion, we have a single uncoupled second order ordinary differential equation for δθ:

δθ -2 h 1 a 1 + 1 4µ 1 a 3 1 h 1 + a 3 1 µ 1 h 1 w 2 0 cos Θ (cos Φ sin(w 0 log λ) + sin Φ cos(w 0 log λ)) λ -1/2 = 0. ( 41
)
This is solved by

δθ(λ) = c 1 + c 2 λ + 4c 3 4 sin(Φ + w 0 log λ)w 2 0 -3 sin(Φ + w 0 log λ) + 8 cos(Φ + w 0 log λ)w 0 λ 3/2 9 + 40w 2 0 + 16w 4 0 , ( 42
)
with c 1 and c 2 undetermined constants, and

c 3 = 2 h 1 a 1 + a 3 1 µ 1 h 1 1 4 + w 2 0 cos Θ. ( 43
)
Thus we have found θ = Θ + δθ up to order λ 3/2 . Clearly we can absorb the constant c 1 into Θ, and the constant c 2 may be fixed by the initial condition on the geodesic that θ| H0 = 0. From (42) we see that terms such as sin(mw 0 log λ) and cos(mw 0 log λ), with positive integer m, arise in the expansion of θ. Likewise, such terms will arise at the same order in the expansions of φ and ψ when we solve those geodesic equations. Similarly this log λ dependence will appear at order λ 2 in r, because the r expansion begins at a higher order than the expansions for the angles.

It should be noted that this log λ dependence will cause lack of smoothness in the metric at the horizon (where λ = 0). This is in addition to the lack of smoothness arising from half-integer powers of λ in the metric components in the Gaussian null coordinate system, which is the cause of the low degree of differentiability in the static case.

A word about notation: we will write O(F (log λ)λ n ) to denote a term that is morally of order λ n (i.e. there is an overall factor of λ n ) although it also contains terms of the form sin(mw 0 log λ) or cos(mw 0 log λ). Thus F (log λ) represents a bounded periodic function of log λ. These terms are badly behaved at the horizon due to the infinite number of periods.

Series ansatz

Now that we have determined that the unusual terms involving trigonometric functions of log λ will appear in the functions for which we are solving, we must include them in our power series ansatz. We now give the general form of this ansatz. We can, of course, use the expansions for the single BMPV case in ( 15) and ( 17) to write down the first few terms just as we did in the previous section. The general form for the ansatz for r is

r = a 1 λ 1/2 + a 3 λ 3/2 + ∞ n=5 a n (Θ, Φ)λ n/2 (44) + ∞ k=4 λ k/2 k-3 m=1 a (m) k (Θ, Φ) sin(mw 0 log λ) + ã(m) k (Θ, Φ) cos(mw 0 log λ) . ( 45
) (46)
After solving the geodesic equations we have found that a n = 0 if n is an even integer, and a

(m) k = ã(m) k = 0
if k and m are both even or both odd.

For the coordinates φ and ψ, the leading term in the ansatz is the logarithmic behaviour we found earlier for the single BMPV black hole, while the leading order term in θ is a constant. Thus we have

θ = Θ + b 2 λ + ∞ N =4 b N (Θ, Φ)λ N/2 (47) + ∞ k=3 λ k/2 k-2 M =1 b (M ) k (Θ, Φ) sin(Mw 0 log λ) + b(M) k (Θ, Φ) cos(Mw 0 log λ) (48) φ = Φ + w 0 log λ + w 2 λ + ∞ N =4 w N (Θ, Φ)λ N/2 (49) + ∞ k=3 λ k/2 k-2 M =1 w (M ) k (Θ, Φ) sin(Mw 0 log λ) + w(M) k (Θ, Φ) cos(Mw 0 log λ) (50) ψ = Ψ + y 0 log λ + y 2 λ + ∞ N =4 y N (Θ, Φ)λ N/2 (51) + ∞ k=3 λ k/2 k-2 M =1 y (M ) k (Θ, Φ) sin(Mw 0 log λ) + ỹ(M) k (Θ, Φ) cos(Mw 0 log λ) . ( 52
)
Again, after solving the geodesic equations, we have that b N = w N = y N = 0 if N is an odd integer, and b

(M ) k = b(M) k = w (M ) k = w(M) k = y (M ) k = ỹ(M) k = 0 if M is
odd and k is even or vice versa. It turns out that we only need terms up to order λ 2 in r(λ) and order λ 3/2 in the angles to determine the degree of differentiability of the metric. For the sake of brevity we have not made explicit the fact that all these coefficients will, in general, depend on the parameters of the solution, that is

a i = a i (µ 1 , J, h n , Φ, Θ) (53)
and likewise for b i , y i and w i . We will continue to leave this dependence on the parameters implicit.

Coefficients

Using computer algebra, we expand each geodesic equation as a series in λ. Then, for each order in λ, we solve for the coefficients in the ansatz. Although we already know what the first few terms are from the single BMPV case, it is a useful check of our procedure to ensure that these terms are reproduced. Therefore we find y 0 in terms of a 1 by solving the ψ geodesic equation at order λ -2 . This is the first non-trivial order (the leading order behaviour of ψ is log λ and the geodesic equations are, of course, second order). Similarly we can find w 0 by solving the φ equation at the same order to find

w 0 = y 0 = - J 2∆ . ( 54
)
as expected from the single BMPV case. The θ equation has non-zero terms at order λ -2 , but the equation has an overall factor of (w 0 -y 0 ) which vanishes as w 0 = y 0 . The r equation at leading order (O(λ -3/2 )) leads to a quartic polynomial in a 1 , and so there are 4 solutions. We reject the two complex solutions as unphysical, and, since we want r > 0 for λ > 0, of the two real solutions we take the positive one:

a 1 = ∆ µ 1 1/2 (55)
which agrees with that found earlier for the single BMPV black hole. The coefficients w 2 , y 2 and b 2 are unfixed by the equations, being set by the initial conditions on the geodesic. In terms of the coefficients this means b 2 = 0

w 2 = y 2 = Jh 0 10µ 3 1 -J 2 4∆ 2 µ 2 1 . ( 56
)
Comparing this with (17) we see this agrees with the O(λ) terms there, upon setting h 0 = 1 to recover the single black hole case. The next few non-vanishing coefficients in the expansion of r are

a 3 = h 0 2µ 3 1 + J 2 4µ 5/2 1 √ ∆ a (1) 4 (Θ, Φ) = - sin(Θ)h 1 ∆ 2J (J cos(Φ) + 2 sin(Φ)∆) -5 cos(Φ)µ 3 1 25µ 6 1 -6J 2 µ 3 1 ã(1) 4 (Θ, Φ) = - sin(Θ)h 1 ∆ 2J (J sin(Φ) -2 cos(Φ)∆) -5 sin(Φ)µ 3 1 25µ 6 1 -6J 2 µ 3 1 . ( 57
)
The next non-trivial coefficients in the expansion of θ are b

(1) 3 (Θ, Φ) = - 3 cos(Θ)h 1 ∆ 3/2 J (J cos(Φ) + sin(Φ)∆) -3 cos(Φ)µ 3 1 µ 5/2 1 (9µ 3 1 -2J 2 ) b(1) 3 (Θ, Φ) = 3 cos(Θ)h 1 ∆ 3/2 3 sin(Φ)µ 3 1 + J (cos(Φ)∆ -J sin(Φ)) µ 5/2 1 (9µ 3 1 -2J 2 ) . ( 58
)
The coefficients b

(1)

3 and b(1) 3 determined above may be used to rewrite the order λ 3/2 term in the θ expansion as functions of sin(Φ + w 0 log λ) and cos(Φ + w 0 log λ). After doing this, we recover the order λ 3/2 term given in (42) which was found by directly solving the θ equation. This is a useful consistency check. Note that the coefficient a 5 (Θ, Φ, Ψ) is not fixed by the r geodesic equation, but by the null condition. The coefficients of the φ and ψ expansions at order λ 3/2 are given in the appendix. We will not list the higher order coefficients, as they are somewhat long and unenlightening.

Lack of smoothness

It is now possible to expand the norm of the Killing vector K = ∂ ψ in λ and thus determine the smoothness of the metric, as discussed earlier. The coefficients listed above are all that is necessary for this purpose. In fact, we only need w 0 , a 1 , b [START_REF] Hartle | Solutions of the Einstein-Maxwell equations with many black holes[END_REF] 3 and b(1) 3 . As can be seen from ( 25), the norm of K is

K 2 = -H(r, θ, φ) -2 ω 2 ψ + H(r, θ, φ)r 2 cos 2 θ. ( 59
)
Using the functions r(λ), θ(λ) and φ(λ) that we have just found, we can expand this in λ to get

K 2 = cos 2 Θµ 1 - J 2 cos 4 Θ 4µ 2 1 + cos 2 Θh 0 (J 2 cos 2 Θ + 2µ 3 1 )∆ 2µ 4 1 λ + O(F (log λ)λ 3/2 ), (60)
where we have used the notation discussed at the end of section 4.2. The order O(F (log λ)λ 3/2 ) term in (60) is

J 2 cos 2 (Θ) sin(Θ)h 1 ∆ 3/2 5J 2 cos 2 (Θ) -(9 cos(2Θ) + 11)µ 3 1 µ 9/2 1 (2J 2 -9µ 3 1 ) sin Φ - J log λ 2∆ λ 3/2 + 3J cos 2 (Θ) sin(Θ)h 1 J 2 cos 2 (Θ) -2µ 3 1 ∆ 5/2 µ 9/2 1 (9µ 3 1 -2J 2 ) cos Φ - J log λ 2∆ λ 3/2 . ( 61
)
Thus K 2 is not twice continuously differentiable. As argued earlier, lack of smoothness in the norm of this Killing vector field implies the metric is not smooth. In particular we now see that the metric is not C 2 . We can recover the result for a line of static black holes [START_REF] Candlish | On the smoothness of static multi-black hole solutions of higher-dimensional Einstein-Maxwell theory[END_REF] by simply setting J = 0. Then the dependence on log λ vanishes, and indeed the coefficients of the above terms also vanish. Then the next troublesome term 2 in the expansion of K 2 will be of order λ 5/2 (with, of course, no log λ dependence), and so the result that a multi-centre static metric is not C 3 is recovered.

Gaussian null metric

We have seen that the metric at the horizon is not C 2 . We will now show that it is C 1 there by transforming the metric to Gaussian null coordinates. This proceeds in exactly the same manner as for the single BMPV solution. We use the functions r(λ), θ(λ), φ(λ) and ψ(λ) to perform the transformation. As before the coordinates in the Gaussian null system are v, λ, Θ, Φ and Ψ. The coordinates Θ, Φ and Ψ are simply the constant terms in the expansions of θ, φ and ψ given in (47). Just as before we define the coordinate v according to

v ≡ t + (H 2 + ω)dλ, (62)
where, to perform the integral, we must of course substitute the functions r(λ), θ(λ) and φ(λ) into H and ω. After doing this we can then write

dt = dv -d (H 2 + ω)dλ . ( 63
)
Clearly this will involve dλ, dΘ and dΦ components. Next, we can use the functions r(λ), θ(λ), φ(λ) and ψ(λ) directly to find the appropriate transformations. For example:

dr = ∂ λ r(λ, Θ, Φ)dλ + ∂ Θ r(λ, Θ, Φ)dΘ + ∂ Φ r(λ, Θ, Φ)dΦ ( 64
)
where we must of course keep in mind that the constants Θ and Φ are our new coordinates.

In practice we cannot take the expansions in section 4.3 to infinite order; we truncate the expansions at the order required to find a C 1 metric (as we now know the metric is not C 2 ). As we are truncating our series expansion ansatz, the metric will only take the Gaussian null form up to a certain order. We assume that inclusion of all higher order terms in the ansatz will allow us to find a truly Gaussian null form for the metric. For this reason we refer to the metric as being in "nearly Gaussian null" form.

The components of the metric in the Gaussian null coordinates are given in the appendix. We have already shown, with the analysis of the norm of the ∂ ψ Killing vector in section 4.5 that the metric cannot be C 2 at the horizon. From the form of the metric given in the appendix we see that all the metric components are C 1 . Therefore we have shown that the metric is C 1 but is not C 2 at the horizon. This is the main result of this paper.

Maxwell Field

The one-form potential for the Maxwell field in this solution is given by

A = √ 3 2 H -1 (dt + ω). ( 65
)
In the Gaussian null coordinate system this will obviously have the general form

A = A v dv + A λ dλ + A Θ dΘ + A Φ dΦ + A Ψ dΨ. ( 66
)
Performing the coordinate transformation using the expansions we have determined for r, θ, φ and ψ, as well as (62), we find

A v = √ 3∆ 2µ 2 1 λ + 3 √ 3h 0 J 2 -2µ 3 1 4µ 4 1 λ 2 + O(F (log λ)λ 5/2 ) A λ = - √ 3µ 2 1 2∆ λ -1 + 3 √ 3h 0 J 2 -2µ 3 1 4∆ 2 + 4 √ 3 sin(Θ)h 1 J 2 -5µ 3 1 √ ∆ √ µ 1 (25µ 3 1 -6J 2 ) F 1 (Φ, log λ)λ 1/2 + 4 √ 3J sin(Θ)h 1 ∆ 3/2 √ µ 1 (25µ 3 1 -6J 2 ) F 2 (Φ, log λ)λ 1/2 + O(F (log λ)λ) A Θ = O(F (log λ)λ 3/2 ) A Φ = O(F (log λ)λ 3/2 ) A Ψ = √ 3J cos 2 (Θ) 4µ 1 - √ 3J cos 2 (Θ)h 0 ∆ 4µ 3 1 λ + O(F (log λ)λ 3/2 ). (67)
The order λ -1 and λ 0 terms in A λ are pure gauge, so the Maxwell field strength is continuous at the horizon but not differentiable as F λΘ = O(F (log λ)λ 1/2 . As the metric is not C 2 we might wonder if Einstein's equation makes sense at the horizon. The fact that the Maxwell field is continuous, however, and the metric is C 1 , means that the stress-energy tensor is continuous at the horizon. Einstein's equation therefore holds at the horizon by continuity.

Parallel propagation of the Riemann tensor

The fact that the metric is not C 2 implies the existence of a curvature singularity at the horizon. We will now show that there is a parallely propagated curvature singularity there. We choose a demonstrative component of the Riemann tensor in an orthonormal frame for this purpose. The leading order behaviour of the R λΘλΘ component of the Riemann tensor for small λ is

R λΘλΘ = - 1 2 ∂ 2 λ g ΘΘ + . . . ( 68
)
where the ellipsis denotes subleading terms. Referring back to (75) we see that this will go as O(F (log λ)λ -1/2 ) and is therefore divergent at the horizon. As the metric is C 1 , no other Riemann tensor components will be more divergent than this. Now we construct an orthonormal basis on the horizon, which we can parallely propagate along the null geodesic with tangent vector U = ∂/∂λ. First, we set e 0 = dλ and e 1 = dv, so we have U = e 1 = e 0 . Finally we take e 2 = g 1/2

ΘΘ dΘ so we have e 0 • e 1 = 1, e 0 • e 2 = 0, (e 0 ) 2 = (e 1 ) 2 = 0 and (e 2 ) 2 = 1 on the horizon. This frame is extended off the horizon by demanding that the basis one-forms are parallely propagated along the null geodesic: U • ∇e 1,2 = 0, which preserves the orthogonality relations. Note that while the basis one-forms given above are valid only on the horizon, e 0 = U everywhere.

In this basis then, we can find a particular component of the Riemann tensor

R 0202 = e λ 0 e Θ 2 e λ 0 e Θ 2 R λΘλΘ + . . . ( 69
)
which gives

R 0202 = g -1 ΘΘ R λΘλΘ = - 1 2µ 1 ∂ 2 λ g ΘΘ + . . . ( 70
)
at leading order. This is clearly divergent as λ → 0 and therefore there is a parallely propagated curvature singularity at the horizon.

Discussion

In this paper we have demonstrated that the supersymmetric rotating five dimensional black hole solution known as the BMPV solution does not exhibit a smooth event horizon in the presence of other extremal static black holes.

For the axial configuration discussed here the horizon has been demonstrated to be C 1 but not C 2 . This is worse than the case for a line of only static black holes, which has a C 2 (but not C 3 ) horizon. The physical interpretation for the lack of smoothness in certain multi-black hole solutions depends on the properties of the solution. In [START_REF] Brill | Testing cosmic censorship with black hole collisions[END_REF] and [START_REF] Chruściel | Nonsmoothness of event horizons of Robinson-Trautman black holes[END_REF] non-smooth horizons seem to be related to the existence of non-smooth gravitational and electromagnetic radiation. In the present case the lack of smoothness would seem to be due to breaking one of the rotational symmetries of the BMPV black hole at the horizon. The physical reason for the lack of smoothness in the static case investigated in [START_REF] Candlish | On the smoothness of static multi-black hole solutions of higher-dimensional Einstein-Maxwell theory[END_REF] remains obscure.

Other non-smooth solutions

A similar situation involving breaking of a rotational symmetry occurs in the solution presented in [START_REF] Bena | Sliding rings and spinning holes[END_REF], which comprises a supersymmetric black ring, along with a BMPV black hole vertically displaced from it with respect to the plane of the ring. The BMPV black hole is aligned with the centre of the ring in the sense that the ψ symmetry is not broken, but the vertical displacement breaks the φ symmetry. In [START_REF] Bena | Sliding rings and spinning holes[END_REF] we indeed see that the rotation one form contains terms involving, for example, cos φ.

The breaking of one of the U (1) isometries of both the BMPV black hole and the black ring implies that this solution will be non-smooth. From the results found for the multi-BMPV metric, we expect problems to arise because of the fact that φ ∼ log λ close to the horizon of the BMPV black hole. In fact in [START_REF] Elvang | A Supersymmetric black ring[END_REF] it can be seen that to continue the supersymmetric black ring metric through the horizon requires transformations of the form

dφ = dφ - C r dr, dψ = dψ - C r dr ( 71
)
where r is a radial coordinate that takes the value r = 0 on the black ring horizon. So for r ∼ λ n where n is some power, the angles must behave as φ ∼ log λ and ψ ∼ log λ, just as we have seen for the BMPV black hole. From this we therefore also expect lack of smoothness at the horizon of the black ring in the solution given in [START_REF] Bena | Sliding rings and spinning holes[END_REF].

It is not surprising that this solution is not smooth, given the results we have found, as it breaks the rotational symmetries at the horizon for both the BMPV black hole and the black ring. (79)

The components g λλ , g λΘ , g λΦ and g λΨ are vanishing in a true Gaussian null coordinate system. The presence of non-zero terms here is purely an artefact of truncating the series ansatz.

1 + sin 2 (Θ)h 0 2µ 3 1 + J 2 sin 2 ( 2 1 2µ 9 / 2 1 (9µ 3 1 -2J 2 )F 1 ( 2 + 1 ∆ 5 F 2 ( 1 + J 2 1 λ+ J 2 1 2µ 9 / 2 1 (9µ 3 1 -2J 2 )F 1 ( 2 + 1 ∆ 5 F 2 ( 1 + cos 2 1 λ+ J 2 1 µ 9 / 2 1 (2J 2 -9µ 3 1 )F 1 2 + 1 ∆ 5 1 -2J 2 )F 2 (F 1 (F 2 (

 1212219212121521212192121215212121921121512212 andg ΦΦ = sin 2 (Θ)µ 1 -J 2 sin 4 (Θ) 4µ 2 sin 3 (Θ)h 1 ∆ 3/2 J 2 (5 cos(2Θ) + 7) -2(9 cos(2Θ) + 13)µ 3 Φ, log λ)λ 3/3J sin 3 (Θ)h 1 J 2 (cos(2Θ) + 3) -8µ 3 Φ, log λ)λ 3/2 + O(F (log λ)λ 2 ) g ΦΨ = -J 2 cos 2 (Θ) sin 2 (Θ) 4µ 2 cos 2 (Θ) sin 2 (Θ)h 0 ∆ 2µ 4 cos 2 (Θ) sin(Θ)h 1 ∆ 3/2 J 2 (5 cos(2Θ) + 1) -6(3 cos(2Θ) + 1)µ 3 Φ, log λ)λ 3/3J cos 2 (Θ) sin(Θ)h 1 J 2 cos 2 (Θ) -3µ 3 Φ, log λ)λ 3/2 + O(F (log λ)λ 2 ) (77)and finallyg ΨΨ = cos 2 (Θ)µ 1 -J 2 cos 4 (Θ) 4µ 2 (Θ)h 0 2µ 3 1 + J 2 cos 2 (Θ) ∆ 2µ 4 cos 2 (Θ) sin(Θ)h 1 ∆ 3/2 5J 2 cos 2 (Θ) -(9 cos(2Θ) + 11)µ 3 (Φ, log λ)λ 3/3J cos 2 (Θ) sin(Θ)h 1 J 2 cos 2 (Θ) -2µ 3 Φ, log λ)λ 3/2 + O(F (log λ)λ 2 ) Φ, log λ) = sin Φ -J log λ 2∆ Φ, log λ) = cos Φ -J log λ 2∆ .

Five dimensional static multi-black holes on a Gibbons-Hawking space were discussed in[START_REF] Kimura | Analyticity of Event Horizons of Five-Dimensional Multi-Black Holes with Non-Trivial Asymptotic Structure[END_REF], where the horizons were shown to be analytic. This is because the solutions are written in terms of harmonic functions on R 3 , and so the dependence of the radial coordinate on an affine parameter along a null geodesic is the same as that in four dimensions. Ensuring each connected component of the horizon has S 3 topology (or a lens space S 3 /Zn) requires placing the black holes on multi-centre Eguchi-Hanson space or multi-Taub-NUT and as such they are no longer asymptotically flat.

In the static case the lack of smoothness is solely because the radial coordinate in the Gaussian null system, λ, appears with half-integer powers (just as it does here). Clearly any term of order λ n where n is an even integer is smooth.

Acknowledgements

The author is supported by the University of Nottingham, and would like to thank H.S. Reall for suggesting the project and many helpful discussions. I am grateful to Masashi Kimura for pointing out the work in [4]. I would also like to thank DAMTP for their hospitality.

A Single BMPV metric in Gaussian null coordinates

The metric for a single BMPV black hole in the Gaussian null coordinate system is given below. The metric components are written as a series expansion in λ around λ = 0. The leading order terms are given.

B Expansion coefficients of φ and ψ at order λ 3/2

The O(λ 3/2 ) coefficients for the φ expansion are

At this order the coefficients for ψ are

(74)

C Metric components

We now give the form of the metric for the multi-centre spacetime (single BMPV and additional static black holes) in our "nearly Gaussian null" coordinate system:

(75)


Bartolomé Coll
email: bartolome.coll@uv.es

Joan Josep Ferrando
email: joan.ferrando@uv.es

Juan Antonio Morales-Lladosa

Positioning systems in Minkowski space-time: from emission to inertial coordinates

Keywords: numbers: 04.20.-q, 45.20.Dd, 04.20.Cv, 95.10.Jk

The coordinate transformation between emission coordinates and inertial coordinates in Minkowski space-time is obtained for arbitrary configurations of the emitters. It appears that a positioning system always generates two different coordinate domains, namely, the front and the back emission coordinate domains. For both domains, the corresponding covariant expression of the transformation is explicitly given in terms of the emitter world-lines. This task requires the notion of orientation of a emitter configuration. The orientation is shown to be computable from the emission coordinates for the users of a 'central' region of the front emission coordinate domain. Other space-time regions associated with the emission coordinates are also outlined.

Introduction

A relativistic positioning system consists of a set of four clocks A (A = 1, 2, 3, 4) broadcasting their respective proper times τ A by means of electromagnetic signals. Then, every event reached by the signals is naturally labelled by the four times {τ A }: the emission coordinates of this event. Elsewhere [START_REF] Coll | Relativistic Positioning Systems: current status[END_REF], we have presented a brief report on relativistic positioning and related issues, providing a background with current references on the subject.

A user of a positioning system that receives the four times {τ A } knows his own coordinates in the emission system. Then, if he wants to know his position in another coordinate system, he must obtain the transformation between both coordinate systems. Thus, we must solve the following important problem in relativistic positioning. Suppose that the world-lines of the emitters γ A (τ A ) are known in a coordinate system {x α }: can the user obtain his coordinates in this system if he knows his emission coordinates {τ A }? Or, slightly more general, can the coordinate transformation x α = κ α (τ A ) be obtained?

The main goal of this paper is to solve this question in Minkowski space-time or, more explicitly, to obtain the coordinate transformation between emission and inertial coordinates for arbitrary world-lines of the emitters. In a two-dimensional approach to relativistic positioning systems this query is rather simple. Indeed, in this case the knowledge of the emitter's world-lines in a chosen null coordinate system gives the coordinate transformation between this null system and the emission coordinates at once [START_REF] Coll | Two-dimensional approach to relativistic positioning systems[END_REF][START_REF] Coll | Positioning with stationary emitters in a twodimensional space-time[END_REF]. This fact facilitates an analytical study of the positioning systems defined by inertial emitters in Minkowski plane [START_REF] Coll | Two-dimensional approach to relativistic positioning systems[END_REF], and those defined by stationary emitters in both Minkowski and Schwarzschild planes [START_REF] Coll | Positioning with stationary emitters in a twodimensional space-time[END_REF].

The general properties of the emission coordinates for the four-dimensional case have been analyzed in [START_REF] Coll | Relativistic positioning systems: the emission coordinates[END_REF]. Nevertheless, in this generic case it is difficult to solve the above stated problem. The study of specific three-dimensional situations [START_REF] Pozo | Constructions in 3D and in 4D Relativistic Coordinates[END_REF] has shed light on the complex geometry of the domains and codomains of the emission coordinates. On the other hand, the transformation between emission and inertial coordinates has recently been obtained for emitters following particular inertial motions in flat space-time, and also considering the immediate vicinity of a Fermi observer in the Schwarzschild geometry [START_REF] Bini | Emission versus Fermi coordinates: applications to relativistic positioning systems[END_REF].

Here we present the solution to this problem for a generic configuration of the emitters in Minkowski space-time. We show that two emission domains exist which are called the front emission coordinate domain and the back emission coordinate domain, and we give the coordinate transformation for each one. The transformation x α = κ α (τ A ) between inertial and emission coordinates is given in a covariant way in terms of the world-lines γ A (τ A ) of the emitters. The compact covariant expression of our result is a powerful tool for use in subsequent applications: to obtain the coordinate transformation for specific configurations of the emitters or, under perturbation methods, modeling more realistic gravitational situations.

The paper is structured as follows. In section 2, we pose the problem to be solved, defining the emission regions and the main relations governing the coordinate transformation: the null propagation conditions and the emission conditions. By choosing one of the emitters arbitrarily as reference emitter, in section 3, we show that the null propagation equations of a positioning system are equivalent to a rank-three linear system and a sole quadratic equation. Then, section 4 is devoted to obtaining the general solution of the linear system, and in section 5 we impose the remaining quadratic equation and we obtain a general form for both solutions. In section 6, we apply the emission conditions which guarantee that the solutions are physically interpretable as emission solutions. In section 7 we define the orientation of the positioning system with respect to an event, a concept which allows us to give the main result of this paper in compact form, namely, the explicit expression of the coordinate transformation between emission and inertial coordinates. The computational and/or observational determination of the orientation of the positioning system is analyzed in section 8. Section 9 deals with the analysis of our results and comments about ongoing work in progress and on further practical applications. Finally, an appendix is devoted to presenting a technical proof of the results given in section 4.

A short communication on this work has been presented at the Spanish Relativity meeting ERE-2008 [START_REF] Coll | Emission coordinates in Minkowski space-time Physics and Mathematics of Gravitation[END_REF].

Statement of the problem. Emission regions and emission relations

The emission region R of a positioning system

Let us consider a positioning system in Minkowski space-time M 4 , and let γ A (τ A ), A = 1, . . . , 4, be the world-lines of its four distinct emitters A (clocks) watching their proper time τ A . These four times, broadcast by means of electromagnetic signals, will reach some region, say the emission region R, of events of M 4 . In the interior of R, the four times {τ A } converging at every event define generically the so called emission coordinates [START_REF] Coll | Two-dimensional approach to relativistic positioning systems[END_REF][START_REF] Coll | Positioning with stationary emitters in a twodimensional space-time[END_REF][START_REF] Coll | Relativistic positioning systems: the emission coordinates[END_REF][START_REF] Coll | Elements for a theory of relativistic coordinate systems[END_REF][START_REF] Coll | Relativistic Positioning Systems A Century of Relativity Physics Proc. of the XXVIII Spanish Relativity Meeting ERE-2005[END_REF].

It is clear that an event P of M 4 belongs to the emission region R if and only if there exists a past directed null geodesic from P to every emitter γ A (τ A ) for some value of τ A . For future discussions, we need to make the ingredients of this assertion more explicit.

Let us denote by x the position vector of P with respect to the origin O of some inertial chart {x α }, x ≡ OP, and by γ A the position vector of the emitters with respect to this chart, γ A ≡ Oγ A (τ A ). Then, in order for P to belong to R, the four vectors

m A ≡ x -γ A , (1)
which represent the trajectories followed by the electromagnetic signals in vacuum issued from the emitters A (see Fig. 1a), must verify the null propagation conditions L :

L : (m A ) 2 = 0 , ∀A .
Furthermore, these four vectors have to be future-pointing or, in other words, must verify the emission conditions E :

E : u • m A < 0 , ∀A ,
where u is any given, everywhere non vanishing, time-like vector field defining the arrow of time and 2 , = ±1, is the metric signature.

The events x where the emission condition E holds must be receivers. But it is worth noting that the null propagation conditions L also apply in the case of 'active' events x, able to send or to reflect null signals to one or more of the emitters. Such other location systems, as the physical realizations of coordinate systems are called here, have been dealt with elsewhere [START_REF] Ehlers | The geometry of free fall and light propagation[END_REF][START_REF] Coll | Epistemic Relativity GraviMAS FEST workshop in honour of Lluís Mas[END_REF][START_REF] Perlick | On the Radar Method in General-Relativistic space-times Lasers, Clocks, and Drag-Free Control[END_REF] but will not be considered here.

Nevertheless, in obtaining our coordinate transformations, we will need to consider, besides the emission condition E, its 'causally dual' reception conditions R :

R : u • m A > 0 , ∀A ,
both condensed in the emission-reception conditions E-R :

E-R : m A • m B < 0 , ∀A , B ,
as it is easy to argue. Now, the above assertion about P and R may be stated as follows: an event P belongs to the emission region R of a positioning system of emitters γ A (τ A ) if, and only if, its position vector x in some inertial chart verifies the null and emission conditions L and E, respectively, for some values of the proper times τ A of the emitters.

The characteristic emission function Θ of a positioning system and the emission coordinate region C of the space-time

It then turns out that a positioning system may be considered as a device Θ that physically associates, to every event of the emission region R, a set of four times {τ A }.

From a formal point of view, this device is nothing but an application Θ : R -→ T , henceforth called the characteristic emission function of the positioning system, that, to every event x of the emission region R of the space-time, associates four times τ A of the grid T of the τ s, Θ :

x -→ (τ A ) = Θ(x).
The grid T is nothing but the Cartesian product T ≡ 4 × {τ } ≈ R 4 of the spaces (real lines) of the variables τ (for details on the concept and role of the grid, see for example [START_REF] Coll | Two-dimensional approach to relativistic positioning systems[END_REF][START_REF] Coll | Positioning with stationary emitters in a twodimensional space-time[END_REF][START_REF] Pozo | Constructions in 3D and in 4D Relativistic Coordinates[END_REF]).

In this grid T , the image Θ(R) of the emission region R by the characteristic emission function Θ, is called the emission co-region of the positioning system, and is denoted by Θ R, Θ R = Θ(R). The points of this region of the grid T are the quadruplets of times that can really be received in the space-time, so that they are related to the space-time events at which they are measured. In this sense, this emission co-region Θ R is the sole region of the grid T which possesses a physical meaning (the other quadruplets of T are a convenient mathematical completion of the emission co-region but with no relation to space-time events).

Obviously, if the world-lines of the four emitters are sufficiently smooth and broadcast their proper time continuously, the emission co-region Θ R ⊂ T is connected and, because of the regularity of the light cones in Minkowski space-time, R is connected too. But this property does not guarantee that the emission function Θ, τ A = Θ A (x), is invertible in R.

For Θ to be invertible, the gradients dτ A , normal to the hypersurfaces τ A = constant, have to be well defined and linearly independent. But, because in Minkowski space-time our light cones are everywhere differentiable up to on their vertices, i.e. on the emitter world-lines γ A , the gradients dτ A are well defined everywhere in R up to on R ∩ ∪ A γ A (τ A ) . Then, on the region R -∪ A γ A (τ A ) , because they are collinear to the above null vectors m A , the linear independence of the dτ A may be expressed by the coordinate condition C:

C : m 1 ∧ m 2 ∧ m 3 ∧ m 4 = 0 .
This condition C is equivalent to say that j Θ (x) = 0, where j Θ (x) is the determinant of the Jacobian matrix J Θ (x) of Θ, so that the locus where the dτ A are linearly dependent is the hypersurface J in R of equation J ≡ {j Θ (x) = 0}. We shall call the regions D ≡ J ∪ (∪ A γ A (τ A )) and C ≡ R -D the emission degenerate region, and the emission coordinate region respectively, R = C ∪ D.

It is to be noted that, the condition C being satisfied on its events, C is an open set. For the same reason, Θ is invertible in C, so that all its events may be locally labeled by the coordinates {τ A }. Nevertheless, this does not means necessarily that C be a coordinate domain of a local chart (C, Θ), because the condition C assures only the local invertibility of Θ. In fact, we will see that C is not a coordinate domain, but the union of two coordinate domains; this is why we have called C the emission coordinate region. The region in the grid T where the characteristic emission function Θ, τ A = Θ A (x), is locally invertible is the region Θ C ≡ Θ(C), and will be called the emission coordinate co-region of the grid T .

The main relations of a positioning system

When the world-lines of the emitters are known, the main relations of a positioning system are the set {L, E} of the null propagation conditions L and the emission conditions E.

In terms of the position vectors γ A of these world-lines γ A (τ A ), they give rise, by (1), to the null propagation equations

L : (x -γ A ) • (x -γ A ) = 0 , ∀A . ( 2
)
and to the emission inequalities

E : u • (x -γ A ) < 0 , ∀A , ( 3
)
where u is any given, everywhere non vanishing, future-pointing time-like vector.

To invert the function Θ in the emission coordinate co-region Θ C is to solve the main relations ( 2), (3) in x, x = κ(τ A ), for values of the τ A 's verifying the coordinate condition C or, by (1), the coordinate inequality:

C : (x -γ 1 ) ∧ (x -γ 2 ) ∧ (x -γ 3 ) ∧ (x -γ 4 ) = 0 . ( 4
)
These solutions x = κ(τ A ) may alternatively be read as the position vectors of the events whose past light cone cuts the emitter world-lines γ A (τ A ) at their times τ A . The components x α = κ α (τ A ) of x then define the coordinate transformation between the emission coordinates {τ A } and the inertial ones {x α }.

The main object of this paper is to obtain this coordinate transformation.

The null propagation equations L

When {τ A } are the emission coordinates of the event x, the emitters are at the events {γ A (τ A )}. These four events define the internal configuration of the emitters for the event x.

One can try to solve the null propagation equations (2) in x, x = κ(τ A ) directly, but it is better to first carefully separate the ingredients intrinsically related to the configuration of the four emitters A, which are independent of the inertial chart {x α }, from the ingredients related to this chart, which is independent of the configuration of the emitters.

For this purpose, we shall arbitrarily choose one of the emitters, say A = 4, as reference emitter and we shall search the solution x under the form: where y is the solution to the null propagation equations ( 2) when the origin is chosen at the position γ 4 of the emitter 4 (see Fig. 1b). Thus, y is submitted to

x = γ 4 + y , ( 5
) x t O P m g (t ) A A A A g o o A x t O P m g (t )
L 4 : (y -e A ) • (y -e A ) = 0 , ∀A , (6)
where

e A = γ A -γ 4 . ( 7
)
Now, splitting these relations for A = 4 and A = a = 1, 2, 3, we have

e 4 = 0 , e A = e a ( 8
)
and, for the null propagation equations ( 6),

y 2 = 0 , A = 4 , y 2 -2 e a • y + (e a ) 2 = 0 , A = a . ( 9
)
Noting that e a are the relative position vectors of the referred emitter A = a with respect to the reference emitter A = 4, and that the half of their squares, represent the world functions [START_REF] Synge | Relativity: The General Theory[END_REF] of the referred and reference emitters, we have:

Lemma 1
The null propagation equations (6) are equivalent to the main linear system

e a • y = Ω a , a = 1, 2, 3 , ( 10
)
and the main quadratic equation

y 2 = 0 , ( 11
)
where the scalars Ω a , world functions of the endpoints of the vectors e a , are given by

Ω a ≡ 1 2 (e a ) 2 , ( 12
)
In the study of the null propagation equations we are supposing that the emitted times {τ A } received at x constitute the emission coordinates of x so that the coordinate inequality (4) fulfils. From the definitions [START_REF] Coll | Emission coordinates in Minkowski space-time Physics and Mathematics of Gravitation[END_REF] this condition writes

C 4 : χ • y = 0 , ( 13
)
where χ is the configuration vector defined by χ ≡ * (e 1 ∧ e 2 ∧ e 3 ) , (14) * being the Hodge dual operator. Note that, under the form C 4 of C, χ is necessarily non vanishing:

χ = 0 , ( 15
)
which means that the configuration {γ A (τ A )} determines a hyperplane. This configuration hyperplane contains the four events {γ A (τ A )} and is orthogonal to the configuration vector χ. The sign of χ 2 provides the causal character (space-like, lightlike or time-like) of this hyperplane. Thus, in the emission coordinate region C one can distinguish three disjoint regions: the space-like configuration region

C s ≡ {x ∈ C | χ 2 < 0}, the null configuration region C ≡ {x ∈ C | χ 2 = 0}, the time-like configuration region C t ≡ {x ∈ C | χ 2 > 0}. with C = C s ∪ C ∪ C t .
The ingredients Ω a and χ of the internal configuration of the emitters make use of the relative position vectors e a and their double exterior product e 1 ∧ e 2 ∧ e 3 . Let us complete them by introducing their simple exterior products in the form E a ≡ * (e a+1 ∧ e a+2 ) , [START_REF] Coll | Emission coordinates in Minkowski space-time: the metric tensor[END_REF] where the operations on the indices obviously have to be understood modulo 3.

In looking for the solutions to the null propagation equations [START_REF] Bini | Emission versus Fermi coordinates: applications to relativistic positioning systems[END_REF], it is convenient to first start solving the main linear system [START_REF] Ehlers | The geometry of free fall and light propagation[END_REF] in y and then constraint it to verify the main quadratic equation [START_REF] Coll | Epistemic Relativity GraviMAS FEST workshop in honour of Lluís Mas[END_REF]. Then, the solution(s) y = ι(τ A ) so obtained, incremented by the position γ 4 of the reference emitter A = 4, will give by (5) the solution(s) to the null propagation equations (2), x = κ(τ A ) = γ 4 + ι(τ A ), i.e. the wanted coordinate transformation between emission coordinates and inertial ones. The next section is devoted to solving the main linear system [START_REF] Ehlers | The geometry of free fall and light propagation[END_REF].

The main linear system

The regularity condition [START_REF] Coll | Emission coordinates in Minkowski space-time: relative formulation[END_REF] for the configuration vector χ means that the rank of the main linear system (10) is exactly 3. As a consequence, its general solution y depends of a sole parameter, say λ, so that denoting by y * a particular solution, one has y = y * +λχ because, by [START_REF] Eisenhart | Continuous Groups of Transformations[END_REF], χ is a vector such that e a • χ = 0. We can thus state: Lemma 2 In the emission coordinate region C, the general solution to the main linear system (10) is of the form y = y * + λχ , [START_REF] Coll | Emission coordinates in Minkowski space-time: a geometric approach[END_REF] where the parameter λ takes arbitrary values, χ is the configuration vector (14) and y * is a particular solution.

A tentative study of explicit expressions of the particular solution y * shows that, for every emission coordinate domain, there is not a sole analytical function, depending exclusively of the elements of the configuration {γ A (τ A )} of the emitters, that is valid throughout the domain. This is why it is necessary to introduce an external element to obtain a sole analytical expression. We shall see that it is sufficient for this external element to be a vector field ξ transversal to the configuration, i.e. such that

ξ • χ = 0 , ( 18
)
and otherwise arbitrary. Denoting by i( ) the interior product, it may then be shown (see Appendix A for the proof):

Proposition 1 In all the emission coordinate region C, the general solution to the main linear system [START_REF] Ehlers | The geometry of free fall and light propagation[END_REF],

e a • y = Ω a , a = 1, 2, 3 , is of the form y = y * + λχ , ( 19
)
where the parameter λ takes arbitrary values, χ is the configuration vector χ ≡ * (e 1 ∧ e 2 ∧ e 3 ) , and y * is the particular solution orthogonal to a chosen transversal vector ξ, ξ • χ = 0, given by

y * ≡ 1 ξ • χ i(ξ)H , ( 20
)
where the bivector H is, like the vector χ, a function of the configuration of the emitters,

H ≡ Ω a E a , E a ≡ * (e a+1 ∧ e a+2 ) . ( 21
)
The direct and simple discussion that has enabled us to reach proposition 1 is the result of a careful search in order to find, in every coordinate domain, a sole, general and covariant expression for the coordinate transformation between emission coordinates {τ A } and inertial ones {x α }.

The last two requirements of generality and covariance enable us to find once and for all and for any inertial system (and in compact form due to the intrinsic vector formalism) the coordinate transformation in question. But they do not suffice to lead by themselves to a sole expression valid for all the configurations, i.e. valid at all the events of every emission coordinate domain. For example, if we were to work directly on the main linear system in the cases of a configuration generating a nonnull vector χ, we would find the solution

y = 1 χ 2 i(χ)H + λχ , ( 22
)
as now directly follows from (20) and ( 19) for the choice ξ = χ. But, meanwhile in fact there is neither physical nor mathematical reason for the solution to lose analyticity, this expression (22) becomes manifestly undefined on the null configurations, χ 2 = 0. This is why it is necessary to introduce an external element to find the sole regularized analytical expression above mentioned. This external element, the transversal vector field ξ, independent of the configuration of the emitters, is otherwise arbitrary. But, meanwhile due to lemma 2 we know that the main linear system depends on a sole parameter λ, the expressions (19), (20) of y apparently seem to indicate that it depends on the (4 + 1 =) 5 arbitrary parameters {ξ; λ}. Thus, what is the precise role played by this transversal field ξ?

From the definition (21) of the bivector H it is clear that its dual * H is orthogonal to χ, i(χ) * H = 0, so that, one has H = χ ∧ a for some vector a. With this expression of H it is easy to prove from ( 19) and (20) that the variation of the solution y with respect to the parameters ξ verifies ∂y ∂ξ

∧ χ = 0 , ( 23
)
which shows that the changes in ξ may be absorbed by λ, i.e. that the parameters ξ α are not essential in the Eisenhart sense [START_REF] Eisenhart | Continuous Groups of Transformations[END_REF]. Nevertheless their elimination gives rise to different expressions for the causally different regions that one can find in the whole emission coordinate region.

The main quadratic equation

Obtaining the transformation x α = κ α (τ A ) between emission coordinates {τ A } and inertial ones {x α } amounts to determine the intersection of the four future light cones emitted by the emitters at the configuration {γ A (τ A )} or, in the dual interpretation, to determine the vertex of the past light cone that contains the emitters in such a configuration.

According to proposition 1, the solutions y of the main linear system (10) depend linearly of the parameter λ, and consequently, describe a straight line in M 4 . The vertex that we are searching for is consequently on this straight line and, according to our choice of reference emitter, is the point on it related to the emitter A = 4 by the null vector m 4 = x -γ 4 = y. This is nothing more than what the main quadratic equation [START_REF] Coll | Epistemic Relativity GraviMAS FEST workshop in honour of Lluís Mas[END_REF] expresses: y 2 = 0. This cone, whose vertex is the emitter A = 4, cuts the straight line (19) at a value of λ given implicitly by:

χ 2 λ 2 + 2(χ • y * )λ + y 2 * = 0 , ( 24
)
and it is this value of λ which, put in the expression (19) of proposition 1, enables us to obtain the coordinate transformation y = ι(τ A ) that we are searching for.

But, before obtaining it, and in order to be sure of its real character, it is convenient to observe that the sole assumption we have made so far to solve the null propagation equations is that of the regularity of the configuration data {γ A (τ A )}, i.e. of the nonvanishing of the configuration vector χ, χ = 0. This assumption is only a strict part of the coordinate condition C, as its form C 4 , χ • y = 0, shows.

The study of degenerate configuration data χ = 0 and of the degenerate coordinate condition χ • y = 0 is interesting to gain more in-depth knowledge of the physical conditions leading to their presence, in order to avoid, control or predict them. But we shall restrict ourselves here, as already stated, to obtaining the coordinate transformation between emission coordinates {τ A } and inertial ones {x α }. For this purpose, from now on, we shall work under the coordinate condition C.

In fact, it is this condition that guarantees the real character of the solutions of the null propagation system. To demonstrate this, we shall separately analyze the cases of null and non null configurations.

Null configurations.

In the region where the configuration vector is null, χ 2 = 0, expression (19) for the solution y, y = y * + λχ, leads to χ • y = χ • y * , so that the last expression [START_REF] Synge | Relativity: The General Theory[END_REF] for the coordinate condition C insures the well defined character of λ in (24),

λ = - y 2 * 2(χ • y * ) , (25)
and we have Proposition 2 Under the coordinate condition C, for null configurations, χ 2 = 0, the null propagation equations (6) admit a real and single solution y given by:

y = y * - y 2 * 2(χ • y * ) χ , ( 26
)
where y * and χ are respectively given by (20) and [START_REF] Eisenhart | Continuous Groups of Transformations[END_REF].

Non-null configurations. In the regions where the configuration vector is non-null, χ 2 = 0, (24) gives:

λ ± = 1 χ 2 -χ • y * ± √ ∆ , ∆ ≡ (χ • y * ) 2 -χ 2 y 2 * . ( 27
)
But, for these non-null configurations, we can choose ξ = χ in the expression (20) of the particular solution y * to the main linear system [START_REF] Ehlers | The geometry of free fall and light propagation[END_REF]. Let us denote by c the corresponding y * :

c = 1 χ 2 i(χ)H . ( 28
)
Because H is antisymmetric, it is obvious that c is orthogonal to χ, so that, for the corresponding value λ c of λ, (27) takes the simple form:

λ c± = ± √ ∆ c χ 2 , ∆ c ≡ -χ 2 c 2 . ( 29
)
Now, expression (19) for the solution y, y = c + λ c χ, leads, by product by χ, to y • χ = λ c χ 2 . Taking its square and substituting in it the value (29) of λ c , one obtains:

(y • χ) 2 = ∆ c , (30)
so that the form (13) of the coordinate condition C insures the strict positiveness of the discriminant ∆ c . Then, for y * = c and the values (29) of λ c , equations (19) lead, after a little analysis, to the following result:

Proposition 3 Under the coordinate condition C, for non-null configurations χ 2 = 0, the null propagation equations (6) admit two real solutions y ± given by

y ± = c ± |c| ν , ν ≡ χ 2 |χ| 3 χ , ( 31
)
where c and χ are respectively given by (28) and ( 14).

In the above proposition and in what follows we denote |v| the modulus of a vector v, |v| ≡ |v 2 |.

We have seen that, for the particular solution y * = c, the discriminant ∆ given by (27) takes the value ∆ c given by (29). To what extent the discriminant ∆, corresponding to other particular solutions y * = c, differs from the value ∆ c ?

The question is pertinent because we want to unify expressions ( 26) and (31) so as to have a single expression for every coordinate domain, irrespective of the causal orientation of the configuration vector χ. To answer this, note that c is the particular solution to the linear system orthogonal to χ and that, from (19), any y * differs from c by a term of the form λχ. More precisely, c = y * -y * •χ χ 2 χ. Then, a straightforward calculation leads to: Lemma 3 Under the coordinate condition C, for non-null configurations χ 2 = 0, the discriminant ∆ and each one of the solutions y ± of the quadratic equation (24), given in (27), are independent of the particular solution chosen and, therefore, of the subsidiary vector ξ too. The discriminant is positive and its invariant value is

∆ = ∆ c = -χ 2 c 2 > 0 , ( 32
)
where c is given by (28).

Unified expression for any configuration.

As already stated, we want to find, for every coordinate domain, a sole expression valid for the whole domain, i.e. valid for all the configurations {γ A (τ A )} that could correspond to the events of the domain. Nevertheless, in principle, by continuously changing of events in the coordinate domain, we can make these configurations to have a vanishing, positive or negative value of χ 2 . When this is the case, none of the two standard expressions (27) of λ for χ 2 = 0 comes down to the sole solution (25) for χ 2 = 0 and, consequently, none of the expressions (31) reduces to expression (26), both of them becoming degenerate. To correct this degeneration, it is sufficient to multiply numerator and denominator of the second member of ( 27) by the conjugate of the numerator. Once this is done, taking into account proposition 1 one has the following result.

Theorem 1 In all the emission coordinate region C, the solutions to the null propagation equations (6) are real and admit the expression:

y ± = y * + λ ± χ , λ ± ≡ - y 2 * (χ • y * ) ± √ ∆ , ∆ ≡ (χ • y * ) 2 -χ 2 y 2 * , ( 33
)
where y * and χ are respectively given by (20) and [START_REF] Eisenhart | Continuous Groups of Transformations[END_REF]. For null configurations χ 2 = 0, the sole solution is given by one of these expressions, the other becoming degenerate.

The emission conditions E

The expressions (33) of theorem 1 give the Minkowski events y that can be related to the configuration events {γ A (τ A )} by means of emission or reception of null signals. But we want y to be the events reached by null signals emitted from the configuration events.

Thus we have to impose on y the emission conditions E. This will select a special class of configurations, which will be called emission configurations.

In a first step we shall impose the intermediate emission-reception condition E-R, guaranteeing that either all the signals at y have been emitted by the configuration, or all the signals at y will be received by the configuration; such a configuration will be called an emission-reception configuration. And in a second step we shall choose with E the first of these two cases.

As we have seen in section 2, the emission-reception condition E-R states that all the null vectors m A joining γ A and y, m A ≡ y -γ A , must have the same (past or future) orientation, i.e. m A • m B < 0 , ∀ A, B . In terms of the relative position vectors e a of the configuration {γ A (τ A )} given by ( 7) and ( 8), one has:

m 4 -m a = γ a -γ 4 = e a m a -m b = γ b -γ a = e b -e a , ( 34
) so that, because (m A -m B ) 2 = -2 m A • m B , we have:
Proposition 4 For a regular configuration {γ A (τ A )} to be an emission-reception configuration it is necessary and sufficient that all their relative positions be space-like. In other words: the null directions m A verify the emission-reception condition E-R if, and only iff,

(E-R) 4 : (e a ) 2 > 0 , (e a -e b ) 2 > 0 . ( 35
)
Let us note that, in particular, this proposition tells us that, in the grid T ≈ R 4 of the τ 's, the domains of emission coordinates {τ A } are in the interior of the region determined by (35), so that the points of the grid T in the complementary region certainly have no physical meaning.

Equations (33) of theorem 1 show us that, under the coordinate condition C, two real and definite solutions to the null propagation system may correspond to every configuration {γ A (τ A )}. Now, under the additional emission-reception condition E-R, one of these definite solutions may be that of emission and the other of reception. This Proposition 7 For a null emission configuration, χ 2 = 0, the non degenerate solution is y + (respectively, y -) if χ is past-pointing (respectively, future-pointing).

It is worth remarking the attachment between the results stated in propositions 5 and 7. If we continuously change to a null configuration coming from a space-like one, the configuration vector χ keeps its future or past orientation. If χ is past-pointing (respectively, future-pointing) we must take the (continuous) expression y + (respectively, y -) in both the null and space-like regions.

The coordinate transformation from emission to inertial coordinates

Front and back emission coordinate domains

Theorem 1 shows that the emission coordinate region C is mapped with two local charts and gives analytical expressions for the transformation between emission and inertial coordinates for the two corresponding coordinate domains.

These expressions of the coordinate transformation and the study of the emission conditions in section 6 show that the causal character of the configuration of the emitters differs for the two emission coordinate domains. Space-like and null configurations only admit one of the solutions given in theorem 1 as emission solution. And, of course, the coordinate domain of this solution contains by continuity a time-like configuration region. The coordinate domain of the other solution only contains a time-like configuration region.

We see that the two coordinate domains differ enough in the causal character of their emitter configurations. We shall call front emission coordinate domain C F the coordinate domain that contains events with the three possible causal configurations. More precisely, C F contains all the space-like configuration region C s , all the null configuration region C and a part C F t of the time-like configuration region C t . We shall call back emission coordinate domain C B the other coordinate domain, that only contains events with time-like configurations. ‡ In fact, C B = C t -C F t . Moreover, the two time-like regions have the same codomain in the grid, namely, Θ(C F t ) = Θ(C B ). This relation is very important. It says that, whatever be the four values {τ A } received by a user in the coordinate region C F t (resp. C B ), another user in the coordinate region C B (resp. C F t ) may receive the same values {τ A }. In other words: a user that only receives four times {τ A } defining a time-like configuration of the emitters is unable to detect in what part of the region C t , in C F t or in C B , he is. ‡ Let us remember that it is usual to call coordinate domain the open set U of any local chart (U, φ) of the atlas defining a differentiable manifold. This appellation requires attention because a coordinate domain is not necessarily a domain, but simply a (not necessarily connected) topological open set. In fact, meanwhile the front emission coordinate domain C F is generically connected, the back emission coordinate domain C B is generically the disjoint union of four connected components [START_REF] Pozo | Constructions in 3D and in 4D Relativistic Coordinates[END_REF].

Orientation of a positioning system

To be able to detect in what of these regions a user is, a notion of orientation of a positioning system is necessary. We give the following one: Definition 1 The orientation of a positioning system with respect to an event of its emission coordinate region C is the sign ˆ of the coordinate condition scalar:

ˆ ≡ sgn * (m 1 ∧ m 2 ∧ m 3 ∧ m 4 ) . ( 38
)
We have seen in section 2.2 that, in the emission region R, the hypersurface J separates the emission coordinate region C in two open sets. We can now identify them with the above front and back coordinate domains C F and C B respectively. Because J is the hypersurface where the coordinate condition C is not verified (vanishing Jacobian j Θ (x) = 0), the non vanishing member of its inequality has a constant sign in every coordinate domain C F and C B , so that we have the simple but important result: Proposition 8 The orientation ˆ of a positioning system is constant in every one of the coordinate domains C F and C B .

The same way that leads to the form (13) of the coordinate condition C shows that ˆ = sgn (y • χ). Because y is necessarily future-pointing for a emitted signal, if χ is time-like or null we can be sure that the sign of (y • χ) is the same that the sign of (u • χ) for any everywhere non vanishing future-pointing time-like vector u, this last sign being plus or minus according to the past-or future-pointing character of χ. Taking into account propositions 5 and 7, one has the following: Proposition 9 In the regions C s and C , the orientation ˆ of a positioning system is given by

ˆ = sgn (u • χ) (39)
for any future-pointing time-like vector u.

Explicit expression of the coordinate transformation from emission to inertial coordinates

We shall comment below in section 8 that for users in the region C t the determination of the orientation needs additional information. But for the moment, Theorem 1 and the above propositions lead us to the following result.

Theorem 2 Let γ A be the position vectors of the world-line equations γ A (τ A ) of the four emitters of a positioning system with respect to an inertial coordinate system {x α }, and {τ A } their emission coordinates. In all the emission coordinate region C = C F ∪ C B , the coordinate transformation x = κ(τ A ) is given by:

x = γ 4 + y * + y 2 * χ (y * • χ) + ˆ (y * • χ) 2 -y 2 * χ 2 (40)
where y * is the quantity given by (20), χ is the configuration vector [START_REF] Eisenhart | Continuous Groups of Transformations[END_REF] and ˆ is the orientation (38) of the positioning system with respect to the event that receives the data {τ A }. This is the main result reached in this paper. It rest to analyze now to what extent this expression (40) may be determined by a user from the data received by him.

The central region of a positioning system. Computational and observational orientation

The world-lines γ A (τ A ) of the emitters in an inertial system {x α }, as well as the spacetime metric in it, are here supposed known 'background' data for any user of the positioning system (the world-lines can be pre-determined initially or broadcast in real time). Thus, any user who receives only the emission data {τ A } is able to compute the quantities γ 4 , y * , χ appearing in (40). He has to follow the four steps:

Step 1. Compute the four position vectors of the emitters, γ A = Oγ A (τ A ), for the received values {τ A }.

Step 2. Choose a reference emitter, say γ 4 , and compute the position vector of the three referred emitters e a = γ a (τ a ) -γ 4 (τ 4 ).

Step 3. Compute the configuration scalars Ω a ≡ 1 2 e 2 a , the configuration vector χ ≡ * (e 1 ∧ e 2 ∧ e 3 ), and the configuration bivectors E a ≡ * (e a+1 ∧ e a+2 ) and H ≡ Ω a E a .

Step 4. Choose a transversal vector ξ, ξ • χ = 0, and compute y * ≡ 1 ξ•χ i(ξ)H. At this level, the user has computed the quantities γ 4 , y * , χ. But he is also able to compute in what of the coordinate regions, C t , C or C s of the positioning system he is; for this, one additional step is sufficient:

Step 5. Determine the sign of χ 2 . Then, according to their definitions, the user is in C t , C or C s if this sign of χ 2 is > 0, = 0 or < 0 respectively. Now, suppose that the user is in C or C s . Another step allows him to compute the orientation ˆ :

Step 6. Determine the sign of u•χ for a future-pointing time-like vector u, arbitrarily chosen. Then, according to proposition 9, ˆ = sgn (u • χ).

We shall call central region of a positioning system the region C C ≡ C s ∪ C . A part of theorem 2 may be then stated as follows.

Proposition 10 Let γ A be the position vectors of the known world-line equations γ A (τ A ) of the four emitters with respect to an inertial coordinate system. Then the users of the central region C C of the positioning system can obtain their position {x α } in the inertial system by computation from their sole emission coordinates {τ A }.

What about the users in C t , i.e. out of the central region C C ? We have seen that everywhere ˆ = sgn (y • χ). Nevertheless, the null vector y involves not only the initial data γ A (τ A ) and the received data τ A , but is the solution we are looking for, so that the quantity y • χ cannot be computed before itself. For this reason proposition 9 is exclusive: only the users of the central region C C are able to compute the orientation ˆ of the positioning system.

Can the users in C t know the orientation of the positioning system? It is possible to show that all the users of the coordinate region C, be them in the central region C C or not, can determine the orientation of the positioning system if, in addition to the reception of the {τ A }, they are able to observe the emitters in their celestial sphere. But this fact will be analyzed in a forthcoming paper.

Discussion and work in progress

In this paper we have obtained the coordinate transformation (40) between emission coordinates and inertial coordinates in Minkowski space-time. We use the intrinsic vector formalism to express the position vector, x ≡ (x α ), of every event in the emitter coordinate region as a function f of the emitter world-lines γ A (τ

A ) : x = f (Oγ A (τ A )) ≡ κ(τ A ).
This general and compact expression of the coordinate transformation will be a powerful tool for subsequent applications. For example, we can particularize it for different choices of the emitter world-lines which model specific physical situations. In doing so, a previous basic task appears to be convenient in many cases: to write our covariant expressions in a 3+1 formalism with respect to an arbitrary inertial observer, which is the goal of another work [START_REF] Coll | Emission coordinates in Minkowski space-time: relative formulation[END_REF]. Moreover, from the expression of the coordinate transformation we can easily obtain the components of the metric tensor in emission coordinates and we can study the region where the emission coordinates are more efficient than the inertial ones [START_REF] Coll | Emission coordinates in Minkowski space-time: the metric tensor[END_REF].

In section 6 we have studied the emission conditions in order to distinguish between emission and reception or mixed configurations. These results are a necessary tool to carry out more in-depth analysis of the domains and co-domains of the emission coordinates. This task will be tackled in another paper [START_REF] Coll | Emission coordinates in Minkowski space-time: a geometric approach[END_REF] where we will also study the geometry of the emitter configurations attending to their different causal character, and we will analyze how this geometry influences the solutions to the null propagation equations. Some preliminary results on this question have been presented in [START_REF] Coll | Emission coordinates in Minkowski space-time Physics and Mathematics of Gravitation[END_REF].

The orientation ˆ of a positioning system with respect to an event (see section 7) is a concept which has allowed us to give an explicit expression of the coordinate transformation in theorem 2. In section 8 we have pointed out that this orientation ˆ may be computed from the emission data {τ A } in the so-called central region. Nevertheless, out of this region, the determination of ˆ demands an observational method which will be analyzed elsewhere.

The development of all this theoretical work paves the way to the study of more realistic, gravitationally influenced, positioning systems. For example, those defined by emitter world-lines modeling a satellite constellation around the Earth in a weak gravitational field. It is worth remarking that for the use and smooth running of a positioning system one needs, not only the coordinate transformation to emission coordinates but also a good understanding of the domains in the space-time and codomains in the grid. The analysis of the degenerate configuration data, χ = 0, and of the hypersurfaces of the emission region R where the Jacobian j Θ (x) vanishes must also be well understood. We already know that the events of vanishing Jacobian are, and only are, those for which any user in them can see the four emitters on a circle in his celestial sphere [START_REF] Pozo | Constructions in 3D and in 4D Relativistic Coordinates[END_REF]. This includes the possibility for the user seeing less than four satellites, when some of them are in the shadows of the others.

Obviously, in realistic situations, the inertial coordinate system considered here must give rise to more useful ones. Namely: the International Celestial Reference System (ICRS) for the positioning system based on four millisecond pulsars, valid for the Solar System, as proposed in [START_REF] Coll | A Galactic Positioning System Proceedings Journées Systèmes de Référence Spatio-Temporels[END_REF]; the Barycentric Celestial Reference System (BCRS), to compare the planet trajectories (Earth at least and already) obtained in this millisecond pulsar system with the ones obtained by standard astronomical observations; the Geocentric Celestial Reference System (GCRS) or related World Geodetic System 84 (WGS84) or International Earth Reference System (ITRS) for the applications of relativistic positioning systems to the Global Navigation Satellite Systems (GNSS). Now, we regularize the main linear system (10) by adding a new equation as follows. Given a vector ξ such that ξ • χ = 0, let us consider a linear system of the form (A.1) for the unknown z, with Consequently, the sole solution to this system may be obtained as stated in lemma 5. Now Ω 4 = 0, and then we only need to calculate the vectors φ a , a = 1, 2, 3, of the dual basis given in (A.3). They take, in this case, the expression: Then, the solution (A.2) becomes now

φ a ≡ 1 3!D aP QR * (φ P ∧ φ Q ∧ φ R ) = 1
z = 1 ξ • χ i(ξ)H , H ≡ Ω a E a . (A.9)
Finally, note that this vector z is a solution to the main linear system and it is orthogonal to ξ. Thus it is the particular solution y * that we are looking for.

Confidential: not for

  distribution. Submitted to IOP Publishing for peer review 24 December 2009

4 Figure 1 .

 41 Figure 1. (a)The light-like vectors m A ≡ x -γ A represent the trajectories followed by the electromagnetic signals between the emitters and an event P in the emission region. x is the position vector of this event with respect to some inertial chart, and γ A , A = 1, 2, 3, 4, are the position vectors of the emitters with respect to this chart. (b) If we choose the emitter 4 as origin (reference emitter), the relative positions of the others (referred emitters) are e a = γ a -γ 4 , a = 1, 2, 3, and the position vector of the event P is y = m 4 .

φ

  A = e a , Ω A = Ω a , A = 1, 2, 3 ;(A.4)φ 4 = ξ , Ω 4 = 0 . (A.5)This linear system is regular since:D ≡ * (φ 1 ∧ φ 2 ∧ φ 3 ∧ φ 4 ) = * (e 1 ∧ e 2 ∧ e 3 ∧ ξ) = - * (ξ ∧ e 1 ∧ e 2 ∧ e 3 ) = -i(ξ) * (e 1 ∧ e 2 ∧ e 3 ) = -ξ • χ = 0 . (A.6)

2 D

 2 abc4 * (e b ∧ e c ∧ ξ)= 1 2 D abc * (ξ ∧ e b ∧ e c ) = -1 2 D abc i(ξ) * (e b ∧ e c ) = 1 ξ•χ i(ξ)E a , (A.7)where E a ≡ * (e a+1 ∧ e a+2 ) . (A.8)

Acknowledgments

This work has been supported by the Spanish Ministerio de Educación y Ciencia, MEC-FEDER project FIS2006-06062 and Spanish Ministerio de Ciencia e Innovación, MICIN-FEDER project FIS2009-07705.

last one finally has to be detected and discarded by means of the emission condition E, which, under the E-R one, reduces to:

where u is any given, everywhere non vanishing, future-pointing time-like vector.

For non-null configurations, χ 2 = 0, the general expression (33) gives the two admissible solutions under the coordinate condition C. These solutions also admit the non-unified expression given in proposition 3. From it we obtain:

Then, (32) implies that if χ 2 > 0 (respectively, χ 2 < 0) then c 2 < 0 (respectively, c 2 > 0 ) and, as a consequence of (37), y + is future-pointing iff y -is future-pointing (respectively, past-pointing). Thus, we have: Lemma 4 If χ is a time-like vector, χ 2 < 0, then only one of the solutions (33) corresponds to an emission configuration.

If χ is a space-like vector, χ 2 > 0, then the two solutions of (33) correspond to either two emission configurations or two reception configurations.

When χ is a time-like vector, can we detect the solution which corresponds to an emission configuration? The answer is affirmative. Indeed, as a consequence of lemma 3, each one of the solutions y ± is independent of ξ. Thus, taking for them the non-unified expression given in lemma 3, we obtain (when χ 2 < 0):

and, consequently, we can state: Proposition 5 For a space-like emission configuration, χ 2 < 0, the sole emission solution is y + (respectively, y -) if χ is past-pointing (respectively, future-pointing).

When χ is a space-like vector, we can know a priori whether the two solutions y ± correspond to emission configurations. Indeed, in this case c is a time-like vector. Then, making use again of the non-unified expression for y ± given in proposition 3, we obtain:

and, consequently, we can state: Proposition 6 For a time-like emission configuration, χ 2 > 0 the solutions y ± correspond to emission (respectively, reception) configurations if c is future-pointing (respectively, past-pointing).

Finally, for a null configuration, χ 2 = 0, lemma 2 gives the sole admissible solution under the coordinate condition C, which of course also reduces to one of the solutions of the general expression (33). In this case ∆ = (y * • χ) 2 and, consequently, the non degenerate solution in (33) is y + (respectively, y -) if ( χ) • y * = χ • y * > 0 (respectively, < 0). Thus, we can state:

Appendix A. Proof of proposition 1

From lemma 2, to prove proposition 1 we must obtain a particular solution of the main linear system [START_REF] Ehlers | The geometry of free fall and light propagation[END_REF]. More precisely, we must obtain the particular solution y * which is orthogonal to a chosen transversal vector ξ, ξ • χ = 0.

In order to obtain such a solution in a covariant way we begin by studying a regular linear system φ A • z = Ω A . In this case, the vectors {φ A } define a base, that is, φ 1 ∧ φ 2 ∧ φ 3 ∧ φ 4 = 0 Then, the sole solution to the system takes the expression z = Ω A φ A , {φ A } being the dual base. More explicitly, we have: