Static and radiating p-form black holes in the higher dimensional Robinson-Trautman class

We study Robinson-Trautman spacetimes in the presence of an aligned p-form Maxwell field and an arbitrary cosmological constant in n>=4 dimensions. As it turns out, the character of these exact solutions depends significantly on the (relative) value of n and p. In odd dimensions the solutions reduce to static black holes dressed by an electric and a magnetic field and whose horizon is an Einstein space (further constrained by the Einstein-Maxwell equations) -- both the Weyl and Maxwell type are D. Even dimensions, however, open up more possibilities. In particular, when 2p=n there exist non-static solutions describing black holes acquiring (or losing) mass by receiving (or emitting) electromagnetic radiation. In this case the Weyl type is II (D) and the Maxwell type can be II (D) or N. Conditions under which the Maxwell field is self-dual (for odd p) are also discussed, and a few explicit examples presented. Finally, the case p=1 is special in all dimensions and leads to static metrics with a non-Einstein transverse space.


Introduction
Gravity in more than four spacetime dimensions has attracted a lot of interest in recent years. In particular, several properties of higher dimensional black holes have been elucidated (see, e.g., the reviews [1][2][3]). In addition to vacuum spacetimes, solutions of various theories with gauge fields have also been extensively investigated (numerous references being given in [1][2][3]). The simplest such theory one can consider is probably n-dimensional Einstein-Maxwell gravity in the presence of a single 2-form field F µν , for which, e.g., the analog of the Reissner-Nordström solution has been known for a long time [4], together with its generalizations admitting an Einstein horizon [5] (by contrast, no n > 4 exact solution is known that extends the Kerr-Newman metric, except in the vacuum limit [6], or in other theories [1][2][3]). However, since extended objects naturally couple to higher-rank forms, theories of gravity with p > 2 forms are also of interest, especially from the standpoint of supergravity and string theory. A direct generalization of Einstein-Maxwell gravity is thus the p-form theory defined by where F 2 = F α1...αp F α1...αp (and κ 0 is a constant keeping into account different possible normalizations found in the literature, cf. also footnote 3), to which we shall restrict throughout the paper. Although the theory (1) is in some respects very similar to standard Einstein-Maxwell gravity, the case p = 2 possesses some distinct features. For example, it has been shown recently that asymptotically flat static black holes cannot couple to electric p-form fields when (n + 1)/2 ≤ p ≤ n − 1 (and thus do not posses dipole hair) [7] and that, for any p > 2, static perturbations of the vacuum Schwarzschild-Tangherlini metric do not exist [8,9]. In addition, results of [10] indicate that electromagnetic radiation may have properties different from those of standard n = 4, p = 2 electrovac general relativity (except possibly for 2p = n), as confirmed in [11] in the case of test fields.
A relatively simple and yet rich class of exact solutions in n = 4 general relativity is given by the Robinson-Tratuman family [12] (cf. the reviews [13,14]), defined by the existence of a geodesic, shear-free, twist-free but expanding null vector field k. It includes static black holes with an arbitrary cosmological constant, their accelerating counterpart (the C-metric) and other spacetimes containing both gravitational and electromagnetic radiation, as well as pure radiation solutions such as the Vaidya metric. Some of these solutions have found useful applications also beyond general relativity, for example to describe 2 + 1 black holes on a brane [15], or in the context of the AdS/CFT correspondence [16]. Electrovac Robinson-Tratuman, in particular, have been thoroughly investigated in the presence of a Maxwell field F µν aligned with k (already starting in [12]) and can be of Petrov type II, D and III (not N and O), while the Maxwell field can be of both types D ("non-null") and N ("null"), but it must be null if the Petrov type is III (see section 28.2 of [13] and section 19.6 of [14] for reviews and for a number of original references). Within this class, solutions of Petrov type D with a null Maxwell field [12] 1 are of special interest, as they describe formation of black holes by gravitational collapse of purely electromagnetic radiation [21] (see [22][23][24] for related earlier studies).
In [25] it was shown how the Robinson-Trautman line-element can be constructed in arbitrary dimension n (see also [26][27][28][29][30] for additional results). In the context outlined above, it is thus of interest to study Robinson-Tratuman solutions of the theory (1) with arbitrary n and p, which is the purpose of the present paper. We observe that the case p = 2 has been already studied in detail in [27]. Similarly as in the vacuum case [25], it turned out that the Robinson-Tratuman class is much more restricted when n > 4, and it essentially contains only static black hole spacetimes of Weyl type D (plus a few special non-static solutions [25][26][27][28]). However, the results of [10,11] mentioned above suggest that it need not be so for the theory (1). As we will work out, this expectation turns out to be correct. More in detail, our results can be summarized as follows.
• In odd n dimensions, Robinson-Trautman spacetimes coupled to a p-form Maxwell field with 2 ≤ p ≤ n − 2 reduce to static black holes specified by four independent parameters related to mass, electric and magnetic field strength, and cosmological constant (section 4.1). These black holes have been already studied in [31], to which we add a few comments. Except for the case p = 2 (p = n − 2) of [4], they cannot be asymptotically flat (so there is no conflict with the results of [7][8][9]), but they can be asymptotically locally (A)dS in some cases. The horizon is an Einstein space but must also obey further constraints following from the field equations. The metric is (55) with (56) and the Maxwell field is (57) (see the text for more details). Both the Weyl and the Maxwell tensor are of type D and share the same pair of doubly aligned (geodesic) null directions.
• In even n dimensions (with 2 ≤ p ≤ n − 2) static black holes as those described above are also present (cf. also [31]) and, again, the horizon geometry is constrained by the field equations (for example, for p = 2 it must be almost-Kähler if the magnetic field is non-zero [27]). Additionally, together with static black holes some exceptional solutions are also possible when 2p = n ± 2 and n ≥ 6 (as first noticed in [27] for the case n = 6, p = 2), cf. section 4.2 for details. Generically these exceptional metrics are non-static and of Weyl type II, while the Maxwell type is still D (but now possesses also a non-geodesic aligned null direction).
• In even n dimensions, the unique rank p satisfying 2p = n (including, in particular, n = 4 with p = 2 [12][13][14]) gives rise to an additional (and the more interesting) new class of solutions (section 5), consisting of the metric (55) with (77) and the Maxwell field (76). The Maxwell field is allowed to be of type II, D or N and in all these cases it can have a radiative term (thus giving concrete examples to the predictions of [10,11]), while the Weyl tensor can be of type II or D (more precisely, II(bd)/II(bcd) or D(bd)/D(bcd)) but with no radiative term for n > 4. However, when the Maxwell type is N and n > 4 the Weyl type can only be D(bd)/D(bcd) (this is a significant difference w.r.t the n = 4, p = 2 case, which can be traced back to the absence of gravitational radiation in the vacuum higher dimensional Robinson-Tratuman class [25]). Similarly as in [21], some of these solutions can be used to describe black hole formation (or white hole evaporation, by time-reversal) by collapse (emission) of electromagnetic radiation. As in the static case, these black holes can be asymptotically locally (A)dS for certain choice of the parameters -again, the horizon is an Einstein space and can be flat, in particular. For odd p, some of these solutions possess a self-dual p-form field, a property which is of interest in supergravity and string theory [32].
• Both in odd and even dimensions, the rank p = 1 (or its dual p = n − 1) has special features and needs to be studied separately (appendix C). This results again in a family of static solutions of Weyl and Maxwell type D. In this case the transverse space cannot be an Einstein space since it "feels" the backreaction of the electromagnetic field (as opposed to the generic case 2 ≤ p ≤ n − 2), which also defines a preferred direction in it. Further, the transverse space cannot be a space with an everywhere non-negative Ricci scalar. There exists no n = 4 counterpart of these solutions.
The rest of paper is organized as follows. In section 2 we describe our assumptions and set up the corresponding general form of the line-element (based on [25]) and of the Maxwell field. Section 3 is devoted to a systematic integration of the resulting Einstein-Maxwell equations -it can well be skipped by readers not interested in those technicalities. Our results are summarized in section 4 for the generic case 2p = n, and in sections 5.2-5.4 for the special case 2p = n (section 5.1 contains the integration of the subset of the Einstein-Maxwell equations that are special when 2p = n). Appendix A, largely based on [25,27,29,30], summarizes certain general properties of Robinson-Trauman spacetimes useful in the paper, but also contains a few new observations (sections A.3.1-A.3.4). Appendix B discusses some properties on the geometry of the transverse metric h ij of (55) (in particular, the black hole horizon) that follow from the Einstein equations (it partly overlaps with [27] when p = 2 or p = n − 2, and also summarizes certain observations of [31]). Appendix C studies the special ranks p = 1 and p = n − 1, also showing that these are forbidden in the four dimensional Robinson-Trautman class.

Notation and conventions
Throughout the paper we focus on n > 4 dimensions, but large part of the results applies also to the n = 4 case, on which we shall comment explicitly when important differences arises. We consider a p-form field (1), satisfies the source-free Maxwell equations d * F = 0, dF = 0 or, in components, The Maxwell field F backreacts on the spacetime geometry via the energy-momentum tensor where (3), Einstein's equations with a cosmological constant R µν − 1 2 Rg µν + Λg µν = 8πT µν (following from (1)) take the form It may be useful to recall the well-known fact that for any solution of (2) and (4) with a given p-form F , a "discrete" duality transformation F → * F gives raise to a dual solution 3 where the same metric is coupled to the (n − p)-form * F , defined by * F b1...bn−p ≡ 1 p! ǫ a1...ap b1...bn−p F a1...ap . In the special case 2p = n, (anti-)self-dual p-form solutions * F = ±F (for which necessarily F 2 = 0 = * F · F , where total contraction is understood) may exist if p is odd (since F is real and the signature Lorentzian, see, e.g., [32]). Instead, for 2p = n with even p (anti-)self-dual p-forms do not existon the other hand, in that case there is a continuous SO(2) duality symmetry (in addition to the discrete one mentioned above) which maps solutions into solutions (cf., e.g., [34] and appendix A of [35]).

Robinson-Trautman geometry with aligned Maxwell fields
We consider a n-dimensional spacetime that admits a non-twisting, non-shearing, expanding geodesic null vector field k. The associated Robinson-Trautman line-element was obtained in adapted coordinates in [25]. The corresponding curvature has been fully computed recently in [30], showing in particular that the spacetime is generically of aligned Weyl type I(b). It is the purpose of this paper to determine Robinson-Trautman spacetimes in the presence of a Maxwell p-form field in the theory (1), i.e., such that the Einstein equations (4) are satisfied, along with the Maxwell equations (2). However, we shall restrict to the case of Maxwell fields that are aligned with k (so that F is of type II or more special). 4 This means that the components of F of b.w. +1 vanish, which by (4) implies that the Ricci tensor components of b.w. +2 and +1 must also vanish, i.e., the Ricci type is II (or more special) aligned with k. With this condition, we can take advantage of a previous results of [25] (summarized in theorem A.1 of appendix A) saying that the spacetimes in question can be represented by the line-element Here the adapted coordinates (u, r, x 1 , . . . , x n−2 ) are used (Latin indices i, j, . . . or i 1 , i 2 , . . . etc. range over 1, . . . , n − 2 and label the spatial coordinates x i , sometimes collectively denoted simply as x) such that where r is an affine parameter along the generator k of the null hypersurfaces u = const. In such coordinates, the assumed alignment condition on F takes the form or, equivalently, F ui1...ip−1 = 0, while the alignment conditions on the Ricci tensor (automatically satisfied by (5)) read R rr = 0 = R ri . By construction, the corresponding components of the Einstein equation (4) are thus identically satisfied (note also that (7) implies T rr = 0 = T ri , cf. (3)).
In the rest of the paper we thus need to study only the remaining Einstein equations for R ij , R ur , R ui and R uu . These will contain terms depending on the Maxwell field. Hence, in order to proceed it will be convenient to first fix the r-dependence of F using Maxwell's equations (2).
For later calculations it is also useful to note that where g ≡ det g µν and h(u, x) ≡ det h ij .
The r-dependence of the Maxwell field is thus completely determined by (9), the second of (13) and (11), and can be summarized as where lowercase symbols b, e or f denote integration functions independent of r. Since we are can be non-zero only for 1 ≤ p ≤ n − 2, and e i1...ip−2 = e [i1...ip−2] for 2 ≤ p ≤ n − 1 (for p = 2 the term e i1...ip−2 obviously reduces to a scalar function e). From now on we will use the convention that indices of b, e and f are raised with the spatial metric h ij so that, e.g., In the special case with 2p = n + 1 (n ≥ 5 odd, p ≥ 3), eq. (16) is replaced by 5 (However, we will see in section 3.3 that the above logarithmic term must in fact vanish.) From (14) we observe, in particular, that the magnetic components F i1...ip are always r-independent, whereas the electric components F uri1...ip−2 become r-independent only in the special case 2p = 2 + n (n even).
Using (15) and (14), the first of (13) and (10) can be rewritten simply as For later purposes, it will be useful to define the following r-independent quantities built out of the electric and magnetic parts of the Maxwell field (which will enter some of the Einstein equations) Clearly E 2 ij = E 2 (ij) and B 2 ij = B 2 (ij) . In the case p = 2 the indices i, j disappear from e i1...ip−2 and E 2 ij and we simply have E = e, while B 2 ij is identically zero for p > n−2.
In particular, the invariant F 2 (useful in the following) can be written as

Einstein equations for R ij and R ur
Knowing the r-dependence of the Maxwell field, we can now consider the remaining Einstein equations. It is convenient to start from the equations for the Ricci components R ij and R ur (essentially those of higher b.w., i.e., zero). From (4) with (5), (14), (15), (22) (using the definitions (20), (21)), these read The component R ij of the metric (5) is reproduced in appendix A as eq. (A4). Comparing this with (23) immediately reveals that β i = 0 (this follows by comparing various powers of r in (23) and (A4), which implies that β i β j is either zero or proportional to h ij -the latter option is however impossible, cf. also [25]). 6 One can further perform a coordinate transformation (at least locally) to set α i = 0 [25]. From now on we shall thus have in (5) and therefore g ri = 0 = g ui , which will simplify several expressions. In particular, the Weyl type will thus be II(d) or more special, aligned with k (cf. theorem A.1). Using (25), the component (A4) now simplifies drastically (cf. (A5)). By comparing its various powers of r with those of (23), after r-integration (and contraction with h ij when necessary) one readily determines the r-dependence of the metric function H where µ is an arbitrary integration function independent of r.
In the special odd dimensional cases 2p = n ± 1 the last two terms of (26) should be replaced, respectively, by In addition, as a further consequence of the Einstein equation for R ij , the following constraints (coming from terms of order r 0 , r, r 2(p+1−n) and r 2(1−p) , respectively) must be satisfied (also when 2p = n ± 1) where R ij is the Ricci tensor associated with the spatial metric h ij , R is its Ricci scalar, and the general identity h ij h ij,u = 2(ln √ h) ,u has been used. For p = 2 eq. (31) becomes an identity. For 2p = n eqs. (31) and (32) are replaced by the following single equation Eq. (33) is satisfied identically in the case n = 4, p = 2. Note that (31), (32) and (33) imply with (23) that in all cases R ij ∝ h ij (as can also be seen from (A5) with (29)). As indicated above, equations (29), (31) and (32) do not hold in the limiting dual cases p = 1 (or p = n − 1) -these are special since the B 2 ij (or E 2 ij ) terms behave as r 0 in (23), (24) (and therefore in (26)), and thus effectively act as sources for the transverse geometry. Since further differences will also arise in the remaining Einstein equations, we present all the results for p = 1, n − 1 in appendix C and from now on we assume p = 1, n − 1.
As in [25], relation (29) means that, at any given u = u 0 = const, the spatial metric h ij (x, u 0 ) must describe a (n − 2)-dimensional Riemannian Einstein space. It is well-known (see, e.g., p.76 of [41]) that for n > 4 (i.e., n − 2 > 2) this implies that R ,i = 0, so that R can depend only on the coordinate u (additionally, for n = 5 the metric h ij must be of constant curvature since it is 3-dimensional and Einstein). For n = 4 eq. (29) is instead an identity.
Equation (30) gives [25] so that h ij can depend on u only via the conformal factor h 1/(n−2) . Eqs. (31) and (32) (or (33)) constrain both the Maxwell field and the metric h ij , and the permitted relation between p and n -some comments are given in appendix B (see also [31], and [27] for the case p = 2).
The component R ur of the metric (5) with (25) is reproduced in appendix A as eq. (A6). Substituting (26) into (A6) and comparing with (24) one finds that the corresponding Einstein equation is identically satisfied (including the cases 2p = n ± 1). We finally observe that (29) (with (25)) further restricts the Weyl tensor to be of type II(bd), aligned with k (see theorem A.1).

Maxwell equations, step two (2p = n)
We can now turn to the remaining set of the Maxwell equations, namely ( ..ip−2 ) ,µ = 0, which were not considered in section 3.1. As it turns out, from now on it will be necessary to consider the case 2p = n separately. Hereafter we thus restrict to the "generic" case 2p = n, while the corresponding analysis for the special case 2p = n will be given later on in section 5.
Using (14), (16), and (25), Note that the last two equations mean (see (16)) With (7) this implies that F is aligned also with the null vector l = −∂ u + H∂ r (and is thus of type D), and will be important in the following (as a consequence, F cannot be of type N, as found in [10]). Eq. (12) thus becomes b i1...ip,u = 0.
Next, from ( √ −g F µri1...ip−2 ) ,µ = 0 one gets (using (36)) which with (34) simply gives Note that the above results apply also in the 2p = n + 1 case (18) with a logarithmic term, which thus in fact vanishes. For later purposes it is useful to observe that (39) and (37) imply, respectively (recall (20) and (21)), which will be useful later on. We observe that above we have not employed eqs. (29)- (32), so that the results of the present section 3.3 apply also to the cases p = 1, n − 1.

Einstein equation for
R ui (2p = n; p = 1, n − 1) As remarked above, we now consider 2p = n. Recalling (25) we have g ui = 0 = g ri . Using (7) and (36) we also obtain F uα1...αp−1 F i α1...αp−1 = 0 (and thus T ui = 0). The corresponding Einstein equation (4) thus simplify considerably to where the explicit Ricci component is given by (A7). Employing (26) shows that this equation contains distinct powers of r, namely r 0 , r −1 , r 2−n , r 2p−2n+1 , r 1−2p . The term of order r 0 vanishes identically thanks to (30), while the remaining terms immediately give, respectively, the following conditions Eq. (42) is an identity due to (29) (as mentioned in section 3.2). One arrives at (42)-(45) also for 2p = n ± 1 after replacing (26) by the corresponding form of H containing the logarithmic terms (eqs. (27), (28)). Thus, generically, the Ricci curvature R of the transverse (n − 2)-dimensional Riemannian space, the "mass" parameter µ, the electric scalar E 2 and the magnetic scalar B 2 must all be independent of the spatial coordinates. However, E 2 and B 2 can depend on the x in the special cases 2p = n + 2 and 2p = n − 2, respectively (n necessarily even) -this will have some consequences in section 3.5. The case n = 4 is also special in that R can depend on the coordinates x -however, since the case n = 4, p = 2 is already well-known [12][13][14], only the cases n = 4, p = 1, 3 remain to be studied. These are precisely the ones dealt with in appendix C, so that from now on we can restrict in the main text to n > 4 with no loss of generality.
However, in the special case 2p = n + 2 with B 2 = 0( = E 2 ) (or 2p = n − 2 with E 2 = 0( = B 2 )) one cannot in general conclude that h(u, x) = U (u)X(x), and one has to consider the general equations (51), (52). If h(u, x) is indeed non-factorized, then this metric describes an Einstein space that admits a conformal (non-homothetic) map on Einstein spaces [26,27] and it is thus further constrained [45] (in particular, it must be of constant curvature when n = 6 and p = 4 or p = 2 in the electric and magnetic case, respectively [27]). One can still normalize R = 0, ±(n − 2)(n − 3).

Metric and Maxwell field
Keeping into account also the concluding observations of section 3.5.1, the line-element is given by (cf. (5) with (25)) where h ij = h ij (x) is the metric of a Riemannian Einstein space of dimension (n − 2) and scalar curvature R = K(n − 2)(n − 3), and ( (26) with (48)) where Λ, µ, E 2 , B 2 and K = 0, ±1 are all constants. The metric thus always admits the Killing vector field ∂ u , and it is static in regions where H > 0, while roots of H(r) define Killing horizons (see also [31]). Recall that when 2p = n ± 1 (n odd), the second line of (56) should be replaced by (27) and (28), respectively. 7 The "Coulombian" Maxwell field is given by (eqs. (7), (14), (15) and (36)) where e i1...ip−2 and b i1...ip are harmonic forms (of respective rank (p − 2) and p) in the transverse geometry h ij , i.e., they obey the Euclidean source-free Maxwell equations in (n − 2) dimensions (cf. (19) and the first and third of (35)). These forms are, however, further constrained by the conditions (31), (32) on the (constant) "square" of the field strengths, i.e., It is worth emphasizing again that conditions (58) do not only constraint the Maxwell field, but also impose severe restrictions on the Einstein metric h ij . For instance, when p = 2 it must also be almost-Kähler if B 2 = 0 and n must be even [27]. See [31] and appendix B for further comments.
The above solutions can be seen as an extension to arbitrary p of the p = 2 (n = 6) solutions studied in [27] (including, when B 2 = 0, the higher-dimensional Reissner-Nordström solution found by Tangherlini [4]) and possess similar qualitative features. In particular, they represent static black holes (at least for certain values of the parameters in (56)) dressed by electric and magnetic Maxwell fields. These solutions were previously obtained (starting from a static ansatz) and analyzed (including their thermodynamics) in [31], so that a detailed discussion is not necessary here. Recalling that in an (n − 2)-dimensional compact Riemannian space of positive constant curvature there exist no non-zero harmonic forms (except for a 0-form or a (n − 2)-form) [42,43], we observe that the metric h ij in (55) cannot describe a round sphere, except when p = 2 and b i1i2 = 0 or, dually, when p = n − 2 and e i1...in−4 = 0. Therefore, these static black holes cannot have a spherical horizon and cannot be asymptotically flat (in agreement with [7][8][9]), except in the electric p = 2 (or magnetic p = n − 2) Reissner-Nordström solution of [4]. A flat and compact horizon metric h ij is instead permitted (giving K = 0 in (56)), in which case the harmonic forms e i1...ip−2 and b i1...ip must be covariantly constant [42,43]. This allows for, e.g., asymptotically locally (A)dS black holes (see also [31]).
Similarly as for the case p = 2 [27], an additional "Vaidya-type" matter field representing pure radiation aligned with k (i.e., adding an extra term to the component T uu only) can easily be included by appropriately [27] modifying (47) and thus allowing for µ ,u = 0 (see also a comment at the end of section 6 of [31]) -in the special case 2p = n this pure radiation can be sourced by the Maxwell field itself, as we show below in section 5.

Weyl and Maxwell types
As discussed in [27], the warped product structure of the metric (55) with H = H(r) implies [28,40] (see also [46] for a compact summary of these results) that the corresponding Weyl type is D(bd) and that are the (unique) double WANDs (no other WANDs -even single ones -exist since the type is D(bd), cf. appendix A.3, in particular footnote 14; l also defines a second Robinson-Trautman null direction, as follows by "time-reflection" symmetry [27,28]). Since the type is D(bd), all the Weyl components are uniquely determined from (eqs. (A12)-(A14) of appendix (A) with (56)) where Cījkl are the components of the Weyl tensor associated with h ij in a frame of h ij , and the notation of [10,46] is employed (with a hat denoting components in a frame of the full spacetime metric g µν , cf. appendix A.3). Since h ij must be Einstein, Cījkl = 0 iff h ij is of constant curvature (which is necessarily the case when n = 5), in which case the Weyl type becomes D(bcd). The scalar invariant C µνρσ C µνρσ = 4Φ 2 (n − 1)/(n − 3) + r −4 CījklCījkl (cf. eqs. (69) and (70) of [47]) also signals a curvature singularity at r = 0. When 2p = n ± 1, the equation replacing (60) can be obtained by recalling (27), (28) and using (A12). k and l are manifestly also aligned null direction of the Maxwell field (57) (k is such by construction, recall (7), and l then due to (36)), which is thus also of type D. It follows that also the Ricci tensor is of (aligned) type D (as can be explicitly verified thanks to R ui = 0 and R uu = 2HR ur , cf. sections 3.4 and 3.5.1) -apart from (59), no other Ricci aligned null directions exist, not even single ones, as can be seen recalling that R ij ∝ h ij . One can easily see that in a frame parallelly transported along k (and adapted to (59), see appendix A.3) the electric and magnetic components of (57) fall off, respectively, with the monopole rate 1/r n−p and 1/r p (as one could expect from a study of test fields [11]).
Examples with p = 2 (or, dually, p = n − 2) are given, e.g., in [4,27]. Several other examples have been constructed in [31]. Using a construction described in [31] (see appendix B for a brief summary) one can produce more solutions, as we now exemplify.
Example (n = 11, p = 3(8)) A magnetic solution with n = 11, p = 3 (or electric with p = 8 after dualization) and a direct product transverse space is given by where h ij and h 4.2 Exceptional case 2p = n ± 2 (n ≥ 6, even) These special ranks of F also fall into the discussion of section 4.1 if the additional assumptions (E 2 ) ,i = 0 (for 2p = n + 2) or (B 2 ) ,i = 0 (for 2p = n − 2) are made (or if the transverse space is assumed to be compact and the electric and magnetic field are both non-zero, cf. section 3.5.3), in which case they again describe static black holes, as in the n = 6, p = 2 (or p = 4 after dualization) example given by eq. (66) of [27] (with D = 6), or as in the following example.
Example (n = 8, p = 5(3)) A solution describing an electric field with n = 8, p = 5 (or a magnetic field with p = 3 after dualization) is Note that here the electric field is r-independent since 2p = 2 + n (first term of (57)). These are locally AdS (for Λ < 0) electric static black holes with a flat horizon. However, more general solutions are now permitted for 2p = n + 2 with B 2 = 0, E 2 = E 2 (u, x) or for 2p = n − 2 with E 2 = 0, B 2 = B 2 (u, x) (or for a non-compact transverse space). The line-element is again given by (55), but H, given by (26), can generically depend on all coordinates (and the metric is thus non-static, in general). The Einstein metric h ij = h ij (u, x) is generically non-factorized, and thus further constrained by the property of admitting conformal maps on Einstein spaces [45]. The Maxwell field is still given by (57) and satisfies the source-free Maxwell equations in the transverse geometry h ij , but E 2 (or B 2 ) can now depend on x and u (eqs. (40)); the form e i1...ip−2 (or b i1...ip ) is still u-independent (eq. (39) with n = 2p + 2, and (37)). The remaining equations to be satisfied are (51), (52) (see section 3.5.3 for more comments). Because (57) holds also here, the Maxwell and Ricci tensors obviously possess the same algebraic properties as discussed in section 4.1 and are thus still of type D, doubly aligned with both null vectors (59). The Weyl type is generically II(bd) (see section 3.2 and appendix (A.3)) and the Weyl components of b.w. 0 (where Φ is necessarily non-zero) are still given by (60). We observe that H ,i = 0 implies that the vector l is non-geodesic (appendix A.3) and not a multiple WAND, as can be seen using the results of appendix A.3.3 with (26) and (51), (52) (and viceversa, i.e., if H ,i = 0 then l is a geodesic multiple WAND, as observed in section 4.1.2 for the case 2p = n ± 2).
It should also be observed that the ranks 2p = n ± 2 are special (even in the subcase of static metrics) because contributions from the electric and the magnetic terms to the energy-momentum components T ij cancel out (thanks to (31), (32)) for 2p = n + 2 and for 2p = n − 2, respectively. Related to this, for even p there can exist forms e i1...ip−2 (for 2p = n + 2) or b i1...ip (for 2p = n − 2) that are self-dual in the transverse geometry h ij , cf. also appendix B.
5 Special case 2p = n (n even): black holes with electromagnetic radiation As we observed, the rank satisfying 2p = n has special properties and need to be studied separately. This is not so surprising since this is the unique rank for which the Maxwell equations are conformally invariant and admit self-dual solutions (for odd p) or a continuous duality symmetry (for even p), cf., e.g., [32,34,35,48].
The results of sections 3.1 and 3.2 are still valid also in the case 2p = n (but recall (33)). Instead, in sections 3.3, 3.4 and 3.5.1 we assumed 2p = n and those results are modified as follows.

Maxwell equations, step two
Certain terms in the Maxwell equations studied in section 3.3 combine since they have the same r-dependence, and the equations of section 3.3 are thus replaced by 8 One also obtains an additional equation ..ip−2,k] ) ,l = 0 (⇔ ⋆ d⋆de = 0), which is however identically satisfied by virtue of (63) and the antisymmetry of b i1...ip .
Note that the r.h.s. of (63) acts as a "current" for the p-form b i1...ip so that b i1...ip and e [i2...ip−1,i1] no longer satisfy the Euclidean source-free Maxwell equations in the transverse space. A fundamental difference with the generic case 2p = n is that now the components F ui1...ip−1 (eq. (16)) can be nonzero. Instead of (40) we now obtain from (65) and (64) (with (34))

Summary and discussion
We first observe that static black hole solutions clearly exist also for the special rank 2p = n (see section 5.3 below), to which a discussion similar to that section 4.1 still applies. However, now there exists also a new family of solutions in the case F uj1...jp−1 = 0. The Maxwell field is given by The forms e i1...ip−2 (u, x) and b i1...ip (u, x) in general are not harmonic in the transverse space, but satisfy the "modified" Euclidean Maxwell equations in (n − 2) dimensions (19) and (63). In addition, they can depend on u, cf. eqs. (64), (65). The latter further tells us that the (p − 1)-form f i1...ip−1 (u, x) is also generically non-harmonic. Notice that the Maxwell equations do not specify the u-dependence of f i1...ip−1 , which can be interpreted as a freedom in the choice of a "wave profile". The line-element is given by (55) with

Maxwell type and self-duality
The Maxwell field (76) is in general of type II (aligned with k by construction) and, in a parallely transported frame adapted to (59) (appendix A.3), peels off as (in agreement with test fields results [11]) where the symbols II and N specify the algebraic type of the corresponding terms, with the "radiative" N term proportional to f i1...ip−1 . The Maxwell type becomes D when there exists a second null direction aligned with F , i.e. (using a null rotation (A21)), iff the following equation admits a solution 10 for the null rotation parameter zī p ≡ r −1 zî see solution (86) below for an example. As a special case, this occurs (with zī 1 = 0) when the frame vector l (59) is aligned with the Maxwell field, i.e., iff f i1...ip−1 = 0 = e [i2...ip−1,i1] (which implies that the radiative term vanishes). On the other hand, the Maxwell type is N iff k is doubly aligned, i.e., e i1...ip−2 = 0 = b i1...ip -these special cases are discuss below in sections 5.3 and 5.4. We observe that for n = 6 (p = 3) only the Maxwell type N is possible (cf. appendix B.3). Let us further notice that the b.w. -2 component 8πT µν l µ l ν = R µν l µ l ν = R uu −2HR ur (with (59) and R rr = 0) of the energy-momentum tensor, expressing the flux of electromagnetic energy along k, equals expression (71) and is characterized by the leading term κ 0 r 2−n F 2 -an invariant quantity taking the same value in any frame parallely transported along k (i.e., it is invariant under a null rotation (A21) of the frame (A11) with zî independent of r, and under spins). When p is odd, straightforward calculations show that the field (76) is self-dual ( * F = F ) iff e = ⋆b, ⋆de = −de and ⋆f = −f (recall the definition of ⋆ in footnote 8), whereas self-duality is not possible for an even p [32] (in particular, the condition e = ⋆b implies p(p − 1)E 2 = B 2 , and that the l.h.s. and r.h.s. of (33) must vanish separately). Examples satisfying these conditions (with E 2 = 0 = B 2 ) are given by (100) and (101) (under the conditions described there).

Examples
where E and B are constants. This spacetime generically represents black holes with a flat horizon in the presence of an electromagnetic field which consists of both a static component and of udependent expanding radiation. 11 Correspondingly, the mass parameter µ (and thus the location of horizons) is u-dependent and monotonically increases or decreases according to the time-orientation of k (thus corresponding to received or emitted radiation, cf. [21]). These metrics are asymptotically locally (A)dS if Λ = 0, but if µ > 0 they clearly possess a horizon also when Λ = 0. The Weyl type is D(bcd) because here H = H(u, r) and h ij is flat [28,40,46] (as can also be seen explicitly from (80)-(83)). Generically the Maxwell field is of genuine type II. Here E and B are independent parameters and, in particular, can vanish independently (in other words, (31) and (32) are satisfied separately). However, since here 2p = n, solutions exist that satisfy only the weaker constraint (33) (i.e., the electric and magnetic field must both be non-zero), as illustrated by the following example.
Example (n = 8, p = 4) To obtain a different solution with n = 8, p = 4, one can specify the metric functions and electromagnetic field by where E is a constant. Clearly (31) and (32) are not satisfied in this case. If E = 0 = f ijk the Maxwell field is of genuine type II. Comments similar to those given for (84) apply also here, in particular the Weyl type is again D(bcd).
Example (n = 8, p = 4) An example in which µ has also a non-trivial x-dependence is given by where B and f 0 are constants (for f 0 = 0 this trivially reduces to a subcase of (84)). Here Φ = 0 = Ψ ′ i whileΦîĵkl = 0 = Ω ′ iĵ (see (80)-(83)), so that the Weyl tensor is of genuine type II(bcd) (by the argument in appendix A.3.4, since here (A24) admits no solutions). The vector field l is clearly not aligned with F , nevertheless the Maxwell type is D, a second aligned null direction being given by a null rotation (A21) with m (1) = r −1 ∂ x1 and the only non-zero parameter z1 = −6 √ 2K 0 B −1 r (cf. (79)).

l aligned with F : static black holes (n ≥ 8)
The null vector field l of (59) is uniquely defined by certain geometrical properties (appendix A.3). Furthermore, as noticed above, it is aligned with F iff which implies that the Maxwell field is of type D and non-radiative (as mentioned above and in appendix B.3, for n = 6 this would lead to F = 0, so we can restrict here to n ≥ 8) -indeed here T µν l µ l ν = 0. Equations (63), (64), (65), (66), (67), (69) and (72) now take exactly the same form as in the generic case 2p = n (in particular, (66), (67) reduce to (40)). Instead of the conditions E 2 ,i = 0 = B 2 ,i found in the 2p = n case, one has the weaker condition (from (70)) However, together with (40) this suffices to show that, again, one can rescale away the u-dependence of h ij (and thus also of µ and e i1...ip−2 ), and the discussion of section 4.1 then applies (with the only difference that (58) are replaced by (33), and that (88) replaces E 2 ,i = 0 = B 2 ,i ). The metric is thus (55) with H(r) given by (77) but without the 2 n−2 (ln √ h) ,u r term (all the coefficients of the powers of r appearing in H are indeed constants) and represents static black holes. Again it follows that the Weyl type is D(bd) (possibly, D(bcd)) and l is also a double WAND. The Maxwell field is given by (57) with 2p = n. Here (78) clearly reduces to F = D r − n 2 . These 2p = n solutions with F of type D were previously studied in [31]. Two examples with n = 8, p = 4 are given by (84) and (85) with f ijk (u) = 0 (and thus µ = const), see [31] for others.

Type N Maxwell field
The field (76) is of type N iff with k being the unique aligned null direction, so that and the peeling (78) becomes simply F = N r 1− n 2 . In this case eqs. (63)-(72) reduce to Eqs. (91) mean that f i1...ip−1 (u, x) effectively defines a Maxwell n 2 − 1 -form in the (n−2)-dimensional transverse space (i.e., it is harmonic).
Here the energy-momentum tensor possesses only the (b.w. -2) component describing a flux of radiation along k 8πT µν l µ l ν = 8πT uu = κ 0 r 2−n F 2 , and thus the Maxwell form acts as an aligned pure radiation field. Therefore these special Robinson-Trautman metrics are contained in the pure radiation family of solutions studied in [25] (but the corresponding Maxwell equations were not considered there). In particular, from appendix A of [25] it immediately follows that the Weyl type is D(bd) (D(bcd) if h ij is of constant curvature), with (59) being the two multiple WANDs and as can also be seen explicitly from (80)-(83) with E 2 = 0 = B 2 .

A special subcase: solutions with a factorized h(u, x) = U (u)X(x)
In the special case of a factorized h(u, x) = U (u)X(x), one can set U (u) = 1 by a coordinate transformation, and thus obtain a special subclass of solutions with metric (55) with h ij = h ij (x) and H(u, r) given by Here necessarily µ ,u = 0. The Maxwell field is given by (90), where f i1...ip−1 must satisfy (91) and, by (93), also (F 2 ) ,i = 0.
These solutions will in general describe black holes in the presence of electromagnetic radiation with non-zero expansion, with a monotonically increasing (or decreasing, according to the choice of time-orientation of k) mass parameter µ. A simple solution can be obtained by taking the transverse metric to be flat, i.e., h ij = δ ij and f i1...ip−1 = f i1...ip−1 (u) (obviously with such a choice (91) and (99) are identically satisfied). For definiteness, we give this explicitly in the case n = 6 in the following example.
Example (n = 6, p = 3) An explicit example with n = 6, p = 3 is given by For Λ < 0 this spacetime represents asymptotically locally AdS black holes with electromagnetic radiation. By a rotation one can always simplify the Maxwell field so as to have the only non-zero components F u12 = f 12 (u), F u34 = f 34 (u), in which case F is self-dual when f 34 (u) = −f 12 (u) (or anti-self-dual when f 34 (u) = +f 12 (u)), cf. section 5.2. Solution (100) is an extension of a solution given in 4D (for Λ = 0) in [12] (also reproduced in eq. (28.43) of [13]) and recently discussed in [21] (see also footnote 1). It should be observed that for n = 6 this example in fact comprises all the possible Robinson-Trautman solutions with a Maxwell field of type N if the transverse space is assumed to be of constant curvature (this follows from (91) and (99) after using coordinate adapted to a constant curvature space, e.g., those employed in [25] -it also follows that the (constant) curvature must necessarily be zero). A similar example with n = 8, p = 4 is given by (84) with E = 0 = B. An n = 6 example where, instead, h ij is not of constant curvature follows.
Example (n = 6, p = 3) As another example in 6D one can take h ij to be a direct product of two S 2 or H 2 , namely As above, (anti-)self-duality holds iff f 34 (u) = ∓f 12 (u). For K = 0 this solutions reduces to (100) (up to a space rotation).

A Robinson-Tratuman spacetimes with an aligned Ricci tensor of type II
For the purposes of the present paper and for possible future reference it is useful to summarize some of the results of [25,27,29,30] in the present appendix. Note that we restrict here to the Robinson-Tratuman spacetimes in which the Ricci tensor is of type II aligned with the privileged vector field k. This includes vacuum solutions as well as solutions with aligned matter content (as we assume in the main text).

A.1 General metric
The line-element and its Weyl type are specified by the following theorem.
Theorem A.1. If a n-dimensional spacetime (n ≥ 4) admits a non-twisting, non-shearing, expanding geodesic null vector field k and the Ricci tensor is of aligned type II, adapted coordinates (u, r, x 1 , . . . , x n−2 ) can be chosen such that [25] where H is an arbitrary function of all coordinates. k is automatically a WAND, such that the Weyl tensor is in general of aligned type I(b). It is a multiple WAND iff β i = 0 [30], in which case the Weyl tensor is of aligned type II(d) (or more special). When β i = 0, one can locally set W i = 0 (after a coordinate transformation giving α i = 0) [25]. The Weyl type further specializes to II(bd) iff h ij is an Einstein metric (with still W i = 0) [30].
The vector field k is the generator of the null hypersurfaces u = const such that k µ dx µ = −du, r is an affine parameter along k, θ is its expansion scalar, and x ≡ (x i ) ≡ (x 1 , . . . , x n−2 ) are spatial coordinates on a "transverse" (n − 2)-dimensional Riemannian manifold. We observe that the condition that the Ricci tensor is doubly aligned with k can be expressed in a frame-independent form as as R µν k ν = αk µ , which in the coordinates of (A1) means R rr = 0 = R ri . For certain calculations it may be useful to observe that 2H = g rr = −g uu and W i = g ri (such that W i = 0 ⇔ g ri = 0 ⇔ g ui = 0).
The first part of the theorem was proven in sections 3.1 and 3.2 of [25]. The results on Weyl alignement follow immediately from eqs. (13)- (15), (17) and (19) (with (24), (25)) of [30] together with (A1). 13 In vacuum or with aligned pure radiation (i.e., the only non-zero energy-momentum tensor component being T uu ) necessarily β i = 0 and h ij is Einstein for any n ≥ 4, and the Weyl tensor further specializes to type D(bd) (possibly, D(bcd) or D(acd)) if n > 4 [25] (the latter result was also rederived in [30] without the coordinate choice α i = 0), the type O being possible only in the trivial case of constant curvature spacetimes. In [29] it was shown that for any n ≥ 4 Robinson-Trautman spacetimes cannot support aligned gyratonic matter (i.e., an energy-momentum tensor with both T uu and T ui -and no other components -being non-zero). Les us finally observe that if one relaxes the assumptions of the theorem by requiring only the aligned Ricci type I (i.e., R rr = 0), one obtains the same form of the metric, except that the W i (u, r, x) are arbitrary functions [25] (the Weyl type remains I(b) [30]).

A.2 Ricci tensor components
The Ricci tensor component R ij for the general metric (A1), (A2) was given explicitly in eq. (A.1) of [27] and reads where R ij is the Ricci tensor associated with the spatial metric h ij (as defined in section 3.2), and a partial derivative w.r.t. (e.g.) x j is simply denoted by a comma followed by j. For the purposes of the present paper we need the remaining Ricci components only in the special case W i = 0, which we now present (but see [29,30] for the Ricci tensor components of the most general Robinson-Tratuman metric, i.e., without not even enforcing R rr = 0 = R ri ).

A.2.1 Case W i = 0 (k is a multiple WAND -Weyl type II(bd))
If one further assumes W i = 0 in (A1) (i.e., k is a multiple WAND, as we indeed find in section 3.2), eq. (A4) and the remaining Ricci components, given in eqs. (26), (27) and (31) of [25], reduce to where in the last expression the first quantity on the r.h.s. has been written in terms of (A6) for brevity and for convenience in the calculations of section 3.5.1, and △ ≡ 1 is the Laplace operator in the (n − 2)-dimensional space with metric of h ij . Note that 2(n − 2)R ur = −r −1 h ij R ij ,r . For general purposes it is useful to observe that the component R ui can be rewritten more compactly as R ui = r 4−n (r n−4 H ,i ) ,r + 1 2 h kl (h ki,u||l − h kl,u||i ) [29], where the lower double bar || denotes a covariant derivative w.r.t. h ij -but in the computations of the present paper the form (A7) can be used more readily.

A.3 Weyl tensor components in the case
For the purposes of the present paper, we further restrict here to the case when the condition R ij ∝ h ij holds for the metric (A1) with W i = 0 (this is also true in various other cases of physical interest, e.g., in vacuum, or whenever the traceless Ricci tensor is of aligned type III or more special), while we refer to [30] for the Weyl tensor components of the most general Robinson-Tratuman metric. By (A5) this restriction is equivalent to so that, in particular, h ij is Einstein and therefore (n − 4)R ,i = 0. (Note that, indeed, these conditions are obtained in section 3.2 as a consequence of the Einstein equations -except in the cases p = 1, n − 1, for which see appendix C.) Using (A9) one can further prove the identity (cf. eq. (144) of [30]) needed in the following. We choose a frame where the functions m j (î) do not depend on r. The spacelike vectors m (î) span the transverse space of constant u and r, and obviouslym (ĩ) ≡ rm (î) = m j (î) ∂ j defines an orthonormal frame for the metric h ij . Before proceeding, it is useful to observe that since k and l span the 2-space (u, r) we which must hold together with and (A16). In this case, for n > 4 the only non-zero Weyl components are (using also (A10)) See [13] for n = 4.
A.3.3 Conditions for l to be a multiple WAND (⇒ Weyl type D(bd)) It is clear from (A13) and (A14) that, in general, the vector field l of (A11) is not a WAND (not even a single one). The conditions for l being a multiple WAND read For n > 4 the latter of these is identically satisfied thanks to (A10). When all these conditions are met the Weyl type becomes D(bd). This happens, for example, in the special case c 2,i = 0.

A.3.4 Conditions for the Weyl type D(bd) when l is not a multiple WAND
Even when the vector field l of (A11) is not a multiple WAND (i.e., (A19), (A20) are not simultaneously satisfied) the Weyl type can still be D(bd) provided a second multiple WAND (in addition to k, and different from l) exists. In order to find conditions for this to happen, it is necessary to perform a null rotation about k, i.e., such that, in the transformed frame, Ψ ′ i → 0,Ψ ′ iĵk → 0 and Ω ′ iĵ → 0. The transformation laws under (A21) for the negative b.w. Weyl components are given by eqs. (2.33)-(2.35) of [10] (while non-negative b.w. components are unchanged under (A21) since k is a multiple WAND). Using the fact that the Weyl type is II(bd), the condition Ψ ′ i → 0 uniquely fixes the parameters zî by except in the case Φ = 0 = Ψ ′ i , for which the zî remain arbitrary. Next, requiringΨ ′ iĵk → 0 and using (A22) further gives (also in the case in the case Φ = 0 = Ψ ′ i ) Finally, imposing Ω ′ iĵ → 0 gives which after using (A22), (A23) (and multiplying by Φ) gives a constraint among the Weyl tensor components 14 Recalling that for the metric (A1) with W i = 0 and (A9) one hasΨ ′ iĵk = 0, and using (A12), eq. (A23) reduces to This is a restriction on the Weyl tensor of h ij that will not be true for a generic Einstein metric h ij , therefore we can conclude that generically the metric (A1) with W i = 0 and (A9) is of genuine type II(bd). However, the "genericity" conditions may be violated if a special choice of H (e.g., in vacuum [25]) or of h ij is made in (A1), in which case the Weyl types D(bd), D(bcd), and D(abd) are all possible (see, e.g., [25,27,30,46]). The specific form of (A25) for spacetimes (A1) with W i = 0 and (A9) can be obtained by substituing (A12)-(A14) andΨ ′ iĵk = 0 into (A25). (31) and (32)

B Comments on the (Einstein) constraints
The b.w. 0 components of F can be divided into an electric and a magnetic part described, respectively, by e i1...ip−2 and b i1...ip (cf. (14), (15)). The latter live in the transverse geometry of h ij and must obey the constraints (31) and (32) (replaced by (33) if 2p = n). In this appendix we discuss some general consequences of those constraints. Recall that the form e i1...ip−2 is defined for 2 ≤ p ≤ n while b i1...ip for 0 ≤ p ≤ n − 2. The cases p = 0, n are trivial (footnote 2) while p = 1, n − 1 require a special discussion (appendix C), so here we restrict for both e i1...ip−2 and b i1...ip to the ranks 2 ≤ p ≤ n − 2. Let us also observe that the equations obeyed by e i1...ip−2 and b i1...ip ((31) and (32)) are identical, so the algebraic constraints derived from them which apply on b i1...ip for a certain p will also automatically apply on e i1...i p ′ −2 for p ′ = p + 2 (and viceversa). Additionally, by duality constraints on b i1...ip for a certain p will also apply on e i1...i p ′ −2 for p ′ = n − p (and viceversa). In the magnetic and electric cases the following ranks p are worth mentioning.

B.1 Magnetic fields
• p = 2: this is the case studied in [27] and b ij = 0 if n is odd, so that n must be even (so that, in particular, the ln r term in 2H (eq. (28)) vanishes for n = 5 [27]). If F is assumed to be regular (and non-zero) on the transverse space, then Maxwell's equations (19), (35) for b ij 14 In passing, let us observe that (A22), (A23) and (A25) give the necessary and sufficient conditions under which any Weyl tensor of type II(bd) admits a second multiple WAND (so being in fact of type D(bd)), since no special features of the Robinson-Tratuman class have been used in their derivation. For the same reason, (A24) alone determines the conditions under which a Weyl tensor of type II(bd) admits a single WAND (in addition to a multiple one). This implies that a Weyl tensor of proper type D(bd) or D(bcd) admits precisely two WANDs (necessarily double) since (A24) has no non-zero solution if Ω ′ iĵ = Ψ ′ iĵk = Ψ ′ i = 0, Φ = 0 (as easily seen after contraction with zĵ ). However, Weyl tensors of type D(abd) may admit an infinity of multiple WANDs, and this occurs precisely when the equationΦlîĵkzl = 0 (cf. (A23)) admits a solution (in this case (A22) and (A25) are satisfied trivially) -see, e.g., [46,50] for examples of such spacetimes. Similarly, an infinity of single WANDs exists for type D(abd) if the weaker condition zkzlΦkîlĵ = 0 is satisfied. We observe that L. Wylleman has obtained more general results on the structure of multiple WANDs for any Weyl type D (private communication), some of which are mentioned in [40,46]. case, this always works when p = n 2 with p even (with d = 2, N = p − 1), see example (84). (For N = 2, m ′ = 1 this reduces to the case p = n 2 + 1 discussed above.) By combining the constructions for electric and the magnetic case, for d = 2 (m = m ′ + 1) one can also construct dyonic fields [31].
• p = n 2 + 2 with p even: these solutions can be constructed as explained for the magnetic forms with p = n 2 in section B.1.

B.3 Case 2p = n (n even)
It should be recalled that in the case p = n 2 , if both an electric and a magnetic field are present then they generically obey the weaker constrain (33) (the corresponding "Maxwell" equations are also generically modified, as discussed in section 5.2). The construction with p = n 2 (with p even) mentioned above in sections B.1 and B.2 still provides examples in the special case when (31) and (32) are satisfied separately (and not just (33)), but more general solutions also exist, see for example (85). Note, however, that in the case n = 6, p = 3, not only (31) and (32) (as discussed in sections B.1 and B.2) but also the weaker constraint (33) can be satisfied only trivially by E 2 = 0 = B 2 , as can be seen by directly substituting into (33) the most general possible form of a 1-form e and a 3-form b (examples are given by (100), (101)). Therefore, solutions 2p = n for which e and b are not both zero require n ≥ 8 (p ≥ 4) (see, e.g., (84), (85)).
C Cases p = 1 and p = n − 1 Here we analyze the special ranks p = 1 and p = n − 1. The results of section 3.1 apply also here and need not be repeated. Just note that there is no electric term e i1...ip−2 for p = 1 (while B 2 ij = b i b j ), and no magnetic term b i1...ip for p = n − 1.
As observed in section 3.2, a first difference appears in the Einstein equation for R ij . Instead of (29) and (32) [(31)] we now have the tracefree equations while (30) remains true, so that, again, h ij = h 1/(n−2) γ ij (x) (with det γ ij = 1.) As noticed, the results of section 3.3 are still valid also here. In particular we have F u = 0 for p = 1 and F ui1...in−2 = 0 for p = n − 1, and (40) are still true.

C.1 Case n > 4
To summarize, for n > 4 the metric is given by (55) with where R 0 , Λ and µ are constants, and h ij is a Riemannian metric satisfying (C1) or (C2) (and thus having a Ricci scalar given by (C7)). The Maxwell field is given by where b i (x) and e i1...in−3 (x) are harmonic forms in the geometry of h ij . In contrast to the case 2 ≤ p ≤ n − 2, here the Maxwell field does not enter the metric function H, but instead "backreacts" on the transverse geometry h ij , which thus cannot be Einstein. The Weyl type is D(bd) (since H = H(r)), the Maxwell type is D, and (59) are doubly aligned null directions for both the Weyl and the Maxwell tensor. The metric is static (at least in regions where H > 0). We further observe that (C1) gives (n − 2)R ij b i b j = B 2 [R + κ 0 (n − 3)B 2 ], which is certainly non-negative if R ≥ 0. Since b i is harmonic, this implies (see [42,51]; also theorem 2.9 of [43]) that h ij cannot describe a compact space having R ≥ 0 everywhere (unless, trivially, R = 0 = B 2 ). Thanks to (C1) and (C2), it is easy to see that any (n − 2)-dimensional solution of the Euclidean Einstein-Maxwell theory (i.e., a metric h ij coupled to a 1-form b i (x) or a (n − 3)-form e i1...in−3 (x)) can be used to generate an n-dimensional Robinson-Trautman spacetime coupled to a 1-form (or a (n − 1)-form). For example, by taking as a "seed" the Euclidean version of the 3D charged BTZ metric (with "J = 0") [52] we obtain the following 5D example.
Example (n = 5, p = 1) where λ = R0 6 , m and b x are constants (respectively, the cosmological constant, mass parameter and field strength of the 3D BTZ solution), and R = 6λ + κ 0 b 2 x ρ −2 . Having a Lorentzian signature requires (at least for a large ρ) that λ < 0. The dualization to p = 4 is obvious. We observe however that the ρ = 0 curvature singularity of the BTZ metric extends also to the full 5D solution also beyond the horizon(s). Euclideanization of this 5D solution can be used, in turn, to produce a 7D Robinson-Trautman solutions, and so on to higher odd dimensions.

C.2 Case n = 4
We observe that for n = 4 the l.h.s. of both (C1) (p = 1) and (C2) (p = 3) is identically zero (since h ij is 2-dimensional), which implies, respectively, b i = 0 and e i = 0. Since we also have F u = 0 for p = 1 and F ui1...in−2 = 0 for p = n − 1, we conclude that F vanishes identically and one is left only with vacuum Robinson-Trautman spacetimes, obeying the standard equation −△R + 4µ ,u + 6µ(ln √ h) ,u = 0 (this contains, in particular, the Schwarzschild metric, which has F = 0 and yet can describe a black hole with non-zero axionic charge in the presence of a nonzero Kalb-Ramond field -in fact it is the only such static and asymptotically flat solution [53]). Therefore in 4D the only electrovac spacetimes with F = 0 are obtained for p = 2, which is the wellknown Einstein-Maxwell case [12][13][14]. We observe that certain Robinson-Trautman 4D solutions with aligned p = 3 forms (axionic black holes) have been discussed in [31], but these involve multiple p-form fields and thus do not contradict our result.