We investigate solutions of new massive gravity with two commuting Killing vectors, one of which is null, with a special emphasis on black hole solutions. Besides extreme BTZ black holes and, for a special value of the coupling constant, massless null warped black holes, we also obtain for a critical coupling a family of massive “log” black holes. These are asymptotic to the extreme BTZ black holes in the sense of log gravity.
1 Introduction
It is well known that threedimensional cosmological Einstein gravity is dynamically trivial, without propagating degrees of freedom. This is cured in topologically massive gravity (TMG) [2] by the addition to the Einstein action of a parityviolating gravitational ChernSimons term, leading after linearization to the propagation of massive quanta. The same goal is achieved in a paritypreserving fashion in the recently proposed new theory of massive gravity (NMG) [3] through the addition of a particular quadratic combination of the curvature tensor components. This theory was found to be unitary in the treelevel in [4], and renormalizable in [5]. New massive gravity was shown in [6] to admit both BTZ black holes [7] and warped black holes [8, 9, 10] as solutions, and the entropy, mass and angular momentum of these black holes were computed. The central charges for these black holes were recently obtained in [11].
In cosmological TMG linearized around a constant curvature background, either the massive gravitons or the BTZ black holes have negative energy, except for a critical, “chiral” value of the ChernSimons coupling constant at which all the masses vanish [12]. However it was shown in [13] that for this special value TMG also admit a family of massive black holes with the extremal BTZ black hole as ground state. These exact “log” solutions are a limiting case of a family of exact solutions of cosmological TMG first constructed in [14].
Similarly, the sign of the energy of massive excitations of NMG linearized around an background is opposite to the sign of the mass of the BTZ black holes, except for the critical value of the quadratic coupling constant, at which all the masses again apparently vanish [15, 16] ^{1}^{1}1Actually it has recently been shown [20] that the situation is more subtle, the massless gravitons being replaced for by massive photons.. The purpose of this paper is to investigate whether the special massive black hole solutions of [13] can be generalized to critical NMG.
The solutions of [13] are actually a special case of wave solutions of TMG [17, 18, 19], reinterpreted as black hole solutions. As the present work was under way, the paper [21] came out, in which wave solutions of NMG are constructed and studied. Our solutions are also special cases of the solutions of [21], however the interpretation is different.
In the next section we construct, using the methods of [14] and [6], solutions of NMG with two commuting Killing vectors, one of which is null. These solutions exist for all real values of the coupling contant , however, as shown in Sect. 3 based on the results of the Appendix, they lead to regular black holes (other than the extreme BTZ black holes) only for the three values , and . The entropy, mass and angular momentum of these black holes are computed in Sect. 4. Only the analogs of the black holes of [13] are massive. Our results are briefly discussed in the last section.
2 Stationary solutions with a null Killing vector
The action of the cosmological new massive gravity theory is [3]
(2.1) 
where is the trace of the Ricci tensor , the quadratic curvature invariant is
(2.2) 
and , and are the Newton constant, quadratic coupling constant and “bare” cosmological constant.
Let us briefly recall the dimensional reduction of this theory as carried out in [6] in the case of two commuting Killing vectors. We choose the parametrisation
(2.3) 
(, ), where is the matrix
(2.4) 
is the Minkowski pseudonorm of the “vector” ,
(2.5) 
and the scale factor allows for arbitrary reparametrizations of the radial coordinate . The scalar product of two vectors and is defined by , and their wedge product by
(2.6) 
(with ). The ansatz (2.3) reduces the equations of NMG to
(2.7)  
and
(2.8) 
The equations (2) are trivially solved by , leading to constant curvature spacetimes
(2.9) 
where and are two linearly independent constant vectors, and the scale of is related through (2) to the bare cosmological constant. A strategy to generate nontrivial solutions of the system (2)(2) is to consider linear deformations of the trivial ansatz (2.9) which solve these equations exactly. Warped black hole solutions were obtained in [6] from the quadratic ansatz , with
(2.10) 
for some real constant , implying
(2.11) 
In this paper, we consider the ansatz [14]
(2.12) 
where the form of the function shall be determined from the field equations.
Before applying this ansatz to NMG, it is instructive to recall its outcome in TMG. The field equations of TMG reduced according to the stationary circularly symmetric ansatz (2.3) are [14]
(2.13)  
(2.14) 
Provided the vectors and satisfy (2.10), the vector equation (2.13) is linearized by (2.12) to
(2.15) 
while the scalar equation (2.14) leads to
(2.16) 
for a negative cosmological constant . Without loss of generality we can choose and , leading to the solution [14] of the master equation (2.15)
(2.17) 
depending on three integration constants , and . Actually the constant is redundant and may be set to zero by a redefinition of the vector , , which does not affect Eq. (2.10). For () or () the solution (2.17) with degenerates to [13]
(2.18)  
(2.19) 
In the case of NMG, the vector equation (2) is again linearized by the ansatz (2.12) (with and satisfying (2.10)) to the fourth order equation
(2.20) 
the scalar equation (2) leading to the constraint
(2.21) 
Assuming , we again choose and
(2.22) 
where the effective curvature parameter is obtained by solving (2.21),
(2.23) 
The solution of the master equation (2.20) then leads to
(2.24) 
now depending on three integration constants , and (again we have discarded a redundant term by a redefinition of the vector ). Note that from (2.23),
(2.25) 
so that the square root in (2.24) is real either for the upper sign in (2.23) and , , or for the lower sign in (2.23) and , . For , (2.24) is replaced by
(2.26) 
(with another integration constant). For the special value (), (2.24) or (2.26) degenerate to
(2.27) 
Finally, (2.24) is also degenerate for the value (), where it must be replaced by
(2.28) 
or, if ,
(2.29) 
( constant).
The choice of basis vectors
(2.30) 
leads to the metric
(2.31) 
which, in the limiting case of a constant M/2, reduces to the extreme BTZ metric with J/ M. The metric (2.31) can be put in the form
(2.32) 
with , , , . This has the obvious Killing vectors and , the latter being null. In the generic case these are the only infinitesimal isometries of (2.32). In the case of the solution (2.24) with , the metric has a third local isometry generated by
(2.33) 
Similarly, in the case of the solution (2.28) () with , the third local Killing vector is
(2.34) 
Finally, in the special cases or with , the metric (2.32) describes respectively extreme BTZ black holes or null warped black holes (see next section), which both admit four local Killing vectors generating the algebra (the Killing vectors for the null warped black hole case are given in Eqs. (6.3) and (6.4) of [9]).
All the results of this section are consistent with the results of [21], so that our solutions (2.32) with given by (2.24) and (2.26)(2.29) are special cases of the waves
(2.35) 
of [21]. For these solutions to describe black holes, the spacelike Killing vector should have closed orbits, which essentially restricts (2.35) to (2.32).
3 Black holes
In the present paper we are interested in regular black hole solutions. The metric (2.31) has a horizon at . As pointed out in [14], all the scalar curvature invariants constructed from this metric are constant (the function does not contribute because is null and orthogonal to ), however the metric may develop nonscalar curvature singularities at the horizon , as well as at , or the horizon can be at geodesic infinity. To elucidate this question, we consider the first integral of the geodesic equation
(3.1) 
with , where and are the constant conjugate momenta to and , and or for timelike, null, or spacelike geodesics. In the discussion of this equation, we can exclude outright the case (2.26), for which the areal radius
(3.2) 
oscillates wildly around zero for , leading to naked closed timelike curves (CTC). For the metric (2.31) reduces under the radial coordinate transformation to the extreme BTZ black hole metric with mass parameter M . If and do not both vanish, a lengthy analysis, carried out in the Appendix, leads to the conclusion that the metric (2.31) leads to regular black holes in only three cases:
1) For , the solution (2.24) with , , and yields a regular black hole (free from naked CTC). After transforming to a rotating frame in which the null Killing vector is , which amounts to replacing the basis vectors (2.30) by
(3.3) 
this leads to a null warped black hole [9, 10] (corresponding to the warped black holes of [6] with )
(3.4) 
This metric is in ADM form with the square lapse and the shift given by , .
2) For , the solution (2.24) with , , , and leads (again after transforming to the basis (3.3)) to the metric
(3.5) 
(). This spacetime is free from naked CTC. The point horizon at hides a timelike causal singularity at . At spacelike infinity, the twodimensional metric reduced relative to goes as
(3.6) 
with , so that spacelike infinity is conformally timelike.
3) For the value (), the “logarithmic” solution (2.28) leads to three black hole subcases:
a) If and , the metric in the basis (3.3) is similar to (3.4) with
(3.7) 
Necessary conditions for the absence of naked CTC are (no CTC at infinity), and (no CTC near the horizon). This metric is “almost” asymptotically (the asymptotic behavior overshoots that of by a logarithmic factor).
b) If and (solution (2.29)), the only difference with the preceding case is that now
(3.8) 
with . This is free from naked CTC provided and .
c) If and , the ADM shift goes at spatial infinity to a constant in the basis (3.3). A metric with an asymptotically vanishing shift function may be obtained by transforming back to the frame (2.30). This metric,
(3.9)  
is free from naked CTC provided . It is asymptotically in the weak sense of log gravity [22, 23, 24], and develops a timelike causal singularity at some negative .
Note that (2.24) with reduces to (2.17), so that the black holes (3.4) and (3.5) also solve the equations of TMG (the fact that these are black holes was overlooked in [14]), while the black holes (3.8) and (3.9) reduce (after appropriate coordinate transformations) to the black hole solutions of TMG (2.18) and (2.19) given in [13].
4 Physical parameters
Now we compute the physical parameters of these black holes. By applying Wald’s general formula [25] the black hole entropy was found in [6] to be given in NMG by
(4.1) 
evaluated on the horizon , with . Because of the possible presence of logarithmic factors, we evaluate this more carefully than in [6]. From the expressions of the Ricci tensor components given there, we find (for )
(4.2) 
For the ansatz (2.12), this leads to
(4.3) 
For the null warped black holes (3.4), this formula gives the BekensteinHawking entropy renormalized by a factor [6]. In the case of the black holes (3.5) and (3.8), the formula (4.3) yields straightforwardly
(4.4) 
The case of of the black holes (3.7) and (3.9) is more delicate. In this case, does not vanish on the horizon but goes to a constant, however this is suppressed by the prefactor which goes to zero as an inverse logarithm, while the first term in (4.3) does not contribute because for this case, so that the net entropy is again zero.
Provided
(4.5) 
the metric (2.31) is asymptotically . In that case we can for large linearize the metric around the BTZ vacuum as
(4.6) 
and use the AbbottDeserTekin (ADT) approach [26, 27] to compute the mass and angular momentum of our black holes. The ADT conserved charge associated with a background Killing vector has been computed for NMG in [16]. Using the coordinatefree parametrization (2.12), such that , , and evaluating the covariant derivatives and Ricci tensor components with the help of the formulas given in Appendix B of [28], we obtain^{2}^{2}2The sign of our is opposite to that used in [16]. from Eqs. (2.25) and (2.27) of [16] the Killing charge
(4.7) 
where is the matrix
(4.8) 
The charge (4.7) is a constant of the motion by virtue of (2.20). For the generic solution (2.24) or (2.26), as well as in the special cases (2.27) and (2.29), the evaluation of this charge leads to
(4.9) 
while for the special solution (2.28) (), we find
(4.10) 
These values of the charge were derived under the assumption (4.5) which is valid, in particular, for the generic solution (2.24) with , provided and for the solution (2.28) () with . We will here assume that they can be extrapolated to the cases with and with . A more satisfactory derivation would require an extension of the ADT approach to the case of massive gravity with nonconstant curvature backgrounds, similar to that carried out in [28] for topologically massive gravity. Choosing to be one of two Killing vectors and , we obtain the mass and angular momentum of the various black holes of the previous section^{3}^{3}3The superangular momentum approach of [29] as applied to NMG in [6] leads to the same results.:
2) For the black holes (3.5),
(4.12) 
3c) For the black holes (3.9),
(4.14) 
Because the absence of naked CTC constrains , the mass is negative for a positive Newton constant , and vanishes as it should in the limit of the extreme BTZ black hole for [6]. The values (4.14) may be compared with the corresponding values for the same black holes as solutions for TMG. Evaluating for the ansatz (2.12) with (2.10) the TMG superangular momentum [28], we obtain
(4.15) 
leading for , given by (2.19) and as in (2.30) to
(4.16) 
(the correction coming from the last three terms of Eq. (3.15) of [28] vanishes in the present case). These values agree with those of [13] (Eq.(24), where is our ) up to a factor 2/3.
Finally the Hawking temperature and the horizon angular velocity, computed from the metric in ADM form, are
(4.17) 
The resulting Hawking temperatures vanish for all our black holes, . The horizon angular velocities vanish for the black holes (3.4) with and the black holes (3.7) and (3.8), while for the black holes (3.4) with and the black holes (3.5), and for the black holes (3.9). It follows that the first law of black hole thermodynamics, which in the case of vanishing Hawking temperature reduces to
(4.18) 
is satisfied trivially (both sides vanish) for the black holes (3.4), (3.5),(3.7), and (3.8), and non trivially () for the black holes (3.9)
5 Discussion
In this paper, we have investigated solutions of new massive gravity with two commuting Killing vectors, one of which is null, with a special emphasis on black hole solutions. In addition to the wellknown extreme BTZ black holes, we found several black hole types. The first of these includes the black holes (3.5) () and (3.8) (), both with (so that, as shown at the end of Sect. 2, they have a third local Killing vector). Because all their physical characteristics (entropy, mass and angular momentum) vanish, these are not genuine black holes. A second family () includes the null warped black holes of [9], with metric (3.4) enjoying a local isometry algebra. These have as null Killing vector, are massless, but have a nonzero angular momentum. The third family (), with metric (3.7), has similar properties, but only two local isometries.
The most interesting fourth black hole type (also , corresponding to the value of the bare cosmological constant from (2.23)) differs from the preceding by the fact that it is asymptotically in the sense of log gravity [22, 23, 24]. This implies that in the basis (3.3) appropriate to the other black hole types, the ADM shift function does not vanish at infinity. After transforming to the basis (2.30) in which (which transforms the null Killing vector to ), we obtained a continuum of black hole states (3.9) with mass and angular momentum , above the massless extreme BTZ family as ground state. These properties are similar to those of the “log” solutions of TMG at the chiral point found in [13]. Further work is needed to understand the implications of these solutions on the consistency of new massive gravity at the critical value .
Appendix
We discuss here the geodesic equation (3.1) in the case of the generic solution (2.24). Assuming that and do not both vanish, we must distinguish between four possibilities for the leading nearhorizon behavior of the effective potential:

and either , or and . The leading term in is , with (if the spacetime is geodesically complete, with naked CTC [14]), so that the affine parameter is proportional to near the horizon. The geodesic equation (3.1) can be rewritten in terms of the adapted radial coordinate as
(A.1) The geodesics can be extended across the horizon if the effective potential in (A.1) is analytical in [30, 31]. A necessary condition for this is that the exponent be integer. However leads to , so that this exponent can be integer only for , corresponding to the special case (2.28) or (2.29) (see below). The conclusion is that generically geodesics terminate at the singular horizon .

and either , or and . The leading term in is constant, so that the adapted radial coordinate is , and the horizon is regular provided the function in (2.24) is analytic. For , this is not possible because is not integer. For , must be integer, . The case corresponds to (see below). In the case (), (2.12) reduces to the quadratic ansatz , with , leading (after an appropriate coordinate transformation) to null warped black holes (Eq. (3.17) of [9] with , , , and in (3.20)). Finally, in the case and ( leads to naked CTC), the affine parameter is for large proportional to , so that geodesics terminate at infinity.

, and . The leading term in is , with ( leads again to a geodesically complete spacetime with naked CTC) so that the affine parameter is proportional to near the horizon. The geodesic equation rewritten in terms of is of the form (A.1) with replaced by . From , we find , so that the exponent is not integer, and the horizon is singular.

, and . Now the leading term in the effective potential is linear in , so that the affine parameter is proportional to near the horizon. The geodesic equation rewritten in terms of is
(A.2) The effective potential is analytical provided , with a positive integer. For (, corresponding to ), we obtain (after transforming to a rotating frame) the metric (3.5). For (), the resulting metric is (again after transforming to a rotating frame)a special case of the null warped black holes of [9] (with in eq. (3.20) of [9]). For , the affine parameter is for large proportional to , so that again geodesics terminate at infinity.
There remains the case of the special solutions (2.27) and (2.28) or (2.29). The solution (2.27) corresponds to the degenerate case , which from the previous analysis cannot possibly lead to regular black holes. In the case of the solution (2.28), the affine parameter is for proportional to near the horizon. The geodesic equation rewritten in terms of will contain only integer powers of and powers of , which is positive on both sides of the horizon, so that geodesics can be continued across the horizon. The same conclusion holds for the solution (2.29) with and .
References
 [1]
 [2] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. Phys., NY 140 (1982) 372.
 [3] E.A. Bergshoeff, O. Hohm and P.K. Townsend, Phys. Rev. Lett. 102 (2009) 201301.
 [4] M. Nakasone and I. Oda, “On unitarity of massive gravity in three dimensions”, arXiv:0902.3531.
 [5] I. Oda, JHEP 0905 (2009) 064.
 [6] G. Clément, Class. Quantum Grav. 26 (2009) 105015.
 [7] M. Bañados, C.Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849; M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48 (1993) 1506.
 [8] K. Ait Moussa, G. Clément and C. Leygnac, Class. Quantum Grav. 20 (2003) L277.
 [9] K. Ait Moussa, G. Clément, H. Guennoune and C. Leygnac, Phys. Rev. D 78 (2008) 064065.
 [10] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, JHEP 0903 (2009) 130.
 [11] W. Kim and E.J. Son, “Central charges in 2D reduced cosmological massive gravity”, arXiv:0904.4538.
 [12] W. Li, W. Song and A. Strominger, JHEP 0804 (2008) 082.
 [13] A. Garbarz, G. Giribet and Y. Vásquez, Phys. Rev. D 79 (2009) 044036.
 [14] G. Clément, Class. Quantum Grav. 11 (1994) L115.
 [15] Y. Liu and Y.W. Sun, JHEP 0904 (2009) 106.
 [16] Y. Liu and Y.W. Sun, “Consistent boundary conditions for new massive gravity in ”, arXiv:0903.2933.
 [17] E. AyónBeato and M. Hassaïne, Ann. Phys. 317 (2005) 175; Phys. Rev. D 73 (2006) 104001.
 [18] G.W. Gibbons, C.N. Pope and E. Sezgin, Class. Quantum Grav. 25 (2008) 205005.
 [19] S. Carlip, S. Deser, A. Waldron and D.K. Wise, Phys. Lett. B 666 (2008) 272.
 [20] E.A. Bergshoeff, O. Hohm and P.K. Townsend, “More on massive 3D gravity”, arXiv:0905.1259.
 [21] E. AyónBeato, G. Giribet and M. Hassaïne, JHEP 0905 (2009) 029.
 [22] D. Grumiller and N. Johansson, JHEP 0807 (2008) 134; Int. J. Mod. Phys. D 17 (2009) 2367.
 [23] M. Henneaux, C. Martinez and R. Troncoso, “Asymptotically antide Sitter spacetimes in topologically massive gravity”, arXiv:0901.2874.
 [24] A. Maloney, W. Song and A. Strominger, “Chiral gravity, log gravity and extremal CFT”, arXiv:0903.4573.
 [25] R.M. Wald, Phys. Rev. D 48 (1993) 3427.
 [26] L.F. Abbott and S. Deser, Nucl. Phys. B 195 (1982) 76.
 [27] S. Deser and B. Tekin, Phys. Rev. Lett. 89 (2002) 101101; Phys. Rev. D 67 (2003) 084009.
 [28] A. Bouchareb and G. Clément, Class. Quantum Grav. 24 (2007) 5581.
 [29] G. Clément, Phys. Rev. D 68 (2003) 024032.
 [30] G. Clément and A. Fabbri, Class. Quantum Grav. 16 (1999) 323.
 [31] K.A. Bronnikov, G. Clément, C.P, Constantinidis and J.C. Fabris, Grav. & Cosm. 4 (1998) 128.