9 can be written in the form, are not invariant under the coordinate transformations (3.11). In particular, under these transformations, we have For spherically symmetric static solutions, the extra term δR$_{ij}$ in general gives rise to an anisotropic fluid with heat flow [38], possibly subjected to some energy conditions [31]. In particular, for the LMP solutions of Eqs. (3.1) and (3.21), the energy density J $^{t}$, defined by R$_{ij}$ → R$_{ij}$ + δR$_{ij}$. (A.1) τ$_{ij}$ = e 2 $^{ν}$p$_{r}$ δ i $_{r}$δ j r + r $^{2}$p$_{θ}$ Ω$_{ij}$ , (A.8) where Ω$_{ij}$ ≡ δ θ i δ θ j + sin 2 θδ φ i δ φ j , and J t = 2 ( N δ L M δN + L M ) , (A.2) ∫ d $^{3}$x √ g ( L K + L $_{V}$) = 8 πG ∫ d $^{3}$x √ g J $^{t}$, (A.3) is given by the Hamiltonian equation, where p$_{r}$ = Λ$_{W}$ e 2( µ − ν ) 64 πGx $^{2}$Δ2 { ξx [ 4 x ΔΔ ′′ − 3 x Δ ′ 2 + 4 β$_{±}$ ΔΔ ′ ] +8Δ [ x Δ ′ − 2(1 − ξ ) β$_{±}$ Δ ] + 8(1 − 4 ξ )Δ 2 } − e − 2 ν 8 πG F$_{rr}$, p$_{θ}$ = Λ$_{W}$ e 2( µ − ν ) 16 πGx $^{2}$Δ2 { (1 − ξ ) x $^{2}$ΔΔ ′′ + 1 4 ξx $^{2}$Δ ′ 2 +(1 − ξ ) β$_{±}$x ΔΔ ′ + 2(1 − 2 ξ ) ( x Δ ′ + β$_{±}$ Δ ) Δ } + Λ$_{W}$ 8 πGx $_{2}$F$_{θθ}$ , (A.9) L K = − Λ$_{W}$ e 2( µ − ν ) 4 x $^{2}$Δ { ξx $^{2}$Δ ′ 2 − 8(1 − ξ ) x ΔΔ ′ − 8(1 − 2 ξ ) x Δ 2 } , L V = Λ$_{W}$ x 2 [ 2 + 3 x 2 + 2 ( 1 − 4 α$_{±}$ ) x 2 β ± ] + Λ$_{W}$ x 8(1 − α ± ) { [ 8 ξα 2 ± − 8(1 − ξ ) ( 1 − α$_{±}$ ) + 3 ] +2 [ 4(1 − ξ )(1 − α$_{±}$ ) − 1 ] x 2(1 − 2 α ± ) +(1 − 2 ξ ) x 4(1 − 2 α ± ) } , (A.4) F$_{rr}$ = Λ$_{W}$ 2 x 2 [ 2 − 2 x − 2 β ± − 3 x 2(1 − β ± ) ] − Λ$_{W}$ 4 x 4 − 2 β ± [ ( 19 − 32 ξ ) +4(3 − 5 ξ ) 2 − 3 β$_{±}$ ) β$_{±}$ − 2 ( 7 − 12 ξ ) x − 2 β ± − ( 5 − 8 ξ ) x − 4 β ± ] , F$_{θθ}$ = 3 2 x 2 − β$_{±}$x 2 β ± + 1 2 x 2 − 4 β ± { (1 + 2 ξ ) +2(11 − 16 ξ ) β$_{±}$ − 2 ξ ( 7 − 6 β$_{±}$ ) β 2 ± − 2 ( 1 − β$_{±}$ ) x − 2 β ± + (1 − 2 ξ ) x − 4 β ± } . (A.10) J i ≡ − N δ L M δN$_{i}$ = e − ( µ + ν $^{)}$( v, 0 , 0 ) , (A.5) with where Δ is given by Eq. (3.22), and Δ ′ ≡ d Δ /dx . The quantity v , defined by which is related to heat flow [38], is given by On the other hand, the corresponding stress part τ ij defined by, Clearly, to interpret the above as an anisotropic fluid with heat flow, some energy conditions [31] might need to be imposed. Since in this paper we are mainly concerned with the nature of singularities, we shall not study these conditions here. v = Λ$_{W}$ e 2( µ − ν ) 32 πGx $^{2}$Δ2 { ξx [ 2 x ΔΔ ′′ + x Δ ′ 2 − 2 β$_{±}$ ΔΔ ′ ] +4Δ [ ξx Δ ′ − 2(1 − ξ ) β$_{±}$ Δ ] − 8 ξ Δ 2 } . (A.6) τ ij = 2 √ g δ ( √ g L M ) δg$_{ij}$ , (A.7) [1] P. Horava, JHEP, 0903 , 020 (2009) [arXiv:0812.4287]; Phys. Rev. D 79 , 084008 (2009) [arXiv:0901.3775]; and Phys. Rev. Lett. 102 , 161301 (2009) [arXiv:0902.3657].