No Scalar-Haired Cauchy Horizon Theorem in Einstein-Maxwell-Horndeski Theories
aa r X i v : . [ h e p - t h ] F e b arXiv:2101.1101v2 [hep-th] No Scalar-Haired Cauchy Horizon Theorem inEinstein-Maxwell-Horndeski Theories
Deniz O. Devecio˘glu ∗ and Mu-In Park † School of Physics, Huazhong University of Scienceand Technology, Wuhan, Hubei, 430074, China Center for Quantum Spacetime, Sogang University, Seoul, 121-742, Korea (Dated: February 4, 2021)
Abstract
Recently, a no inner (Cauchy) horizon theorem for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories. In this paper, we extend the theorem to thestatic black holes in Einstein-Maxwell-Horndeski theories. We study the black hole interior geometryfor some exact solutions and find that the spacetime has a (space-like) curvature singularity wherethe black hole mass gets an extremum and the Hawking temperature vanishes. We discuss furtherextensions of the theorem, including general Horndeski theories from disformal transformations.
Keywords: No hair theorem, Cauchy horizon, Black hole singularity, Horndeski theory ∗ E-mail address: [email protected] † E-mail address: [email protected], Corresponding author non-trivial scalar hairs in Einstein-Maxwell-scalar theories.From the simplicity and quite generic results, which do not depend on the details of the scalarpotentials as well as the spacetime structure at the short distance, it’s applicability would bequite far-reaching. In this respect, it would be important to classify all possible extensions ofthe theorem.In this paper, as the first step towards that goal, we consider an extension of the theorem tothe black holes in Einstein-Maxwell-Horndeski (EMH) theories, with the lowest-order Horndeskiterms [5]. To this ends, we consider the D -dimensional EMH action, with the U (1) gauge filed A µ and charged scalar field ϕ , I = Z d D x √− g (cid:20) κ R + L m (cid:21) , (1) L m = − Z ( | ϕ | )4 κ F µν F µν − ( αg µν − γG µν ) ( D µ ϕ ) ∗ ( D ν ϕ ) − V ( | ϕ | ) , where F µν ≡ ∇ µ A ν − ∇ ν A µ and D µ ≡ ∇ µ − iqA µ with a scalar field charge q and κ ≡ πG .Here, we have introduced Z and V as arbitrary functions of | ϕ | in which the constant shiftsymmetry of the Horndeski terms, with the Einstein tensor coupling as well as the usual minimalcoupling, may be broken generally.Let us then consider the general static ansatz, ds = − N ( r ) dt + 1 f ( r ) dr + r d Ω D − ,k , (2) ϕ = ϕ ( r ) , A = A t ( r ) dt, where d Ω D − ,k is the metric of ( D − k = 1 , , − D = 4 case for simplicity. But its higher-dimensional generalizationis straightforward (for the details, see Appendix A ) [4, 6] and we will discuss about this later]2re given by (the prime ( ′ ) denotes the derivative with respect to r ), E A ≡ κ (cid:18) Z ( | ϕ | ) √ fN r A ′ t (cid:19) ′ − q | ϕ | A t f (cid:18) √ fN (cid:19) (cid:2) αr + γ ( k − f ) − γrf ′ (cid:3) = 0 , (3) E ϕ ≡ (cid:18) N √ f f (cid:20) αr + γ ( k − f ) − γrf (cid:18) N ′ N (cid:19)(cid:21) ϕ ′ (cid:19) ′ + (cid:18) N √ f (cid:19) q A t ϕN (cid:2) αr + γ ( k − f ) − γrf ′ (cid:3) + r (cid:20) κ √ fN ˙ ZA ′ t − N √ f ˙ V (cid:21) ϕ = 0 , (4) E N ≡ κ ( f − k + rf ′ ) + r V ( | ϕ | ) + f (cid:2) αr + γ ( k + f ) + 3 γrf ′ (cid:3) | ϕ ′ | +2 γrf (cid:0) | ϕ ′ | (cid:1) ′ + 12 N (cid:18) κ Zf r A ′ t + 2 q | ϕ | A t (cid:2) αr + γ ( k − f ) − γrf ′ (cid:3)(cid:19) = 0 , (5) E f ≡ κ (cid:20) f − k + rf (cid:18) N ′ N (cid:19)(cid:21) + r V ( | ϕ | ) − f (cid:20) αr + γ ( k − f ) − γrf (cid:18) N ′ N (cid:19)(cid:21) | ϕ ′ | + 12 N (cid:18) κ Zf r A ′ t − q | ϕ | A t (cid:2) αr + γ ( k − f ) (cid:3)(cid:19) + γq N (cid:18) | ϕ | (cid:20) − rf A t (cid:18) N ′ N (cid:19) + 2 rf ( A t ) ′ (cid:21) + 2 rf A t ( | ϕ | ) ′ (cid:19) = 0 , (6)where the first two equations (3), (4) are the equations for the gauge field and scalar fieldrespectively and the last two (5), (6) are for the metric. In particular, (3) and (5) correspondto the Gauss’s law constraint and
Hamiltonian constraint equations in canonical formulation,respectively. Even though it is difficult to solve the coupled non-linear PDE generally, severalexact solutions are known for some specific choices of couplings and potentials. We will studythe black hole interior geometry for those exact solutions in details. But before this, we firstprove the “no scalar-haired Cauchy horizon theorem” quite generally, without knowing thedetails of the interior geometry, following [4].
No Scalar-Haired Cauchy Horizon Theorem : The proof depends on whether the scalar fieldis charged or not. Let us first consider the charged scalar field case . Then, suppose that therewere a finite-temperature ( i.e. , non-extremal) black hole with the outer and inner (Cauchy)horizons at r + and r − ( < r + ), respectively (Fig. 1), i.e. N ( r + ) = f ( r + ) = 0 , N ′ ( r + ) , f ′ ( r + ) > , (7) N ( r − ) = f ( r − ) = 0 , N ′ ( r − ) , f ′ ( r − ) ≤ . (8) We may take a real scalar field because the phase field ω of the “complex” scalar field with a charge ϕ = | ϕ | e iω can be consistently fixed to be zero in the equations of motion (3) - (6), without affecting the coupling to gaugefield. We may further consider f ( r ) ≡ N e W ( r ) with a smooth (finite) function W ( r ) at the horizons, for theconvenience of the proof without loss of generality [4]. The holography-inspired metric in the literature [3, 4]can be also obtained from the coordinate transformation, r = 1 /z, N ( r ) = e − χ ( z ) ˜ f ( z ) /z , f ( r ) = ˜ f ( z ) /z . (cid:2879) (cid:2200) (cid:2878) (cid:84)(cid:18) FIG. 1: Typical plots of N ( r ) and f ( r ) for a finite-temperature , charged black hole with the inner(Cauchy) horizon r − and outer (event) horizon r + . Due to a non-zero Hawking temperature, we have N = f = 0 , N ′ , f ′ > r + , whereas N = f = 0 , N ′ , f ′ ≤ if existed (orange, solid line). In the absence of an inner horizon, N ′ , f ′ can havearbitrary values but there is no inner point of N = f = 0 (blue, dashed line). On the other hand, inthe absence of outer horizon but now with an inner horizon, the situation is similar for an observersitting inside the inner horizon which becomes now an event horizon (green, dashed line). Now, in order to have a “non-trivial”, i.e. , finite charged scalar field ϕ at the two horizons, theequations of motion (3 - 6) imply the conditions [4], A t ( r + ) = A t ( r − ) = 0 , (9)which being one of the key ingredients in the proof.Another key ingredient is the existence of radially conserved charge, Q = ( D − r D − (cid:20) N p f (cid:18) κ + γ f | ϕ ′ | (cid:19) (cid:18) N ′ N − r (cid:19) − κ √ fN ZA t ′ + √ fN γq A t | ϕ | (cid:18) A ′ t A t − N ′ N − r (cid:19) + √ fN γq A t | ϕ | ′ (cid:21) + k ( D − D − Z r dx x D − " γq A t | ϕ | p N f + N √ f (cid:18) κ − γf | ϕ ′ | (cid:19) , (10)where we have recovered the dimensional dependencies. From the equations of motion (3) - (6),one can prove that Q is radially conserved, i.e., r-independent , due to a (remarkable) relation This phenomena which represents a “repulsion between the charged scalar field and the gauge field”, similarto
Meissner effect in superconductivity. This seems to be quite generic [4] and might reflect the black holehorizon superconductivity . , Q ′ = 2 √ fN (cid:20) r (cid:18) N f E f ′ + 12 N ′ f E f (cid:19) + (cid:18) ( D − N f + r N ′ f (cid:19) E N + r N ϕ ′ E ϕ + 12 (( D − A t + rA ′ t ) E A (cid:21) = 0 , (11)where E ψ ’s represent the equations of motion for the fields ψ = ( N, f, ϕ, A t ) which can beobtained directly from the action and related to E ψ ’s in (3) -(6) as E N = − E N r D − / , E f = E f r D − / , E ϕ = 2 E ϕ /N √ f , E A = − E A N/ √ f .For the planar topology, k = 0, the first line of (10) is a Noether charge associated with aparticular scaling symmetry for a neutral scalar field with q = 0 [6, 7]. For other topologies k = 0, the local parts in the first and second lines are not conserved but the non-local , integratedterms are needed in order that the non-conserved terms are canceled [4].Now, due to the r -independence, the charge satisfies Q r + = Q r − at the two horizons, inparticular, where the boundary conditions (7), (8), and (9) hold, we have the relation ( D − κ (cid:20) r D − √ fN N ′ (cid:12)(cid:12)(cid:12) r + − r D − − √ fN N ′ (cid:12)(cid:12)(cid:12) r − (cid:21) = k ( D − D − Z r + r − dx x D − " − γq A t | ϕ | p N f + N √ f (cid:18) − κ + γf | ϕ ′ | (cid:19) . (12)From the assumed inner horizon r − , i.e. , N ′ √ f /N | r − ≤
0, as well as the outer horizon r + witha non-vanishing Hawking temperature or surface gravity ∼ N ′ √ f /N | r + >
0, the left handside of (12) is positive . On the other hand, the integrand on the right hand side is negative forthe “positive” Horndeski coupling constant γ > r + and r − , where N , f <
0, so that (12) is inconsistent unless the hyperbolic topology k = − no smooth inner (Cauchy) horizon with the outer horizon canbe formed for k = 0 or k = 1 and this generalize the theorem of [3, 4] to the EMH theories.Here, it interesting to note that the theorem may also imply the no outer horizon with an innerhorizon as described in Fig.1 (green, dashed line). This completes the proof of the theorem forthe charged scalar field with q = 0 and note that the proof is quite generic without assumingany specific form of the scalar potential. In the black hole interior, where r is a time coordinate, this becomes a dynamical relation whose detailedphysical meaning is still unclear and needs to be investigated. For D = 3, the result is essentially the same as in the planar topology due to the absence of spatial curvatureso that the theorem applies always (we thank the authors of [4] for the discussion about this matter). For D = 2, the charge Q of (10) vanishes trivially so that there is no constraint on D = 2 black holes from(12). This is due to a triviality of the D = 2 Einstein gravity and a separate consideration is needed for anappropriate treatment. But, since the proper D = 2 gravity, which being a dilaton gravity , can be obtainedfrom a dimensional reduction of D = 3 gravity, the result is not much different from that of D = 3 for theconventional Kaluza-Klein reduction (for an explicit computation, see Appendix B ).
5n the other hand, for the k = − can be consistent with the innerhorizon but it can be still consistent even without the inner horizon, i.e. , N ′ √ f /N | r − > k = − q = 0), the vanishing gauge field (9) can not be justified anymore and thereis no constraint on gauge field from the regularity at the horizons so that the novel relation(12) can not be used for the proof and we need a separate consideration as described below.However, we note that the same theorem also applies to neutral black holes, i.e. , in the absenceof U (1) gauge field A µ , since the essential parts of (12) do not depend on it.For the proof of neutral scalar field, following [3], we multiply ϕ ∗ to (4) and obtain0 = Z r + r − (cid:18) N √ f f (cid:20) αr + γ ( k − f ) − γrf (cid:18) N ′ N (cid:19)(cid:21) ϕ ∗ ϕ ′ (cid:19) ′ = Z r + r − N √ f (cid:26) f (cid:20) αr + γ ( k − f ) − γrf (cid:18) N ′ N (cid:19)(cid:21) | ϕ ′ | − r (cid:20) κ fN ˙ ZA ′ t − ˙ V (cid:21) | ϕ | − q A t | ϕ | N (cid:2) αr + γ ( k − f ) − γrf ′ (cid:3)(cid:27) . (13)For the neutral scalar field with q = 0, where the last term in the third line is absent, theintegrand in the second line is non-positive if the quantities in [ ] for first and second terms inthe second line are “positive”. In this case, the second line is consistent with the first line onlyfor the vanishing scalar field ϕ = 0 in the range of ( r − , r + ), which means the impossibility ofinner (Cauchy) horizon with a non-trivial scalar field ( i.e., hair ) and an outer (event) horizon,or similarly the impossibility of an outer horizon with a non-trivial scalar field and an innerhorizon, as in the charged case. For the Einstein-Maxwell-scalar theories with α > , γ = 0 asin [3, 4], the required condition for the theorem is ˙ ZA ′ t f /N κ − ˙ V >
0. With the Horndeskiterms with γ >
0, we may need a further condition, like αr + γ ( k − f ) − γrf N ′ /N >
0. Thisproduces the constraint on the range, r < N [ αr + γ ( k − f )] /f N ′ , near r + (or equivalentlynear r − ) with N ′ > could imply the non-vanishingscalar field in the black hole interior near r + . On the other hand, in the presence of a charge q , the last term gets a positive contribution even if the above conditions are satisfied so thatone can not generally rule out the inner (or outer) horizon formations, as noted in [4], unlesswe need more conditions on the metric and matter fields or electric charge. An Example of Interior Geometry with Inner Horizon
Now, in order to study explicitly the interior geometry of a charged black hole with a non-trivial scalar field, we consider an exact solution for α = 0 (no minimal scalar coupling), q = 0(neutral scalar field), Z = 1 (minimal gauge coupling), V = 2Λ /κ (cosmological constant term,no self-scalar-coupling) in D = 4, for simplicity (for k = 1, see [6, 8, 9] and also [10, 11]) but For the k -dependent terms, there is a k -factor difference with [8]. This gives the different results for k = − k = 0, where the solution (17) is not defined, we need a different solution [8]. (cid:2879) (cid:2200) (cid:1499) (cid:84) (cid:2200) (cid:2878) FIG. 2: Plots of N ( r ) (blue line) and f ( r ) (red line) for a charged black hole solution (14), (15) with k = − , M < <
0. Due to a Horndeski coupling γ = 0, the metric function f is divergent at r ∗ ≈ .
18 where a (space-like) curvature singularity occurs, in addition to the usualsingularity at r = 0. Moreover, even with the inner horizon r − , no outer horizon r + is formed (red,blue solid lines), in contrast with two horizons in the pure Einstein-Maxwell theories (dashed line).Here, we have considered Λ = − . , q = 1 , M = − . γ = 0 (non-vanishing Einstein tensor-scalar coupling), N = k (cid:18) k + Λ¯ q (cid:19) − Mr + k q r − k
192 ¯ q r − k Λ3 r + k Λ r , (14) fN = 64 r (4Λ r − kr + ¯ q ) , (15) A t = A (0) t − k ¯ qr + ¯ q r − ¯ q Λ2 r, (16) ϕ ′ = s r + ¯ q /r − γkκf , (17)where M and ¯ q represent the two integration constants, associated with the black hole massand charge, respectively. Here, we note that the solution can be defined only when γ = 0 isconsidered though not manifest in the metric and gauge field solutions. But interestingly, thesolution for k = ±
1, pure Einstein-Maxwell theories with a scalar field, i.e. , α = γ = 0, can bealso read with a certain truncation of higher-order terms, N = f = k − Mr + 14 ¯ q r − Λ3 r , (18) A t = A (0) t − k ¯ qr , (19)which seems to imply a smooth limit of γ → r ∗ = s − k ∓ p k − Λ¯ q / − Λ (20)in which f /N in (15) diverges, as well as the usual point-like singularity at r = 0 (see Fig. 2,for the case k = − , M < r H reaches to the singular radius r ∗ , i.e. r H = r ∗ , the black hole mass parameter M , M = k (cid:18) k + Λ¯ q (cid:19) r H + k q r H − k
192 ¯ q r H − k Λ3 r H + k Λ r H (21)becomes an extremum value, M ∗ = [ k (Λ ¯ q − k Λ¯ q − k ) ∓ k + Λ¯ q ) p k − Λ¯ q/ √ Λ( k ± p k − Λ¯ q / / . (22)Moreover, at the extreme horizon radius r H = r ∗ , the Hawking temperature , T H = ~ π (cid:18) N √ f (cid:19) N ′ (cid:12)(cid:12)(cid:12)(cid:12) r H = ~ k πr H (cid:0) k r H − k ¯ q − k Λ r H (cid:1) (23)becomes zero.For the hyperbolic case with k = − , M <
0, there is only a “cosmological” horizon r − , which used to be the inner (Cauchy) horizon in the purely Einstein-Maxwell theories withouta (neutral) scalar field (dashed line in Fig. 2). In this case, the outer horizon is not formeddue to Horndeski coupling γ of scalar field , which behaves as (with a vanishing integrationconstant) ϕ = 2 s rg ( r )2 γκ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − √ r − − r + O ( r − − r ) / , (24) g ( r ) ≡ r + ¯ q /r , (25)near the inner horizon r − , implying the vanishing and non-analytic scalar field at the innerhorizon when g ( r − ) /γ >
0. In the exterior region of the (inner) horizon, r > r − , where r is In this convention of the Hawking temperature T H , we have T H > r < r − and k = − r > r + and k = +1. This may be also compared with the Hawking temperature forthe pure Einstein-Maxwell case, T H = ~ k πr H (cid:0) k r H − q − r H (cid:1) . For the spherical case with k = +1 and M > , Λ <
0, the inner horizon is not formed instead and there isonly a black hole horizon r + . From the non-analyticity, ϕ ′ in (17) is singular at the horizon r H , where f ( r H ) = 0. But a coordinate-invariantquantity g µν ∂ µ ϕ∂ ν ϕ = f ( r ) ϕ ′ is regular for the whole region, including the singular point r ∗ . r ∗ , ϕ = − s k − k ¯ q /r γkκN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ∗ ( r ∗ − r ) / + O ( r ∗ − r ) / (26)when g ( r ) /γ < i.e. , r − < ˜ r < r with g (˜ r ) = 0 and ϕ ′ ( r ∗ ) = 0. On the other hand, the gaugefield is regular and analytic at the (inner) horizon and has a vanishing electric field, A ′ t = ¯ qr N √ f at r ∗ .As a summary, we have extended the “no scalar-haired inner (Cauchy) horizon theorem” tothe black holes in EMH theories. As an explicit example of the black hole interior geometrywith inner horizon, we studied some exact solutions with an inner horizon but no outer horizondue to non-trivial scalar field (hairs). Several further remarks about remaining challengingproblems are in order.1. We have considered a special case of Horndeski terms, with G , G in the usual termi-nology. By considering disformal transformations , ˜ g µν = A ( ϕ, X ) g µν + B ( ϕ, X ) ∇ µ ϕ ∇ ν ϕ , with X ≡ − g µν ∇ µ ∇ ν ϕ one can get the general form of the Horndeski action but with the specialcoefficients depending on A and B [12]. Hence, for the obtained Horndeski actions, one canalso argue the same no-Cauchy-horizon theorem. It would be interesting to see whether wehave the same result for the fully general Horndeski actions or not.2. The theorem implies that the instability of Cauchy horizon under non-trivial scalarfield (linear or non-linear) perturbations. However, the (dynamical) stability of the non-trivialhairy geometry is not be guaranteed generally and its explicit check would be an importantcomplementary for the theorem.3. In this paper, we have considered the Cauchy horizon for a “spherically symmetric”,charged black hole. It would be a challenging question whether the theorem can be generalizedto the non-spherically symmetric case also, if the removal of Cauchy horizon by the non-trivialscalar hairs is a real one. One outstanding case would be rotating black holes and it would beinteresting to see whether an angular velocity Ω and spinning matter fields “repel” each other“at the horizon”, like the gauge field and charged scalar fields in the charged black holes (seefootnote No.2), due to similarity of angular velocity and gauge connection [13].
Acknowledgments
We would like to thank Rong-Gen Cai, Li Li, Tsutomu Kobayashi, and Kyung Kiu Kimfor helpful discussions. DOD was supported by the National Natural Science Foundationof China under Grant No. 11875136 and the Major Program of the National Natural Sci-ence Foundation of China under Grant No. 11690021. MIP was supported by Basic Sci-ence Research Program through the National Research Foundation of Korea (NRF) funded bythe Ministry of Education, Science and Technology (2016R1A2B401304, 2020R1A2C1010372,2020R1A6A1A03047877). 9 ppendix A: Full Dimension-Dependent Equations of Motion
In this Appendix, we present the full dimension-dependent equations of motion for D = 4case (3) - (6) in the text: E A ≡ κ (cid:18) Z ( | ϕ | ) √ fN r D − A ′ t (cid:19) ′ − q r D − | ϕ | A t f (cid:18) √ fN (cid:19) (cid:20) αr + ( D − D − γ ( k − f ) − ( D − γrf ′ (cid:21) = 0 , (A1) E ϕ ≡ (cid:18) N √ f r D − f (cid:20) αr + ( D − D − γ ( k − f ) − ( D − γrf (cid:18) N ′ N (cid:19)(cid:21) ϕ ′ (cid:19) ′ + (cid:18) N √ f (cid:19) q r D − A t ϕN (cid:20) αr + ( D − D − γ ( k − f ) − ( D − γrf ′ (cid:21) + r D − (cid:20) κ √ fN ˙ ZA ′ t − N √ f ˙ V (cid:21) ϕ = 0 , (A2) E N ≡ ( D − D − κ (cid:18) f − k + rf ′ D − (cid:19) + r V ( | ϕ | )+ f (cid:20) αr + ( D − D − γ ( k + f ) + 3( D − γrf ′ (cid:21) | ϕ ′ | + ( D − γrf (cid:0) | ϕ ′ | (cid:1) ′ + 12 N (cid:18) κ Zf r A ′ t + 2 q | ϕ | A t (cid:20) αr + ( D − D − γ ( k − f ) − ( D − γrf ′ (cid:21)(cid:19) = 0 , (A3) E f ≡ ( D − D − κ (cid:20) f − k + rfD − (cid:18) N ′ N (cid:19)(cid:21) + r V ( | ϕ | ) − f (cid:20) αr + ( D − D − γ ( k − f ) − D − γrf (cid:18) N ′ N (cid:19)(cid:21) | ϕ ′ | + 12 N (cid:18) κ Zf r A ′ t − q | ϕ | A t (cid:20) αr + ( D − D − γ ( k − f ) (cid:21)(cid:19) + ( D − γq N (cid:18) | ϕ | (cid:20) − rf A t (cid:18) N ′ N (cid:19) + 2 rf ( A t ) ′ (cid:21) + 2 rf A t ( | ϕ | ) ′ (cid:19) = 0 , (A4) Appendix B: Lower-Dimensional Dilaton Gravity from Dimension Reduction
Here, we present the ( D − D -dimensional EMH theory. Let us consider the Kaluza-Klein reduction of the D -dimensionalmetric g µν as ds = g µν dx µν ≡ e σ (cid:0) dz + h ab dx a dx b (cid:1) , (B1)10here the dilaton field σ and metric h ab depend only on ( D − x a ( a = 0 , , , · · · , ( D − , one can find that R ab = e − σ n ˜ R ab − h ab (cid:16) ˜ ∇ σ + ( D − ∇ σ ) (cid:17) + ( D − (cid:16) ˜ ∇ a σ ˜ ∇ b σ − ˜ ∇ a ˜ ∇ b σ (cid:17)o , (B2) R = e − σ n ˜ R − ( D − D − ∇ σ ) − D −
1) ˜ ∇ σ o , (B3) G ab ≡ R ab − g ab R = e − σ (cid:26) ˜ R ab − h ab ˜ R + ( D − (cid:20) ˜ ∇ a σ ˜ ∇ b σ − ˜ ∇ a ˜ ∇ b σ + h ab (cid:18) ˜ ∇ σ + ( D − ∇ σ ) (cid:19)(cid:21)(cid:27) , (B4)where R = R ab g ab = e σ R ab h ab and ˜ ∇ a , ˜ R ab , ˜ R denote the covariant derivatives, Ricci tensorand scalar with respects to metric h ab . Then, for the case of A = A a ( x b ) dx a , (B5) i.e. , the absence of gauge potential A z along the compactification direction z , one can easilyfind that D -dimensional EMH theory (1) reduces to, up to boundary terms, I ( D − = 2 π Z dx D − √− h e ( D − σ (cid:26) ˜ R + ( D − D − ∇ σ ) − Z ( | ϕ | )4 κ e − σ ˜ F ab ˜ F ab − αh ab ( ˜ D a ϕ ) ∗ ( ˜ D b ϕ ) + γe − σ (cid:20) ˜ R ab − h ab ˜ R + ( D − (cid:18) ˜ ∇ σ + ( D − ∇ σ ) (cid:19) | ˜ Dϕ | +( D − (cid:16) ˜ ∇ a σ ˜ ∇ b σ − ˜ ∇ a ˜ ∇ b σ (cid:17) ( ˜ D a ϕ ) ∗ ( ˜ D b ϕ ) i − e σ V ( | ϕ | ) o , (B6)where ˜ F ab ≡ ˜ ∇ a A b − ˜ ∇ b A a , ˜ D a ≡ ˜ ∇ a − iqA a and set the length scale of compact dimension R dz ≡ π . The reduced action corresponds the ( D − e σ ≡ r, h tt ≡ − N /r , h rr ≡ /f r with anidentification of the azimuthal angular coordinate θ ≡ z in D dimensions. Note that therelevant conserved charge of ( D − D -dimensional Q in (10). This indicates the validity of “NoScalar-Haired Cauchy Horizon Theorem” for the dimensionally-reduced dilaton gravity also.In particular, from the dimensional reduction of D = 3 case, the D = 2 dilaton gravity withMaxwell and Horndeski couplings is given by I (2) = 2 π Z dx √− h e σ (cid:26) ˜ R + 2( ˜ ∇ σ ) − Z ( | ϕ | )4 κ e − σ ˜ F ab ˜ F ab − αh ab ( ˜ D a ϕ ) ∗ ( ˜ D b ϕ )+ γe − σ h ˜ ∇ σ | ˜ Dϕ | + (cid:16) ˜ ∇ a σ ˜ ∇ b σ − ˜ ∇ a ˜ ∇ b σ (cid:17) ( ˜ D a ϕ ) ∗ ( ˜ D b ϕ ) i − e σ V ( | ϕ | ) o , (B7) We thank Kyung Kiu Kim for sharing his unpublished work about this subject. G ab ≡ ˜ R ab − h ab ˜ R = 0 in two dimensions. By identifying the σ = ϕ , i.e. , a neutral scalar field ϕ , (B8) reduces further to I (2) = 2 π Z dx √− h e σ (cid:26) ˜ R + (2 − α )( ˜ ∇ σ ) − Z ( σ )4 κ e − σ ˜ F ab ˜ F ab + γe − σ h ˜ ∇ σ ( ˜ ∇ σ ) + (cid:16) ˜ ∇ a σ ˜ ∇ b σ − ˜ ∇ a ˜ ∇ b σ (cid:17) ( ˜ ∇ a σ )( ˜ ∇ b σ ) i − e σ V ( σ ) o , (B8)which covers a very general class of D = 2 dilaton gravities [14].Similarly, from the dimensional reduction of D = 4 case, D = 3 dilaton gravity with Maxwelland Horndeski couplings is given by I (3) = 2 π Z dx √− h e σ (cid:26) ˜ R + 6( ˜ ∇ σ ) − Z ( | ϕ | )4 κ e − σ ˜ F ab ˜ F ab − αh ab ( ˜ D a ϕ ) ∗ ( ˜ D b ϕ )+ γe − σ (cid:20) ˜ R ab − h ab ˜ R + 2 (cid:18) ˜ ∇ σ + 12 ( ˜ ∇ σ ) (cid:19) | ˜ Dϕ | + 2 (cid:16) ˜ ∇ a σ ˜ ∇ b σ − ˜ ∇ a ˜ ∇ b σ (cid:17) ( ˜ D a ϕ ) ∗ ( ˜ D b ϕ ) (cid:21) − e σ V ( | ϕ | ) (cid:9) . (B9) [1] B.P. Abbott et al., Phys. Rev. Lett. , 241103 (2016).[2] K. Akiyama et al., Astrophys. J. Lett. , L1 (2019).[3] S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos, arXiv:2006.10056 [hep-th];arXiv:2008.12786 [hep-th].[4] R. G. Cai, L. Li and R. Q. Yang, arXiv:2009.05520 [gr-qc].[5] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974).[6] X. H. Feng, H. S. Liu, H. L¨u and C. N. Pope, Phys. Rev. D , 044030 (2016).[7] S. S. Gubser and A. Nellore, Phys. Rev. D , 105007 (2009).[8] A. Anabalon, A. Cisterna and J. Oliva, Phys. Rev. D , 084050 (2014).[9] A. Cisterna and C. Erices, Phys. Rev. D , 084048 (2012).[11] E. Babichev and C. Charmousis, JHEP , 106 (2014).[12] D. Bettoni and S. Liberati, Phys. Rev. D , 084020 (2013).[13] J. H. Yoon, Class. Quant. Grav., , 1863 (1999).[14] T. Banks and M. O’Loughlin, Nucl. Phys. B , 649 (1991); R. B. Mann, Phye. Rev. D , 4438(1993); A. V. Frolov, K. R. Kristjansson and L. Thorlacius, Phys. Rev. D , 124036 (2006). G.Kunstatter, H. Maeda, and T. Taves, Class. Quant. Grav., , 105005 (2016); K. Takahashi andT. Kobayashi, Class. Quant. Grav. , 095003 (2019)., 095003 (2019).