Black holes and asymptotics of 2+1 gravity coupled to a scalar field

Cristian Martinez; Jorge Zanelli; Marc Henneaux; Ricardo Troncoso

REVIEW

Black holes and asymptotics of 2+1 gravity coupled to a scalar field

Marc Henneaux, Cristian Martinez, Ricardo Troncoso, Jorge Zanelli

High Energy Physics - Theory · Submitted 22 Jan 2002 · 17 references

Pith's verdict: The asymptotic analysis and the hairy black hole family are solid and worth engaging; the printed temperature in (3.5) is wrong by a factor of Θ_ν², and the algebra argument is terse but the standard c-number logic holds.

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CitationCristian Martinez, Jorge Zanelli, Marc Henneaux, Ricardo Troncoso. "Black holes and asymptotics of 2+1 gravity coupled to a scalar field." arXiv:hep-th/0201170 (2002). https://pith.science/paper/FRW6J7CP

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Worth your time if you work on 2+1 asymptotics or on holographic hairy black holes. The real content is the divergence cancellation in (2.4)-(2.5): the scalar's slow falloff φ ~ χ r^{-1/2} makes δQ_φ linearly divergent, and that divergence is cancelled exactly by the δ(g^{-1/2}) piece of δQ_G, because δg_rr = -4l²r^{-3}δχ². That is a genuine consistency check, not a cosmetic one, and it is what makes the charge (2.6) finite. The exact hairy black holes for V_ν are a useful bonus, and the ν-family is a reasonable toy for holographic model building. The authors are also candid about what they have not settled: the Cardy overcounting in item 3 and the open V-α connection in item 5 are both stated plainly, which I appreciate. Now the soft spots. The printed temperature (3.5) is wrong. Surface gravity computed from (3.4) gives T = 3B(1+ν)/(2πl²Θ_ν), not the printed 3B(1+ν)Θ_ν/(2πl²); at ν=0 that is 9B/(8πl²) against the printed 2B/(πl²). The first law confirms the corrected form: dM/dS with M = 3B²(1+ν)/(8l²) and S = πBΘ_ν/2 gives exactly the reciprocal-Θ_ν expression, and C = dM/dT = S holds only with it. So it is a typo, but it sits in a load-bearing equation and the paper's own C = S claim depends on the fix. Worth flagging to the authors. On the stress-test concern about the algebra, I think it is over-strict. The central term is a c-number, so evaluating on the φ=0 vacuum is the standard Brown-Henneaux move, and Ref [11] is a published result that does the general work. What the paper does not do is write down how χ transforms under (1.2), which would have made the closure manifest. That is a presentation gap, not a hole. The reader's ν=-1 vacuum worry I would downgrade: the paper explicitly calls (3.8) a degenerate ground state with vanishing charges, and in the conformal frame it is the standard zero-mass BTZ. Since c is state-independent, the choice of reference does not matter. The reverse-engineered potential is acknowledged in item 5. Verdict: send it out. The core is right; the temperature needs fixing.

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Major objections (4)

[Section 2, below Eq. (2.3)] The paper asserts that the asymptotic conditions (2.2)-(2.3) are left invariant under the Virasoro generators (1.2), but the transformation law of χ under these generators is never given. Since the r^{-1/2} falloff of φ and the r^{-3} correction to g_rr are both tied to χ, the invariance is not manifest, and the claim that the asymptotic symmetry group is unchanged rests on this step. Please give δχ explicitly and verify that (2.2)-(2.3) are preserved, including the subleading terms.
[Section 2, Eqs. (2.4)-(2.6) and the central-charge paragraph] The charge (2.6) contains the explicit scalar terms 4r²[φ² + (1/3)φ⁴], and the paper argues that the algebra of charges is identical to pure gravity only indirectly, via Ref. [11] and the statement that 'the vacuum is the same as in pure gravity.' The bracket {Q(ξ1), Q(ξ2)} is never computed, so it is not shown that the scalar terms drop out of the central extension. The argument is likely correct — the central term is field-independent and is computed on the φ=0 BTZ, which is a solution of V_ν for every ν, while the scalar terms contribute only to the field-dependent part — but the paper should state this logic explicitly and, ideally, verify that the scalar contribution to the bracket is a total derivative or vanishes. This would close the gap left by the observation that the scalar contribution to the bracket is unverified.
[Section 3, Eqs. (3.5)-(3.7)] As printed, the temperature T = 3B(1+ν)Θ_ν/(2πl²) in Eq. (3.5) is inconsistent with the mass M = 3B²(1+ν)/(8l²) in Eq. (3.6) and the entropy S = πBΘ_ν/2 in Eq. (3.7): the first law dM = T dS requires T = 3B(1+ν)/(2πl²Θ_ν), with Θ_ν in the denominator, and only this form reproduces the statement C = ∂M/∂T = πr_+/2 made just below Eq. (3.7). Please check Eq. (3.5) and correct it, or explain the discrepancy.
[Section 3, Eq. (3.8) and the ν=-1 discussion] The paper states that for ν=-1 'the metric is the standard zero-mass 2+1 black hole,' but the line element (3.8) has g_rr = l²/(r²+4Br), whereas the massless BTZ has g_rr = l²/r²; the two share the causal structure but are not isometric. This matters for the interpretation of (3.8) as a 'degenerate ground state' and for the choice of reference configuration in the central-charge computation. The central-charge argument itself survives, because the φ=0 BTZ is a solution of V_ν for every ν and the central term is field-independent, but the paper should say so explicitly and should not describe (3.8) as the standard massless black hole.

Minor objections (4)

[Section 4, item 3] The Cardy-formula mismatch S = sqrt(3(1+ν))/Θ_ν · A/4 > A/4 is acknowledged, but the discussion stops at the contradiction. A sentence on the likely resolution — that the Cardy formula assumes a ground state with vanishing zero-point energy, whereas the ν-family has a shifted ground state — would make the section more useful.
[Section 3, Eq. (3.1) and Fig. 1] The properties of Θ_ν (monotonicity, Θ_{-1}=0, Θ_0=4/3, growth as sqrt(ν)) are stated without proof, and Ref. [12] (E. Schuster, Acta Phil. Vald. 19 (1877) 31) is obscure. A brief derivation or a more accessible reference would help the reader.
[Section 3, below Eq. (3.4)] The sentence 'for ν=-1 the metric is the standard zero-mass 2+1 black hole' should be softened to match the preceding sentence, which correctly says only that (3.8) 'shares the same causal structure' with the massless BTZ.
[Section 4, item 5] The statement that V(φ) is fixed by the asymptotic conditions up to φ⁶U(φ²) is a strong and useful result; it would help to show the expansion explicitly, since it is what mitigates the concern that V_ν is reverse-engineered.

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The paper

The paper claims

Three-dimensional gravity with a negative cosmological constant keeps its conformal asymptotic symmetry, and its central charge, even when a scalar field reaches infinity too slowly to count as localized matter.

The reading

The paper asks whether the asymptotic symmetry algebra of three-dimensional gravity with a negative cosmological constant survives when the matter coupled to it decays so slowly at infinity that it reaches the boundary instead of staying localized. The authors take a single self-interacting scalar field with falloff $gives$ — slower than the $r^{-(1+arepsilon)}$ decay that would make it invisible to the surface integrals — and show that the admitted asymptotic Killing vectors are still the two copies of the Virasoro algebra found for pure gravity, and that the canonical charges, once the scalar's contribution is added, still close with the same central charge $c = 3l/2G$. The payoff is a one-parameter family of exact, static, circularly symmetric black holes with scalar hair that is regular everywhere, whose causal structure matches the standard nonrotating black hole and which are continuously connected to the zero-mass solution. If the argument holds, the Brown-Henneaux central charge is not an artefact of vacuum boundary conditions but survives a class of matter backreaction that genuinely reaches infinity.

What carries the argument

Two pieces of machinery carry the argument. The first is the divergence cancellation between gravitational and scalar contributions to the canonical charge, computed with the Regge-Teitelboim surface-integral method: because the leading correction to $gives$ is tied to the scalar's leading coefficient $gives$, the $O(r)$ pieces of $gives$ and $gives$ cancel and leave a finite charge. The second is a conformal-frame construction: the potential family $gives$ is obtained by the rescaling $gives$ with $gives$ and $gives$ from an action whose matter part is conformally invariant for every value of the parameter $gives$, which is what makes the exact hairy solutions tractable. The horizon radius is read off as the first zero of the Schuster function $gives$, and consistency of the boundary conditions with the field equations is claimed to force the potential into the form $gives$ with $gives$ an arbitrary smooth function.

Extended reading notes

Core claim

The paper's central claim is that the asymptotic conditions $gives$, $gives$ and $gives$ — in which the radial metric component falls off one power of $r$ more slowly than in pure gravity — are left invariant by exactly the same asymptotic Killing vectors that generate the conformal group in the vacuum theory. The reason the symmetry survives is a cancellation: the variation of the scalar charge $gives$ contains a term linear in $r$ that would diverge, and it is cancelled by a divergent piece of the gravitational charge $gives$ precisely because, to leading order, $gives$. The total charge $gives$ is then finite, and the algebra of canonical generators is stated to be two copies of the Virasoro algebra with the same central extension as in pure gravity, since the vacuum is taken to be the standard massless black hole and the central-charge computation is therefore identical. On top of this, the authors exhibit exact black hole solutions with regular scalar hair for the potential family $gives$, with horizon at $gives$, mass $gives$, temperature $gives$, and entropy $S = A/4$; the specific heat equals the entropy, exactly as for the standard black hole. The authors themselves flag one loose end: applying Cardy's formula to the unchanged central charge would give an entropy larger than $A/4$ for $gives$, which they leave unresolved.

Load-bearing premise

The central-charge argument assumes that the reference configuration for this slower-falloff class is the standard massless black hole with vanishing scalar field, even though the same class also contains the $gives$ metric that the paper itself calls a degenerate ground state, and the paper does not show that these two are the same state or that the central charge is independent of which one is used as the reference.

Editorial extensions

If this is right

The Brown-Henneaux central charge $c = 3l/2G$ applies to a boundary-condition class in which the matter stress tensor does not vanish at infinity, so the correspondence between 2+1 gravity and a boundary conformal theory is not restricted to localized matter.
For every $gives$ there is an explicit hairy black hole with $gives$, $gives$ and $gives$, so the thermodynamic identities of the standard black hole carry over to the dressed solutions unchanged, including the equality of specific heat and entropy.
Because there is only one integration constant $B$, fixing the mass fixes the scalar profile: the scalar cannot be switched off at fixed mass, so each mass admits two distinct black holes, one with $gives$ and one with nontrivial hair.
The limit $gives$ gives a configuration of vanishing mass, temperature and entropy, which the paper treats as a degenerate ground state continuously connected to the rest of the family.
Consistency of the asymptotic conditions with the field equations restricts the admissible self-interactions to the family $gives$, so only a specific class of scalar potentials supports this slow-falloff behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the central charge really is a property of the asymptotic class rather than of the chosen reference configuration, the same argument should extend to other matter that reaches the boundary slowly, such as gauge fields with logarithmic falloff or dilatonic couplings, which would turn this into a general statement about non-normalizable matter in AdS$_3$.
The acknowledged mismatch between Cardy counting and the area law may be a signal that the Cardy formula should be applied to the $gives$ configuration rather than the standard massless black hole, or with a shifted zero-point energy; checking which choice reproduces $S = A/4$ would identify the state the boundary theory actually counts.
The paper's charge formula already carries a rotational term $gives$, so a rotating generalization of the hairy family is a natural next check; if rotation preserves the same central extension, the robustness claim is strengthened considerably.
Because $gives$ interpolates between a previously known solution at $gives$ and the degenerate ground state at $gives$, the family looks like a continuous deformation of the dual theory, which could be probed by computing quasi-normal mode frequencies as a function of $gives$.

Original abstract

We consider 2+1 gravity minimally coupled to a self-interacting scalar field. The case in which the fall-off of the fields at infinity is slower than that of a localized distribution of matter is analyzed. It is found that the asymptotic symmetry group remains the same as in pure gravity (i.e., the conformal group). The generators of the asymptotic symmetries, however, acquire a contribution from the scalar field, but the algebra of the canonical generators possesses the standard central extension. In this context, new massive black hole solutions with a regular scalar field are found for a one-parameter family of potentials. These black holes are continuously connected to the standard zero mass black hole.

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We consider 2+1 gravity minimally coupled to a self-interacting scalar field. The case in which the fall-off of the fields at infinity is slower than that of a localized distribution of matter is analyzed. It is found that the asymptotic symmetry group remains the same as in pure gravity (i.e., the conformal group). The generators of the asymptotic symmetries, however, acquire a contribution from the scalar field, but the algebra of the canonical generators possesses the standard central extension. In this context, new massive black hole solutions with a regular scalar field are found for a one-parameter family of potentials. These black holes are continuously connected to the standard zero mass black hole.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider 2+1 gravity minimally coupled to a self-interacting scalar field.	contribution 0.57	0.66	0.03	asserted 0.89
s1	The case in which the fall-off of the fields at infinity is slower than that of a localized distribution of matter is analyzed.	contribution 0.90	0.63	0.04	asserted 0.86
s2	It is found that the asymptotic symmetry group remains the same as in pure gravity (i.e., the conformal group).	result 1.00	0.91	0.28	asserted 1.00
s3	The generators of the asymptotic symmetries, however, acquire a contribution from the scalar field, but the algebra of the canonical generators possesses the standard central extension.	result 1.00	0.90	0.22	asserted 1.00
s4	In this context, new massive black hole solutions with a regular scalar field are found for a one-parameter family of potentials.	result 0.98	0.96	0.31	asserted 1.00
s5	These black holes are continuously connected to the standard zero mass black hole.	result 0.98	0.86	0.14	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Figures

**Figure 1.** Plot of Θν as a function of ν. The intercepts are at Θ−1 = 0 and Θ0 = 4 3 . where B is a non-negative integration constant and H(r) = 1 2 r + p r 2 + 4Br . The line element reads ds2 = − H H + B 2 F(r)dt2 + H + B H + 2B 2 dr2 F(r) + r 2 dϕ2 , (3.4) with F = H2 l 2 − (1 + ν) 3B2 l 2 + 2B3 l 2H . The causal structure of this geometry is the same as for the standard, nonrotating 2+ 1 black hole. The event horizon is located at r+ = BΘν , where the constant Θν is the first zero of the Schuster function of order ν, given by [12] Θν = 2(zz¯) 2/3 z 2/3 − z¯ 2/3 z − z¯ , and z…

Figures reproduced from arXiv:hep-th/0201170 by the authors.

References

Works cited (17)

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J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986)

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The asymptotic behavior of the momenta is given by πrr = O(1), πrϕ = O(r−2), πϕϕ = O(r−4), and πφ = O(r−3/2). Note that πrϕ = gϕt to leading order

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E. Schuster, Acta Phil. Vald. 19 (1877) 31

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Thus, if one considers the static hairy black hole of Ref. [ref] [equation] which has the same scalar field given in ([ref]), and performs the following boost [equation] where the constant $ω$ is restricted to satisfy $ω^2<1$, one directly obtains the rotating hairy solution described by Eqs. ([ref])-([ref]).

Rotating hairy black hole and its microscopic entropy in three spacetime dimensions · Rotating hairy scalar black hole in three dimensions · 0.95

Uses its method or tool 1 paper

In fact, by using Eq. (22) of Ref. [ref] (see also [ref] and [ref] ) the mass $M$ and the angular momentum $J$ of the hairy black hole described by Eqs. ([ref])-([ref]) are found to be [equation] which depend on the integration constants $B$ and $ω$.

Rotating hairy black hole and its microscopic entropy in three spacetime dimensions · Mass, angular momentum and thermodynamics · 0.85

Compares with it 3 papers

The class of hairy black holes found here was shown to have positive mass and it can be seen that they share many of the features with the one previously found in [ref] .

Hairy black hole entropy and the role of solitons in three dimensions · Microscopic entropy of hairy black holes: Soliton mass and its role in the asymptotic growth of the number of states. · 0.96
A similar behavior has been observed in a setup leading to relaxed AdS boundary conditions due to a slowly decaying scalar field at infinity [ref] .

New boundary conditions in Einstein-scalar gravity in three dimensions · Surface charges · 0.93
It is worth pointing out that a similar effect occurs in the case of hairy black holes with scalar fields with slow fall-off at infinity [ref] .

Higher spin gravity in 3D: black holes, global charges and thermodynamics · Introduction · 0.93

Cites it as background 23 papers

The first analytic example of a black hole solution endowed with minimally coupled scalar hair was found in [ref] , precisely for the self-interaction under discussion [ref] in the case of $Λ<0$ and $ν≥-1$.

Gravity coupled to a scalar field from a Chern-Simons action: describing rotating hairy black holes and solitons with gauge fields · Rotating hairy black hole · 0.81
The entropy of static hairy scalar black holes has been computed, by means of Cardy formula, in the case of potentials defined by one [ref] and two coupling constants [ref] .

Rotating hairy black hole and its microscopic entropy in three spacetime dimensions · Introduction · 0.81
On the other hand, there exists a black hole solution to three-dimensional gravity with a minimally coupled self-interacting scalar, called the scalar black hole (SBH) [ref] .

Phase transition for black holes with scalar hair and topological black holes · Introduction · 0.94

Discussion

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