REVIEW
Black hole hair: twenty--five years after
General Relativity and Quantum Cosmology · Submitted 28 May 1996 · 21 references
Pith's verdict: A useful, honest synthesis of the no-scalar-hair case, but the abstract's 'new theorems' framing outruns what the body actually delivers.
T0 · AI review
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Referee report
The paper
The paper claims
Bekenstein argues that scalar fields are almost never allowed to hang around a spherical black hole, and states a conjecture that would close the last gap.
The reading
What carries the argument
Two devices do the work. The first is a conformal transformation of the metric, $0$, which absorbs the $0$ term into a rescaled theory with an effective positive energy density $0$; because the rescaled metric stays regular and of the right signature from horizon to infinity (for $0$, and for $0$ given the causality restriction), the existing positive-energy no-hair theorem applies in the new frame and forbids neutral scalar hair in the old one. The second is the local behaviour of the scalar field near the horizon: writing the static spherical metric with $0$, the Maxwell equation forces the electric potential to grow monotonically inward without diverging, and the scalar equation then has the generic near-horizon solution $0$, an oscillation in $0$ that cannot be zero at the horizon unless it is zero everywhere. The BBM solution sits outside these arguments: it has hair only as a two-valued sign choice, has no nonextremal continuous extension, and is known to be linearly unstable.
Extended reading notes
Core claim
The paper's central claim is a conjecture plus the theorems that nearly establish it. The conjecture: the only asymptotically flat, spherically symmetric, truly static black hole solutions of the Abelian scalar action (Einstein–Hilbert–Maxwell plus a complex scalar with nonminimal coupling $0$ and potential $V$) are the Reissner–Nordström family and the generalized BBM family. Three results carry the weight. First, for a real scalar field with positive-definite energy ($0$ and $0$ for positive argument), the only static spherically symmetric black hole is Reissner–Nordström with no scalar field at all. Second, a conformal rescaling of the metric turns the nonminimally coupled neutral problem into a minimally coupled one with positive energy, so the same exclusion applies for $0$ and for $0$, the latter under a causality condition on the Poynting vector of the scalar's stress tensor. Third, for a charged complex scalar, a near-horizon analysis of the Maxwell and scalar equations shows the field must oscillate logarithmically in $0$ and therefore cannot vanish on the horizon unless it vanishes identically — no charged scalar hair, for any coupling $0$ and any regular potential. What remains is the neutral case in the window $0 < 1/2$, which the author explicitly flags as the missing step.
Load-bearing premise
The load-bearing premise is the positive-energy stipulation on the scalar action ($0$ and $0$ for positive argument), together with the causality condition on the Poynting vector used when $0$; the paper itself shows this is not decorative, since flipping the sign of the scalar action at $1/6$ with $0$ produces a hairy black hole for $0$. The author also states plainly that the neutral window $0 < 1/2$ is not covered, and two cross-references in the conformal-frame section appear as unresolved '$0$' markers, so that step leans on cited work rather than on an argument reproduced in the text.
Editorial extensions
If this is right
- If the conjecture holds, a spherical static black hole in Einstein–Maxwell-scalar theory is fully specified by mass, electric charge and angular momentum, with the scalar field contributing at most a discrete sign label.
- The BBM black hole is demoted to a marginal case: it carries no continuously tunable scalar charge, admits no nonextremal generalization, and is linearly unstable, so it does not restore a scalar hair parameter.
- The absence of scalar hair parameters keeps the identification of black hole entropy with the logarithm of a count of internal microscopic states on the table, rather than forcing entropy to track exterior field configurations.
- The nonabelian gauge-field black holes discovered in the early 1990s are consistent with this picture, because gauge invariance lets a gauge potential convey charge information outward even from inside a horizon, whereas a scalar field has no such Gauss-law escape route.
- A proof covering the neutral window $0 < 1/2$ would convert the conjecture into a theorem and complete the classification of static spherical black holes in this theory.
Reading between the lines
- The exclusion argument turns on positivity and causality conditions rather than on the field equations alone, which suggests the interesting question is not 'can a scalar field sit outside a horizon' but 'which energy conditions make such a configuration impossible to form by collapse'.
- Because the charged-scalar result holds for every coupling $0$, the remaining obstruction is specifically the neutral, nonminimally coupled case; a numerical search in the window $0 < 1/2$ with a positive-definite potential is the natural way to test whether the gap is real or empty.
- The same conformal-frame strategy should transfer to any theory whose nonminimal coupling can be absorbed by a field-dependent rescaling of the metric, so the technique may be more general than the particular scalar action treated here.
Original abstract
Originally regarded as forbidden, black hole ``hair'' are fields associated with a stationary black hole apart from the gravitational and electromagnetic ones. Several stable stationary black hole solutions with gauge or Skyrme field hair are known today within general relativity. We formulate here a ``no scalar--hair'' conjecture, and adduce some new theorems that almost establish it by ruling out - for all but a small parameter range - scalar field hair of spherical black holes in general relativity, whether the field be self--interacting, coupled to an Abelian gauge field, or nonminimally coupled to gravity.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Originally regarded as forbidden, black hole ``hair'' are fields associated with a stationary black hole apart from the gravitational and electromagnetic ones. Several stable stationary black hole solutions with gauge or Skyrme field hair are known today within general relativity. We formulate here a ``no scalar--hair'' conjecture, and adduce some new theorems that almost establish it by ruling out - for all but a small parameter range - scalar field hair of spherical black holes in general relativity, whether the field be self--interacting, coupled to an Abelian gauge field, or nonminimally coupled to gravity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Originally regarded as forbidden, black hole ``hair'' are fields associated with a stationary black hole apart from the gravitational and electromagnetic ones. | background 1.00 | 0.04 | 0.03 | asserted 0.99 |
| s1 | Several stable stationary black hole solutions with gauge or Skyrme field hair are known today within general relativity. | background 1.00 | 0.06 | 0.05 | asserted 1.00 |
| s2 | We formulate here a ``no scalar--hair'' conjecture, and adduce some new theorems that almost establish it by ruling out - for all but a small parameter range - scalar field hair of spherical black holes in general relativity, whether the field be self--interacting, coupled to an Abelian gauge field, or nonminimally coupled to gravity. | contribution 0.95 | 0.95 | 0.87 | speculative 0.49 |
What was examined, and what was not
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- P abstract
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References
How the literature uses it
118 arXiv papers cite this in 151 sentences
Papers citing it · 80 sentences without a confident label
Builds on it 1 paper
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Interestingly, our novel no-hair theorem, to be presented below, is valid for generic values of the physical parameter $ξ$ which characterizes the non-minimal field-curvature coupling in the composed black-hole-massive-scalar-field system (In particular, in the present paper we shall extend the important no-hair theorems of [ref] to the regime of nonminimally coupled massive neutral scalar fields with $0<ξ≤1/4$).
No hair for spherically symmetric neutral black holes: nonminimally coupled massive scalar fields · Introduction · 1.00
Compares with it 3 papers
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Finally, we would like to emphasize the interesting fact that, while existing no-hair theorems for the composed black-hole-massive-scalar-field system are valid in the restricted physical regimes $ξ<0$ and $ξ≥1/2$ [ref] , the no-hair theorem presented in this paper for the composed compact-reflecting-star-massive-scalar-field system is valid for generic values of the dimensionless nonminimal coupling parameter $ξ$.
No hair for spherically symmetric neutral reflecting stars: nonminimally coupled massive scalar fields · Summary · 0.96
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Our results are also of mathematical interest, as they establish continuous non-uniqueness of de Sitter black holes, in contrast to the asymptotically flat case [ref] .
Spinning Black Binaries in de Sitter space · Introduction · 0.87
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This result matches the no-hair theorems of [ref] (which are, however, more general).
Revisiting the analogue of the Jebsen-Birkhoff theorem in Brans-Dicke gravity · Conclusions · 0.94
Cites it as background 61 papers
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The influential no-scalar-hair theorems of Bekenstein [ref] and Teitelboim [ref] (see also [ref] and references therein) have revealed the interesting fact that asymptotically flat classical black-hole spacetimes, which are characterized by absorbing one-way membranes (event horizons), cannot support static non-linear matter configurations (external `hair') made of spatially regular massive scalar fields with minimal coupling to gravity [ref] .
No hair for spherically symmetric neutral reflecting stars: nonminimally coupled massive scalar fields · Introduction · 0.84
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Following the no-hair conjecture (also known as the no-hair theorem (NHT)) [ref] which precludes the existence of asymptotically flat black holes solutions of Einstein's field equations with additional parameters (the so called hair) besides the mass, angular momentum and charge of the black hole, there have been several generalizations of this conjecture, particularly in other theories of gravity such as Brans-Dicke theory, scalar tensor theories [ref] and Horndeski gravity [ref] .
A no-hair theorem for spherically symmetric black holes in $R^2$ gravity · Introduction · 0.98
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It was found that the static massive scalar fields cannot exist in asymptotically flat black holes, for references see [ref] - [ref] and a review see [ref] .
Studies of no light scalar hair behaviors for compact reflecting stars in a box · Introduction · 0.89
Discussion
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