New 'hairy' black hole and soliton solutions to
anti de-Sitter Einstein-Yang-Mills theories
21st International Conference on General Relativity and Gravitation,
Columbia University, New York (2016).
J. ERIK BAXTER
E . B A X T E R @ S H U. A C . U K
Acknowledgements:
Prof. J. Stewart
Prof. E. Winstanley
Outline
 A brief history of black holes and solitons in Einstein-Yang-Mills theory (i.e. gravity + nonAbelian gauge theory);
 A whistlestop tour of some recently published results;
 Relevance to diverse questions in physics, including Condensed Matter Physics (CMP) and the
‘Black hole information paradox’.
What is ‘black hole hair’?
 Uniqueness theorems in the 70s indicated that all (known) black holes were of the general KerrNewman form. “A black hole has no hair”. (Problem: where has the information ‘gone’?)
 Then – Bizon, Bartnik and McKinnon found an infinite (discrete) family of so-called ‘coloured’
black holes and solitons for 𝔰𝔲(2) EYM theory in asymptotically flat space, classified by the
integer 𝑛, with the single gauge function 𝜔(𝑟) having 𝑛 zeroes.
 Since then, many generalisations of the original model have been considered.
Einstein-Yang-Mills theory (I)
 Essentially, we have a Lie group gauge theory over spacetime; or, a principal fibre bundle with 𝐺-valued fibres over a
4D Lorentzian manifold 𝑀 as the base space. We have 2 ansätze: the metric and the gauge potential.
 The metric is a ‘Schwarzschild-type’, e.g.:
𝑑𝑠 2 = −𝜇𝑆 2 𝑑𝑡 2 + 𝜇−1 𝑑𝑟 2 + 𝑟 2 𝑑𝜃 2 + 𝑟 2 sin2 𝜃 𝑑𝜙 2 .
 A general one-form potential is:
𝒜 = 𝐴 + 𝑊1 𝑑𝜃 + 𝑊2 sin 𝜃 + 𝑊3 cos 𝜃 𝑑𝜙,
where 𝐴 (the ‘electric sector’) is defined only on the 𝑡, 𝑟 part of the manifold, and 𝑊1 , 𝑊2 , 𝑊3 (the ‘magnetic
sector’) satisfies the Wang equations
𝑊2 , 𝑊3 = 𝑊1 ,
𝑊3 , 𝑊1 = 𝑊2 .
Einstein-Yang-Mills theory (II)
 We begin with the EYMH action 𝒮 =
1
2
𝑅 − 2Λ + Tr 𝐹𝜇𝜈 𝐹𝜇𝜈
𝑑𝑥 4 .
 Varying w.r.t. 𝑔𝜇𝜈 and 𝐴𝜇 (where 𝒜 = 𝐴𝜇 𝑑𝑥 𝜇 ) we get 2 sets of coupled field equations:
1
𝐺𝜇𝜈 + Λ𝑔𝜇𝜈 = Tr 2𝐹𝜇 𝜆 𝐹𝜆𝜈 + 2 𝐹𝜇𝜈 𝐹𝜇𝜈 ,
𝛻𝜆 𝐹 𝜆𝜈 + 𝐴𝜆 , 𝐹 𝜆𝜈 = 0,
with 𝑞 = 𝑐 = 4𝜋𝐺 = 1, and
𝐹𝜇𝜈 = 𝜕𝜇 𝐴𝜈 − 𝜕𝜈 𝐴𝜇 + 𝐴𝜇 , 𝐴𝜈 .
 Plug in the ansätze and require asymptotic regularity, and a regular non-extremal event horizon or a regular origin.
Why look at anti-de Sitter space (𝚲 < 𝟎)?
 Asymptotically AdS solutions tend to exist in a continuum; asymptotically dS or flat solutions exist in
a discretum.
 AdS space has a boundary, whereas other cases are ‘open’ geometries: much harder to find a stable
balance between gravity and hair – in general these systems are unstable (Brodbeck and Straumann,
1994).
 Related to this is that dS/flat solutions always possess one or more ‘nodes’, which indicate unstable
modes; however AdS solutions may be nodeless in the general case and hence (possibly) stable.
 AdS therefore has a richer solution space and ‘nicer’ analytical properties.
 Also – AdS/CFT!
Relevance to physics
Bizon’s modified “No-Hair” theorem: “Within a given matter model, a stable black
hole is characterised by a finite number of global charges.”
 For a given model:




Do solutions to the field equations even exist, for some values of the initial parameters?
If so, what does the solution space look like?
Are these classically/thermodynamically stable?
What do they look like ‘asymptotically’ – can we define ‘global charges’?
Some recent references:
 Shepherd, B.L., Winstanley, E. “Characterizing asymptotically anti-de Sitter black holes with
abundant stable gauge field hair”. Class. Quant. Grav. 29, 155004 (2012).
 Nolan, B.C., Winstanley, E. “On the existence of dyons and dyonic black holes in Einstein–Yang–
Mills theory”. Class. Quant. Grav. 29, 235024 (2012).
 Nolan, B.C., Winstanley, E. “On the stability of dyons and dyonic black holes in Einstein-YangMills theory”. Class. Quant. Grav. 33, 045003 (2016).
 Baxter, J.E., Winstanley, E. “On the stability of soliton and hairy black hole solutions of SU(N)
Einstein-Yang-Mills theory with a negative cosmological constant”. Jour. Math. Phys. 57, 022506
(2016).
Relevance to physics
 AdS/CFT correspondence (Maldacena, 1997): Gravitational theories in the bulk of (𝑁 + 1)dimensional AdS space can translate to 𝑁-dimensional particle theories on the boundary.
 Example: 𝔰𝔲(2) planar dyonic black holes may be used to model holographic superconductors (Cai et al.
2015).
 ‘Black hole information paradox’: Hawking’s recent work suggests that black hole hair may be able
to represent the “lost information”.
𝔰𝔲(𝑁) Topological Dyonic solutions
 Baxter, J. E. “Existence of topological hairy dyons and dyonic black holes in anti de-Sitter SU(N) Einstein-Yang-Mills
theory.” Jour. Math. Phys. 57, 022505 (2016).
 “Topological” - instead of foliating spacetime by spheres, we foliate by surfaces of constant Gaussian curvature. Due to van der Bij
and Radu (2001) in the 𝔰𝔲(2) case.
 “Dyons/dyonic black holes” – contain both ‘magnetic’ and ‘electric’ gauge sectors. (Only available for Λ < 0.)

Metric: 𝑑𝑠 2 = −𝜇𝑆 2 𝑑𝑡 2 + 𝜇 −1 𝑑𝑟 2 + 𝑟 2 𝑑𝜃 2 + 𝑟 2 𝑓𝑘 𝜃
𝜇 𝑟 =𝑘−

2𝑚 𝑟
𝑟
−
Λ𝑟 2
3
2
𝑑𝜙 2 , with
sin 𝜃
𝜃
and 𝑓𝑘 𝜃 =
sinh𝜃
𝔰𝔲(𝑁)-valued potential: 𝒜 = 𝐴𝑑𝑡 + 𝐵𝑑𝑟 +
1
2
𝐶 − 𝐶 𝐻 𝑑𝜃 −
𝑖
2
for 𝑘 = 1
for 𝑘 = 0
for 𝑘 = −1
(′𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙′ : 𝐾 > 0),
(′𝑝𝑙𝑎𝑛𝑎𝑟 ′ : 𝐾 = 0),
′ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 ′ : 𝐾 < 0 ;
𝐶 + 𝐶 𝐻 𝑓𝑘 (𝜃) + 𝐷
with complex matrices 𝐴 to 𝐷. NB: Static solutions (all functions of r alone).
𝑑𝑓𝑘
𝑑𝜃
𝑑𝜙,
Topological dyonic field equations

With a certain choice of gauge we can take 𝐵 = 0, 𝐶 ∈ ℝ; so we have 𝑁 − 1 magnetic gauge functions 𝜔𝑗 (𝑟), 𝑁 − 1 (independent)
electric gauge functions 𝛼𝑗 (𝑟), and 2 metric functions which can be taken as 𝑚(𝑟) and 𝑆(𝑟) (all real functions).
2 Einstein eqns:
𝑚′ =
𝑟2𝜂
4𝑆 2
+
𝜁
4𝜇𝑆 2
𝑆′
𝑆
+ 𝜇𝐺 + 𝑃,
=
2𝐺
𝑟
+
𝜁
;
2𝜇 2 𝑆 2 𝑟
2𝑁 − 2 Yang-Mills eqns:
𝑟 2 𝜇𝛼𝑗′′ +
𝑟 2 𝜇𝜔𝑗′′ + 𝑟 2 𝜇
with
𝜂=
𝑁
𝑗=1
𝑆′
𝑆
+
𝜇′
𝜇
2
𝑟
−
𝜔𝑗′ +
2
𝛼𝑗 ′ ,
𝜇=𝑘−
𝑆′
𝑆
2
𝛼𝑗′ + 𝜔𝑗−1
𝛼𝑗−1 − 𝛼𝑗 − 𝜔𝑗2 𝛼𝑗 − 𝛼𝑗+1 = 0,
𝑟2
𝜔′
4𝜇𝑆 2 𝑗
𝛼𝑗 − 𝛼𝑗+1
𝜁=
2𝑚
𝑟
+
𝑟2
,
ℓ2
2
𝑁−1 2
𝑗=1 𝜔𝑗
𝑃=
+ 𝜔𝑗 𝑘 − 𝜔𝑗2 +
2
𝛼𝑗 − 𝛼𝑗+1 ,
1
4𝑟 2
𝑁
𝑗=1
1
2
2
2
𝜔𝑗−1
+ 𝜔𝑗+1
𝐺=
= 0;
𝑁−1
𝑗=1
2
2
𝜔𝑗2 − 𝜔𝑗−1
− 𝑘(𝑁 + 1 − 2𝑗) .
2
𝜔𝑗 ′ ,
Existence proof
 We prove that nodeless non-trivial (soliton and black hole) solutions to the field equations exist in
some neighbourhood of known embedded (trivial) solutions, and in the limit Λ → ∞, by proving the
following.
1. There are nodeless ‘trivial’ (embedded) solutions to the field equations, whose existence is obvious or
has been proven already, and some of which are nodeless – including a unique trivial solution in the limit
Λ → ∞…
𝑟 = 𝑟0
𝑟=∞
Nodeless trivial solutions
 Letting 𝛼𝑗 = 0, we recover the purely magnetic system.
 Letting 𝑘 = 1, we recover the spherical system.
 Letting 𝛼𝑗 = 0, 𝜔𝑗 =
𝑗(𝑁 − 𝑗) gives the Schwarschild-AdS solution.
 Rescaling all quantities we find embedded 𝔰𝔲(2) solutions.
Existence proof
 We prove that nodeless non-trivial (soliton and black hole) solutions to the field equations exist in
some neighbourhood of known embedded solutions, and in the limit Λ → ∞, by proving the
following.
1. There are nodeless ‘trivial’ (embedded) solutions to the field equations, whose existence is obvious or
has been proven already, and some of which are nodeless – including a unique trivial solution in the limit
Λ → ∞.
2. Prove that solutions exist in some neighbourhood of the boundaries and as Λ → ∞…
𝑟 = 𝑟0
𝑟=∞
Local existence theorem
 Coddington, E.A., Levinson, N. “The theory of ordinary differential equations”. McGraw-Hill, New York, 1955.
 “Consider a system of differential equations for 𝑛 + 𝑚 functions 𝑎 = (𝑎1 , 𝑎2 , … , 𝑎𝑛 ) and 𝑏 = (𝑏1 , 𝑏2 , … , 𝑏𝑚 ) of the
form
𝑥
𝑑𝑎𝑖
𝑑𝑥
= 𝑥 𝑝𝑖 𝑓𝑖 𝑥, 𝐚, 𝐛 ,
𝑥
𝑑𝑏𝑖
𝑑𝑥
= −𝜆𝑖 𝑏𝑖 + 𝑥 𝑞𝑖 𝑔𝑖 (𝑥, 𝐚, 𝐛)
with constants 𝜆𝑖 > 0 and integers 𝑝𝑖 , 𝑞𝑖 ≥ 1 and let 𝐶 be an open subset of ℝ𝑛 such that the functions 𝑓𝑖 and 𝑔𝑖 are
analytic in a neighbourhood of 𝑥 = 0, 𝐚 = 𝐜, 𝐯 = 0, ∀𝐜 ∈ 𝐶. Then there exists an 𝑛-parameter family of solutions of the
system such that
𝑎𝑖 𝑥 = 𝑐𝑖 + 𝑂 𝑥 𝑝𝑖 ,
𝑏𝑖 𝑥 = 𝑂(𝑥 𝑞𝑖 )
where 𝑎𝑖 𝑥 and 𝑏𝑖 𝑥 are defined for 𝐜 ∈ 𝐶, 𝑥 < 𝑥0 (𝐜) and are analytic in 𝑥 and 𝐜.”
 Apply this at all boundaries: 𝑟 = 0 or 𝑟 = 𝑟ℎ , and as 𝑟 → ∞. 𝑟 = 𝑟ℎ and 𝑟 → ∞ not too hard - 𝑟 = 0 a nightmare!
Existence proof
 We prove that nodeless non-trivial (soliton and black hole) solutions to the field equations exist in
some neighbourhood of known embedded solutions, and in the limit Λ → ∞, by proving the
following.
1. There are ‘trivial’ (embedded) solutions to the field equations, whose existence is obvious or has been
proven already, and some of which are nodeless – including a unique trivial solution in the limit Λ → ∞.
2. Prove that local solutions exist in some neighbourhood of the boundaries and as Λ → ∞.
3. Prove that as long as 𝜇(𝑟) > 0 throughout the range, then local solutions may be integrated out
arbitrarily far from the event horizon (or origin)…
𝑟 = 𝑟0
𝑟=∞
Existence proof
 We prove that nodeless non-trivial (soliton and black hole) solutions to the field equations exist in
some neighbourhood of known embedded solutions, and in the limit Λ → ∞, by proving the
following.
1. There are ‘trivial’ (embedded) solutions to the field equations, whose existence is obvious or has been
proven already, and some of which are nodeless – including a unique trivial solution in the limit Λ → ∞.
2. Prove that local solutions exist in some neighbourhood of the boundaries and as Λ → ∞.
3. Prove that as long as 𝜇(𝑟) > 0 throughout the range, then local solutions may be integrated out
arbitrarily far from the event horizon (or origin).
4. Prove that the field equations remain regular in the asymptotic limit 𝑟 → ∞.
𝑟 = 𝑟0
𝑟=∞
Existence proof
 We prove that nodeless non-trivial (soliton and black hole) solutions to the field equations exist in
some neighbourhood of known embedded solutions, and in the limit Λ → ∞, by proving the
following.
1. There are ‘trivial’ (embedded) solutions to the field equations, whose existence is obvious or has been
proven already, and some of which are nodeless – including a unique trivial solution in the limit Λ → ∞.
2. Prove that local solutions exist in some neighbourhood of the boundaries and as Λ → ∞.
3. Prove that as long as 𝜇(𝑟) > 0 throughout the range, then local solutions may be integrated out
arbitrarily far from the event horizon (or origin).
4. Prove that the field equations remain regular in the asymptotic limit 𝑟 → ∞.
 We use these to prove that any solution to the field equations exists in an open set in the parameter
space: therefore we can find genuinely non-trivial solutions ‘nearby’ the trivial solutions that we
found.
Topological purely magnetic black holes:
Numerical analysis
 Baxter, J. E., Winstanley, E. “Topological black holes in SU(N) Einstein-Yang-Mills theory with a
negative cosmological constant”. Phys. Lett. B 753, 268–273 (2016).
 Having analytically proven existence of topological, purely magnetic, hairy black holes (Baxter,
2015), we investigated the solution space of 𝔰𝔲 3 topological black holes, with 2 gauge
functions 𝜔1 and 𝜔2 .
 NB - There are no regular topological solitons: if 𝑘 ≠ 1, 𝑅 blows up at the origin.
 We are interested in examining the solution space, to verify our results.
 This information also comes in useful when proving stability, in case we have stability
conditions.
Topological purely magnetic black holes:
Numerical analysis
 Note:
 Initial conditions are 𝑣1 and 𝑣2 .
 Above the dashed lines we do not have a
regular non-extremal event horizon; below
solid lines we have solutions.
 Embedded SU(2) solutions are where 𝑣1 = 𝑣2.
These are monotonic, hence nodeless.
 As |Λ| gets larger:
1.
the space of nodeless solutions increases;
2.
the space of no solutions decreases;
3.
all existing solutions are nodeless.
(This accords with our stability results for the
system.)
Solution space of 𝔰𝔲 3 black holes with 𝑘 = 0.
Topological purely magnetic black holes:
Numerical analysis
 Note:
 Initial conditions are 𝑣1 and 𝑣2 .
 Above the dashed lines we do not have a
regular non-extremal event horizon; below
solid lines we have solutions.
 Embedded SU(2) solutions are where 𝑣1 = 𝑣2.
These are monotonic, hence nodeless.
 As |Λ| gets larger:
1.
the space of nodeless solutions increases;
2.
the space of no solutions decreases;
3.
all existing solutions are nodeless.
(This accords with our stability results for the
system.)
Solution space of 𝔰𝔲 3 black holes with 𝑘 = −1.
Open Questions:
 Can we further refine/confirm Bizon’s theorem?
 How do 𝔰𝔲 𝑁 dyons apply to CMP, analogous to the dyonic planar 𝔰𝔲 2 solutions? Given that the
dual CFT will contain observables sensitive to then presence of hair, what is gained?
 How about other gauge groups? Higher dimensions? (AdS5 is an obvious one.)
 Is 𝑘 = −1 ‘of use’?
 Can this work be compared with Hawking’s to yield insights on the Information Paradox?