first order formalism for non-supersymmetric multi black hole configurations
DESCRIPTION
FIRST ORDER FORMALISM FOR NON-SUPERSYMMETRIC MULTI BLACK HOLE CONFIGURATIONS. A.Shcherbakov LNF INFN Frascati (Italy). in collaboration with A.Yeranyan. Supersymmetry in Integrable Systems - SIS'12. Purpose. - PowerPoint PPT PresentationTRANSCRIPT
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FIRST ORDER FORMALISM FOR
NON-SUPERSYMMETRIC MULTI
BLACK HOLE CONFIGURATIONS
A . S h c h e r b a k o vLNF INFN Frascati
(Italy)
i n c o l l a b o r a t i o n w i t h A . Ye r a n y a n
S u p e r s y m m e t r y i n I n t e g r a b l e S y s t e m s - S I S ' 1 2
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Purpose
In the framework of N=2 D=4 supergravity, construct the first order equation formalism governing the dynamics of the graviton, scalar and electromagnetic fields in the background of extremal black hole(s)
1. multiple black hole configuration2. supersymmetric and non-supersymmetric3. rotating black holes
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3
Why equations and not solutions?
The main goal – to find a solution.
The equations of motion are coupled non-linear differential equations of the second order.
The known solutions are just particular ones.
Why not to rewrite the equations of motion in an easier-to-solve manner?
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Results
EquationsTwo possible cases
A.
B.
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Setup
Einstein gravity coupled to electromagnetic fields
in a stationary background
With N=2 D=4 SUSY, the σ-model metric Gaā and couplings μΛΣ and νΛΣ are expressed in terms of a holomorphic prepotential F=F(z).
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Reduction to three dimensions
Reduction is performed in Kaluza-Klein manner
metric
vector-potentialThree dimensional vector potentials a and w can be dualized in scalars
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If the three dimensional space is flat, the equations of motion read
with an additional constraint
These equations contain the following objects
Equations of motion
divergenless
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Black hole potential
In the case of a single non-rotating black hole
tensorial black hole potential reduces to a singlet
For N=2 D=4 SUGRA
where
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Black hole potential
Single non rotating BH General case
rotation & Maxwell
Recall
hints to introduce
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Equations of motion (summary)
The equations of motion has the following form
with the constraint
where
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Present state of art
supersymmetric, single center supersymmetric, multi center
non-supersymmetric, single center non-supersymmetric, multi center
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Supersymmetric single o multi-center
Single center
Natural splitting
Entropy
Multi center S.Ferrara, G.Gibbons, R.Kallosh ‘97 F.Denef ‘00
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Non supersymmetric single center
Analogous description for non-BPS black holes
Entropy
A.CeresoleG. Dall’Agata ‘07
S.Bellucci, S.Ferrara,A.Marrani, A.Yeranyan ‘08
Example of a fake superpotential
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Constructing the first order equations
General form of the first order equations
plus other equations (if any).The algebraic constraint
imposes a relation
What functions W, Pi and li are equal to?
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As a starting point, let us consider the spatial infinity and the supersymmetric flow.
Wi and Pi are defined by ADM mass M, NUT charge N and scalar charges πAt spatial infinity
Phase restoration
Constructing flow-defining functions
G.Bossard’11
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Now let us generalize the consideration for the whole space:
Constructing flow-defining functions
To pass to a non-supersymmetric solution, “charge flipping” is needed.
G.Bossard’11D0
D2D2D2D4D4D4
D6
Toy example:
1. Composite 2. Almost BPS
A.Yeranyan ‘12
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Composite
Full set of equations
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Almost BPS
Full set of equations
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Properties
We showed that solutions1. Rasheed-Larsen black holes2. magnetic/electric multi-black hole satisfy the corresponding equations of motion.Let us stress that all these solutions are particular ones and not general.Appearance of the phases demonstrates how the concept of “flat directions” gets generalized for multi-black hole configurations.
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THANK YOU!
- I think you should be more explicit here in step two…