BLACK HOLE CONFIGURATIONS

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16 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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FI RST ORDER FORMALI SM FOR

NON
-
SUPERSYMMETRI C MULTI
BLACK HOLE CONFI GURATI ONS

A.S h c h e r b a k o v

LNF INFN Frascati
(Italy)


i n c ol l abor at i on wi t h A.Yer anyan


S upe r s y mmet r y i n I nt e g r a bl e Sy s t e ms
-

S I S'12

Purpose

In

the

framework

of

N=
2

D=
4

supergravity,

c
onstruct

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

configuration

2.
supersymmetric

and

non
-
supersymmetric

3.
rotating

black

holes

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

2

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?



S u p e r s y mme t r y

i n I n t e g r a b l e S y s t e ms 2 0 1 2

3

Results

Equations

Two

possible

cases

A.


B.


S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

4

Setup

Einstein

gravity

coupled

to

electromagnetic

fields

in

a

stationary

background

With

N=
2

D=
4

SUSY
,

the

σ
-
model

metric

G


and

couplings

μ
ΛΣ

and

ν
ΛΣ

are

expressed

in

terms

of

a

holomorphic

prepotential

F=F(z)
.



S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

5

Reduction to three dimensions

Reduction

is

performed

in

Kaluza
-
Klein

manner

metric

vector
-
potential

Three

dimensional

vector

potentials

a


and

w

can

be

dualized

in

scalars


S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

6

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

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

7

divergenless

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

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

8

Black hole potential

Single

non

rotating

BH

General

case

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

9

Recall

hints to introduce


Equations of motion (summary)

The

equations

of

motion

has

the

following

form

with

the

constraint

where

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

10

Present state of art



supersymmetric,

single

center

supersymmetric,

multi

center






non
-
supersymmetric,

single

center

non
-
supersymmetric,

multi

center

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

11

Supersymmetri c si ngl e o mul ti
-
center

Single

center





Natural

splitting

Entropy

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

12


Multi center

S.Ferrara, G.Gibbons, R.Kallosh ‘97

F.Denef ‘00

Non supersymmetric single center

Analogous

description

for

non
-
BPS

black

holes

Entropy

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

13

A.Ceresole

G. Dall’Agata ‘07

S.Bellucci, S.Ferrara,

A.Marrani, A.Yeranyan ‘08

Example

of

a

fake

superpotential

Constructi ng the fi rst order equati ons

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?

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

14

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

π

At

spatial

infinity

Phase

restoration


Constructing flow
-
defining functions

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

15

G.Bossard’11

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
.




S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

16

G.Bossard’11

Toy example:

1. Composite

2. Almost BPS

A.Yeranyan ‘12

Composite

Full

set

of

equations

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

17

Almost BPS

S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

18

Full

set

of

equations

Properties

We

showed

that

solutions

1.
Rasheed
-
Larsen

black

holes

2.
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
.

S u p e r s y mme t r y

i n
I n t e g r a b l e

S y s t e ms 2 0 1 2

19

THANK YOU!


-

I

think

you

should

be

more

explicit

here

in

step

two


S u p e r s y mme t r y i n I n t e g r a b l e S y s t e ms 2 0 1 2

20