Hawking radiation for a Proca fieldHawking radiation for a Proca field

Mengjie Wang （王梦杰 ）

In collaboration with Carlos Herdeiro & Marco Sampaio

Mengjie WangMengjie Wang 王梦杰

Based on: PRD85(2012) 024005

Outline

Introduction

Hawking radiation in D dimensions

Hawking radiation on the brane

Discussion & Conclusions4.

3.

2.

1.

Introduction (I)

What ?

Hawking radiation is the most prominent quantum effect for quantum fields in a background spacetime with an event horizon.

Intuitive picture

BH ××-E EHawking radiation

×E real particle

virtual pair creation of particles near the event horizon

Methodology

QFT in curved spacetime Path-integral derivation

TunnelingGravitational anomaly ... ...

Introduction (II)

Why ?From the Brane-World scenario, black holes can be produced in colliders or in cosmic ray interactions. We can detect black hole events via Hawking radiation, and we can read the extra dimension from it.

SM particles are confined on a 4-dimensional Brane.

Generalization & Improving current black hole event generators.

Constructing a kind of systematic numerical method to deal with the coupledOrdinary Differential Equations(ODEs), as well as Partial DifferentialEquations(PDEs).

Generally speaking, the Equation of Motion in curved space-time cannot be decoupled, or variables cannot be separated.

Introduction

How ?How to get the EOMs?

How to solve the EOMs?

from the second to the first define scattering matrix

provide physical prescription choose scattering matrix

transmission factor matrix

background geometry m n M N H

line element 2 2 2ds ( ) ( )M N a bMN ab ng dz dz g y dy dy r y d 2 i j

n ijd dx dx

Hodge decomposition theoremSuppose to be a compact Riemannian manifold, any dual vector field on can be uniquely decomposed as

( , )nijH

nH

( ) ( )ˆ s ti i iv D v v ( )ˆ 0t i

iDv

( )sv ( )tivis a scalar field is a transverse vector

arXiv: 0712.2703

Hawking radiation in D dimensions

The Lagrangian for Proca field, which describe the Z and W particles in the standard model, is

†1=

2W W

L

W W W F is the electromagnetic field strength tensor

Equations of motion for Proca field2 0W M W iqW F

The gravitational background

2 2 2( ) ( )a bab nds h y dy dy r y d

with Einstein symmetric spaces2 ( ) i jn ijd x dx dx

{ }ay spanning the m-dimensional space with metric abh

{ }ix spanning the n-dimensional Einstein space

(m+n)-dimensional spacetime whose manifold structure is locally a warped product type m n M N H

2M †W W

†iqW W F

Hawking radiation in D-dimensions

Decomposition of the vector field in tensorial typesfor aW

20

ˆ( ) 0ak W 20 ( 1)k l l n ̂ is the Laplacian operator in Einstein space

for , which can be decomposed into a scalar and a transverse vectoriW

ˆ Ti i iW D W ˆ ˆ 0Ti

iDW 20

ˆ( ) 0k

21

ˆ( ) 0Tik W 2

1 ( 1) 1k l l n

The above decompositons and conditions allow for an expansion of the form

( ) ( )a aW w y x y ( ) ( )y x

y ,( ) ( )Ti iW q y x

y

Hawking radiation in D-dimensions

Equations of motion in Schwarzschild spacetimenow we specialize to Schwarzschild case, i.e.

1

1( ) 1tt n

rr

h V rh r

1( 1)n

Hr

Modes with 0 0k 2

2 2 2 2 '02 2

1[ ( ) ( ) ] 0n

n

kd dV r M V i V

dr r dr r

222 2 '0

2

2[ ( ) ( ) ] ( ) 0n

n

kV d d Vr M V i V

r dr dr r r

22 2 21

2 2

ˆ /[ ( ) ( ) ] 0n

n

k R nV d dr V M V

r dr dr r

Modes 0 0k (0)

2 2[ ( ) 1] 0

n

n

V d r V d

r dr M V dr

(0) (0)

2 2

i V d

M V dr

22 2 0

2 2[ ( ) ] 0n

n

kd V dV r Vdr r dr r

22

( )nn

iV dr

r dr

massive coupled

transverse

massless

Hawking radiation in D-dimensions

Boundary conditions at the horizonwe can rewrite our equations as

2

2[ ( ) ( ) ( )] ( ) 0

d dA r B r C r D r

dy dy

2

2[ ( ) ( ) ( )] ( ) 0

d dA r B r C r D r

dy dy

making use of Frobenius method0

jj

j

y y

0

jj

j

y y

we get recurrence relations

1

i

n

0 0 3 2 1 1 0 0 1

12 0

( ( 1) )

( 1)

a b c d d

a c

2 2( 1) ( )( 1) ( 1)

j j jj j

n j j i nf f

g g

2 2( 1) ( )( 1) ( 1)j j j

j j

n j j i nf f

g g

2 2 2 2 2( 1) ( ( 1) ( )( 1))jg n n j j

2 11

[( ( )( 1) ( ) ) ]j

j m m m j m m j mm

f a j m j m b j m c d

2 11

[( ( )( 1) ( ) ) ]j

j m m m j m m j mm

f a j m j m b j m c d

Hawking radiation in D-dimensions

Asymptotic behavior at infinity

To understand the asymptotic behavior of the coupled modes at infinity, we study the following asymptotic expansion

0

jr p

j j

ae r

r

0

jr p

j j

be r

r

2 2,2, 1 / 2 , ( ) / (2 )nik p n i k k

2 2 2k M

The asymptotic form for 1 1

0 0/2 1

1[( ...) ( ...) ]i i

n

a aa e a e

r r r

0 0/2 1

1[( ( ) ...) ( ( ) ...) ]i i

n

k c k ca e a e

r r r

logkr r 2 2

2 2 2 20 ,2 ,3 ,2

(2 )[ (2 )( ) ( ( ) ) ]

2 4 2 2n n n

i n n M Mc k M i

k k

Hawking radiation in D-dimensions

The first order equationsFor the numerical convenience, we rewrite the coupled equations in the first orderformdefine a vector V ( , , , )T d d

Vdr dr

coupled equations can be written as matrix formdV

XVdr

define another vector 1 1 0 0( , , , )T a a a a

from the above asymptotic expansion, we have relation V T

from

dVXV

dr

V T1( )

d dTT XT

dr dr

Hawking radiation in D-dimensions

Definition of transmission factorWe know that a general solution is parameterized by 4 independent coefficients inone of the asymptotic regions, either at the horizon or at infinity. Because of the linearity of the coupled equations, we can use a matrix to relate the coefficients atthe horizon and at infinity.We denote the ingoing and outgoing wave coefficients at the horizon

( , )i ih h h

at infinity

( , )i iy y y

y Sh y S S h

y S S h

impose an ingoing boundary condition 0h s sy S h

y S h

y S h + + 1= yy S S Ry （ ） †1T R R

transmission factor=eigenvalue(T)

Hawking radiation in D-dimensions

Physical prescription

There is still some freedom in the definition of the asymptotic coefficients

s s sy M y Hh M h new reflection matrix?

for a single decoupled field with definite energy, the transmission factor is2 2 2 2

2 2=inHin

Y Y Y YT

Y Y

（ ） （ ） （ ） （ ）

（ ） （ ）

（ - ） （ - ） FF

the definition of flux |n

rr tS

d T F

the energy momentum tensor for complex neutral Proca field

† 2 †1( . )

2 2

gT W W M W W c c

L

the flux at infinity †( )coupled y Ty F 2 2

0 0 0 1( ) / / ( )Ty k k a k M xa ika

the flux at the horizon †( )coupledH h h F 2

0 0 0 1( )Th k x z † 1( )T S S

Hawking radiation in D-dimensions

Hawking radiation in D-dimension

The number and energy fluxes are2

/

d { , } 1 {1, }

2 1HTl

N Ed T

dtd e

( 2 1)( 2)!

( 1)! !S

n l n ld

n l

( 2 1)( 1)( 3)!

( 1)( 1)!( 2)!V

n l n l n ld

l l n

1

4H

nT

Results

Hawking radiation in D-dimension

Comparison between small mass and exact zero mass

Hawking radiation on the Brane

Specialize to charged braneNow we generalize the previous work to brane caseconsidering the background

2 2 2 2 22

1ds Vdt dr r d

V

2

1 21

n

QV

r r

22 22 2 2 '0

2 2 2 2 2 2 2[ ( ) ] [ ] 0

kd QqV d iQqV i QqV M V i V

dr M r dr r r M r

22 ' 2 2

2 2 2 '02 2 2 2 2 2 2 2 2 2

2[ ( ) ( ) ] [ ( ) ] 0

kV d d QqV d QqV iQqV d V iQqVr M V i V

r dr dr M r dr r M r M r dr r r

perform the same procedure, we get the following equations of motioncoupled modes with 0k

transverse mode2

2 212

ˆ / 2[ ( ) ( ) ] 0

k Rd dV V M Vdr dr r

0k mode2 2 2 2 2

2 2 2 2 2 2 4 2 2

1 1[ ( ) ( ) ] 0

( ) t

d M r V d d Qq M Q qW

r dr M V dr r dr M V V r M V

Hawking radiation on the Brane

Hawking radiation on the brane

Hawking radiation on the brane

Hawking radiation on the brane

Conlusion

We have used a numerical strategy to solve the coupled wave equations for Procafield in D-dimensional Schwarzschild black hole. Our results show some expectedfeatures, such as the mass suppression of the Hawking fluxes as the Proca massis increased, but also some novel features, such as the nonzero limit of thetransmission factor, for vanishing spatial momentum, in n=2,3. Moreover, a precise study of the longitudinal degrees of freedom was carried out.

We have shown the charge effects on the transmission factor. We found there is contribution for nonzero limit of transmission factor from the charge. For one component of the coupled transmission factor, it is increased through the field charge vary from the negative to positive, the other component is reverse.

We have shown the difference of transmission factor between in the bulk and on the brane. We found the the nonzero limit of the transmission factor is existedfor all n.