Hawking radiation for a proca field
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Mengjie Wang (王梦杰 ) In collaboration with Carlos Herdeiro & Marco Sampaio. Hawking radiation for a Proca field. Based on: PRD85(2012) 024005. 王梦杰. Mengjie Wang. Mengjie Wang. Outline. 1. Introduction. 2. Hawking radiation in D dimensions. 3. Hawking radiation on the brane. 4.

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Hawking radiation for a Proca field

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Mengjie Wang(王梦杰 )

In collaboration with Carlos Herdeiro & Marco Sampaio

Hawking radiation for a Proca field

Based on: PRD85(2012) 024005


Mengjie Wang

Mengjie Wang





Hawking radiation in D dimensions


Hawking radiation on the brane


Discussion & Conclusions

Introduction (I)

What ?

Hawking radiation is the most prominent quantum effect for quantum fields in a

background spacetime with an event horizon.

Intuitive picture


virtual pair creationof particles near the event horizon


real particle






Hawking radiation


QFT in curved spacetime

Path-integral derivation

... ...

Gravitational 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.

Generally speaking, the Equation of Motion in curved space-time cannot be

decoupled, or variables cannot be separated.

Constructing a kind of systematic numerical method to deal with the coupled

Ordinary Differential Equations(ODEs), as well as Partial Differential



How ?

How to get the EOMs?

arXiv: 0712.2703

background geometry

line element

Hodge decomposition theorem

Suppose to be a compact Riemannian manifold, any dual vector field on

can be uniquely decomposed as

is a scalar field

is a transverse vector

How to solve the EOMs?

from the second to the first

define scattering matrix


factor matrix

provide physical prescription

choose scattering matrix

Hawking radiation in D dimensions

The Lagrangian for Proca field, which describe the Z and W particles

in the standard model, is

is the electromagnetic field strength tensor

Equations of motion for Proca field

The gravitational background

(m+n)-dimensional spacetime whose manifold structure is locally a warped

product type

with Einstein symmetric spaces

spanning the m-dimensional space with metric

spanning the n-dimensional Einstein space

Hawking radiation in D-dimensions

Decomposition of the vector field in tensorial types


is the Laplacian operator in Einstein space

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

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

Hawking radiation in D-dimensions

Equations of motion in Schwarzschild spacetime

now we specialize to Schwarzschild case, i.e.

Modes with






Hawking radiation in D-dimensions

Boundary conditions at the horizon

we can rewrite our equations as

making use of Frobenius method

we get recurrence relations

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

The asymptotic form for

Hawking radiation in D-dimensions

The first order equations

For the numerical convenience, we rewrite the coupled equations in the first order


define a vector V

coupled equations can be written as matrix form

define another vector

from the above asymptotic expansion, we have relation


Hawking radiation in D-dimensions

Definition of transmission factor

We know that a general solution is parameterized by 4 independent coefficients in

one 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 at

the horizon and at infinity.

We denote the ingoing and outgoing wave coefficients

at the horizon

at infinity

impose an ingoing boundary condition

transmission factor=eigenvalue(T)

Hawking radiation in D-dimensions

Physical prescription

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

new reflection matrix?

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

the definition of flux

the energy momentum tensor for complex neutral Proca field

the flux at infinity

the flux at the horizon

Hawking radiation in D-dimensions

Hawking radiation in D-dimension

The number and energy fluxes are


Hawking radiation in D-dimension

Comparison between small mass and exact zero mass

Hawking radiation on the Brane

Specialize to charged brane

Now we generalize the previous work to brane case

considering the background

perform the same procedure, we get the following equations of motion

coupled modes with

transverse mode


Hawking radiation on the Brane

Hawking radiation on the brane

Hawking radiation on the brane

Hawking radiation on the brane


We have used a numerical strategy to solve the coupled wave equations for Proca

field in D-dimensional Schwarzschild black hole. Our results show some expected

features, such as the mass suppression of the Hawking fluxes as the Proca mass

is increased, but also some novel features, such as the nonzero limit of the

transmission 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 difference of transmission factor between in the bulk and

on the brane. We found the the nonzero limit of the transmission factor is existed

for all n.

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.

Thank You !

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