1. Graviton Emission in The Bulk from a Higher Dimensional Black Hole
in collaboration with S. Creek , P. Kanti , K. Tamvakis
hep-th/

2. Plan of the talk: Brief introduction to the brane world scenario
Write down the equations for graviton emission through Hawking radiation from a Schwarzschild-like black hole in the presence of extra dimensions
Solve the equations – compute energy rates for the emission
Conclusions

3. Introduction According to the Brane-World scenario :
SM particles confined in a 4-dimensional Brane
Gravity propagates on (4+n) dimensions (Bulk). Extra dimensions: compact, spacelike
Hierarchy problem can be solved
Black holes can be produced in colliders or in cosmic ray interactions

4. In the last few years there has been considerable interest in theoretical studies of these higher dimensional black holes that may be created in near future experiments (perhaps even LHC) . One important property of them :
Hawking Radiation
→ Black holes are not completely black!
→ They emit radiation in a thermal spectrum , characterized by a temperature TH , such as a blackbody. → They deviate from the blackbody radiation by a factor called greybody factor. For a spherically symmetric BH

5. Because of the curved geometry outside a BH : some of the particles created outside the horizon will backscatter into the BH
Greybody factor measures the probability of particle escaping to infinity
Depends on particles energy, spin, angular momentum and on space dimensionality
AND HAS BEEN PROVEN TO BE PROPORTIONAL TO THE INCOMING ABSORPTION PROBABILITY!

6. In the simplest case, that of higher dimensional Schwarzschild black holes, there have been both analytical and numerical studies for the case of Standard Model particle radiation. For example see :
P. Kanti hep-ph/
C.Harris ,P. Kanti hep-ph/

7. Gravitons emitted from a black hole :
But until recently, there have been little studies for the case of graviton emission in the bulk !
Gravitons emitted from a black hole :
will “see” the entire (4+n) dimensional space-time (Bulk)
will account for an energy loss from the brane where the BH is located, to the bulk
their emission rate will depend from the dimensionality of space-time, namely n

8. Our plan : ↓ ↓ Write the equations for graviton emission
Analytically solve the equations, compute the absorption probability ↓
Compute the energy emission rate and its dependence on n

9. FRAMEWORK: Higher Dimensional Schwarzschild BH
line element (R. Myers, M. Perry):
with
and dΩ2n+2 the line element in a (n+2) dimensional unit sphere
The temperature of the black hole associated with Hawking radiation is

10. H. Kodama, A. Ishibashi hep-th/0305147
1. The equations
The equations for graviton emission from a Schwarzschild BH in (4+n) dimensions are known
H. Kodama, A. Ishibashi hep-th/
How to produce them :
→ perturb the metric outside a BH , use Einstein equations
→ we get 3 independent wave equations describing emitted gravitational waves , analyzed in terms of partial waves with specific angular momentum.These equations are called
Scalar perturbation equation
Vector >>
Tensor >>
They describe the emitted gravitons that correspond to each type of metric perturbation. All three can be written in the form of a master equation for the partial waves

11. With V having different form for every perturbation
With V having different form for every perturbation. For the case of tensor and vector type we have:
With l being the angular momentum quantum number and k= -1 (3) for tensor (vector) type .
For the Scalar case the potential V is a bit more complicated

13. 2. Solving the equations
We will use an approximate method , valid at the low energy limit of the equations. What we will do is:
Solve the equations close to the horizon rH
Solve the equations far away from the horizon
Stretch the two solutions and match them in the intermediate zone

14. there are no outgoing waves in the limit r→ rH
Boundary condition
“Nothing can escape from the black hole after it crosses the horizon”
So in the solution we will have we must impose the condition
there are no outgoing waves in the limit r→ rH

15. A. Tensor and Vector type
We will first look for the near horizon solution . We change the variables from r→ f (r) = 1 - (rH / r) n+1 ,and after making the field redefinition
and taking the limit r→ rH ,the equation can be brought to the following form
for the following choice for the parameters :
This equations has known solutions, the hypergeometric functions

16. So the near horizon solution for tensor-vector mode gravitons is :
With F the hypergeometric function and A1,A2 constants. By expanding this solution in the limit r→ rH and taking the boundary condition we get A2=0
We then solve the equation in the far field regime , that is far away from the horizon . The solution after taking r>>rH is easily obtained in terms of the Bessel functions :
We now have to match the two solutions in the intermediate zone

17. We take the limit r>>rH in the near horizon solution
We take the limit ωr<<1 in the far field solution
We look in the low energy regime ωrH<<1
we compute the relation between the constants B1 , B2
We construct a solution for the whole domain of r , valid for low energies

18. We want the absorption probability ,so we expand the FF solution for r→ in terms of ingoing/outgoing spherical waves, and we only need to compute the ratio of their amplitudes . The result put in a compact form, is:
the same formula holds for the scalar type as well
the only dependence from the type of perturbation is in G , which is different for every type

21. 3. Energy emission rates
State multiplicities : there is a number of states that correspond to the same angular momentum number ℓ .
→ it depends on ℓ , n
This has been computed in the literature for every type of perturbation (Rubin&Ordonez J. Math. Phys 26,65(1985)

22. Multiplicities of states with angular momentum ℓ

23. We are finally ready to compute the energy emission rate !
Putting everything together we have
We can now plot the above emission rate for all 3 types, and for every n .

26. Results
Vector mode gravitons are the dominant graviton mode to be emitted by the BH for every value of n
Relative magnitude of the emission rate for scalar and tensor mode gravitons depends on n
Energy emission rate in the bulk decreases as n increases, although we would expect the opposite

27. Conclusions
We studied graviton emission in the bulk from a spherically symmetric (4+n) dimensional Schwarzschild BH
The differential equations for the graviton emission were analytically solved for low energies using a matching technique
The equation solutions were found in the “near horizon” and “far field” regime and were stretched and matched in the intermediate zone

28. We thus computed the absorption probability for all 3 types of perturbations , and written down the equations describing the BH’s Hawking radiation spectrum .
We computed the energy emission rate for low energy gravitons in the bulk , which encodes information about the number of extra dimensions that might exist in nature