Absorption probabilities associated to spin-3/2 particles
near N dimensional Schwarzschild black holes
Gerhard Harmsen
Supervisor: Prof. Alan Cornell
Collaborators: C. H. Chen, H. T. Cho.
University of the Witwaterrand
[email protected]
9th February 2016
Gravitino fields in N dimensional Schwarzschild black hole spacetimes
Gerhard Harmsen (Wits)
Absorption probabilities
9th February 2016
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Overview
1
Introduction
Black hole absorption probability
Rarita-Schwinger equation
2
N dimensional Schwarzschild black holes
N dimensional metric
Radial equation
Potential function
3
Approximation methods
Unruh method
Particle potentials
WKB method
4
Results
5
Conclusion
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Absorption probabilities
9th February 2016
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Black hole absorption
Field theory allows for particle scattering off a black hole.
This scattering implies “Grey body”, not pure “Black body”,
behaviour.
Pioneered by Unruh in 1976.
Unruh ideas describe Hawking radiation.
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Absorption probabilities
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Rarita-Schwinger equation
The relativistic field equation of spin-3/2 particles.
Rarita-Schwinger equation
γ µνα ∇ν Ψα = 0
where γ µνα = γ µ γ ν γ α − γ µ g να + γ ν g να − γ α g µν .
Gravitino is predicted to have a spin of 3/2.
Lightest supersymmetic particle.
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Absorption probabilities
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Schwarzschild metric
N dimensional Schwarzschild metric
ds 2 = −f (r )dt 2 +
with f (r ) = 1 −
2M N−3
r
1
dr 2 + r 2 dΩ2N−2 ,
f (r )
.
N dimensional spherical metric
dΩ2N = dθ2 + sin2 (θ)dΩ2N−1 ,
with dΩ22 = dθ2 + sin2 (θ)dφ2 .
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Absorption probabilities
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Radial equation
Radial equation
−
where
d
dr∗
d2
d2
2
φ̄
+
V
φ̄
=
ω
φ̄
;
−
φ̄2 + V2 φ̄2 = ω 2 φ̄2 ,
1
1
1
1
dr∗2
dr∗2
d
= f (r ) dr
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Absorption probabilities
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Potential function
N dimensional potential function
V1,2 = ±f
dW
+ W 2,
dr
with,
W =
j+
N−3
2
√
r
Gerhard Harmsen (Wits)

f 

2
N−2
2
2
N−2
N−4 2M N−3 
j + 2N−5
− N−2 r
2


1
2N−5
2M N−3
j−2 j+ 2
+ r
j−
2
1
2
Absorption probabilities
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Unruh method
Consider three regions:
0.4
Near Region:
0.3
Central Region:
Far Region:
f(r)®0 V(r)pΩ
VHrL
f(r)®1
0.2
0.1
0.0
0
2
4
6
8
10
r
Figure: Potential for the 4 dimensional Schwarzschild black hole
Gerhard
Harmsen
(Wits)
Gravitino
fields
in
probabilities
9th February 2016
SchwarzschildAbsorption
black hole
spacetimes (arXiv:1504.02579)
8 / 18
Unruh method
The absorption probability is given as,
Unruh absorption probability
|Aj (ω)|2 = 4πC 2 ω 2j+1 1 + πC 2 ω 2j+1
−2
≈ 4πC 2 ω 2j+1 ,
where,
C=
j + 23
1
,
22j+1 Γ(j + 1) j − 12
where Γ is the gamma function and ω < 1.
Gerhard Harmsen (Wits)
Absorption probabilities
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Potential near black hole
7
N=9
6
VHrL
5
N=8
4
N=7
3
2
N=6
N=5
1
N=4
0
0
2
4
6
8
10
r
Figure: Potential function for spin-3/2 particles in 4 and higher dimensional
Schwarzschild backgrounds with angular momentum j=3/2.
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Absorption probabilities
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Potential near black hole
j=13/2
7
6
j=11/2
VHrL
5
4
j=9/2
3
j=7/2
2
j=5/2
1
j=3/2
0
0
2
4
6
8
10
r
Figure: Absorption probability for spin-3/2 particles in 4 dimensional
Schwarzschild background with angular momentum j=3/2 to 13/2.
Gerhard Harmsen (Wits)
Absorption probabilities
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WKB method
WKB absorption probability
|Aj (ω)|2 =
1
,
1 + e 2S(ω)
with,
1 2
15 2
3
S(ω) = πk
z +
b − b4 z04
2 0
64 3 16
315 2
35 2 35
5
1/2 1155 4
+ πk
b −
b b4 +
b + b3 b5 − b6 z06
2048 3 256 3
128 4 64
32
3
7
+ πk −1/2
b4 − b32
16
64
1365
525 2
85 2 95
25
−1/2
4
− πk
b −
b b4 +
b + b3 b5 − b6 z02 ,
2048 3 256 3
128 4 64
32
1/2
Gerhard Harmsen (Wits)
Absorption probabilities
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Results
1.0
0.8
AHΩL¤2
0.6
j=3/2
j=5/2 j=7/2 j=9/2 j=11/2 j=13/2
0.4
0.2
0.0
0
1
2
3
4
Ω
Figure: Absorption probability for spin-3/2 particles in 4 dimensional
Schwarzschild backgrounds for angular momentum j=3/2 to 13/2.
Gerhard Harmsen (Wits)
Absorption probabilities
9th February 2016
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Results
1.0
0.8
AHΩL¤2
0.6
j=3/2
j=5/2
j=7/2
j=9/2
j=11/2
j=13/2
0.4
0.2
0.0
0
1
2
3
4
Ω
Figure: Absorption probability for spin-3/2 particles in 5 dimensional
Schwarzschild backgrounds with angular momentum j=3/2 to 13/2.
Gerhard Harmsen (Wits)
Absorption probabilities
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Results
1.0
0.8
AHΩL¤2
0.6
N=4 N=5 N=6 N=7 N=8
N=9
0.4
0.2
0.0
0
1
2
3
4
Ω
Figure: Absorption probabilities for spin-3/2 particles with an angular momentum
of j=3/2 in 4 and higher dimensional Schwarzschild backgrounds.
Gerhard Harmsen (Wits)
Absorption probabilities
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Results
1.0
0.8
AHΩL¤2
0.6
N=4
N=5
N=6
N=7
N=8
N=9
0.4
0.2
0.0
0
1
2
3
4
Ω
Figure: Absorption probabilities for spin-3/2 particles with an angular momentum
of j=5/2 in 4 and higher dimensional Schwarzschild backgrounds.
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Absorption probabilities
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Concluding remarks
An increase in quantum number j results in an increase in the
required particle energy for absorption.
Increase in space time dimensions results in an increase in the
required particle energy for absorption.
Higher dimensional black holes exhibit a slower changes from total
reflection to total absorption
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Absorption probabilities
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Acknowledgements to:
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Absorption probabilities
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