Black holes and extra
dimensions
Finn Ravndal, UiO
• Schwarzschild black hole
• Planck units
• Thermodynamics
• Hawking radiation
• Extra dimensions
• Production of black holes
HEP-coll, 13/2 - 2009
Black holes
GM
Newton gravitational potential: Φ = −
r
1
Escape velocity
mv 2 + mΦ ≥ 0 or
2
2GM
2
v ≥
r
2
2
v
≤
c
But
so that
Schwarzschild radius:
r ≥ R where
2GM
R=
c2
Minkowski metric
R
ds2 = dt2 − dr2 − r2 dΩ2
changed to Schwarzschild metric
2
dr
ds2 = (1 − R/r)dt2 −
− r2 dΩ2
1 − R/r
Horizon area:
!
"
2 2
A = 4πR = 16π GM/c
2
Planck units
L
!!!!"! h/Mc
L
P
R = GM/c^2
mass
M
M
Black hole when
P
h̄
GM
>
2
c
Mc
or M > MP
!
h̄c
= 10−5 g = 1019 GeV /c2
G
Planck mass:
MP =
Planck length:
h̄
LP =
MP c
Planck time:
LP
tP =
c
= 10−33 cm
= 10−44 sec
Black hole
thermodynamics
Area of Schwarzschild black hole:
h̄ = c = kB = 1
A = 4πR2 = 16π(GM )2
Adding small mass δM increases area
δA = 32πG2 M δM
κ
4GM
δA with surface gravity κ =
or δM =
8πG
A
Rotating black hole (Kerr) with spin J has area
!
#
"
A = 8π(GM )2 1 + 1 − (J/GM 2 )2
and surface gravity
4GM !
κ=
1 − (J/GM 2 )2
A
κ
1) Energy conservation: δM =
δA + ωδJ
8πG
2) Hawking area theorem:
from Einstein gravity alone!
δA ≥ 0
Thermodynamics:
1) δU = T δS + ωδJ
2) δS ≥ 0
Bekenstein (1972):
Black hole entropy:
Black hole temperature:
S∝A
T ∝κ
Bekenstein (1973):
Entropy:
S = f A/L2p
Temperature:
T = κL2P /8πGf
Hawking radiation
S. Hawking (1974) quantum field theory around black hole:
1
=⇒ f =
4
Schwarzschild black hole temperature
and entropy
S = 4πGM
2
1
T =
8πGM
Black hole particle emission:
! !
"#$%&'()#*+
,-)#.-$
Thermal spectrum:
Γω
nω = ω/T
e
±1
Emission rate as for black body
1
dM
4
= σT A =
−
dt
15π · 210 (GM )2
π2
where approximate Stefan-Boltzmann constant σ ≈
60
=⇒ M (t) = M
!
Lifetime of black hole
or:
1−
t
5π · 210 G2 M 3
"1/3
t0 = 5π · 210 G2 M 3
! M "3
!
"3
tP ! M/M! × 1071 sec
t0 = 5120π
MP
M
Decay of black hole mass:
t
t0
Hawking emission power:
P
tU ! 1018 sec
t0 ≤ tU ⇒ M ≤ 1015 g
t
t0
Extra dimensions
GM
Newton’s law Φ = −
from ∇2 Φ = 4πGρ
r
GM
Gravitational acceleration g = −∇Φ = − 2
r
follows also from Gauss’ law.
Now define gravitational constant Gn with
n
Gn M
extra dimensions by g = −∇Φ = − 2+n
r
Thus we have for gravitational potential in spacetime with
D = 4+n dimensions:
Gn M
Φ=−
(n + 1)rn+1
Schwarzschild metric in this spacetime same form as for
D = 4:
2
dr
ds2 = (1 + 2Φ)dt2 −
− r2 dΩ2
1 + 2Φ
Radius of horizon given by
O. Aursjø, MSc 2007
2Gn M
=0
1−
n+1
(n + 1)R
Thus
R=
!
2Gn M
n+1
"1/n+1
Planck mass Mn in extra-dimensional spacetime
again determined by R ! λ = 1/M or:
Gn Mnn+2 = 1
What is size of extra dimensions?
!"#$%
& !'($)$*+
& !(
&
,
GM
Gn M
= 2
n+2
L
L
or
Gn = GLn
In terms of Planck masses
G = 1/MP2
and
Gn = 1/Mnn+2 follows
MP2 = Ln Mn2+n
No quantum gravity for E < 1 TeV, i.e. Mn ≥ 1TeV
Thus
L ≤ 1032/n × 10−16 mm
n = 1:
L ≤ 1016 mm
L ≤ 1 mm
L ≤ 1 nm
n = 2:
n = 3:
Ruled out!
Most interesting!
Hopeless to test?
With n = 2 extra dimensions, gravitational potential:
! " d3 k
eik·r
Φ(r) = −4πGM
3 k 2 + (πn /L)2 + (πn /L)2
(2π)
1
2
n ,n
1
2
"
GM !
1 + αe−r/λ + · · ·
=−
r
Black hole production
E
2R n
Schwarzschild radius
2E
E
Rn = (Gn M )1/n+1
expressed in terms of extradimensional Planck mass
Gn = 1/Mnn+2
98
gives production cross-section for BH with mass M = 2E:
!
"2/n+1
π
E
2
σ = πRn = 2
M
M
Chapter
5. n
Black Hole Production
n
4
MD = 1 TeV
σ̂I × (TeV)2
3
PSfrag replacements
2
1
D =7
D =9
0
10
20
30
√ 40 50
s/TeV
D
D
D
D
=
=
=
=
60
70
80
6
8
10
11
Figure 5.1: The black hole cross-section σ̂I from Eq.(5.2) as a func-
Does a black hole at LHC pose any dangers?
Giddings and Mangano, PR D78 (2008)
J. Ellis et al., CERN-PH-Th/2008-136
NO!!