BLACK HOLES ANDTHE GENERALIZED UNCERTAINTY PRINCIPLE
Bernard CarrQueen Mary, University of London
Macroscopic MicroscopicBLACK HOLES
BLACK HOLE FORMATION
RS = 2GM/c2 = 3(M/MO) km => ρS = 1018(M/MO)-2 g/cm3
Higher dimensions => TeV quantum gravity => larger minimum?
10-5g at 10-43s (minimum)MPBH ~ c3t/G = 1015g at 10-23s (evaporating now) 1MO at 10-5s (maximum)
Small “primordial” BHs can only form in early Universecf. cosmological density ρ ~ 1/(Gt2) ~ 106(t/s)-2g/cm3=> PBHs have horizon mass at formation
Good evidence that BHs form at present or recent epochs.Stellar BH (M~101-2MO), IMBH (M~103-5MO), SMBH (M~106-9MO)
Planck scale
PBH
SMBHIMBH
Stellar BH
Accel’ BH
WHEN BLACK HOLES FORM
PBH EVAPORATION
Black holes radiate thermally with temperature
T = ~ 10-7 K (Hawking 1974) => evaporate completely in time tevap ~ 1064 y
M ~ 1015g => final explosion phase today (1030 ergs)
!
M
M0
"
# $
%
& '
(1
!
M
M0
"
# $
%
& '
3
!
!
hc3
8"GkM
γ-ray bgd at 100 MeV => ΩPBH(1015g) < 10-8(Page & Hawking 1976)
=> explosions undetectable in standard particle physics model
PBHs important even if never formed!
Microlensing searches => MACHOs with 0.5 MOPBH formation at QCD transition?Pressure reduction => PBH mass function peak at 0.5 MOBut microlensing => < 20% of DM can be in these objects1026-1033g PBHs excluded by microlensing of LMC1017-1020g PBHs excluded by femtolensing of GRBsAbove 105M0 excluded by dynamical effectsBut no constraints for 1016-1017g or 1020-1026g or above 1033gStable Planck-mass relics of evaporated BHs?
Will measurements of gamma-ray bursts, like the one shown sterilizing a planet in this artist's rendering, reveal the existence of tiny black holes? We may know soon.
DETECTION OF 1017G PBHS BY FEMTOLENSING?
Marani et al. (1999)
What Would Happen if a Small Black Hole Hit the Earth?by IAN O'NEILL on FEBRUARY 17, 2008
Khriplovich et al. (2008)Long tube of radiatively damaged material recognisable for geological time
Could Primordial Black Holes Deflect Asteriods on a Collision Course with Earth?by IAN O'NEILL on FEBRUARY 22, 2008
Shatskiy (2008)Earth-mass PBHs could deflect asteroids onto Earth every 190M years
BRANE COSMOLOGY (Bowcock et al. 2000, Mukohyama et al. 2000) Brane can be viewed as moving through 5th dimension in static bulk described by 5D Schwarzschild-anti de Sitter :
(5)ds2 = -F(R)dT2 + F(R)-1dR2 + R2 [(1-Kr2) -1dr2 + r2dΩ2], F(R) = K - m/R2 + (R/L) 2
5th dimension is identified with cosmic scale factor R=a(t)0, +1, -1 mass of BH cosm const scale
m=0, a=const => Randall-Sundrum 1-braneK=1 (closed) and m non-zero => event horizon at “radial” distance
5D BH F(R)dT2+F(R)-1dR2 4D FRW -dT2+R2 (1-Kr2) -1dr2
Universe emerges from 5D black hole as a(t) passes through
Rh = (1/2)[(L4+4mL2)1/2-L2] = m1/2 for Rh
1015g
1012g
10-5g
1 MO
109 MO QSO
Stellar
exploding
LHC
Planck Universal
BLACK HOLES
extra dimensions
M-theory Higher dimensions Multiverse
106 MO MW
1021glunar
1022 MO
102 MO IMBH
1025gterrestrial
L. Modesto & I. Premont-Schwarz,Self-dual Black holes in LQG: Theoryand Phenomenology, Phys. Rev. D. 80, 064041 (2009).
B. Carr, L. Modesto & I. Premont-Schwarz, Generalized UncertaintyPrinciple and Self-Dual Black Holes, arXiv: 1107.0708 [gr-qc] (2011).
B. Carr, Black Holes, the Generalized Uncertainty Principle and HigherDimensions, Phys. Lett. A 28, 134001 (2013).
B. Carr, L. Modesto & I. Premont-Schwarz, Loop Black Holes and theBlack Hole Uncertainty Principle Correspondence (2013).
GENERALIZED UNCERTAINTY PRINCIPLE - link with loop quantum gravity
See talks by M. Isi, A. Kempf, R. Casadio, A.Bonanno, F.Scardigli
!
"#x >h
(2)#p
UNCERTAINTY PRINCIPLE
Photon of momentum p determines position to precision
!
"x > # = h / p
!
"p ~ pbut imparts momentum
!
RP
= Gh /c3~ 10
"33cm,
!
MP = hc /G ~ 10"5g,
!
"P =c5
h2G~ 10
94gcm
#3
BLACK HOLE EVENT HORIZON
(Schwarzschild radius)R < RS= 2GM/c2
Intersect at Planck scales
!
" RC
=h
Mc
Particle production for QFT
!
R < RC"
(Compton wavelength)
!
h" h
QUANTUM DOMAIN
RELATIVISTIC DOMAIN
even
t hori
zon
log (R/cm)
log (M/gm)
Planck scale
-5
proton
LHC
-24 -20 15
-13
-18
-33
Compton wavelength
CLASSICAL DOMAIN
time f
unny
space funny
evaporating black hole
!
RC
=h
Mc
!
RS
=2GM
c2
Elementary Black particles holes
QUANTUM GRAVITY REGIMES
!
" R <3M
4#$P
%
& '
(
) *
1/ 3
~ (M /MP)1/ 3RP
!
R < RP• (spacetime foam)
• (above Planck density)!
"R3(#$)2
8%G>hc
R"$ & R#$ >
hcG
R
!
"g
g#$
c2
>RP
R
Energy density in gravitational field
!
"g =(#$)2
8%G
=> spacetime distortion
Eg. black hole or big bang singularity!
" > "P
?
What happens to Compton and Schwarzschild lines near MP…
…is important feature of theory of quantum gravity.
Newtonian heuristic argument
Photon of frequency ω approaching to distance R induces=> acceleration over time=> uncertainty in momentum and in position
!
a ~ Gh" /(cR)2
!
t ~ R /c
!
"xg > at2~ Gh# /c 4 ~ G"p /c 3 ~ RP
2
"p /h
!
"p ~ p ~ h# /c
Einstein heuristic argument
Metric fluctuation δgµν on scale R is
!
"gµ#R2
=8$G
c4
%
& '
(
) * pc
R3
!
"#xg ~ R$gµ% ~ G#p /c3~ RP
2
#p /h
!
"x >h
"p+ RP
2 "p
h> 2RPBoth suggest
GENERALIZED UNCERTAINTY PRINCIPLE
(GUP)
Adler, Am. J. Phys. 78, 925 (2010)
!
"x >h
2"p1#
$2
2("p)2 +O($4 )
%
& '
(
) *
Polymer corrections in loop quantum gravity
GUP in string theory
(Hossain et al. 2010)
(Veneziano 1986, Witten 1996)
!
"x >h
"p+#
"p
hstring tension
!
" ~ (10RP)2
Experimental probes of GUP (Pikowski et al 2012))
Minimal length considerations (Maggiore 1993)
Principle of relative locality (Doplicher 2010)
Modifications of commutator [x,p] (Magueio & Smolin 2002)
Link with back holes (Scardigli 1999, Calmet et al. 2004)
!
"x >h
"p+#RP
2 "p
h
!
=h
"p1+#
"p
cMP
$
% &
'
( )
2*
+ , ,
-
. / /
!
"x# R
!
h
Mc1+"
M
MP
#
$ %
&
' (
2)
* + +
,
- . .
Compton irrelevant for M>>MP since RC
GENERALIZED EVENT HORIZON
!
R > RS
/
=2GM
c21+ "
MP
M
#
$ % %
&
' ( (
2)
*
+ +
,
-
.
.
!
RC
/
" RS
/
#
!
2GM /c2
!
h /(Mc)
For M>>MP we obtain generalized event horizon
This becomes Compton wavelength for M
GUPGEH
HUP
Interesting alternative
α < 0 => cusp rather than smooth minimum
!
h" 0#
!
G" 0# asymptotic safety (Bonanno & Reuter 2006)classical theory (Scardigli et al. 2009)
!
"x >h
"p
#
$ %
&
' (
2
+ )RP2 "p
h
#
$ %
&
' (
2
!
" RC
/
=h
Mc
#
$ %
&
' (
2
+)GM
c2
#
$ %
&
' (
2
!
"h
Mc1+
# 2
2
M
MP
$
% &
'
( )
4*
+ , ,
-
. / /
!
"x >h
"p
#
$ %
&
' (
n
+ )RP2 "p
h
#
$ %
&
' ( n*
+ , ,
-
. / /
1/ n
!
" RC
/
=h
Mc
#
$ %
&
' ( n
+)GM
c2
#
$ %
&
' ( n*
+ ,
-
. /
1/ n
!
"h
Mc1+
# n
n
M
MP
$
% &
'
( )
2n*
+ , ,
-
. / /
Root-mean-square error would give
More generally
DO GUP UNCERTAINTIES ADD LINEARLY?
!
" RS
/
=2GM
c2
#
$ %
&
' ( n
+)h
Mc
#
$ %
&
' ( n*
+ ,
-
. /
1/ n
!
"2GM
c21+
#
n
n
MP
M
$
% & &
'
( ) )
2n*
+
, ,
-
.
/ /
!
Rmin
= 21/ n
2"RP,
!
Mmin
= " /2MP
Minimum at
!
" RS
/
=2GM
c2
#
$ %
&
' (
2
+)h
Mc
#
$ %
&
' (
2
!
"2GM
c21+
#
8
2
MP
M
$
% & &
'
( ) )
4*
+
, ,
-
.
/ /
n=1
n=20
n=2
• Can there be black holes for M
Pikowski et al. Nature Phys. 8 39 (2012)Probing Planck-scale physics with quantum optics
“Generalized Uncertainty and Self-dual Black Holes” Carr, Modesto & Premont-Schwarz, arXiv: 1107.0708
At large r
Metric
implies
LOOP BLACK HOLES
Immirzi parameter 1
=> another asymptotic infinity (r=0) with BH mass MP2/m
Physical radial coordinate
!
" RS
=2Gm
c2
#
$ %
&
' (
2
+c2ao
2Gm
#
$ %
&
' (
2
)!
R = H(r) = r2
+ao
r2
2
!
3"#
4
h
mc
Introduce dual coordinates
!
r = ao/r" R = r
2+ r
2
!
t = tr*
2
/ao
!
2Gm
c2
Metric has self-duality with dual radius
!
r = r = ao
This removes the singularity, permits existence of black holeswith m MP)
!
(m < MP)
Penrose diagrams for Schwarzschild and LBH metric
cf. Reissner-Nordstrom
L. Modesto & I. Premont-Schwarz, Phys. Rev. D. 80, 064041 (2009)
GUP AND BLACK HOLE THERMODYNAMICS
Heuristic argument
!
kTBH ="c#p ="hc
#x="hc 3
2GM~MP
2
Mη=1/(4π)
!
" TBH
=#Mc 2
$k1% 1%
$MP
2
M2
&
' ( (
)
* + + ,
hc3
8-GkM1+
$MP
2
4M2
.
/ 0
1
2 3
Complex for
!
"p ~ T
!
"x ~ GM /c2
=> Planck mass relics.
Putting and in linear GUP (Adler & Chen)
!
M < "MP
!
"2GM
c2
=#hc
kT+$R
P
2
kT
h#c
!
(M >> MP)
!
(M >> MP)
Quadratic GUP
!
" TBH
=2#Mc 2
$k1% 1%
$ 2MP
4
4M4
&
' ( (
)
* + +
1/ 2
,hc
3
8-GkM1+
$ 2MP
4
32M4
.
/ 0
1
2 3 (M >> MP )
!
"2GM
c2
=#hc
kT
$
% &
'
( ) 2
+*R
P
2kT
h#c
$
% &
'
( )
2+
,
- -
.
/
0 0
1/ 2
Complex for
!
M < " /2MP
Minus sign gives T>TPwhich may be unphysical
-
!
Tmax
="MPc2/ #
=> smaller relics
Quadratic GUP + GEH
!
"2GM
c2
#
$ %
&
' ( 2
+h)
Mc
#
$ %
&
' ( 2*
+ ,
-
. /
1/ 2
=0hc
kT
#
$ %
&
' ( 2
+1R
P
2kT
h0c
#
$ %
&
' (
2*
+
, ,
-
.
/ /
1/ 2
!
" TBH
=2#Mc 2
$k1+
% 2MP
4
4MP
4& 1+
(2% 2 &$ 2)MP
4
4M4
+% 4M
P
8
16MP
8
'
( ) )
*
+ , ,
1/ 2
Real for all M if
!
" < 2#
!
"Mc 2
#k1+
$ 2 % 4# 2
2# 4&
' (
)
* + M
4
MP
4
,
- . .
/
0 1 1
!
"hc 3
2GkM1+
# 2 $ 4% 2
32
&
' (
)
* + M
P
4
M4
,
- . .
/
0 1 1
!
" TBH#
Regard α are β as independent
BHUP => α = 2β
!
" TBH
=h#c 3
2kGMor
!
TBH
="
#Mc
2 Exact!
!
(M >> MP)
!
(M
SURFACE GRAVITY ARGUMENT
!
T "GM
RS
2"
M3n / 2
M2n
+ (# /2)n MP
2n
$
% &
'
( )
2 / n
"
!
M"1(M >> M
P)
!
M3(M different prediction in sub-Planckian range!
(n=2)
(area)
+
!
"#R
BH
#rBH
$
!
(M /MP)"2(M
More precise surface gravity argument
!
" 2 = #gµ$ gµ$%µ&'%$ &
(
!
r"#$ g00"1$ % µ = (1,0,0,0)$
!
"# =4G
3m3c4P4
16G4m4P8
+ aoc8
!
"+ =4G
3m3c4
16G4m4
+ aoc8
!
r" 0# g00" r
*
4/a
0
2= P 4 (m /MP )
4 # $ µ = P%2(m /MP )%2(1,0,0,0)#
!
T" = T0P2(m /M
P)2
This reconciles to the two arguments!
T = hκ/(2πkc)
Three mass regimes
!
M > P"2M
P# r
P< r" < r+ # T$ %M
"1,T0%P
"6M
"3
!
M < MP" r# < r+ < rP " T$ %M
3,T0%P
2M
!
MP
< M < P"2M
P# r" < rP < r+ # T$ %M
"1,T0%P
2M
our space other space
CAN SUB-PLANCKIAN RELICS PROVIDE DARK MATTER?
cooler than CMB for
!
M <TCMB
TP
"
# $
%
& '
1/ 3
MP ~ 10(16g
!
T "M3#
!
dM
dt"R
#2T4"M
10$ M(t)" t
#1/ 9
=> never evaporate but current mass
!
M ~tP
t0
"
# $
%
& '
1/ 9
MP ~ 10(12g
=> current temperature
!
T ~tP
to
"
# $
%
& '
1/ 3
TP~ 10
12K
=> same observational effects as PBHs with M~1015g!
stableunstable
BLACK HOLES AS A PROBE OF HIGHER DIMENSIONS
BLACK HOLES AND EXTRA DIMENSION
Higher dimensions => MDn +2Vn ~ Mp2
Vn is volume of compactified or warped space
Standard model => Vn ~ MP-n , MD ~ Mp, Large extra dimensions => Vn >> MP-n, MD
Forming black holes by collisions
Cross-section σ(ij BH) = πrS2Θ(E - MBHmin)Schwarzschild radius rS= MP-1(MBH/MP)1/(1+n)Temperature TBH = (n+1)/rS < 4D caseLifetime τBH =MP-1(MBH/MP)(n+3)/(1+n) > 4D case
centre of mass energy
BLACK HOLES AND HIGHER DIMENSIONS
Gauss law
!
Fgrav =GDm1m2
R2+n
!
Fgrav =G m
1m2
R2
!
G =G
D
RC
2
(RRC)
3D black hole smaller than RC for
!
M < MC
=c2RC
2G
Black hole condition
!
R <
Intersects Compton boundary at higher dimensional Planck scale
!
MD
= MP
RP
RC
"
# $
%
& '
n /(n+2)
,
!
RD
= RP
RC
RP
"
# $
%
& '
n /(n+2)
Assume D=3+n dimensions for R MC)
!
(M < MC)
QUANTUM DOMAIN
RELATIVISTIC DOMAIN
6D 5D 4D
3D
even
t hori
zon
log (R/cm)
log (M/gm)Planck scale
revised Planck scale
-5
LHC
-24 -20 15
-13
-18
-33
Compton wavelength
CLASSICAL DOMAIN
scale of extra dimensions
BH radius RS= MP-1(MBH/MP)1/(1+n)
Hierarchy of compactified dimensions
Ri=αiRP => (RP’/RP)(d+2)/(d+1) = α11/2 α21/6.....αd1/d(d+1)
αi = α => RP’ = RPαd(/d+2) !
" RC
/ #h
Mc1+
M
MP
$
% &
'
( )
(n+2)/(n+1)*
+ , ,
-
. / /
cf. 2D black holes (Mureika & Nicolini 2013)
DETECTABLE AT LHC?
!
MD~ TeV " R
C~ 10
(32 / n )#17cm ~
1016 cm (n=1)10-1 cm (n=2)10-6 cm (n=3)10-13 cm (n=7)
Dark energy
!
"U ~ 10#120"U ~ 10
#29gcm
#3
=> intersects Compton line at
!
M ~ MPtP
tU
"
# $
%
& '
1/ 2
~ 10(30MP ~ 10
(35g,
!
R ~ RURU~ 100µm
excludeddark energy?
QUANTUM DOMAIN
RELATIVISTIC DOMAIN
log (R/cm)
log (M/gm)
-5
collapsing star
-24 -20 15
-13
-18
-33
CLASSICAL DOMAIN
HIGHER-DIMENSIONAL DOMAIN
4D5D6D
Planck d
ensity
DARK ENERGY M~10-35g, R~10-2cm M~1055g, R~1027cm
CONCLUSIONS
• Both black holes and Generalized Uncertainty Principleprovide important link between micro and macro physics.
• They are themselves linked and black holes with sub-Planckian mass may play a role in quantum gravity.
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