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.