Parthasarathi Majumdar - Saha Institute of Nuclear Physics

Critical mass of Neutron Star : an Entropic View
Parthasarathi Majumdar,
Saha Institute of Nuclear Physics, Kolkata, India
Neutron Stars : inside and outside
Kolkata, 18-19 October, 2012
October 18, 2012

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 2
Neutron stars with masses>> ξ M ⊙ , ξ = O(1) not observed
⇒ M ns > M crit ≃ ξ M ⊙ → unstable wrt gravitational collapse to black
hole
How does one determine M crit Most contemporary efforts → variations
on the theme of the Chandrasekhar bound for White Dwarfs
Chandrasekhar’s Nobel Lecture December 1983 : (adapted to deg neutron
cores)
( ) c 3/2
M ns < M crit = ξ m −2 n ≃ ξ M ⊙
G
Hydrost equil betweenP core due to gravity andP deg the Fermi pressure of
relativistic degenerate neutrons :
Solve Einstein eq G ab = 8πG
c 4 T ab with sph symm ansatz for perfect
barotropic fluid→Tolman-Oppenheimer-Volkov eq

Created with pptalk Slide 3
[ ]
dP
dr = −(P +ρ) 4πr 3 P +2m
r(r−2m)
where, m(r) ≡ ∫ r
0
dr ′ 4πr ′2 ρ(r ′ ) , G = c = 1. Assuming uniform ρ(r) =
ρ 0 ⇒ m(r) = M core (r/R) 3 ⇒
⎡
(
P core = P(r = 0) = ρ 0 c 2 1− R )
⎤
1/2−1
S
⎢ R ⎣ (
1−3 1− R )
⎥
1/2 ⎦
S
R
where,R S = 2GM core /c 2
Degenerate neutron gas : Fermi pressure
P deg =
∫ pF
0
n tot (p) v(p) dp
where,n tot (p F ) ≡ ∫ p F
0
n(p) dp = (8π/3 3 ) p 3 F

Created with pptalk Slide 3
[ ]
dP
dr = −(P +ρ) 4πr 3 P +2m
r(r−2m)
where, m(r) ≡ ∫ r
0
dr ′ 4πr ′2 ρ(r ′ ) , G = c = 1. Assuming uniform ρ(r) =
ρ 0 ⇒ m(r) = M core (r/R) 3 ⇒
⎡
(
P core = P(r = 0) = ρ 0 c 2 1− R )
⎤
1/2−1
S
⎢ R ⎣ (
1−3 1− R )
⎥
1/2 ⎦
S
R
where,R S = 2GM core /c 2
Degenerate neutron gas : Fermi pressure
P deg =
∫ pF
0
n tot (p) v(p) dp
where,n tot (p F ) ≡ ∫ p F
0
n(p) dp = (8π/3 3 ) p 3 F

Created with pptalk Slide 3
[ ]
dP
dr = −(P +ρ) 4πr 3 P +2m
r(r−2m)
where, m(r) ≡ ∫ r
0
dr ′ 4πr ′2 ρ(r ′ ) , G = c = 1. Assuming uniform ρ(r) =
ρ 0 ⇒ m(r) = M core (r/R) 3 ⇒
⎡
(
P core = P(r = 0) = ρ 0 c 2 1− R )
⎤
1/2−1
S
⎢ R ⎣ (
1−3 1− R )
⎥
1/2 ⎦
S
R
where,R S = 2GM core /c 2
Degenerate neutron gas : Fermi pressure
P deg =
∫ pF
0
n tot (p) v(p) dp
where,n tot (p F ) ≡ ∫ p F
0
n(p) dp = (8π/3 3 ) p 3 F

Created with pptalk Slide 4
Sp Rel : v(p) < c
P deg < 1 8
Hydrostat equilP core = P deg ⇒
M crit < ξ
( c
G
where,M P = (c/G) 1/2 ∼ 10 19 Gev
(
ρ0
m n
) 4/3
) 3/2
m −2 n = M2 P
m 2 n
M P associated with scale of quantum grav (Planck length l P =
(G/c 3 ) 1/2 = 10 −33 cm)
Why does M P emerge from cl grav + QM Is there a hint that quantum
gravity is somehow sneaking in

Created with pptalk Slide 4
Sp Rel : v(p) < c
P deg < 1 8
Hydrostat equilP core = P deg ⇒
M crit < ξ
( c
G
where,M P = (c/G) 1/2 ∼ 10 19 Gev
(
ρ0
m n
) 4/3
) 3/2
m −2 n = M2 P
m 2 n
M P associated with scale of quantum grav (Planck length l P =
(G/c 3 ) 1/2 = 10 −33 cm)
Why does M P emerge from cl grav + QM Is there a hint that quantum
gravity is somehow sneaking in

Created with pptalk Slide 4
Sp Rel : v(p) < c
P deg < 1 8
Hydrostat equilP core = P deg ⇒
M crit < ξ
( c
G
where,M P = (c/G) 1/2 ∼ 10 19 Gev
(
ρ0
m n
) 4/3
) 3/2
m −2 n = M2 P
m 2 n
M P associated with scale of quantum grav (Planck length l P =
(G/c 3 ) 1/2 = 10 −33 cm)
Why does M P emerge from cl grav + QM Is there a hint that quantum
gravity is somehow sneaking in

Created with pptalk Slide 4
Sp Rel : v(p) < c
P deg < 1 8
Hydrostat equilP core = P deg ⇒
M crit < ξ
( c
G
where,M P = (c/G) 1/2 ∼ 10 19 Gev
(
ρ0
m n
) 4/3
) 3/2
m −2 n = M2 P
m 2 n
M P associated with scale of quantum grav (Planck length l P =
(G/c 3 ) 1/2 = 10 −33 cm)
Why does M P emerge from cl grav + QM Is there a hint that quantum
gravity is somehow sneaking in

Created with pptalk Slide 4
Sp Rel : v(p) < c
P deg < 1 8
Hydrostat equilP core = P deg ⇒
M crit < ξ
( c
G
where,M P = (c/G) 1/2 ∼ 10 19 Gev
(
ρ0
m n
) 4/3
) 3/2
m −2 n = M2 P
m 2 n
M P associated with scale of quantum grav (Planck length l P =
(G/c 3 ) 1/2 = 10 −33 cm)
Why does M P emerge from cl grav + QM Is there a hint that quantum
gravity is somehow sneaking in

Created with pptalk Slide 5
Stability upper bound
• For densitiesρ > 10 13 gm/cc , EoS P = P(ρ) poorly known
• Need model for strong nucleonic interactions involving also hyperons/resonances
and perhaps also quarks
• LE effective models based on SRQFT in flat Minkowski space. P core
computed using GR → inconsistent ! Fortuitous : Chandrasekhar’s
derivation agrees with most observations !
• Chandrasekhar limit for Neutron stars : Right answer using ‘invalid’
theory Similar to Mitchell’s (1784) derivation of Schwarzschild rad
R S = 2GM/c 2 before GR; or Bohr’s derivation of Bohr radius a 0 =
2 /me 2 before QM. Also, Bethe’s derivation of Lamb Shift in H-atom
using NRQM. Pointers to the right theory discovered subsequently.
• But GRQFT → formidable, since many SRQFT properties (e.g., Spin-
Statistics Theorem) invalid in general.

Created with pptalk Slide 5
Stability upper bound
• For densitiesρ > 10 13 gm/cc , EoS P = P(ρ) poorly known
• Need model for strong nucleonic interactions involving also hyperons/resonances
and perhaps also quarks
• LE effective models based on SRQFT in flat Minkowski space. P core
computed using GR → inconsistent ! Fortuitous : Chandrasekhar’s
derivation agrees with most observations !
• Chandrasekhar limit for Neutron stars : Right answer using ‘invalid’
theory Similar to Mitchell’s (1784) derivation of Schwarzschild rad
R S = 2GM/c 2 before GR; or Bohr’s derivation of Bohr radius a 0 =
2 /me 2 before QM. Also, Bethe’s derivation of Lamb Shift in H-atom
using NRQM. Pointers to the right theory discovered subsequently.
• But GRQFT → formidable, since many SRQFT properties (e.g., Spin-
Statistics Theorem) invalid in general.

Created with pptalk Slide 5
Stability upper bound
• For densitiesρ > 10 13 gm/cc , EoS P = P(ρ) poorly known
• Need model for strong nucleonic interactions involving also hyperons/resonances
and perhaps also quarks
• LE effective models based on SRQFT in flat Minkowski space. P core
computed using GR → inconsistent ! Fortuitous : Chandrasekhar’s
derivation agrees with most observations !
• Chandrasekhar limit for Neutron stars : Right answer using ‘invalid’
theory Similar to Mitchell’s (1784) derivation of Schwarzschild rad
R S = 2GM/c 2 before GR; or Bohr’s derivation of Bohr radius a 0 =
2 /me 2 before QM. Also, Bethe’s derivation of Lamb Shift in H-atom
using NRQM. Pointers to the right theory discovered subsequently.
• But GRQFT → formidable, since many SRQFT properties (e.g., Spin-
Statistics Theorem) invalid in general.

Created with pptalk Slide 5
Stability upper bound
• For densitiesρ > 10 13 gm/cc , EoS P = P(ρ) poorly known
• Need model for strong nucleonic interactions involving also hyperons/resonances
and perhaps also quarks
• LE effective models based on SRQFT in flat Minkowski space. P core
computed using GR → inconsistent ! Fortuitous : Chandrasekhar’s
derivation agrees with most observations !
• Chandrasekhar limit for Neutron stars : Right answer using ‘invalid’
theory Similar to Mitchell’s (1784) derivation of Schwarzschild rad
R S = 2GM/c 2 before GR; or Bohr’s derivation of Bohr radius a 0 =
2 /me 2 before QM. Also, Bethe’s derivation of Lamb Shift in H-atom
using NRQM. Pointers to the right theory discovered subsequently.
• But GRQFT → formidable, since many SRQFT properties (e.g., Spin-
Statistics Theorem) invalid in general.

Created with pptalk Slide 5
Stability upper bound
• For densitiesρ > 10 13 gm/cc , EoS P = P(ρ) poorly known
• Need model for strong nucleonic interactions involving also hyperons/resonances
and perhaps also quarks
• LE effective models based on SRQFT in flat Minkowski space. P core
computed using GR → inconsistent ! Fortuitous : Chandrasekhar’s
derivation agrees with most observations !
• Chandrasekhar limit for Neutron stars : Right answer using ‘invalid’
theory Similar to Mitchell’s (1784) derivation of Schwarzschild rad
R S = 2GM/c 2 before GR; or Bohr’s derivation of Bohr radius a 0 =
2 /me 2 before QM. Also, Bethe’s derivation of Lamb Shift in H-atom
using NRQM. Pointers to the right theory discovered subsequently.
• But GRQFT → formidable, since many SRQFT properties (e.g., Spin-
Statistics Theorem) invalid in general.

Created with pptalk Slide 5
Stability upper bound
• For densitiesρ > 10 13 gm/cc , EoS P = P(ρ) poorly known
• Need model for strong nucleonic interactions involving also hyperons/resonances
and perhaps also quarks
• LE effective models based on SRQFT in flat Minkowski space. P core
computed using GR → inconsistent ! Fortuitous : Chandrasekhar’s
derivation agrees with most observations !
• Chandrasekhar limit for Neutron stars : Right answer using ‘invalid’
theory Similar to Mitchell’s (1784) derivation of Schwarzschild rad
R S = 2GM/c 2 before GR; or Bohr’s derivation of Bohr radius a 0 =
2 /me 2 before QM. Also, Bethe’s derivation of Lamb Shift in H-atom
using NRQM. Pointers to the right theory discovered subsequently.
• But GRQFT → formidable, since many SRQFT properties (e.g., Spin-
Statistics Theorem) invalid in general.

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 6
EoS-independent approach to critical neutron mass Rhoades & Ruffini 1974
Begin with TOV equations relating P(r) and m(r). Total M =
∫ R
0
4πρ(r)r 2 dr.
Assume EoS known everywhere except ρ ∈ [ρ 0 ,ρ 1 ] and corresponding
pressures.
Divide up M = M 1 (ρ c ,ρ 1 )+ ∫ ρ 1
ρ 0
4πr 2 ρ(dr/dρ)dρ+M 1 (rho 0 )
MaximiseM subject to constraintv 2 s ≡ dP/dρ ∈ [0,1]
Match allowed region with Harrison-Wheeler EoS for ρ ∼ 10 14 g/cc
M < 3.2M ⊙

Created with pptalk Slide 7
Reexpress critical bound as
( )
Mcrit
M P
< ξ
(
λCn
l P
) 2
Planck scalel P appears nonperturbatively : rhs ր as l P ց
Contrast with perturbative QG effects∼ O(l P ) !
Reminiscent of black hole entropy :
S bh = A hor
4l 2 P
+quantum corr.
• Is the mass bound linked to quantum gravity Derivation used GR
+ Sp Rel QM
• Are the critical mass and S bh related

Created with pptalk Slide 7
Reexpress critical bound as
( )
Mcrit
M P
< ξ
(
λCn
l P
) 2
Planck scalel P appears nonperturbatively : rhs ր as l P ց
Contrast with perturbative QG effects∼ O(l P ) !
Reminiscent of black hole entropy :
S bh = A hor
4l 2 P
+quantum corr.
• Is the mass bound linked to quantum gravity Derivation used GR
+ Sp Rel QM
• Are the critical mass and S bh related

Created with pptalk Slide 7
Reexpress critical bound as
( )
Mcrit
M P
< ξ
(
λCn
l P
) 2
Planck scalel P appears nonperturbatively : rhs ր as l P ց
Contrast with perturbative QG effects∼ O(l P ) !
Reminiscent of black hole entropy :
S bh = A hor
4l 2 P
+quantum corr.
• Is the mass bound linked to quantum gravity Derivation used GR
+ Sp Rel QM
• Are the critical mass and S bh related

Created with pptalk Slide 7
Reexpress critical bound as
( )
Mcrit
M P
< ξ
(
λCn
l P
) 2
Planck scalel P appears nonperturbatively : rhs ր as l P ց
Contrast with perturbative QG effects∼ O(l P ) !
Reminiscent of black hole entropy :
S bh = A hor
4l 2 P
+quantum corr.
• Is the mass bound linked to quantum gravity Derivation used GR
+ Sp Rel QM
• Are the critical mass and S bh related

Created with pptalk Slide 7
Reexpress critical bound as
( )
Mcrit
M P
< ξ
(
λCn
l P
) 2
Planck scalel P appears nonperturbatively : rhs ր as l P ց
Contrast with perturbative QG effects∼ O(l P ) !
Reminiscent of black hole entropy :
S bh = A hor
4l 2 P
+quantum corr.
• Is the mass bound linked to quantum gravity Derivation used GR
+ Sp Rel QM
• Are the critical mass and S bh related

Created with pptalk Slide 8
Reexpress stability upper bound as instability lower bound wrt gravitational
collapse
( )
Mcrit
M P
> ξ
(
λCn
l P
) 2
= ξ
( )
ACn
A P
Maximum mass for neutron stars is minimum mass for black hole (horizon)
formation
⇒ cond for instability wrt formation of event horizon
Mass ratio related to area ratio : recallS bh = ξ (A hor /A P )
Speculate about possible entropic origin of Chandrsekhar mass bound :
black hole formation is quantum gravitational

Created with pptalk Slide 8
Reexpress stability upper bound as instability lower bound wrt gravitational
collapse
( )
Mcrit
M P
> ξ
(
λCn
l P
) 2
= ξ
( )
ACn
A P
Maximum mass for neutron stars is minimum mass for black hole (horizon)
formation
⇒ cond for instability wrt formation of event horizon
Mass ratio related to area ratio : recallS bh = ξ (A hor /A P )
Speculate about possible entropic origin of Chandrsekhar mass bound :
black hole formation is quantum gravitational

Created with pptalk Slide 8
Reexpress stability upper bound as instability lower bound wrt gravitational
collapse
( )
Mcrit
M P
> ξ
(
λCn
l P
) 2
= ξ
( )
ACn
A P
Maximum mass for neutron stars is minimum mass for black hole (horizon)
formation
⇒ cond for instability wrt formation of event horizon
Mass ratio related to area ratio : recallS bh = ξ (A hor /A P )
Speculate about possible entropic origin of Chandrsekhar mass bound :
black hole formation is quantum gravitational

Created with pptalk Slide 8
Reexpress stability upper bound as instability lower bound wrt gravitational
collapse
( )
Mcrit
M P
> ξ
(
λCn
l P
) 2
= ξ
( )
ACn
A P
Maximum mass for neutron stars is minimum mass for black hole (horizon)
formation
⇒ cond for instability wrt formation of event horizon
Mass ratio related to area ratio : recallS bh = ξ (A hor /A P )
Speculate about possible entropic origin of Chandrsekhar mass bound :
black hole formation is quantum gravitational

Created with pptalk Slide 8
Reexpress stability upper bound as instability lower bound wrt gravitational
collapse
( )
Mcrit
M P
> ξ
(
λCn
l P
) 2
= ξ
( )
ACn
A P
Maximum mass for neutron stars is minimum mass for black hole (horizon)
formation
⇒ cond for instability wrt formation of event horizon
Mass ratio related to area ratio : recallS bh = ξ (A hor /A P )
Speculate about possible entropic origin of Chandrsekhar mass bound :
black hole formation is quantum gravitational

Created with pptalk Slide 8
Reexpress stability upper bound as instability lower bound wrt gravitational
collapse
( )
Mcrit
M P
> ξ
(
λCn
l P
) 2
= ξ
( )
ACn
A P
Maximum mass for neutron stars is minimum mass for black hole (horizon)
formation
⇒ cond for instability wrt formation of event horizon
Mass ratio related to area ratio : recallS bh = ξ (A hor /A P )
Speculate about possible entropic origin of Chandrsekhar mass bound :
black hole formation is quantum gravitational

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 9
New Perspective (work in progress) : Outline
• Insensitive to low energy eff theory of strong nucleonic interaction
• Explicit use of classical spacetime metrics unnecessary
• Horizon formation not abrupt, but more like a nucleation process in a
first order phase transition
• Quantum sptm fluctuations at scales ∼ l P lead to formation of a tiny
(‘embryonic’) Trapping (Dynamical) horizon hidden deep inside neutron
star. [Trapping horizons are splk (or tmlk) 3-hypersurfaces whose
sptial foliations are 2 dim closed outer trapped surfaces]
• Interpret instability lower bound as condition for stability and growth
of hidden horizons
• This stability criterion obtained from Thermal Holography and canon
ensemble of isolated horizons using aspects of Loop Quantum Gravity

Created with pptalk Slide 10
TIME
SPACE

Created with pptalk Slide 11
TIME
SPACE

Created with pptalk Slide 12
I + 0
i
I
−

Created with pptalk Slide 13
Assume small energy loss during collapse to black hole
⇒ M crit ≃ M hor = M(A hor )
Core Collapse pushes energy into Hidden Horizon⇒ A hid hor ր
Stops when A hid hor ր A hor ⇒ A hor > A hid hor
Actually a sequence of inequalities, expressed in terms of area of quasiequil
IHs interpolating between trapping horizons
A hhor0 < A hhor1 < A hhor2 < ... < A hor
Actual origin of initial trapping surface probably quantum gravitational

Created with pptalk Slide 13
Assume small energy loss during collapse to black hole
⇒ M crit ≃ M hor = M(A hor )
Core Collapse pushes energy into Hidden Horizon⇒ A hid hor ր
Stops when A hid hor ր A hor ⇒ A hor > A hid hor
Actually a sequence of inequalities, expressed in terms of area of quasiequil
IHs interpolating between trapping horizons
A hhor0 < A hhor1 < A hhor2 < ... < A hor
Actual origin of initial trapping surface probably quantum gravitational

Created with pptalk Slide 13
Assume small energy loss during collapse to black hole
⇒ M crit ≃ M hor = M(A hor )
Core Collapse pushes energy into Hidden Horizon⇒ A hid hor ր
Stops when A hid hor ր A hor ⇒ A hor > A hid hor
Actually a sequence of inequalities, expressed in terms of area of quasiequil
IHs interpolating between trapping horizons
A hhor0 < A hhor1 < A hhor2 < ... < A hor
Actual origin of initial trapping surface probably quantum gravitational

Created with pptalk Slide 13
Assume small energy loss during collapse to black hole
⇒ M crit ≃ M hor = M(A hor )
Core Collapse pushes energy into Hidden Horizon⇒ A hid hor ր
Stops when A hid hor ր A hor ⇒ A hor > A hid hor
Actually a sequence of inequalities, expressed in terms of area of quasiequil
IHs interpolating between trapping horizons
A hhor0 < A hhor1 < A hhor2 < ... < A hor
Actual origin of initial trapping surface probably quantum gravitational

Created with pptalk Slide 13
Assume small energy loss during collapse to black hole
⇒ M crit ≃ M hor = M(A hor )
Core Collapse pushes energy into Hidden Horizon⇒ A hid hor ր
Stops when A hid hor ր A hor ⇒ A hor > A hid hor
Actually a sequence of inequalities, expressed in terms of area of quasiequil
IHs interpolating between trapping horizons
A hhor0 < A hhor1 < A hhor2 < ... < A hor
Actual origin of initial trapping surface probably quantum gravitational

Created with pptalk Slide 14
Digression : In any quantum GR theory
Ĥ = Ĥv }{{}
blk
+ Ĥb }{{}
bdy
|Ψ〉 = ∑ v,b
c vb |ψ v 〉
}{{}
blk
|χ b 〉
}{{}
bdy

Created with pptalk Slide 14
Digression : In any quantum GR theory
Ĥ = Ĥv }{{}
blk
+ Ĥb }{{}
bdy
|Ψ〉 = ∑ v,b
c vb |ψ v 〉
}{{}
blk
|χ b 〉
}{{}
bdy

Created with pptalk Slide 14
Digression : In any quantum GR theory
Ĥ = Ĥv }{{}
blk
+ Ĥb }{{}
bdy
|Ψ〉 = ∑ v,b
c vb |ψ v 〉
}{{}
blk
|χ b 〉
}{{}
bdy

Created with pptalk Slide 15
‘Quantum Einstein EQ’ (bulk)
Ĥ v |ψ v 〉 = 0
Z = ∑ b
⎛
⎝ ∑ v
|c vb
| 2 || |ψ v 〉 || 2 ⎞
⎠〈χ b
|exp−βĤbdy |χ b 〉
≡ Z bdy
Bulk states decouple! → Thermal holography ! (PM 2007, 2009)
Weaker version of holography cf ‘Holographic Hypothesis’ ’t Hooft 1993; Susskind
1995
Canonical Ensemble of (isolated) horizons (as sptm bdy) : States characterized
by A n ∼ n l 2 P , n ∈ Z (LQG)

Created with pptalk Slide 15
‘Quantum Einstein EQ’ (bulk)
Z = ∑ b
⎛
⎝ ∑ v
Ĥ v |ψ v 〉 = 0
⎞
|c vb
| 2 || |ψ v 〉 || 2
⎠〈χ b
|exp−βĤbdy |χ b 〉
≡ Z bdy
Bulk states decouple! → Thermal holography ! (PM 2007, 2009)
Weaker version of holography cf ‘Holographic Hypothesis’ ’t Hooft 1993; Susskind
1995
Canonical Ensemble of (isolated) horizons (as sptm bdy) : States characterized
by A n ∼ n l 2 P , n ∈ Z (LQG)

Created with pptalk Slide 15
‘Quantum Einstein EQ’ (bulk)
Z = ∑ b
⎛
⎝ ∑ v
Ĥ v |ψ v 〉 = 0
⎞
|c vb
| 2 || |ψ v 〉 || 2
⎠〈χ b
|exp−βĤbdy |χ b 〉
≡ Z bdy
Bulk states decouple! → Thermal holography ! (PM 2007, 2009)
Weaker version of holography cf ‘Holographic Hypothesis’ ’t Hooft 1993; Susskind
1995
Canonical Ensemble of (isolated) horizons (as sptm bdy) : States characterized
by A n ∼ n l 2 P , n ∈ Z (LQG)

Created with pptalk Slide 15
‘Quantum Einstein EQ’ (bulk)
Z = ∑ b
⎛
⎝ ∑ v
Ĥ v |ψ v 〉 = 0
⎞
|c vb
| 2 || |ψ v 〉 || 2
⎠〈χ b
|exp−βĤbdy |χ b 〉
≡ Z bdy
Bulk states decouple! → Thermal holography ! (PM 2007, 2009)
Weaker version of holography cf ‘Holographic Hypothesis’ ’t Hooft 1993; Susskind
1995
Canonical Ensemble of (isolated) horizons (as sptm bdy) : States characterized
by A n ∼ n l 2 P , n ∈ Z (LQG)

Created with pptalk Slide 15
‘Quantum Einstein EQ’ (bulk)
Z = ∑ b
⎛
⎝ ∑ v
Ĥ v |ψ v 〉 = 0
⎞
|c vb
| 2 || |ψ v 〉 || 2
⎠〈χ b
|exp−βĤbdy |χ b 〉
≡ Z bdy
Bulk states decouple! → Thermal holography ! (PM 2007, 2009)
Weaker version of holography cf ‘Holographic Hypothesis’ ’t Hooft 1993; Susskind
1995
Canonical Ensemble of (isolated) horizons (as sptm bdy) : States characterized
by A n ∼ n l 2 P , n ∈ Z (LQG)

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 16
Hor partition fct (G = c = k B = 1) Das, Bhaduri, PM 2001; Chatterjee, PM 2004; PM 2007; Majhi,
PM 2011
Z(β) = ∑ n
g(M(A n ))exp−βM(A n )
Canon entropy
≃ exp[S(A hor )−βM(A hor )]·∆ −1/2 (A hor )
S can (A hor ) = S(A hor )+ 1 2 log∆
Stable thermal equil
⇒ S can > 0 ⇒ ∆ > 0
Criterion for Thermal Stability PM 2007, Majhi & PM 2011
M(A hor )
M P
> S(A hor)
k B
= A hor
4A P
+∆ q S( A hor
A P
)
No direct use of classical metrics(Majhi & PM) ButS(A hor ) = Need quantum
gravity

Created with pptalk Slide 17
Horizon deg of freedom & dynamics (Ashtekar et. al. 1997-2000; Basu, Kaul, PM 2009,; Kaul,
PM 2010; Basu, Chatterjee, Ghosh 2010; Engel et. al. 2009-10)
• IH is a null inner boundary ⇒ h ab dx a dx b = 0 , h ab → induced metric
on IH
•⇒ deth = 0 ⇒ theory on IH cannot be ∫ √
h 3 R(h), or any theory
requiring invertingh ab .
• Theory on IH must be topological⇒S IH ≠ S IH [h]
• Gravitational canon DoF in bulk are SU(2) gauge potentials A i a and
densitised triadsEi a after gauge fixing local Lorentz boosts.
• On IH, the boundary gauge potentials are described by anSU(2) Chern
Simons theory of A a i
coupled to Σ ab
ij
≡ E [a
i Eb] j
with coupling k ≡
(A IH /8πlP 2 )
Gravity-gauge theory (topol) link derived

Created with pptalk Slide 17
Horizon deg of freedom & dynamics (Ashtekar et. al. 1997-2000; Basu, Kaul, PM 2009,; Kaul,
PM 2010; Basu, Chatterjee, Ghosh 2010; Engel et. al. 2009-10)
• IH is a null inner boundary ⇒ h ab dx a dx b = 0 , h ab → induced metric
on IH
•⇒ deth = 0 ⇒ theory on IH cannot be ∫ √
h 3 R(h), or any theory
requiring invertingh ab .
• Theory on IH must be topological⇒S IH ≠ S IH [h]
• Gravitational canon DoF in bulk are SU(2) gauge potentials A i a and
densitised triadsEi a after gauge fixing local Lorentz boosts.
• On IH, the boundary gauge potentials are described by anSU(2) Chern
Simons theory of A a i
coupled to Σ ab
ij
≡ E [a
i Eb] j
with coupling k ≡
(A IH /8πlP 2 )
Gravity-gauge theory (topol) link derived

Created with pptalk Slide 17
Horizon deg of freedom & dynamics (Ashtekar et. al. 1997-2000; Basu, Kaul, PM 2009,; Kaul,
PM 2010; Basu, Chatterjee, Ghosh 2010; Engel et. al. 2009-10)
• IH is a null inner boundary ⇒ h ab dx a dx b = 0 , h ab → induced metric
on IH
•⇒ deth = 0 ⇒ theory on IH cannot be ∫ √
h 3 R(h), or any theory
requiring invertingh ab .
• Theory on IH must be topological⇒S IH ≠ S IH [h]
• Gravitational canon DoF in bulk are SU(2) gauge potentials A i a and
densitised triadsEi a after gauge fixing local Lorentz boosts.
• On IH, the boundary gauge potentials are described by anSU(2) Chern
Simons theory of A a i
coupled to Σ ab
ij
≡ E [a
i Eb] j
with coupling k ≡
(A IH /8πlP 2 )
Gravity-gauge theory (topol) link derived

Created with pptalk Slide 17
Horizon deg of freedom & dynamics (Ashtekar et. al. 1997-2000; Basu, Kaul, PM 2009,; Kaul,
PM 2010; Basu, Chatterjee, Ghosh 2010; Engel et. al. 2009-10)
• IH is a null inner boundary ⇒ h ab dx a dx b = 0 , h ab → induced metric
on IH
•⇒ deth = 0 ⇒ theory on IH cannot be ∫ √
h 3 R(h), or any theory
requiring invertingh ab .
• Theory on IH must be topological⇒S IH ≠ S IH [h]
• Gravitational canon DoF in bulk are SU(2) gauge potentials A i a and
densitised triadsEi a after gauge fixing local Lorentz boosts.
• On IH, the boundary gauge potentials are described by anSU(2) Chern
Simons theory of A a i
coupled to Σ ab
ij
≡ E [a
i Eb] j
with coupling k ≡
(A IH /8πlP 2 )
Gravity-gauge theory (topol) link derived

Created with pptalk Slide 17
Horizon deg of freedom & dynamics (Ashtekar et. al. 1997-2000; Basu, Kaul, PM 2009,; Kaul,
PM 2010; Basu, Chatterjee, Ghosh 2010; Engel et. al. 2009-10)
• IH is a null inner boundary ⇒ h ab dx a dx b = 0 , h ab → induced metric
on IH
•⇒ deth = 0 ⇒ theory on IH cannot be ∫ √
h 3 R(h), or any theory
requiring invertingh ab .
• Theory on IH must be topological⇒S IH ≠ S IH [h]
• Gravitational canon DoF in bulk are SU(2) gauge potentials A i a and
densitised triadsEi a after gauge fixing local Lorentz boosts.
• On IH, the boundary gauge potentials are described by anSU(2) Chern
Simons theory of A a i
coupled to Σ ab
ij
≡ E [a
i Eb] j
with coupling k ≡
(A IH /8πlP 2 )
Gravity-gauge theory (topol) link derived

Created with pptalk Slide 17
Horizon deg of freedom & dynamics (Ashtekar et. al. 1997-2000; Basu, Kaul, PM 2009,; Kaul,
PM 2010; Basu, Chatterjee, Ghosh 2010; Engel et. al. 2009-10)
• IH is a null inner boundary ⇒ h ab dx a dx b = 0 , h ab → induced metric
on IH
•⇒ deth = 0 ⇒ theory on IH cannot be ∫ √
h 3 R(h), or any theory
requiring invertingh ab .
• Theory on IH must be topological⇒S IH ≠ S IH [h]
• Gravitational canon DoF in bulk are SU(2) gauge potentials A i a and
densitised triadsEi a after gauge fixing local Lorentz boosts.
• On IH, the boundary gauge potentials are described by anSU(2) Chern
Simons theory of A a i
coupled to Σ ab
ij
≡ E [a
i Eb] j
with coupling k ≡
(A IH /8πlP 2 )
Gravity-gauge theory (topol) link derived

Created with pptalk Slide 18
Spin network : Quantum Space

Created with pptalk Slide 19
Area operator (also volume, length) have bded, discrete spectrum
s I
 S ≡
N∑
I=1
∫
S I
det 1/2 [ 2 g(Ê)]
a(j 1 ,...,j N ) = 1 4 γl2 P
N∑
p=1
lim
N→∞ a(j 1,....j N ) ≤ A cl +O(l 2 P )
Equispaced∀j p = 1/2
√
j p (j p +1)

Created with pptalk Slide 20
Eff Quantum Horizon : Loop Quantum Gravity

Created with pptalk Slide 21
( ) )
kB −1 S = A hor
4lP
2 − 3 2 log A hor 4l
2
4lP
2 + O(
P
A hor
Corrections to area law (Kaul, PM 1998, 2000) are signature LQG effects
Corollary :
End of Digression
β = β Haw
(1+ 6l2 P
A hor
+...
Summary : Thermal stability criterion
M IH
> S IH
= A IH
− 3 ( )
AIH
M P k B 4A P 2 log +···
4A P
Reflects dominance of mass (energy) driven vs entropy driven processes.
In the neutron star context, hydrostatic pressure in gravitational collapse is
analogous to an energy driven process while Fermi degeneracy pressure is
analogous to an entropy driven process
)

Created with pptalk Slide 21
( ) )
kB −1 S = A hor
4lP
2 − 3 2 log A hor 4l
2
4lP
2 + O(
P
A hor
Corrections to area law (Kaul, PM 1998, 2000) are signature LQG effects
Corollary :
End of Digression
β = β Haw
(1+ 6l2 P
A hor
+...
Summary : Thermal stability criterion
M IH
> S IH
= A IH
− 3 ( )
AIH
M P k B 4A P 2 log +···
4A P
Reflects dominance of mass (energy) driven vs entropy driven processes.
In the neutron star context, hydrostatic pressure in gravitational collapse is
analogous to an energy driven process while Fermi degeneracy pressure is
analogous to an entropy driven process
)

Created with pptalk Slide 21
( ) )
kB −1 S = A hor
4lP
2 − 3 2 log A hor 4l
2
4lP
2 + O(
P
A hor
Corrections to area law (Kaul, PM 1998, 2000) are signature LQG effects
Corollary :
End of Digression
β = β Haw
(1+ 6l2 P
A hor
+...
Summary : Thermal stability criterion
M IH
> S IH
= A IH
− 3 ( )
AIH
M P k B 4A P 2 log +···
4A P
Reflects dominance of mass (energy) driven vs entropy driven processes.
In the neutron star context, hydrostatic pressure in gravitational collapse is
analogous to an energy driven process while Fermi degeneracy pressure is
analogous to an entropy driven process
)

Created with pptalk Slide 21
( ) )
kB −1 S = A hor
4lP
2 − 3 2 log A hor 4l
2
4lP
2 + O(
P
A hor
Corrections to area law (Kaul, PM 1998, 2000) are signature LQG effects
Corollary :
End of Digression
β = β Haw
(1+ 6l2 P
A hor
+...
Summary : Thermal stability criterion
M IH
> S IH
= A IH
− 3 ( )
AIH
M P k B 4A P 2 log +···
4A P
Reflects dominance of mass (energy) driven vs entropy driven processes.
In the neutron star context, hydrostatic pressure in gravitational collapse is
analogous to an energy driven process while Fermi degeneracy pressure is
analogous to an entropy driven process
)

Created with pptalk Slide 21
( ) )
kB −1 S = A hor
4lP
2 − 3 2 log A hor 4l
2
4lP
2 + O(
P
A hor
Corrections to area law (Kaul, PM 1998, 2000) are signature LQG effects
Corollary :
End of Digression
β = β Haw
(1+ 6l2 P
A hor
+...
Summary : Thermal stability criterion
M IH
> S IH
= A IH
− 3 ( )
AIH
M P k B 4A P 2 log +···
4A P
Reflects dominance of mass (energy) driven vs entropy driven processes.
In the neutron star context, hydrostatic pressure in gravitational collapse is
analogous to an energy driven process while Fermi degeneracy pressure is
analogous to an entropy driven process
)

Created with pptalk Slide 21
( ) )
kB −1 S = A hor
4lP
2 − 3 2 log A hor 4l
2
4lP
2 + O(
P
A hor
Corrections to area law (Kaul, PM 1998, 2000) are signature LQG effects
Corollary :
End of Digression
β = β Haw
(1+ 6l2 P
A hor
+...
Summary : Thermal stability criterion
M IH
> S IH
= A IH
− 3 ( )
AIH
M P k B 4A P 2 log +···
4A P
Reflects dominance of mass (energy) driven vs entropy driven processes.
In the neutron star context, hydrostatic pressure in gravitational collapse is
analogous to an energy driven process while Fermi degeneracy pressure is
analogous to an entropy driven process
)

Created with pptalk Slide 22
Critical mass
With small energy loss due to gravitational radiation during collapse
(
Mcrit
M P
)min
Recall Thermal stability bound
(
MIH
Therefore
(
Mcrit
M P
)min
≃
(
MIH
M P
)min
= ξ A IH
A P
,ξ = O(1)
≃ ξ
M P
)min
A IH
= ξ A IHA Cn
A P A Cn A P
However, A IH /A Cn >> 1 ! Even though there is a critical upper bound
of mass for a neutron star to collapse, it is far far larger than the Chandrasekhar
limit ! Unacceptable

Created with pptalk Slide 22
Critical mass
With small energy loss due to gravitational radiation during collapse
(
Mcrit
M P
)min
Recall Thermal stability bound
(
MIH
Therefore
(
Mcrit
M P
)min
≃
(
MIH
M P
)min
= ξ A IH
A P
,ξ = O(1)
≃ ξ
M P
)min
A IH
= ξ A IHA Cn
A P A Cn A P
However, A IH /A Cn >> 1 ! Even though there is a critical upper bound
of mass for a neutron star to collapse, it is far far larger than the Chandrasekhar
limit ! Unacceptable

Created with pptalk Slide 22
Critical mass
With small energy loss due to gravitational radiation during collapse
(
Mcrit
M P
)min
Recall Thermal stability bound
(
MIH
Therefore
(
Mcrit
M P
)min
≃
(
MIH
M P
)min
= ξ A IH
A P
,ξ = O(1)
≃ ξ
M P
)min
A IH
= ξ A IHA Cn
A P A Cn A P
However, A IH /A Cn >> 1 ! Even though there is a critical upper bound
of mass for a neutron star to collapse, it is far far larger than the Chandrasekhar
limit ! Unacceptable

Created with pptalk Slide 22
Critical mass
With small energy loss due to gravitational radiation during collapse
(
Mcrit
M P
)min
Recall Thermal stability bound
(
MIH
Therefore
(
Mcrit
M P
)min
≃
(
MIH
M P
)min
= ξ A IH
A P
,ξ = O(1)
≃ ξ
M P
)min
A IH
= ξ A IHA Cn
A P A Cn A P
However, A IH /A Cn >> 1 ! Even though there is a critical upper bound
of mass for a neutron star to collapse, it is far far larger than the Chandrasekhar
limit ! Unacceptable

Created with pptalk Slide 22
Critical mass
With small energy loss due to gravitational radiation during collapse
(
Mcrit
M P
)min
Recall Thermal stability bound
(
MIH
Therefore
(
Mcrit
M P
)min
≃
(
MIH
M P
)min
= ξ A IH
A P
,ξ = O(1)
≃ ξ
M P
)min
A IH
= ξ A IHA Cn
A P A Cn A P
However, A IH /A Cn >> 1 ! Even though there is a critical upper bound
of mass for a neutron star to collapse, it is far far larger than the Chandrasekhar
limit ! Unacceptable

Created with pptalk Slide 22
Critical mass
With small energy loss due to gravitational radiation during collapse
(
Mcrit
M P
)min
Recall Thermal stability bound
(
MIH
Therefore
(
Mcrit
M P
)min
≃
(
MIH
M P
)min
= ξ A IH
A P
,ξ = O(1)
≃ ξ
M P
)min
A IH
= ξ A IHA Cn
A P A Cn A P
However, A IH /A Cn >> 1 ! Even though there is a critical upper bound
of mass for a neutron star to collapse, it is far far larger than the Chandrasekhar
limit ! Unacceptable

Created with pptalk Slide 23
What went wrong with the scenario, even though it is physically plausible
Observe S hor due to quantum grav description of horizon - completely
missed effects of matter (recall Fermi degeneracy pressure of neutrons in
Chandrasekhar’s derivation)
Need to consider entanglement entropy due to entanglement of matter
states with gravitational states at the horizon : S ent = −Trρ red logρ red
where ρ red is reduced density matrix
Entanglement entropy of neutron star with horizon inside Bombelli et. al. 1986;
Srednicki 1993
S ent = ξ e
A hor
A c
+ξ n
A NS
A c
where ξ e , ξ n = O(1) ;A c = a 2 is an area related to the UV cutoff used in
the theory
TotalS = S grav (A hor /A P )+S ent (A hor /A c )+S ent (A NS /A c )+S bulk,n

Created with pptalk Slide 23
What went wrong with the scenario, even though it is physically plausible
Observe S hor due to quantum grav description of horizon - completely
missed effects of matter (recall Fermi degeneracy pressure of neutrons in
Chandrasekhar’s derivation)
Need to consider entanglement entropy due to entanglement of matter
states with gravitational states at the horizon : S ent = −Trρ red logρ red
where ρ red is reduced density matrix
Entanglement entropy of neutron star with horizon inside Bombelli et. al. 1986;
Srednicki 1993
S ent = ξ e
A hor
A c
+ξ n
A NS
A c
where ξ e , ξ n = O(1) ;A c = a 2 is an area related to the UV cutoff used in
the theory
TotalS = S grav (A hor /A P )+S ent (A hor /A c )+S ent (A NS /A c )+S bulk,n

Created with pptalk Slide 23
What went wrong with the scenario, even though it is physically plausible
Observe S hor due to quantum grav description of horizon - completely
missed effects of matter (recall Fermi degeneracy pressure of neutrons in
Chandrasekhar’s derivation)
Need to consider entanglement entropy due to entanglement of matter
states with gravitational states at the horizon : S ent = −Trρ red logρ red
where ρ red is reduced density matrix
Entanglement entropy of neutron star with horizon inside Bombelli et. al. 1986;
Srednicki 1993
S ent = ξ e
A hor
A c
+ξ n
A NS
A c
where ξ e , ξ n = O(1) ;A c = a 2 is an area related to the UV cutoff used in
the theory
TotalS = S grav (A hor /A P )+S ent (A hor /A c )+S ent (A NS /A c )+S bulk,n

Created with pptalk Slide 23
What went wrong with the scenario, even though it is physically plausible
Observe S hor due to quantum grav description of horizon - completely
missed effects of matter (recall Fermi degeneracy pressure of neutrons in
Chandrasekhar’s derivation)
Need to consider entanglement entropy due to entanglement of matter
states with gravitational states at the horizon : S ent = −Trρ red logρ red
where ρ red is reduced density matrix
Entanglement entropy of neutron star with horizon inside Bombelli et. al. 1986;
Srednicki 1993
S ent = ξ e
A hor
A c
+ξ n
A NS
A c
where ξ e , ξ n = O(1) ;A c = a 2 is an area related to the UV cutoff used in
the theory
TotalS = S grav (A hor /A P )+S ent (A hor /A c )+S ent (A NS /A c )+S bulk,n

Created with pptalk Slide 23
What went wrong with the scenario, even though it is physically plausible
Observe S hor due to quantum grav description of horizon - completely
missed effects of matter (recall Fermi degeneracy pressure of neutrons in
Chandrasekhar’s derivation)
Need to consider entanglement entropy due to entanglement of matter
states with gravitational states at the horizon : S ent = −Trρ red logρ red
where ρ red is reduced density matrix
Entanglement entropy of neutron star with horizon inside Bombelli et. al. 1986;
Srednicki 1993
S ent = ξ e
A hor
A c
+ξ n
A NS
A c
where ξ e , ξ n = O(1) ;A c = a 2 is an area related to the UV cutoff used in
the theory
TotalS = S grav (A hor /A P )+S ent (A hor /A c )+S ent (A NS /A c )+S bulk,n

Created with pptalk Slide 24
Need to generalize thermal stability formula incorporatingS ent
Propose, for any spatial slice corresponding to t, the horizon is stable provided
( Mhor,t A hor,t A hor,t
= ξ g +ξ e
A P A c,t
M NS = M hor,t +M ext,t
M P
)min
How does M ext,t relate to S ent (A NS,t /A c,t )+S bulk,n given that both decrease
as the horizon grows
Thermal stability criterion ⇒ Schwarzschild black hole is unstable wrt
Hawking radiation. How does this impact on M crit

Created with pptalk Slide 24
Need to generalize thermal stability formula incorporatingS ent
Propose, for any spatial slice corresponding to t, the horizon is stable provided
( Mhor,t A hor,t A hor,t
= ξ g +ξ e
A P A c,t
M NS = M hor,t +M ext,t
M P
)min
How does M ext,t relate to S ent (A NS,t /A c,t )+S bulk,n given that both decrease
as the horizon grows
Thermal stability criterion ⇒ Schwarzschild black hole is unstable wrt
Hawking radiation. How does this impact on M crit

Created with pptalk Slide 24
Need to generalize thermal stability formula incorporatingS ent
Propose, for any spatial slice corresponding to t, the horizon is stable provided
( Mhor,t A hor,t A hor,t
= ξ g +ξ e
A P A c,t
M NS = M hor,t +M ext,t
M P
)min
How does M ext,t relate to S ent (A NS,t /A c,t )+S bulk,n given that both decrease
as the horizon grows
Thermal stability criterion ⇒ Schwarzschild black hole is unstable wrt
Hawking radiation. How does this impact on M crit

Created with pptalk Slide 24
Need to generalize thermal stability formula incorporatingS ent
Propose, for any spatial slice corresponding to t, the horizon is stable provided
( Mhor,t A hor,t A hor,t
= ξ g +ξ e
A P A c,t
M NS = M hor,t +M ext,t
M P
)min
How does M ext,t relate to S ent (A NS,t /A c,t )+S bulk,n given that both decrease
as the horizon grows
Thermal stability criterion ⇒ Schwarzschild black hole is unstable wrt
Hawking radiation. How does this impact on M crit

Created with pptalk Slide 24
Need to generalize thermal stability formula incorporatingS ent
Propose, for any spatial slice corresponding to t, the horizon is stable provided
( Mhor,t A hor,t A hor,t
= ξ g +ξ e
A P A c,t
M NS = M hor,t +M ext,t
M P
)min
How does M ext,t relate to S ent (A NS,t /A c,t )+S bulk,n given that both decrease
as the horizon grows
Thermal stability criterion ⇒ Schwarzschild black hole is unstable wrt
Hawking radiation. How does this impact on M crit

Created with pptalk Slide 25
Pending Issues
• Need to establish firm relation betweenA Cn andA hid hor
• Need to justify scenario in detail
• Need to go properly incorporateS ent into stability criterion

Created with pptalk Slide 25
Pending Issues
• Need to establish firm relation betweenA Cn andA hid hor
• Need to justify scenario in detail
• Need to go properly incorporateS ent into stability criterion

Created with pptalk Slide 25
Pending Issues
• Need to establish firm relation betweenA Cn andA hid hor
• Need to justify scenario in detail
• Need to go properly incorporateS ent into stability criterion