Download General Relativity, Black Holes and Quantum Field Theory in curved

General Relativity, Black Holes and
Quantum Field Theory in curved space-times
Dr. Davide Batic
Department of Mathematics, University of West Indies at Mona
Kingston, Jamaica
[email protected], [email protected]
1. Introduction
The Heun hierarchy
General Relativity and Black Holes
In General Relativity the gravitational field of a massive object can be thought
as the curvature of a four dimensional space-time and it is described by a metα . The Einstein field equations
ric tensor gµν and the Riemann tensor R βγδ
provide a connection between the space-time geometry and the matter distribution
1
G µν = 8 πTµν , G µν = R µν − gµν R
2
where R µν is the Ricci tensor, R is the scalar curvature and Tµν is the energymomentum tensor describing the matter distribution. One of the most spectacular prediction of General Relativity are black holes. When a sufficiently
massive star runs out of fuel, it is unable to support itself against its own
gravitational attraction, and it collapses into a black hole, i.e. into an object
whose mass concentration is so dense that even light would be trapped. The
collapse of a rotating star can give rise to a spinning black hole, whose external gravitational field is described by means of the vacuum Einstein equation
G µν = 0. A rotating neutral black hole is described by the Kerr metric [1].
In Boyer-Lindquist coordinates (t, r, θ, φ) such a solution is given by the line
element
∆
ds 2 = (dt − a sin2 θdφ)2 − U
U
)
(
[
sin2 θ
dr 2
2
+ dθ −
adt −
∆
U
(
) ]
2
2
2
r + a dφ
The following hierarchy of linear second order ODEs plays a crucial role
in QFT in curved space-time and in Quantum Mechanics, namely
• Hypergeneralized Heun equation (6 regular singular points) [9]
d2F
+
dz 2
4
√ M 2 − a 2.
The Cauchy-horizon is a light-like boundary of the domain of validity of a
Cauchy problem (a particular boundary value problem of the theory of PDEs),
whereas the event horizon is a boundary in space-time beyond which events
cannot effect an outside observer. Depending on the values of the parameters
a and M the Kerr metric goes over into the following space-times
M = |a| = 0 Minkowski metric
M > |a| = 0 Schwarzschild black hole
M = |a| > 0 extreme Kerr black hole
M > |a| > 0 non extreme Kerr black hole
The Dirac equation in curved space-time
The Dirac equation is a relativistic quantum mechanical wave equation providing a description of elementary spin 1/
/ 2 particles (Fermions), such as
electrons. The Dirac equation can be written in curved space-time using a
vierbein, i.e. a local frame that enables to define Dirac matrices at every point
of the space-time under consideration, as follows
µ
− iγ a ea D µ Ψ + m Ψ = 0,
µ
a, µ = 0, · · · , 3.
Here, ea is the vierbein, m is the rest mass of the Fermion, γ a are the usual
Dirac matrices and D µ is the covariant derivative defined as
i
c σ ab,
D µ = ∂µ − ηacωbµ
4
i
σ ab = [γa , γb]
2
i=0
• Generalized Heun equation (3 regular singular points and∞ irregular singular point) [10]
[
]
d2F
1 − µ0 1 − µ1 1 − µ2
dF β0 + β1z + β2z 2
+
+
+ µ3
+
F (z ) = 0
+
z
z− 1
z− b
dz
z (z − 1)(z − b)
dz 2
• Heun equation characterized by four regular singular points (Heun, 1889)
d2F
+
dz 2
(
c = ec ∂ eν + ec eσ Γν
ωbµ
ν µ b
ν b σµ
with Γνσµ the Christoffel symbols, describing the effect of parallel transport in
curved space-time. By means of the Chandrasekhar ansatz [2]
Ψ(t, r, ϑ, ϕ ) = eiωt
(
)
ei (k+ 2 )ϕ ,
1
where ω and k ∈ Ζ denote the energy and the azimuthal quantum number
of the particle, the Dirac equation in the Kerr metric decouples into two systems of linear first order ODEs for the radial and angular components of the
spinorial wave function Ψ, namely
uS :=
with
(
–imr − λ
√ ∆ D−
imr − λ √ ∆ D +
( ) ( ) (
d
δϑ
0 1
−1 0
D± =
S1
S2
∂ _
– iK ,
∂r | ∆
+
)( )
R1
R2
= 0,
− am cosϑ − Q(ϑ)
− Q(ϑ)
am cosϑ
)( ) ( )
K := K (r) = ω(r 2 + a2) +
S1
S2
= λ
( )
k+
S1
S2
1
a
2
and Q(ϑ) = aω sin ϑ + ( k + 1 / 2) cscϑ. To conclude, the Dirac equation can
b e brought int o t he form of a Schrödinger equat ion
i
∂Ψ
= HD Ψ
∂t
with Hamiltonian H D formally self-adjoint with respect to the positive scalar
product [3]
(Ψ, Φ) =
∫
+∞
−∞
du
∫
+1
−1
d(cosϑ)
∫
2π
0
dϕ
(
)
)
γ
dF
δ
∈
αβ0 − q
+
+
+
F (z ) = 0
z z − 1 z − b dz
z (z − 1)(z − a)
• Hypergeometric equation with three regular singular points (Gauss, 1813)
z (z − 1)
d2F
dF
− τ2τ3F (z ) = 0
+
[τ
−
(τ
+
τ
+
1)
z
]
1
2
3
dz
dz 2
2. Results
• In [4] we proved that in L 2(0, π)2 the angular operator u with domain of
definition C0∞ (0, π)2 is essentially self-adjoint. Its spectrum is discrete and
simple. The eigen values λ depend analytically on the parameters µ = am
and ν = aω and satisfy the first order quasi-linear PDE
(µ − 2νλ)
∂λ
∂λ
+ ( ν − 2µλ) + 2 κµ + 2 µν = 0 ,
∂µ
∂ν
κ := k +
1
2
whose characteristic equation can be reduced to a Painleve III equation. In
addition, we derived a power series expansion for the eigenvalues in terms
of ν − µ and ν + µ and we gave a recurrence relation for the coefficients. Finally, we showed that the angular eigenfunctions satisfy a generalized Heun
equation.
• In [3] we proved the completeness of the Chandrasekhar ansatz for the Dirac
equation in the Kerr-Newman metric. Based on this we derived an integral
representation for smooth compactly supported functions, which in turn we
used to derive an integral representation for the propagator (i.e. the timeevolution operator) of solutions of the Cauchy problem with initial data in
C0∞ (R × S )4, namely
Ψ(t, x ) = eitH D Ψ(0, x ) =
∫
+∞
−∞
dω eiωt
∑
kj
kj
Ψ ω (Ψ ω, Ψω)
k,j ∈ Z
with x = ( u, ϑ, ϕ).
c is the spin connection
where ηac is the Minkowski metric andωbµ
R 1(r )S 1(ϑ)
R 2(r)S 2(ϑ)
R 2(r)S 1(ϑ)
R 1(r)S 2(ϑ)
1 − µi dF B 0 + B 1z + B 2z 2 + B 3z 3
+
F (z) = 0
z − z i dz
(z − z 0) · · · (z − z 3)
∑
with U := U(r, θ) = r 2 + a 2 cos2 θ and ∆ := ∆( r ) = r 2 − 2M r + a 2, where
M is the mass and a = J/M is the angular momentum per unit mass of the
black hole. The zeros of ∆, here denoted by r − and r + , define the Cauchy
horizon and the event horizon, respectively
r± = M ±
3. Next goals
a2∆ sin 2 ϑ
1− 2
ΨΦ,
(r + a2)2
where u is the so-called tortoise coordinate defined by du/dr = ( r 2 + a2)/ ∆.
• In [5] and [6] we derived an integral representation of the time-evolution
operator for the Dirac equation in the extreme Kerr metric and we proved
that an atomic system made of a microscopic extreme Kerr black hole as
its nucleus surrounded by an electronic cloud cannot occur in our universe.
• Starting with the Dirac equation outside the event horizon of a non-extreme
Kerr black hole and applying the integral representation for the Dirac propagator derived in [3] we developed a time-dependent scattering theory for
massive Dirac particles. In [7] we proved the existence of Dollard-modified
wave operators at infinity
(∞ )
± Ψ
W(∞
) 0
= s−
(∞ )
lim e− itH D I ∞ eitH ∞ eiδ(t) Ψ0
t±∞
by implementing a time-dependent logarithmic phase shift δ(t) from the free
dynamics to offset the long range terms in the full Hamiltonian due to the
presence of the gravitational force and by introducing a smooth bounded
identifying operator I ∞ . Moreover, we computed explicitly the wave operators (Dollard-modified) at infinity and classical (i.e. without phase shift)
at the event horizon.
• In Differential Geometry the Petrov classification describes the possible algebraic symmetries of the Weyl tensor (i.e. the traceless part of the Riemann curvature tensor) at each point in a pseudo-Riemannian manifold.
According to Petrov there exist six possible types of symmetries and the
so-called type-D metrics admitting two double principal null directions belong to such classification. Type-D metrics contain all relevant black hole
solutions. In [11] we proved that in any type-D metric the separated Teukolsky equation (describing gravitational, electromagnetic, scalar and neutrino
field perturbations of a type-D metric) reduces to a Heun equation.
• In the theory of special functions the generalized Heun equation is very important since it contains the ellipsoidal wave equation as well as the Heun
equation as special cases. In [8] we related the separated radial part of a
massive Dirac equation in the Kerr-Newman metric and the static perturbat ions for t he non-ext remal Reissner-Nordst röm solut ion t o a GHE. Surprisingly, GHE has been scarcely studied. Moreover, there exist PDEs such as
Rarita-Schwinger, Dirac, Klein-Gordon, Teukolsky equantions which can be
separated in a general class of black hole metrics, thus giving rise to certain
second order linear ODEs charcterized by a certain number of singularities.
All these equations still need to be reduced to one or to some confluent form
of the equations of the Heun hierarchy. This is a joint work with Davian
Chin (UWI) and Marek Nowakowski (Universidad de Los Andes).
• In [4] we derived a PDE for the eigenvalues of the Chandrasekhar-Page angular equation. Is it possible to do the same for the generalized spheroidal
equation
[
]
dy
(m + sx )2
d
2
2
2
(1 − x ) − p (1 − x ) + 2 pβx +
y = λy,
dx
dx
1 − x2
x ∈ (− 1, 1)
or other eigenvalues problems? If the answer is positive, this new approach
would represent an alternative method for studying a certain class of eigenvalue problems. This is a joint work with Monika Winklmeier (Universidad de Los Andes).
• Quantum Gravity has not yet converged into a single theory and at present
several rival theories co-exist (String Theory, Loop Quantum Gravity and
Noncommutative Geometry). Nevertheless, all these candidates reveal certain common or global features like noncommutativity of the coordinates at
Planckian scales. This quantum aspect of space-time has been recently applied to the final state of a black hole [13]. The starting point of these new
developments is the commutation relation between coordinate operators.
Based on such a commutation relation one can show that the amplitude
(x, z ) which in the standard quantum mechanical prescription is proportional to the Dirac delta function, becomes now a Gaussian distribution.
Therefore, one of the effects of the quantum geometry is the replacement
of a sharp distance by a Gaussian smearing. This observation gave rise to
new models of mini black holes [12] where the final singularity is replaced
by a self-gravitating droplet. Although the issue of a smeared distance
might not be the only fingerprint of the quantum geometry, these models
explicitly reveal its importance. For instance, at Planckian scales where the
central singularity is replaced by a black hole remnant (droplet) the metric
has unusual properties which might resemble those of wormholes. The final
answer lies in the derivation of the maximal atlas covering the manifold of
the final stage of a black hole as inspired by Noncommutative Geometry. In
other words, we look for coordinate charts avoiding the singularity at the
event horizons as one usually does in deriving the Kruskal-Szekeres coordinates for the enlarged Schwarzschild manifold. This is a joint project with
Marek Nowakowski (Universidad de Los Andes) and Piero Nicolini
(University of Frankfurt).
4. References
[1 ] R. P. KERR, Gravitational Field of a spinning mass as an example of algebraically special metrics , Phys. Rev. Lett. 11, 237 (1963).
[2 ] S. CHANDRASEKHAR, The solution of Dirac's equation in Kerr geometry,
Proc. Roy. Soc. London A 349, 571 (1976).
[3 ] D. BATIC AND H. SCHMID, The Dirac propagator in the Kerr-Newman
metric, Prog. Theor. Phys. 116, 517 (2006).
[4 ] D. BATIC, H. SCHMID AND M. WINKLMEIER, On the eigen values of the
the Chandrasekhar-Page angular equation, J. Math. Phys. 46, 012504 (2005).
[5 ] D. BATIC AND H. SCHMID, The Dirac propagator in the extreme Kerr
metric, J. Phys. A 40, 13443 (2007).
[6 ] D. BATIC AND M. NOWAKOWSKI, On the bound states of the Dirac equation in the extreme Kerr metric, Class. Quant. Grav. 25, (2008).
[7 ] D. BATIC, Scattering for massive Dirac fields on the Kerr metric, J. Math.
Phys. 48, 22502 (2007).
[8 ] D. BATIC, H. SCHMID AND M. WINKLMEIER, The generalized Heun
equation in QFT in curved space-times , J. Phys. A 39, 12559 (2006).
[9 ] D. BATIC AND M. SANDOVAL, The hypergeneralized Heun eqin QFT in
curved space-times, accepted for publication in Central Journal of Physics
(2009).
[10 ] R. SCHÄFKE AND D. SCHMIDT, The connection problem for general
linear differential equations, SIAM J. Math. Anal. 11, 848 (1980).
[11 ] D. BATIC AND H. SCHMID, Heun equation, Teukolsky equation and type-D
metrics, J. Math. Phys. 48, 042502 (2007).
[12 ] I. ARRAUT, D. BATIC AND M. NOWAKOWSKI, A non commutative model
for a mini black hole Class. Quant. Grav. 26, 245006 (2009).
[13 ] P. NICOLINI, Noncommutative black holes, the final appeal to quantum
gravity: A Review, I nt. J. Mod. Phys. A24, 1229 (2009