Is Black Hole an elementary particle?Is Black Hole an elementary particle?

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Hoi-Lai YuHoi-Lai YuIPAS, Oct 30, 2007IPAS, Oct 30, 2007

Basic Questions in searching the Truths of Nature:Basic Questions in searching the Truths of Nature:

What is Space? What is Space?

What is Time?What is Time?

What is the meaning of being somewhere?What is the meaning of being somewhere?

What is meaning of “moving”?What is meaning of “moving”?

Is motion to be defined with respect to objects or with respect to space?Is motion to be defined with respect to objects or with respect to space?

Can we formulate Physics without referring to time or space?Can we formulate Physics without referring to time or space?

What is matter?What is matter?

What is causality?What is causality?

What is the role of Observer in Physics?What is the role of Observer in Physics?

What disappears in GR is precisely the What disappears in GR is precisely the background space time that Newton believed background space time that Newton believed to have been able to detect.to have been able to detect.

Reality is not made by particles and fields on Reality is not made by particles and fields on Spacetime.Spacetime.Reality is made by particles and fieldsReality is made by particles and fields(incldue gravity), that can only be localized with (incldue gravity), that can only be localized with respect to each other. respect to each other.

Active Diffeormophism invariantActive Diffeormophism invariant

(1)(1)Black holesBlack holes entropy entropy

(2)(2)CosmologyCosmology:: Big Big BBangang

SingularitySingularity

Puzzles in general relativityPuzzles in general relativity

GAS

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Results on Loop Quantum GravityResults on Loop Quantum Gravity• Non Perturbative GravityNon Perturbative Gravity• Canonical Analysis in ADM variablesCanonical Analysis in ADM variables• Using the new variables: triad formailism, Using the new variables: triad formailism,

Ashtekar-Barbero variablesAshtekar-Barbero variables• Geometric interpretation of the new variablesGeometric interpretation of the new variables• Quantization of triad, area, volume,…Quantization of triad, area, volume,…• Results: Non commutativty of the geometry, Results: Non commutativty of the geometry,

inflation, Black hole thermodynamics, ringing inflation, Black hole thermodynamics, ringing modes frequencies, Bekenstein-Mukhanov modes frequencies, Bekenstein-Mukhanov effecteffect

Canonical Analysis in ADM variablesCanonical Analysis in ADM variables

Time has 2 aspects:Time has 2 aspects:(1)(1) Instant of time Instant of time → t=constant spacelike surfaces→ t=constant spacelike surfaces

(2) Time evolutions(2) Time evolutions

Geometric theories of gravity and fields:Geometric theories of gravity and fields:Foliation of spacetimeFoliation of spacetime ( ( x R) x R) into: space-like 3 dim into: space-like 3 dim surfaces surfaces

→ → a “timelike” vector fielda “timelike” vector field

Gravity as a gauge theory:Gravity as a gauge theory:

How can one works only with a gauge field without How can one works only with a gauge field without metric?metric?

In Hamiltonian language point of view: In Hamiltonian language point of view: space-time manifold of the form space-time manifold of the form x R x R describing by: describing by:

Astekhar connection, Astekhar connection, AAaaii

and its conjugate momentum, and its conjugate momentum, EEaaii

where where a a is a 3is a 3d d spatial index and spatial index and i i is valued in a lie is valued in a lie algebra, algebra, GG

We have Poisson brackets:We have Poisson brackets:

{ A{ Aaaii (x) , E (x) , Ebb

jj (y) } = (y) } = aabb ii

jj 33 (y,x) (y,x)

In a hamiltonian formulation of a gauge In a hamiltonian formulation of a gauge theory :theory :oneone constraint for each independent constraint for each independent gauge transform. The gauge invariance of a gauge transform. The gauge invariance of a gravitational theory include at least gravitational theory include at least 4 diffeomorphisms, per point4 diffeomorphisms, per point. .

(1) (1) HHaa generates the diffeomorphisms of generates the diffeomorphisms of ,,

(2)(2) HH is the Hamiltonian constraint that generates the rest of the is the Hamiltonian constraint that generates the rest of the diffeomorphism group of the spacetime (and hence changes of the diffeomorphism group of the spacetime (and hence changes of the slicing of the spacetime into spatial slices), slicing of the spacetime into spatial slices),

(3) (3) GGii generates the local gauge Transformations,generates the local gauge Transformations,

(4) (4) h h terms in the Hamiltonian that are not proportional to terms in the Hamiltonian that are not proportional to constraints. However, there is a special feature of gravitational constraints. However, there is a special feature of gravitational theories, which is there is no way theories, which is there is no way locally locally to distinguish the changes in to distinguish the changes in the local fields under evolution from their changes under a the local fields under evolution from their changes under a diffeomorphism that changes the time coordinate. Hence diffeomorphism that changes the time coordinate. Hence hh is always is always just a boundary term, in a theory of gravity.just a boundary term, in a theory of gravity.

(1) From Yang-Mills theory the constraint that generates local gauge (1) From Yang-Mills theory the constraint that generates local gauge transformations under the Poisson bracket is just Gauss's law:transformations under the Poisson bracket is just Gauss's law:

GGii = = DDaa E Eaiai = 0= 0

(2) T(2) Three constraints per point that generate the differomorphisms hree constraints per point that generate the differomorphisms of the spatial slice.of the spatial slice. Infinitesimally these will look like coordinate transformations, hence Infinitesimally these will look like coordinate transformations, hence the parameter that gives the infinitesimal change is a vector field. the parameter that gives the infinitesimal change is a vector field. Hence these constraints must multiply a vector field, without using a Hence these constraints must multiply a vector field, without using a metric.metric.Thus these constraints are the components of a one form. It should Thus these constraints are the components of a one form. It should also be invariant under ordinary gauge transformations, as they also be invariant under ordinary gauge transformations, as they commute with differomorphisms. We can then ask what is the commute with differomorphisms. We can then ask what is the simplest such beast we can make using simplest such beast we can make using AAii

aa andand EEaaii ? ?

The answer is The answer is HHaa = = EEbb

ii F Fiiabab = 0 = 0

where where FFiiabab is the Yang-Mills field strength.is the Yang-Mills field strength.

(3) O(3) One constraint per point, which generates changes in ne constraint per point, which generates changes in the time coordinate,the time coordinate, in the embedding of in the embedding of M M = = x x R R.. This is the Hamiltonian constraint. This is the Hamiltonian constraint. Since its action is locally indistinguishable from the effect Since its action is locally indistinguishable from the effect of changing the time coordinate, of changing the time coordinate, it does contain the it does contain the dynamics dynamics since the parameter it multiplies is proportional since the parameter it multiplies is proportional to the local change in the time coordinate. It must be to the local change in the time coordinate. It must be gauge invariant and a scalar. But it could also be a density, gauge invariant and a scalar. But it could also be a density, so we have the freedom to find the simplest expression that so we have the freedom to find the simplest expression that is a density of some weight. It turns out there are no is a density of some weight. It turns out there are no polynomials in our fields that have density weight zero, polynomials in our fields that have density weight zero, without using a metric. But two expressions have density without using a metric. But two expressions have density weight two. The two simplest such terms that can be weight two. The two simplest such terms that can be written, which are lowest order in derivatives, written, which are lowest order in derivatives,

In fact these two terms already give Einstein's equations, In fact these two terms already give Einstein's equations, so long as we take the simplest nontrivial choice for so long as we take the simplest nontrivial choice for GG, , which is which is SO(3). SO(3). Thus, we take for the Hamiltonian Thus, we take for the Hamiltonian constraint:constraint:

being the cosmological constant. AAaa is a connection and so has dimensions of inverse length. It will turn out that EEaa is related to the metric and so we should make the unconventional choice that it is dimensionsless.

In fact, what we have here is Euclidean general relativity. If we want the Lorentziantheory, we need only modify what we have by putting an “i” into the commutation relations

SEMICLASSICAL LIMIT, HAMILTON-JACOBI EQUATIONS, SEMICLASSICAL LIMIT, HAMILTON-JACOBI EQUATIONS, AND SCHWARZSCHILD BLACK HOLESAND SCHWARZSCHILD BLACK HOLES

We may furthermore map precisely the equation of motion for plane We may furthermore map precisely the equation of motion for plane wave solutions to classical black holes by studying the wave solutions to classical black holes by studying the correspondence of the classical initial data to the Hamilton-Jacobi correspondence of the classical initial data to the Hamilton-Jacobi theory. To achieve this, we note that the familiar Schwarzschild theory. To achieve this, we note that the familiar Schwarzschild metric is:metric is:

So, now we have seen that a black hole is really an elementary partice in superspace with definite dispersion relation:

K+ K- = 1

RINDLER SUPERSPACERINDLER SUPERSPACE

The Minkowski Bessel modes can in fact be understood as the “rapidity Fourier transform" of plane wave solutions

The Dirac equation which is first-order in superspace intrinsic time on the Rindler wedge,

QUANTUM UNITARITY DESPITE THE PRESENCE OF APPARENT QUANTUM UNITARITY DESPITE THE PRESENCE OF APPARENT CLASSICAL SINGULARITIESCLASSICAL SINGULARITIES

An analogous situation happens in free non-relativistic An analogous situation happens in free non-relativistic quantum mechanics wherein hermiticity of the momentum quantum mechanics wherein hermiticity of the momentum operator requires a physical Hilbert space of suitable operator requires a physical Hilbert space of suitable wavepackets which vanish at spatial infinity, and rule out wavepackets which vanish at spatial infinity, and rule out plane wave states with infinitely sharp momentum. From this plane wave states with infinitely sharp momentum. From this perspective the boundary condition guaranteeing quantum perspective the boundary condition guaranteeing quantum unitarity in our present context of spherically symmetric unitarity in our present context of spherically symmetric gravity holds for rather generic wavepackets.gravity holds for rather generic wavepackets.

Replacing the “Quantum censorship” by BC with nonzero wave-function at the classical singular, zero-volume three geometry (but allowing the topological fluctuations) which may correspond to Big Bang singularity may give non-trivial example interpretation of Hawking-Hartle boundary without boundary condition of creating the Universe from nothing!

Summary:Summary:(1) The resultant arena for quantum geodyanmics is two dimernsion of signature (+,-), non-singular – intrinsic time and R radial coordinate time are monotonic function of each other.

(2) Black holes are elementary particles in superspce.

(3) The boundary of the Rindler wedge corresponds to physical horizons and singularities of Black Holes.

(4) Hamilton_Jacboi semi-classical limits consists of plane wave solution can be matched previously to interiors of Schwarzchild black holes with straight line trajectories of free motion in flat superspace – quantum Birkhoff theorem.

(5) Sperspace is free of singularity even in continuum.

(6) Positive definite current can be obtained for wave-functions on superspace for Dirac equation associated with WdW equation.

(7) Hermiticity of Dirac Hamiltonian and thus Unitary of quantum theory is translated BC on the space like hypersurfaces of Rindler Wedge which is better for Hartle-Hawking mechanism for creation of Universe.