Is Black Hole an elementary particle? By Hoi-Lai Yu IPAS, Oct 30, 2007

Basic Questions in searching the Truths of Nature: Basic Questions in searching the Truths of Nature: What is Space? What is Time? What is the meaning of being somewhere? What is meaning of “moving”? Is motion to be defined with respect to objects or with respect to space? Can we formulate Physics without referring to time or space? What is matter? What is causality? What is the role of Observer in Physics?

What disappears in GR is precisely the background space time that Newton believed to have been able to detect. Reality is not made by particles and fields on Spacetime. Reality is made by particles and fields (incldue gravity), that can only be localized with respect to each other. Active Diffeormophism invariant

(1)Black holes entropy (2)Cosmology: Big Bang Singularity Singularity Puzzles in general relativity

Results on Loop Quantum Gravity Non Perturbative GravityNon Perturbative Gravity Canonical Analysis in ADM variablesCanonical Analysis in ADM variables Using the new variables: triad formailism, Ashtekar-Barbero variablesUsing the new variables: triad formailism, Ashtekar-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, inflation, Black hole thermodynamics, ringing modes frequencies, Bekenstein-Mukhanov effectResults: Non commutativty of the geometry, inflation, Black hole thermodynamics, ringing modes frequencies, Bekenstein-Mukhanov effect

Canonical Analysis in ADM variables Time has 2 aspects: (1)Instant of time → t=constant spacelike surfaces (2) Time evolutions Geometric theories of gravity and fields: Foliation of spacetime (  x R) into: space-like 3 dim surfaces  → a “timelike” vector field

Gravity as a gauge theory: How can one works only with a gauge field without metric? In Hamiltonian language point of view: space-time manifold of the form  x R describing by: Astekhar connection, A a i and its conjugate momentum, E a i where a is a 3d spatial index and i is valued in a lie algebra, G

We have Poisson brackets: { A a i (x), E b j (y) } =  a b  i j  3 (y,x) In a hamiltonian formulation of a gauge theory : one constraint for each independent gauge transform. The gauge invariance of a gravitational theory include at least 4 diffeomorphisms, per point.

(1) H a generates the diffeomorphisms of , (2) H is the Hamiltonian constraint that generates the rest of the diffeomorphism group of the spacetime (and hence changes of the slicing of the spacetime into spatial slices), (3) G i generates the local gauge Transformations, (4) h terms in the Hamiltonian that are not proportional to constraints. However, there is a special feature of gravitational theories, which is there is no way locally to distinguish the changes in the local fields under evolution from their changes under a diffeomorphism that changes the time coordinate. Hence h is always just a boundary term, in a theory of gravity.

(1) From Yang-Mills theory the constraint that generates local gauge transformations under the Poisson bracket is just Gauss's law: G i = D a E ai = 0 (2) Three constraints per point that generate the differomorphisms of the spatial slice. Infinitesimally these will look like coordinate transformations, hence the parameter that gives the infinitesimal change is a vector field. Hence these constraints must multiply a vector field, without using a metric. Thus these constraints are the components of a one form. It should also be invariant under ordinary gauge transformations, as they commute with differomorphisms. We can then ask what is the simplest such beast we can make using A i a and E a i ? The answer is H a = E b i F i ab = 0 where F i ab is the Yang-Mills field strength.

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

In fact these two terms already give Einstein's equations, so long as we take the simplest nontrivial choice for G, which is SO(3). Thus, we take for the Hamiltonian constraint:  being the cosmological constant. A a E a A a is a connection and so has dimensions of inverse length. It will turn out that E a 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 Lorentzian theory, we need only modify what we have by putting an “i” into the commutation relations

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

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

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