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In physics, there are two theoretical lengths
• Classical size
• Classical radius of an object given by its classical theory
• Quantum size
• Compton wavelength of a particle given by quantum mechanics
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General Criterion
• If the classical radius of an object is larger than its Compton wavelength, then a classical description is sufficient.
• If the Compton wavelength of an object is larger than its classical size, then a quantum description is necessary.
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Black Holes
• Schwarzschild radius:
• Proportional to mass
• Compton wavelength:
• Proportional to inverse mass
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c
22 mc
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Planck Mass
• At the Planck mass, the Schwarzschild radius is equal to the Compton wavelength and the quantum black hole is formed.
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Planck Length
• Quantum black holes are the smallest and heaviest conceivable elementary particles. They have a microscopic size but a macroscopic mass.
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Dual Nature• Quantum black holes are at the
boundary between classical and quantum regions.
• They obey the macroscopic Laws of Thermodynamics and they decay into elementary particles.
• They can have a semi-classical description.
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Quantum Gravity?
• There is a total lack of evidence of any quantum nature of gravity, despite intensive efforts to develop a quantum theory of gravity.
• Is is possible that quantum gravity is not necessary?
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In General Relativity
• Spacetime is a macroscopic concept.
• Is Einstein’s equation similar in nature to Navier-Stokes equation in fluid mechanics as a macroscopic theory?
ds g x dx dx2 ( )
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Nuclear Force
• Energy levels are quantized in nuclei, but nuclear force is not a fundamental force.
• The fundamental theory is Quantum Chromodynamics of quarks and gluons.
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Graviton
• A hypothetical spin-2 massless particle.
• The existence of the graviton itself in nature remains to be seen.
• At best it propagates in an a priori background spacetime.
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Wave Equation
• The gravitational wave equation, from which the graviton idea is developed, is inherently a weak field approximation in general relativity.
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Detectability
• It is physically impossible to detect a single graviton of energy .
• Detector size has to be less than the Schwarzschild radius of the detector.
RRS
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Classical Gravity
• We take the practical point of view that gravitation is entirely a classical theory, and that general relativity is valid down to the Planck scale.
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Spacetime
• This means that spacetime is continuous as long as we are above the Planck scale.
• At the Planck scale, quantum black holes will appear and they act as a natural cutoff to spacetime.
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What is an elementary particle?
An elementary particle is a logical
construction.
• Are black holes elementary particles?
• Are they fermions or bosons?
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Present Goal
• To construct various fundamental quantum black holes as elementary particles, using the results in general relativity.
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Black Hole Theorems:
1. Singularity Theorem 1965
2. Area Theorem 1972
3. Uniqueness Theorem 1975
4. Positive Energy Theorem 1983
5. Horizon Mass Theorem 2005
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Horizon Mass Theorem
For all black holes: neutral, charged or rotating, the horizon mass is always equal to twice the irreducible mass observed at infinity.
Y.K. Ha, Int. J. Mod. Phys. D14, 2219 (2005)
M r M irr( ) 2
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Black Hole Mass
• The mass of a black hole depends on where the observer is.
• The closer one gets to the black hole, the less gravitational energy one sees.
• As a result, the mass of a black hole increases as one gets near the horizon.
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Asymptotic Mass
• The asymptotic mass is the mass of a neutral, charged or rotating black hole including electrostatic and rotational energy.
• It is the mass observed at infinity.
M
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Horizon Mass
• The horizon mass is the mass which cannot escape from the horizon of a neutral, charged or rotating black hole.
• It is the mass observed at the horizon.
M r( )
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Irreducible Mass
• The irreducible mass is the final mass of a charged or rotating black hole when its charge or angular momentum is removed by adding external particles to the black hole.
• It is the mass observed at infinity.
M irr
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Surprising Consequence !
• The electrostatic and the rotational energy of a general black hole are all external quantities.
• They are absent inside the black hole.
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Charged Black Hole
• A charged black hole does not carry any electric charges inside.
• Like a conductor, the electric charges stay at the surface of the black hole.
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Rotating Black Hole
• A rotating black hole does not rotate.
• It is the external space which is undergoing rotating.
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Significance of Theorem
• The Horizon Mass Theorem is crucial for understanding Hawking radiation.
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Energy Condition
• Black hole radiation is only possible if the horizon mass is greater than the asymptotic mass since it takes an enormous energy for a particle released near the horizon to reach infinity.
M r M( )
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Photoelectric Effect
• The incident photon must have a greater energy than that of the ejected electron in order to overcome binding.
maxkEhf
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Hawking Radiation
• No black hole radiation is possible if the horizon mass is equal to the asymptotic mass.
• Without black hole radiation, the Second Law of Thermodynamics is lost.
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Quantum Black Holes
• Mass - Planck mass
• Radius - Planck length
• Lifetime - stable & unstable
• Spin - integer & half-integer
• Type - neutral & charged
• Other - Area & intrinsic entropy
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Black Hole Types
Spin-0
unstable
Spin-1/2
unstable
Spin-1
unstable
Planck-charge
stable
M r M( ) M r M( )
M r M( ) M r M( )
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Spin-0
• A Planck-size black hole created in ultra-high energy collisions or in the Big Bang.
• Disintegrates immediately after it is formed and become Hawking radiation.
• Observable signatures may be seen from its radiation.
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Planck-Charge
• A Planck-size black hole carrying maximum electric charge but no spin.
• It is absolutely stable and cannot emit any radiation.
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Spin-1/2
• A Planck-size black hole carrying angular momentum and charge
and magnetic moment .
• It is unstable and it will decay into a burst of elementary particles.
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3 2Q P l /
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Spin-1
• A Planck-size rotating black hole with angular momentum but no charge.
• It will also decay into a burst of elementary particles
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Micro Black Holes
• Microscopic black holes with higher mass and larger size may be constructed from the fundamental types.
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Quantization
• Quantization of the area of black holes is a conjecture, not a proof.
• Unphysical spins (transcendental and imaginary numbers) not found in quantum mechanics would appear.
• Integer and half-integer spins do not result in quantization of area.
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Ultra-High Energy Cosmic Rays
Theoretical Upper Limit
• K. Greisen, End to the Cosmic Ray Spectrum, Phys. Rev. Lett 16 (1966) 748
• G.T. Zatsepin and V.A. Kuzmin, Upper Limit of the Spectrum of Cosmic Rays, JETP Lett. 4 (1966) 78.
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GZK Effect
• Interaction of protons with cosmic microwave background photons would result in significant energy loss.
• Energy spectrum would show flux suppression above eV.6 1 0 1 9
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Cosmic Ray Experiments
AGASA
• A dozen events above GZK limit possibly detected.
Hi-Res
• GZK effect observed.
• There is no correlation with nearby sources.
Pierre-Auger
• GZK effect observed.
• Correlation with AGN sources.
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GZK Paradox
• Why are some cosmic ray energies theoretically too high if there are no near-Earth sources?
• Quantum black holes in the neighborhood of the Galaxy could resolve the paradox posed by the GZK limit on the energy of cosmic rays from distant sources.
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Annihilation
• Quantum black holes carrying maximum charges are absolutely stable.
• They can annihilate with opposite ones to produce powerful bursts of elementary particles in all directions with very high energies.
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Dark Matter
• Planck-charge quantum black holes could act as dark matter in cosmology without having to resort to new interactions and exotic particles because they are non-interacting particles.
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Planck-Charge Black Holes
• Their electrostatic repulsion exactly cancels their gravitational attraction.
• There is no effective potential between them at any distance.
• The net energy outside the black hole is identically zero.
• They behave like a non-interacting gas.
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Conclusion
• Quantum black holes could have a real existence and play a significant role in cosmology.
• They would be indispensable to understanding the ultimate nature of spacetime and matter.
• Their discovery would be revolutionary