Published on December 1, 2007
The Swarzschild Radius: The Swarzschild Radius Einstein’s could only obtain an approximate solution to his own equations of relativity - the mathematics was so complex that he thought an exact solution was unlikely to be found Swarzschild showed how the curvature of space varied as a function of distance along a radial line from the centre of an object. But… he also predicted that at a critical distance on approaching the object the curvature of space was so strong that nothing - not even light - could escape Towards black holes: Towards black holes This critical distance is known as the Swarzschild Radius: G is the gravitational constant, M is the mass of the object and c the velocity of light If a sufficiently massive object existed which satisfied the Swarzschild criterion the curvature of space would be so extreme that nothing would be able to resist the self-gravity The object would collapse indefinitely and all the matter would be compressed to a singularity - a single point at the centre However to satisfy the Swarzschild criterion an object the size of Earth would have to be compressed to the size of a pea - an increase in density of 1 000 000 000 000 000 000 000 000 000 times “On continued gravitational collapse”: “On continued gravitational collapse” Robert Oppenheimer (1904-1967) continued the work on the Swarzschild geometry, and published a paper with the above title in 1939 In the Oppenheimer-Snyder model a star, in its later stages of life, would burn out and collapse under its own gravity The gravitational collapse would continue until eventually the surface of the star reaches the critical radius Einstein didn’t like this result! “On continued gravitational collapse”: “On continued gravitational collapse” In the Oppenheimer-Snyder model Space curvature would be so severe that light rays emitted from the star’s surface would bend into the star’s interior, sealing off events from external observers Light rays would be infinitely red-shifted (ie light would have no energy) A one-way event horizon would form in which particles and radiation could enter the star but nothing could be emitted A spacetime singularity would form, not at the critical radius but at the centre of the star But, for an observer falling in with the collapsing star’s surface all physics would appear to be continuous A voyage into a black hole: A voyage into a black hole Black holes as tunnels: Black holes as tunnels However the two openings of the black hole must be at different points in spacetime A black hole is therefore a tunnel through both space and time…... Such tunnels are called wormholes ...which does not necessarily just connect points within the same universe He worked on the concept, predicted by General Relativity, that a black hole should have “two ends” and therefore “another side” Wheeler also considered the possibility of tiny wormholes much smaller than an atom Wormholes: Wormholes Time travel: Time travel A surprising consequence of the marriage of space and time within General Relativity is that time travel is permitted Developments in Relativity Theory in the 1980’s suggested that none of the laws of physics are violated by time travel Neighbouring Universes: Neighbouring Universes Other universes could be existing in hyperspace alongside our own Our universe itself may have been born out of the collapse of another universe 15 billion years ago, which tunneled through hyperspace to create our Big Bang………. Stephen Hawking and black holes: Stephen Hawking and black holes Like Einstein, Hawking had a “happy thought” in 1970 which changed our understanding of black holes. He realised, literally in flash of inspiration, that the surface area of a black hole could never decrease: The rays of light that form the event horizon, the boundary of the black hole, can never approach one another Consequently, the area of the event horizon (ie the surface of the black hole) cannot decrease. Even if two black holes combine the total surface area will always be equal to or greater than the sum of the two, never less. Black holes and entropy: Black holes and entropy Hawking’s law, “the area can never decrease”, is reminiscent of the second law of thermodynamics “the entropy of a system can only stay the same or increase, never decrease” Over a very short period in 1972 Hawking and coworkers used relativity to derive a set of equations for black hole mechanics which bore an uncanny resemblance to the laws of thermodynamics with: S (entropy) = k1 A (surface area of black hole) T (temperature) = k2 G (surface gravity of black hole) with k1 and k2 as constants Each black hole law turned out to be identical to a law of thermodynamics if the substitutions above were made. But if surface area and entropy are equivalent then, according to the laws of thermodynamics, black holes must radiate energy Summary: Summary By taking General Relativity to its logical conclusion first Swarzschild and then Opennheimer established that at very high mass density, for example following gravitational collapse, a black hole would be formed The black hole is an extreme curvature of four dimensional space time from which nothing, not even light can escape. A characteristic feature of the black hole is the event horizon, within which the black hole is effectively cut off from observers in the outside universe However, although time slows down in intense gravitational fields, the laws of physics still hold at the surface of a black hole - only at the very centre is there an intractable singularity Summary: Summary Wheeler, who coined the phrase “black hole”, considered the prediction of Relativity that a black hole should have “another side” - in a different point of spacetime and perhaps in another Universe The link between the two sides of the black hole is called a worm hole, and it represents a “short cut” through spacetime. The same laws of Physics that give rise to the phenomena of black holes do not prevent the concept of time travel - indeed the laws of physics are invariant with respect to the flow of time and worm holes represent a route through time The rather mysterious and esoteric properties of black holes were given a real physical significance in the 1970s when Hawking considered their thermodynamic properties and found that black holes - rather than been separate from the Universe - were a very real part of it. Black holes and entropy: Black holes and entropy Hawking’s law, “the area can never decrease”, is reminiscent of the second law of thermodynamics “the entropy of a system can only stay the same or increase, never decrease” Over a very short period in 1972 Hawking and coworkers used relativity to derive a set of equations for black hole mechanics which bore an uncanny resemblance to the laws of thermodynamics with: S (entropy) = k1 A (surface area of black hole) T (temperature) = k2 G (surface gravity of black hole) with k1 and k2 as constants Each black hole law turned out to be identical to a law of thermodynamics if the substitutions above were made. But if surface area and entropy are equivalent then, according to the laws of thermodynamics, black holes must radiate energy The very big and the very small: The very big and the very small Hawking sought a mechanism by which black holes could possibly radiate He found a solution in the most surprising of places - Quantum Mechanics All previous black hole calculations had ignored Quantum Mechanics - the physics of the very small - as it was assumed their effects would be negligible on a cosmological scale Hawking focussed upon the boundary between the black hole and the vacuum of interstellar space But remember Dirac - empty space is not empty - it is a seething mass of virtual particles (particles and antiparticles) which pop in and out of existence on a time scale allowed by the Uncertainty Principle. Hawking examined the effects of extreme gravity upon these virtual particles……. Black hole radiation: Black hole radiation Near the event horizon of a black hole there will be many virtual particles Each virtual particle is composed of a particle and an antiparticle Normally, the virtual particles live only for a short time (defined by the Heisenberg Uncertainty Principle) before the particle-antiparticle pairs recombine and annihilate Near an event horizon, however, one of the two particles of the pair can be attracted into the black hole (negative energy) reducing the mass of the black hole, while the other particle (positive energy) escapes into space Hawking had successfully combined General Relativity and Quantum Mechanics for the very first time Virtual particles near the event horizon: Virtual particles near the event horizon the event horizon the event horizon The new black holes: The new black holes In Hawking’s new picture black holes were neither mathematical abstractions nor bottomless pits - but real physical objects that emit radiation according to the theories of thermodynamics Also black holes are not absolutely permanent but will eventually evaporate into pure radiation Hawking’s marriage of Quantum Mechanics and General Relativity has been compared to the giant (quantum leap) of Max Planck His equation S=kA looks like Planck’s equation E=hf, although we are still as far away from understanding Hawking’s equation as Planck was from understanding quantum theory in 1900 However it is believed that Hawking’s equation will emerge as a central feature of the yet unborn theory that will ultimately unify gravitation, quantum mechanics and thermodynamics Towards unification: Towards unification The next step: The next step In the early 1980s Hawking began work on the physics of the early universe - work that is continuing today He realized that the Uncertainty Principle invalidates General Relativity at the moment of the Big Bang But because General Relativity is a Classical theory he began to search for a combined quantum mechanical and general relativistic description of the early universe Hawking developed the No Boundary Proposal - in which he suggests that space and time are closed up on themselves without boundaries or edges If this theory is correct there would be no singularities and the laws of science would hold everywhere - including at the beginning of the Universe Quantum cosmology: Quantum cosmology Hawking is attempting to apply quantum theory to the instant of the creation of the universe (ie at time t=0 when it was the size of a quantum particle). Hawking and coworkers are assuming that at the instant of creation the Universe is entirely within the quantum state, and they are therefore trying to determine the wave function of the whole Universe based upon the No Boundary Proposal The theory involves the concept of complex time in which time is divided into two components, one real and one imaginary (ie related to the square root of a negative number) The real component vanishes at t=0, but the imaginary component does not - enabling standard quantum mechanical procedures to be used at the singularity. Applying Schroedinger’s equation to the Universe: Applying Schroedinger’s equation to the Universe Just as Schroedinger replaced classical electron orbits with wave functions, so Hawking and his coworker Hartle assign a wave function to different cosmological models that indicate the probability that the Universe has one particular geometry or another Some Universes expand from a point to maximum size and then collapse to a point again Some Universes expand for ever Some Universes expand differently in different directions but all obey General Relativity By choosing only those Universes with No Boundaries Hawking and Hartle obtain results consistent with observations of our own Universe A closed and uniform Universe: A closed and uniform Universe Closed Universes provide the best solutions They are finite but have no edges in space or time They have a beginning and an end - but these result in boundaries only in the real component of time. The imaginary component is continuous. The singularities at the beginning and end of real time therefore disappear from the calculations Hawking and Hartle also demonstrate that uniform Universes are the most probable - matter is distributed uniformly throughout space - consistent with our observations. Inflation and Quantum Fluctuations: Inflation and Quantum Fluctuations Current models of the Universe suggest that our Universe expanded from an initial state smaller than a proton to a macroscopic size about 10 metres across in a fraction of a second When the Universe was infinitesimally small it was also uniform and this uniformity was preserved through the rapid inflation. When matter and radiation decoupled about 300,000 years later the Universe was still very uniform However the inflation that smoothed out the Universe also preserved the quantum fluctuations that occur even in empty space These quantum fluctuations were the seeds that established the small density variations which appear as ripples of mass and energy in spacetime ...and perhaps seeded the galaxies themselves Ripples in space time : Ripples in space time The ripples of mass and energy in spacetime predicted by Hawking should be imprinted upon this cosmic background It was suggested as early as 1965 by George Gamow the Universe may be filled with a cosmic background radiation of ancient photons released by the big bang Calculations predicted that the background radiation would be like black body radiation with a characteristic temperature of about 3 degrees absolute (ie in the microwave region of the spectrum) Penzias and Wilson discovered the radiation in 1965 and were awarded the Nobel Prize for providing experimental confirmation of the Big Bang COBE: COBE Between 1989 and 1992 the Cosmic Background Explorer Satellite, COBE, mapped the background radiation throughout the Universe The predicted temperature variations (+/-0.005%) resulting from the density fluctuations were observed The end ?: The end ? We now seem to have gone full circle: At the end of the 19th century, problems with thermodynamics, radiation and matter on a microscopic scale brought about the quantum revolution which overthrew classical physics Throughout the 20th century General Relativity, a classical theory, profoundly changed our view of the Universe on a cosmological scale but problems with thermodynamics, radiation and matter were still evident Once again it has been necessary to invoke Quantum Theory in attempts to resolve these problems However we are not as arrogant as the 19th century physicists - we have much more to do than add the sixth decimal place There is still an unimaginable amount of work to be done to unify relativity and quantum mechanics and ultimately obtain the Theory of Everything Strings and sealing wax: Strings and sealing wax The great physicists of today are continuing to try to put together a comprehensive theory to explain the fabric of the Universe and the physical laws by which it functions They continue to probe the inner and outer Universe Theories are now emerging in which the smallest entities are loops and strands of vibrating “string”. These strings are at the very heart of inner space Strings and sealing wax: Strings and sealing wax Strings and sealing wax: Strings and sealing wax The great physicists of today are continuing to try to put together a comprehensive theory to explain the fabric of the Universe and the physical laws by which it functions They continue to probe the inner and outer Universe Theories are now emerging in which the smallest entities are loops and strands of vibrating “string”. These strings are at the very heart of inner space 1020 of these strings, laid end to end, would stretch across the diameter of a single proton - their length is associated with the Planck length (10-35m) Everything may be made of strings, which are themselves made of 10 dimensional pieces of spacetime, rolled up so that only 4 dimensions are visible Recommended web site http://www.superstringtheory.com Strings and sealing wax: Strings and sealing wax Everything may be made of strings, which are themselves made of 10 dimensional pieces of spacetime, rolled up so that only 4 dimensions are visible Recommended web site http://www.superstringtheory.com Strings and sealing wax: Strings and sealing wax Everything may be made of strings, which are themselves made of 10 dimensional pieces of spacetime, rolled up so that only 4 dimensions are visible Recommended web site http://www.superstringtheory.com Strings and sealing wax: Strings and sealing wax Everything may be made of strings, which are themselves made of 10 dimensional pieces of spacetime, rolled up so that only 4 dimensions are visible Recommended web site http://www.superstringtheory.com The four forces: The four forces The four fundamental forces of nature are The Electromagnetic Force The Weak Nuclear Force The Strong Nuclear Force Gravity So far all attempts to unify the four forces within a single theory have failed However, a string theory which explains the first three using open strings automatically includes closed loops which explain gravity String theory may eventually explain everything, including the structure of spacetime - but it will take many years Watch this space…. Some revision (1): Some revision (1) Lecture 1 Classical Determinism and the role of Newton and Maxwell in defining the cornerstones of classical physics: light and electromagnetic radiation, energy and motion The distinct character, within classical physics, of waves and particles and the nature of electromagnetic radiation The beginning of the end for determinism with the introduction of probabilistic concepts to describe gases by Maxwell, and the devlopment of statistical theories of thermodynamics by Boltzmann (the Second Law and Entropy) The three crucial experiments for which Classical Physics had no explanation - Blackbody Radiation, the Photoelectric Effect and Bright Line Optical Spectra Some revision (2): Some revision (2) Lecture 2 Black Body Radiation and the failure of first Wien and then Rayleigh and Jeans to explain it. The Ultraviolet catastrophe Planck’s model of black body radiation (and his reliance on the concepts of Boltzmann)- introduction of the energy quantum -and its implications for the equipartition of energy The photoelectric effect and the problems it posed for Classical Physics. Einstein’s Nobel prize winning explanation in terms of quantised light (ie photons) Introduction of the problem of Bright Line Optical spectra. Balmer’s strictly numerical approach to the problem. Emission and absorption spectra Some revision (3): Some revision (3) Lecture 3 The atom - from Kelvin and Thomson to Rutherford. The significance of the alpha particle scattering experiments. The instability of the Rutherford atom. Niels Bohr and quantised angular momentum - and hence quantised orbitals - the consequences for the model of the atom and for bright line and absorption spectra. Beyond the Bohr model. Three additional quantum numbers including Pauli’s electron spin. An explanation of the periodic table. The Particle-Wave Duality of light Some revision (4): Some revision (4) Lecture 4 The properties of waves - Prince Louis de Broglie and particle wave duality of matter. The significance of the wavelength of a quantum particle and its implications for the model of the atom Heisenberg - and the Uncertainty Principle - the ultimate collapse of classical determinism The properties of waves - interference and superposition Schroedinger - and wave mechanics - Probability waves, the wave function and the atom. Quantum mechanical tunnelling etc Two quantum theories: Some revision (5): Some revision (5) Lecture 5 Complementarity and the Copenhagen Interpretation - The “collapse of the wavefunction” and Schroedinger’s Cat The first complete quantum theory - Dirac - antimatter and “empty space” The importance of quantum physics and the strength of quantum theory Some revision (6): Some revision (6) Lecture 6 Einstein - and the paradoxes associated with speed of light travel. The speed of light as the fastest possible speed of interaction The principle of relativity (after Galileo) and the relativity of simultaneity The speed of light and the aether Some revision (7): Some revision (7) Lecture 7 Distance and Time as non absolutes - the Lorentz transforms and Special Relativity (ie relativity of uniform motion) Time dilation and length contraction - the Twin Paradox - a photon’s view of the Universe Some revision (8): Some revision (8) Lecture 8 Einstein happiest thought and the nature of mass (gravitational and inertial) and the Principle of Equivalence General Relativity and the replacement of gravitational force curvature of space. Proof of the curvature of space and spacetime Special relativity and mass increase. The impossibility of accelerating an object to the speed of light E=mc2 -its origin and meaning Some revision (9): Some revision (9) Lecture 9 Hawking and black hole thermodynamics. Incorporation of quantum mechanics into cosmology. Black hole radiation. More on curved space and the implications for time The No Boundary Proposal and the wavefunction of the Universe. Strings and things The Swarzschild critical radius and Oppenheimer on gravitational collapse and black hole singularities. The event horizon . Wormholes.