The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

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Linear and nonlinear waves

In the current standard viewpoint small black holes are believed to emit black body radiation at the Hawking temperature, at least until they approach Planck size, after which their fate is open to conjecture A cogent argument against the existence of remnants is that, since no evident quantum number prevents it, black holes should radiate completely away to photons and other ordinary stable particles and vacuum, like any unstable quantum system Here we argue the contrary, that the generalized uncertainty principle may prevent their total evaporation in exactly the same way that the uncertainty principle prevents the hydrogen atom from total collapse: the collapse is prevented, not by symmetry, but by dynamics, as a minimum size and mass are approached

The Generalized Uncertainty Principle and Black Hole Remnants

Now more than ever, Gravitation and Spacetime, Second Edition, by Hans C. Ohanian and new coauthor Remo Ruffini, deserves John Wheeler's praise as "the best book on the market today of 500 pages or less on gravitation and general relativity." Gravitation and Spacetime has been thoroughly updated with the most exciting finds and hottest theoretical topics in general relativity and cosmology. Highlights of the revision include the rise and fall of the fifth force, principles and applications of gravitational lensing, COBE's spectacular confirmation of the blackbody spectrum of the cosmic thermal radiation, theories of dark matter and inflation, and the early universe as a testing ground for particle physicists' unification theories, and much, much more. The ideal choice for a graduate-level introduction to general relativity, Gravitation and Spacetime is also suitable for an advanced undergaduate course.

Gravitation and spacetime.

Heisenberg showed in the early days of quantum theory that the uncertainty principle follows as a direct consequence of the quantization of electromagnetic radiation in the form of photons. As we show here the gravitational interaction of the photon and the particle being observed modifies the uncertainty principle with an additional term. From the modified or gravitational uncertainty principle it follows that there is an absolute minimum uncertainty in the position of any particle, of order of the Planck length. A modified uncertainty relation of this form is a standard result of superstring theory, but the derivation given here is based on simpler and rather general considerations with either Newtonian gravitational theory or general relativity theory.

On Gravity and the Uncertainty Principle

Einstein offered the principle of general covariance as the fundamental physical principle of his general theory of relativity and as responsible for extending the principle of relativity to accelerated motion. This view was disputed almost immediately with the counter-claim that the principle was no relativity principle and was physically vacuous. The disagreement persists today. This article reviews the development of Einstein's thought on general covariance, its relation to the foundations of general relativity and the evolution of the continuing debate over his viewpoint.

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General covariance and the foundations of general relativity: eight decades of dispute

Gravitation and Spacetime

1 The Rise and Fall of Absolute Space.- 1.1 Definition of Relativity.- 1.2 Newton's Laws.- 1.3 The Galilean Transformation.- 1.4 The Set of All Inertial Frames.- 1.5 Newtonian Relativity.- 1.6 Newton's Absolute Space.- 1.7 Objections to Newton's Absolute Space.- 1.8 Maxwell's Ether.- 1.9 Where is Maxwell's Ether?.- 1.10 Lorentz's Ether Theory.- 1.11 The Relativity Principle.- 1.12 Arguments for the Relativity Principle.- 1.13 Maxwellian Relativity.- 1.14 Origins of General Relativity.- 1.15 Mach's Principle.- 1.16 Consequences of Mach's Principle.- 1.17 Cosmology.- 1.18 Inertial and Gravitational Mass.- 1.19 The Equivalence Principle.- 1.20 The Semistrong Equivalence Principle.- 1.21 Consequences of the Equivalence Principle.- 2 Einsteinian Kinematics.- 2.1 Basic Features of Special Relativity.- 2.2 On the Nature of Physical Laws.- 2.3 An Archetypal Relativistic Argument.- 2.4 The Relativity of Simultaneity.- 2.5 The Coordinate Lattice.- 2.6 The Lorentz Transformation.- 2.7 Properties of the Lorentz Transformation.- 2.8 Hyperbolic Forms of the Lorentz Transformation.- 2.9 Graphical Representation of the Lorentz Transformation.- 2.10 World-picture and World-map.- 2.11 Length Contraction.- 2.12 Length Contraction Paradoxes.- 2.13 Time Dilation.- 2.14 The Twin Paradox.- 2.15 Velocity Transformation.- 2.16 Proper Acceleration.- 2.17 Special Relativity without the Second Postulate.- 3 Einsteinian Optics.- 3.1 The Drag Effect.- 3.2 The Doppler Effect.- 3.3 Aberration and the Visual Appearance of Moving Objects.- 4 Spacetime and Four-Vectors.- 4.1 Spacetime.- 4.2 Three-Vectors.- 4.3 Four-Vectors.- 4.4 Four-Tensors.- 4.5 The Three-Dimensional Minkowski Diagram.- 4.6 Wave Motion.- 5 Relativistic Particle Mechanics.- 5.1 Domain of Sufficient Validity of Newton's Laws.- 5.2 Why Gravity Does not Fit Naturally into Special Relativity.- 5.3 Relativistic Inertial Mass.- 5.4 Four-Vector Formulation of Relativistic Mechanics.- 5.5 A Note on Galilean Four-Vectors.- 5.6 Equivalence of Mass and Energy.- 5.7 The Center of Momentum Frame.- 5.8 Relativistic Billiards.- 5.9 Threshold Energies.- 5.10 Three-Force and Four-Force.- 5.11 De Broglie Waves.- 5.12 Photons. The Compton Effect.- 5.13 The Energy Tensor of Dust.- 6 Relativity and Electrodynamics.- 6.1 Transformation of the Field Vectors.- 6.2 Magnetic Deflection of Charged Particles.- 6.3 The Field of a Uniformly Moving Charge.- 6.4 The Field of an Infinite Straight Current.- 7 Basic Ideas of General Relativity.- 7.1 Curved Surfaces.- 7.2 Curved Spaces of Higher Dimensions.- 7.3 Riemannian Spaces.- 7.4 A Plan for General Relativity.- 7.5 The Gravitational Doppler Effect.- 7.6 Metric of Static Fields.- 7.7 Geodesics in Static Fields.- 8 Formal Development of General Relativity.- 8.1 Tensors in General Relativity.- 8.2 The Vacuum Field Equations of General Relativity.- 8.3 The Schwarzschild Solution.- 8.4 Rays and Orbits in Schwarzschild Space.- 8.5 The Schwarzschild Horizon, Gravitational Collapse, and Black Holes.- 8.6 Kruskal Space and the Uniform Acceleration Field.- 8.7 A General-Relativistic "Proof" of E = mc2.- 8.8 A Plane-Fronted Gravity Wave.- 8.9 The Laws of Physics in Curved Spacetime.- 8.10 The Field Equations in the Presence of Matter.- 8.11 From Modified Schwarzschild to de Sitter Space.- 8.12 The Linear Approximation to GR.- 9 Cosmology.- 9.1 The Basic Facts.- 9.2 Apparent Difficulties of Prerelativistic Cosmology.- 9.3 Cosmological Relativity: The Cosmological Principle.- 9.4 Milne's Model.- 9.5 The Robertson-Walker Metric.- 9.6 Rubber Models, Red Shifts, and Horizons.- 9.7 Comparison with Observation.- 9.8 Cosmic Dynamics According to Pseudo-Newtonian Theory.- 9.9 Cosmic Dynamics According to General Relativity.- 9.10 The Friedmann Models.- 9.11 Once Again, Comparison with Observation.- 9.12 Mach's Principle Reexamined.- Appendices.- Appendix I: Curvature Tensor Components for the Diagonal Metric.- Appendix II: How to "Invent" Maxwell's Theory.- Exercises.

Essential Relativity: Special, General, and Cosmological

Instead of the scalar “dilaton” field that is usually adopted to construct conformally invariant Lagrangians for gravitation, we here propose a hybrid construction, involving both a complex dilaton scalar and a Weyl gauge-vector, in accord with Weyl’s original concept of a non-Riemannian conformal geometry with a transport law for length and time intervals, for which this gauge vector is required. Such a hybrid construction permits us to avoid the wrong sign of the dilaton kinetic term (the ghost problem) that afflicts the usual construction. The introduction of a Weyl gauge-vector and its interaction with the dilaton also has the collateral benefit of providing an explicit mechanism for spontaneous breaking of the conformal symmetry, whereby the dilaton and the Weyl gauge-vector acquire masses somewhat smaller than $$\textit{m}_\textit{P}$$
 by the Coleman–Weinberg mechanism. Conformal symmetry breaking is assumed to precede inflation, which occurs later by a separate GUT or electroweak symmetry breaking, as in inflationary models based on the Higgs boson.

Hans C. Ohanian

Papers

Gravitation and spacetime.

Gravitation and Spacetime

Gravitation and spacetime.

Essential Relativity: Special, General, and Cosmological

Weyl gauge-vector and complex dilaton scalar for conformal symmetry and its breaking