Dynamic incremental scale transition with application to physics and cosmology

This is an exceptionally fine book which can be used for different goals and on different track levels. As in the very comprehensive and well known book, Gravitation by Misner, Thorne and Wheeler, the current book by Ciufolini and Wheeler has two tracks. The first track is on a lower level and is intended as a more descriptive introduction to general relativity. Track 2 covers topics such as gravitation, inertia, and tests of relativity on a deeper level. The second track is set off in square brackets for easy identification. This dualistic approach makes at least parts of the book accessible to readers with very little background in differential geometry. The authors have written a book which can be used as an introduction to Einstein's general relativity or geometrodynamics. Alternatively, it may be used as a guide to past, current, and proposed experimental tests for geometrodynamics. In addition, it may be used as a monograph on the origin and nature of inertia. The reader wishing a fairly comprehensive introduction to geometrodynamics will find this book exactly what is needed. It should be mentioned that the authors have included a mathematical appendix which will be quite helpful to readers that already have some knowledge of differential geometry, but do not have a strong background in this field. The authors give good descriptions of various equivalence principles with the very strong equivalence principle as fundamental to geometrodynamics. The reader will find all of the basic building blocks of relativity such as the field equation, a number of exact solutions, conservation laws, and the geodesic deviation equation (i.e. the Jacobi equation) as well as the PPN formalism. They consider the implications of assuming that gravity is a metric theory. The authors also stress that the metric tensor in geometrodynamics is determined by boundary conditions as well as by the distribution of mass--energy and mass--energy currents in the universe which enter the theory through the field equation. For the reader with a background in general relativity and wishing good discussions of experimental results as well as of proposed experiments, this is a very good source with a large number of references. The authors give descriptions of a number of experiments and carefully explain why each is important. In particular, they discuss the importance of cosmic background radiation results, geodetic and Lense--Thirring effect experiments, as well as gravitational wave experiments. They include discussions of the COBE (Cosmic Background Explorer) satellite results on the background microwave radiation. They discuss various current and projected experiments one might use to detect gravitational waves. Thus, they discuss the bar experiments, as well as LIGO (Laser-Interferometer Gravitational-wave Observatory), MIGO (Michelson Millimetre Wave Interferometer Gravitational-Wave Observatory), LAGOS (Laser Gravitational-Wave Observatory in Space), and LINE (Laser Interferometer Near Earth). They also discuss LAGEOS (Laser Geodynamics Satellite) and the possibility of important measurements of the Lense--Thirring effect using a second such satellite. A major feature of this book is the emphasis on inertia. In gravitation, inertia plays a key role and, as the name of the book suggests, the authors have made this a central theme. Among other things, the origin of inertia is still somewhat cloudy. This work clarifies the status of our present knowledge of inertia and deals with the possible origins of inertia. The authors stress that there is a fundamental difference in understanding the origin of inertia in universes which are spatially `open' (i.e. have non-compact Cauchy surfaces) as opposed to universes which are spatially `closed' (i.e. have compact Cauchy surfaces). This is an excellent book that will be of real value to anyone seriously interested in general relativity. It is clearly written with a very well connected development of many topics which are not covered in the other books on the subject. The book is self-contained and one does not need any previous background in relativity.

The theoretical framework of cosmology is mainly defined by gravity, of which general relativity is the current model. Recent tests of general relativity within the Lambda Cold Dark Matter (ΛCDM) model have found a concordance between predictions and the observations of the growth rate and clustering of the cosmic web. General relativity has not hitherto been tested on cosmological scales independently of the assumptions of the ΛCDM model. Here we report an observation of the gravitational redshift of light coming from galaxies in clusters at the 99 per cent confidence level, based on archival data. Our measurement agrees with the predictions of general relativity and its modification created to explain cosmic acceleration without the need for dark energy (the f(R) theory), but is inconsistent with alternative models designed to avoid the presence of dark matter.

SPACETIME STRUCTURE Einstein's general theory of relativity gives a mathematical description of space, time and gravitation which is extraordinarily concise, subtle and accurate. It has, however, the appearance of being concise only to those who are already familiar with the mathematical formalism of Riemannian geometry. To someone who is not familiar with that body of mathematical theory – a theory which, though remarkably elegant, is undoubtedly sophisticated, and usually becomes extremely complicated in detailed application – Einstein's General Relativity can seem inaccessible and bewildering in its elaborate structure. But the complication and sophistication lie only in the details of the formalism. Once that mathematics has been mastered, the precise formulation of Einstein's physical theory is, indeed, extremely compact and natural. Although a little of this formalism will be needed here, it will be given in a compact form only that should be reasonably accessible. The mathematical theory of Riemannian geometry applies to smooth spaces of any (positive whole) number N of dimensions. Such a space M is referred to as an N - manifold , and to be a Riemannian manifold it must be assigned a metric , frequently denoted by g , which assigns a notion of ‘length’ to any smooth curve in M connecting any two points a, b . (See Figure 3.1.) For a strictly Riemannian manifold, this length function is what is called positive definite which means that the length of any such curve is a positive number, except in the degenerate situation when a = b and the curve shrinks to a point, for which the length would be zero.

Analysis of cosmic data is the only way to determine whether General Relativity is the law of gravity also on the largest scales in our Universe. The current standard model of cosmology, ΛCDM, is based on General Relativity, and fits all currently available data flawlessly. However, theoretical dissatisfaction with ΛCDM exists: cosmological data probe gravitational interactions, and ΛCDM fits the data only because it introduces two components of startling gravitional behaviour, the cosmological constant, Λ, and cold dark matter (CDM). The cosmological constant has a suspiciously small value when regarded from the perspective of quantum field theories, and cold dark matter has so far not been detected in any experiment of particle physics. This thesis examines the cosmological standard model from the vantage point of statistics. A non-Gaussian likelihood approximation is presented and the need of an unbiased mechanism for dealing with estimated covariance matrices is addressed. Concerning neutrinos, a previously existent parameterization bias in the analysis of the cosmic microwave background is resolved. Using weak lensing and type Ia supernova data of the next generation, it is estimated how much can be learned about dark energy from these future data sets.

This thesis is dedicated to certain nonlocal aspects of conformal quantum field theories (CFTs). Two main directions are the study of CFTs on a particular globally-nontrivial spacetime corresponding to finite temperature, and the study of particular nonlocal CFT observables localized on light-rays. Specifically, we introduce bootstrap techniques for determining finite-temperature data of CFTs, and make novel predictions for the 2+1-dimensional Ising model. We propose the “stringy equivalence principle”, stating that coincident gravitational shocks commute, as a generalization of the strong equivalence principle of Einstein’s General Relativity that must hold in all consistent theories of gravity. We prove it in Anti-de Sitter (AdS) spacetimes by studying light-ray operators in the holographically dual CFT. We also derive an operator product expansion (OPE) for light-ray operators in CFT, by which two light-ray operators on the same light-sheet can be expanded as a sum of single light-ray operators. Light-ray operators model detectors — such as calorimeters. We use the light-ray OPE to compute energy event shape observables suitable for conformal collider physics. An additional part of this thesis determines the low-energy vacua of two-dimensional maximal super-Yang-Mills theory, which describes the dynamics of stacks of D-strings in Type IIB string theory. By computing an invariant of the renormalization group (RG) flow from high to low energy — a modified thermal partition function named the refined elliptic genus — we prove the existence of multiple vacua, and identify the superconformal field theories capturing their dynamics. The vacua correspond to bound states of (p,q)-strings in Type IIB string theory. Our computation serves as a check of the strong-weak S-duality of the Type IIB string.

Four-dimensional Einstein's General Relativity is shown to arise from a gauge theory for the conformal group, SO(4,2). The theory is constructed from a topological dimensional reduction of the six-dimensional Euler density integrated over a manifold with a four-dimensional topological defect. The resulting action is a four-dimensional theory defined by a gauged Wess-Zumino-Witten term. An ansatz is found which reduces the full set of field equations to those of Einstein's General Relativity. When the same ansatz is replaced in the action, the gauged WZW term reduces to the Einstein-Hilbert action. Furthermore, the unique coupling constant in the action can be shown to take integer values if the fields are allowed to be analytically continued to complex values.

Three major predictions of Einstein's General Relativity — black holes, wormholes, and time machines — are reviewed for their development and current status.

I present briefly some critical remarks on recent applications of gravity theories in cosmology. Einstein’s general relativity (GR) is exceptional among physical theories: it is formally merely a point in the huge space of all existing and conceivable theories of gravitation. Most interesting of them are metric nonlinear gravity (NLG) theories which differ from GR only in equations of motion: gravity is described by one unifying tensor field which in the initial formulation of the theories is interpreted as a spacetime metric with a Lagrangian L = f(gαβ ,R αβμν) ,w heref is any smooth scalar function. The simplest of them, restricted NLG theories based on L = f(R), have attracted most attention. In cosmology they are used to provide a generalized Friedmann equation for the cosmic scale function in spatially flat Robertson–Walker (R–W) spacetime to account for the accelerated expansion of the universe without resorting to dark energy. Why to employ to this aim NLG theories instead of assuming that the cosmic acceleration is driven in GR by a kind of self-interacting scalar, vector or tensor field? The standard response is that one must then introduce ad hoc some self-interaction of the field, yet applying an NLG theory one may find approximate solutions which well fit the cosmic evolution by making appropriate approximations in the underlying Lagrangian. Let L =1 /R + R + R 2 , then in the early universe with strong curvature the R 2 term is dominant giving rise to some kind of inflation [1]. In the intermediate curvature period (radiation and galactic eras) the R term dominates and the standard cosmological model is valid. At present and in future the decreasing curvature makes the 1/R term dominant what results in an accelerated expansion. This kind of argument is explicitly or implicitly present in most works on NLG theories. This line of argument, though suggestive, is actually misleading. One should investigate any NLG theory from the viewpoint of field theory and search for its predictions in the whole realm of gravitational physics. A success of a theory only in flat R–W spacetime is by far not enough.

The standard model of cosmology assumes that the Universe can be described to hover around a homogeneous-isotropic solution of Einstein's general theory of relativity. This description needs (sometimes hidden) hypotheses that restrict the generality, and relaxing these restrictions is the headline of a new physical approach to cosmology that refurnishes the cosmological framework. Considering a homogeneous geometry as a template geometry for the in reality highly inhomogeneous Universe must be considered a strong idealization. Unveiling the limitations of the standard model opens the door to rich consequences of general relativity, giving rise to effective (i.e. spatially averaged) cosmological models that may even explain the longstanding problems of dark energy and dark matter. We explore in this talk the influence of structure formation on average properties of the Universe by discussing: (i) general thoughts on why considering average properties, on the key-issue of non-conserved curvature, and on the global gravitational instability of the standard model of cosmology; (ii) the general set of cosmological equations arising from averaging the scalar parts of Einstein's equations, the generic property of structure formation interacting with the average properties of the Universe in a scale-dependent way, and the description of cosmological backreaction in terms of an effective scalar field.

Cosmology and particle astrophysics have enjoyed tremendous progress throughout the past two decades. Healthy interplays between observational/experimental data from numerous projects and theoretical concepts have greatly enhanced our fundamental understanding of the universe. Such advancement in turn triggers new questions to be further addressed. It is fare to say that we are currently living in an era of renaissance in cosmology and particle astrophysics. Under this light, it is clear that a brief overview talk cannot possibly cover all the major advancements nor can it serve justice to all the important works in this field. The situation is further worsened due to the author’s limited expertise and partial knowledge in these burgeoning fields. At the risk of being partial and incomplete, we showcase the progress by a few selected topics with the hope that they would provide a good sampler to the much fuller tapestry of the advancement of these very exciting fields. Modern cosmology began nearly 100 years ago with Einstein’s finalizing his general theory of relativity (GR) and applying it to cosmology on the theory side and Hubble’s discovery of cosmic expansion on the experimental side. Within a century, especially in the past 20 years, a standard model of cosmology has become mature, based on which we are now able to search for the remaining missing pieces. Yet certain aspects of this overall successful picture of the universe are puzzling enough that they may require new concepts that would go beyond this standard model. Our overview focuses on four topics: early universe, dark energy (DE), dark (DM) , and highenergy cosmic neutrinos. We will highlight the progress of each in the following sections. In the areas of early universe and the dark side of the universe, the recently released Planck data has provided refinement on key parameters in cosmology and tighter constraints on theoretical models. The nature of DE, however, remains illusive. We will review a few theoretical attempts of addressing this issue.

Chapter 5, entitled Cosmology and General Relativity: Mathematical Description of the Universe, provides a full-fledged introduction to Relativistic Cosmology. The chapter begins with a long mathematical interlude on the geometry of coset manifolds. These notions are necessary for the mathematical formulation of the Cosmological Principle, stating homogeneity and isotropy, but have a much wider spectrum of applications. In particular they will be very important in the subsequent chapters about Supergravity. Having prepared the stage with this mathematical preliminaries, the next sections deal with homogeneous but not isotropic cosmologies. Bianchi classification of three dimensional Lie groups is recalled, Bianchi metrics are defined and, within Bianchi type I, the Kasner metrics are discussed with some glimpses about the cosmic billiards, realized in Supergravity. Next, as a pedagogical example of a homogeneous but not isotropic cosmology, an exact solution, with and without matter, of Bianchi type II space-time is treated in full detail. After this, we proceed to the Standard Cosmological Model, including both homogeneity and isotropy. Freedman equations, all their implications and known solutions are discussed in detail and a special attention is given to the embedding of the three type of standard cosmologies (open, flat and closed) into de Sitter space. The concept of particle and event horizons is next discussed together with the derivation of exact formulae for read-shift distances. The conceptual problems (horizon and flatness) of the Standard Cosmological Model are next discussed as an introduction to the new inflationary paradigm. The basic inflationary model based on one scalar field and the slow rolling regime are addressed in the following sections with fully detailed calculations. Perturbations, the spectrum of fluctuations up to the evaluation of the spectral index and the principles of comparison with the CMB data form the last part of this very long chapter.

The origin of inertia has intrigued scientists and philosophers for centuries. Inertial frames of reference permeate our daily life. The inertial and centrifugal forces, such as the pull and push that we feel when our vehicle accelerates, brakes and turns, arise because of changes in velocity relative to uniformly moving inertial frames. A classical interpretation ascribed these forces to acceleration relative to some absolute frame independent of the cosmological matter, whereas an opposite view related them to acceleration relative to all the masses and 'fixed stars' in the Universe. An echo and partial realization of the latter idea can be found in Einstein's general theory of relativity, which predicts that a spinning mass will 'drag' inertial frames along with it. Here I review the recent measurements of frame dragging using satellites orbiting Earth.

Lovelock theory is a natural extension of the Einstein theory of general relativity to higher dimensions in which the first and second orders correspond, respectively, to general relativity and Einstein-Gauss-Bonnet gravity. We present exact black hole solutions of $D\geq 4$-dimensional spacetime for first-, second-, and third-order Lovelock gravities in a string cloud background. Further, we compute the mass, temperature, and entropy of black hole solutions for the higher-dimensional general relativity and Einstein-Gauss-Bonnet theories and also perform thermodynamic stability of black holes. It turns out that the presence of the Gauss-Bonnet term and/or background string cloud completely changes the black hole thermodynamics. Interestingly, the entropy of a black hole is unaffected due to a background string cloud. We rediscover several known spherically symmetric black hole solutions in the appropriate limits.

Black holes are one of the most remarkable predictions of Einstein's general relativity. In recent years, ideas in brane-world cosmology, string theory and gauge/gravity duality have motivated studies of black holes in more than four dimensions, with surprising results. In higher dimensions, black holes exist with exotic shapes and unusual dynamics. Edited by leading expert Gary Horowitz, this exciting book is the first devoted to this new field. The major discoveries are explained by the people who made them: Rob Myers describes the Myers–Perry solutions that represent rotating black holes in higher dimensions; Ruth Gregory describes the Gregory–Laflamme instability of black strings; and Juan Maldacena introduces gauge/gravity duality, the remarkable correspondence that relates a gravitational theory to nongravitational physics. Accessible to anyone with a standard course in general relativity, this is an important resource for graduate students and researchers in general relativity, string theory and high energy physics.

Einstein general relativity (GR) can be formally represented by a Lagrangian in a flat Minkowski space-time where light speed is not a universal constant and vanishes at the horizon of black holes. These appear as virtual singularities where no matter or energy can enter. We revisit the fundamental concepts of GR. Due to the principle of mass-energy equivalence, certain physical quantities, such as rest mass-energy, time, and light speed, must be rescaled according to the gravitational potential. We obtain a Lagrangian for a static gravitational potential (called germinal) different from the Schwarzschild Lagrangian but with the same black hole singularity. We develop a method that transforms a germinal Lagrangian into a fully relativistic Lagrangian. Then, the electrostatic Lagrangian becomes the well-known electromagnetic Lagrangian. The same method shows that the obtained germinal Lagrangian of gravitation generates a fully relativistic Lagrangian where gravitation is represented by a relativistic 4×4 tensor potential related to the Einstein stress-energy tensor. This theory is not equivalent to GR. In the weak gravitation limit, the Lagrangian generated by a stationary source only depends on a four-potential formally identical to that of the Heaviside theory of gravitation but different. As an application, we use this result in a simple model explaining the observed anomalies in the galaxy's stellar velocities without requiring hypothetical dark matter.