An Introduction to General Relativity and Cosmology

E=mc² is known as the most famous but least understood equation in physics. This two-volume textbook illuminates this equation and much more through clear and detailed explanations, new demonstrations, a more physical approach, and a deep analysis of the concepts and postulates of Relativity. Volume II contains, notably: In Special Relativity: complementary explanations, alternative demonstrations relying on more advanced means and revealing other aspects. Further topics: accelerated objects and the Relativistic force, nuclear reactions, the use of hyperbolic trigonometry, the Lagrangian approach, the Relativistic Maxwell’s equations. In General Relativity: tensors, the affine connection, the covariant derivative, the geodesic equation, the Schwarzschild solution with two of its great consequences: black holes and the bending of light; further axiomatic considerations on time, space, matter, energy and light speed. In Cosmology: the FLRW Metric, the Friedman equation, the cosmological constant, the four ideal cosmological Models. These subjects are presented in a concrete and incremental manner, and illustrated by many case studies. The emphasis is placed on the theoretical aspects, with rigorous demonstrations based on a minimum set of postulates. The mathematical tools dedicated to Relativity are carefully explained for those without an advanced mathematical background. Both volumes place an emphasis on the physical aspects of Relativity to aid the reader’s understanding and contain numerous questions and problems (147 in total). Solutions are given in a highly detailed manner to provide the maximum benefit to students. This textbook fills a gap in the literature by drawing out the physical aspects and consequences of Relativity, which are otherwise often second place to the mathematical aspects. Its concrete focus on physics allows students to gain a full understanding of the underlying concepts and cornerstones of Relativity.

The Symmetries of Quantum and Classical Information. The Ressurected “Ether" of Quantum Information

(1969). Conjugate Diameters and the Special Theory of Relativity. Mathematics Magazine: Vol. 42, No. 5, pp. 227-232.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

Two radically different views about time are possible. According to the first, the universe is three dimensional. It has a past and a future, but that does not mean it is spread out in time as it is spread out in the three dimensions of space. This view requires that there is an unambiguous, absolute, cosmic-wide “now” at each instant. According to the second view about time, the universe is four dimensional. It is spread out in both space and time—in space-time in short. Special and general relativity rule out the first view. There is, according to relativity theory, no such thing as an unambiguous, absolute cosmic-wide “now” at each instant. However, we have every reason to hold that both special and general relativity are false. Not only does the historical record tell us that physics advances from one false theory to another. Furthermore, elsewhere I have shown that we must interpret physics as having established physicalism—in so far as physics can ever establish anything theoretical. Physicalism, here, is to be interpreted as the thesis that the universe is such that some unified “theory of everything” is true. Granted physicalism, it follows immediately that any physical theory that is about a restricted range of phenomena only, cannot be true, whatever its empirical success may be. It follows that both special and general relativity are false. This does not mean of course that the implication of these two theories that there is no unambiguous cosmic-wide “now” at each instant is false. It still may be the case that the first view of time, indicated at the outset, is false. Are there grounds for holding that an unambiguous cosmic-wide “now” does exist, despite special and general relativity, both of which imply that it does not exist? There are such grounds. Elsewhere I have argued that, in order to solve the quantum wave/particle problem and make sense of the quantum domain we need to interpret quantum theory as a fundamentally probabilistic theory, a theory which specifies how quantum entities—electrons, photons, atoms—interact with one another probabilistically. It is conceivable that this is correct, and the ultimate laws of the universe are probabilistic in character. If so, probabilistic transitions could define unambiguous, absolute cosmic-wide “nows” at each instant. It is entirely unsurprising that special and general relativity have nothing to say about the matter. Both theories are pre-quantum mechanical, classical theories, and general relativity in particular is deterministic. The universe may indeed be three dimensional, with a past and a future, but not spread out in four dimensional space-time, despite the fact that relativity theories appear to rule this out. These considerations, finally, have implications for views about the arrow of time and free will.

The mainstream approach of gravitational field is the development of Geometric theories of gravitation and the application of the Dynamics of General Relativity (GR). Besides, the Generalized Special Relativity (GSR) contains the fundamental parameter (ξI) of Theories of Physics (TPs). Thus, it expresses at the same time Newtonian Physics (NPs) for ξI→0 and Einstein Relativity Theory (ERT) for ξI=1. Moreover, the Equivalence Principle (EP) in the context of GSR, has two possible interpretations: mG = m (1), or mG=γ(ξI,β)m (2), where β=υ/c and mG, m, γ are the gravitational mass, inertial rest mass and Lorentz γ-factor, respectively. In this paper we initially present a new central scalar potential V=V(k,r), where k=k(ξI) and r is the distance from the center of gravity. We demand that ‘this new GSR gravitational field in accordance with EP (1), gives the same precession of Mercury’s orbit as Schwarzschild Metric (SM) does’ and we obtain k=6-ξI2. This emerges Einsteinian SR-horizon at r=5rS, while NPs extends the horizon at six Schwarzschild radius (6rS).We can also explain the Gravitational Red Shift (GRS), if only the proposed GSR Gravitational field strength g=g(k,r) is combined with EP (2). We modify the aforementioned central scalar potential as V=V(h,k,r), where h=h(r). The combination of the above with MOND interpolating functions, or distributions of Dark Matter (DM) in galaxies, provides six different functions h=h(r). Thus, we obtain a new GSR central Gravitational field strength g=g(h,k,r), which not only explains the Precession of Mercury’s Perihelion, but also the Rotation Curves in Galaxies, eliminating Dark Matter.

Chapter 12 - Curvature, Strong Gravity, and Gravitational Waves

The conference brought together people working in mathematical general relativity, a field which lies at the interface of analysis, differential geometry and physics. There was a mix of established workers in the subject with participants at the start of their careers and researchers from neighbouring fields. The Einstein equations are at the heart of general relativity theory. They form a system of evolution equations and their solutions are conveniently parametrized by initial data. These initial data are required to satisfy constraint equations and these were the topic of several lectures at the conference. The talks of Corvino and Pollack were concerned with recent progress in methods for constructing solutions of the constraints. An important concept for solutions of the constraints is that of mass, including quasilocal mass. This was a central theme of the talks of Degeratu, Huisken and Shi. Huisken introduced a striking new approach to the definition of mass based on isoperimetric inequalities. Dain presented a variational characterization of the extreme Reissner–Nordström solution, thus relating the study of the constraint equations to the theory of black holes. The talk of Wohlfarth was rather outside the main area of the conference. He described a new concept of geometry motivated by string theory which may come to enrich the circle of ideas within mathematical relativity. Many of the talks at the conference were on evolution equations related to general relativity. Dafermos and Finster presented results related to the dynamical stability of black holes. Dafermos described new results on the rate of decay of solutions of the wave equation on the Schwarzschild spacetime while Finster explained applications of methods of functional analysis to the wave equation on the Kerr spacetime. Struwe's talk concerned uniqueness for supercritical nonlinear wave equations which from the point of view of the Einstein equations are an important example to compare with. The Maxwell equations are another important comparison system and Bauer showed how they can be used to understand more about radiation formulae. Velázquez gave an introduction to singularity formation in the Keller–Segel model, a parabolic system coming from mathematical biology. This is a possible source of insight for obtaining a rigorous understanding of critical collapse in general relativity. The talks of Andréasson, Choptuik and Lindblom dealt with various aspects of the application of numerical techniques to the study of the Einstein evolution equations. Choptuik showed impressive new simulations of coalescing black holes due to Pretorius which could hardly have been imagined just a year ago. Cosmology is at present a very active area of research in general relativity. This is in part due to the challenge of understanding the observed accelerated expansion of our universe. Mathematics is beginning to make its mark in this subject and this was reflected by talks of Heinzle, Rendall and Tod. Mathematical relativity is a meeting point for many ideas and the abstracts which follow give some idea of the variety of the subject. In fact the spectrum of topics discussed by the participants at the conference was much wider than those for which talks could be scheduled. By limiting the number of presentations it was possible to leave plenty of time for people to exchange insights. The lively interactions observed make us hopeful that this conference has given a boost to the development of the subject in the next few years.

The treatment of the coordinate systems is briefly reviewed in the Newtonian mechanics, in the special theory of relativity, and in the general relativistic theory, respectively. Some reference frames and coordinate systems proposed within the general relativistic framework are introduced. With use of the ideas on which these coordinate systems are based, the proper reference frame comoving with a system of mass-points is defined as a general relativistic extension of the relative coordinate system in the Newtonian mechanics. The coordinate transformation connecting this and the background coordinate systems is presented explicitly in the post-Newtonian formalism. The conversion formulas of some physical quantities caused by this coordirate transformation are discussed. The concept of the rotating coordinate system is reexamined within the relativistic framework. A modification of the introduced proper reference frame is proposed as the basic coordinate system in the astrometry. The relation between the solar system barycentric coordinate system and the terrestrial coordinate system is given explicitly.

Setna rocznica sformułowania przez Alberta Einsteina ogólnej teorii względności jest doskonałą okazją do ponownego postawienia pytań o wpływ wspomnianej teorii na filozofię. Interesującą perspektywę oferują badania z zakresu historii nauki i historii filozofii, które pozwalają dokładniej przyjrzeć się procesowi recepcji wspomnianej teorii. Proces ten jest waż- nym źródłem wiedzy o kształtowaniu się relacji nauk przyrodniczych i filozofii, gdyż poprzez analizę wskazywanych trudności i kontrowersji pozwala zidentyfikować wiele ukrytych założeń o charakterze filozoficznym, które trudno dostrzec na innych etapach rozwoju nauki. Dzięki temu analiza procesu recepcji nowych teorii naukowych staje się wyróżnionym przedmiotem analiz relacji matematyki, nauk przyrodniczych i filozofii. Z tych względów warto przyjrzeć się interesują- cym dyskusjom i polemikom, które miały miejsce w obrębie nauki polskiej.Szczególnie interesujące polemiki i dyskusje wokół teorii względności rozegrały się w Krakowie i we Lwowie przed rokiem 1925. Wspomniane ośrodki zostały wybrane również z tego względu, że w początkach XX wieku, wykorzystując możliwości polityczno-kulturalne powstałe w związku z istnieniem stosunkowo szerokiego zakresu autonomii Galicji, stały się one najważniejszymi centrami rozwoju nauki polskiej. Sytuacja ta zadecydowała również o połączeniu wymienionych ośrodków stosunkowo silną siecią powiązań, widocznych szczególnie na gruncie fizyki. W obu miastach powstała również specyficzna (choć z zachowaniem pewnych lokalnych odrębności) odmiana refleksji nad nauką, którą dogodnie jest określić przy pomocy pojęcia „filozofii w nauce” stworzonego przez M. Hellera.Opracowanie rozpoczyna się od krótkiego przybliżenia historii recepcji szczególnej teorii względności w Krakowie. W procesie tym najistotniejszą rolę odegrał August Witkowski, który już w 1905 roku docenił znaczenie pierwszych prac Einsteina. Kolejne punkty zwrotne w procesie recepcji teorii Einsteina wyznaczyły dwa kolejne Zjazdy Lekarzy i Przyrodników Polskich, które odbyły się kolejno we Lwowie w 1907 r. i w Krakowie w 1911 r. Pierwszy z nich zadecydował o popularyzacji idei relatywistycznych pośród przyrodników polskich, a kolejny – o popularyzacji ich wśród filozofów. Dla zrozumienia lokalnej specyfiki filozoficznych rozważań prowadzonych w ośrodku lwowskim odwołamy się do rozwoju recepcji wokół podstaw mechaniki. Zagadnienie to szczególnie mocno interesowało lwowskich uczonych z tej racji, że Lwów był do 1915 roku jedynym polskim ośrodkiem akademickim rozwijającym nauki techniczne i koncentrował w naturalny sposób większość polskich uczonych zajmujących się nauką mechaniki. To właśnie środowiska związane z lwowską Szkołą Politechniczną odegrały ważną rolę w późniejszych dyskusjach wokół szczególnej i ogólnej teorii względności.W dalszej kolejności szkicowo zaprezentowana zostanie lwowska polemika wokół teorii względności i dla kontrastu ukazana zostanie o wiele krótsza polemika rozgrywająca się w Krakowie. Szczególna uwaga zostanie poświęcona dorobkowi Zygmunta Zawirskiego, który w swej działalności podjął problematykę poruszoną w obu ośrodkach i przedstawił najbardziej interesujące rozważania ukazujące nowoczesną wizję filozofii uprawianej w ścisłym kontakcie z naukami przyrodniczymi.Prezentowany styl filozofii w nauce rozwijany w Krakowie i Lwowie został również skrótowo skonfrontowany z konkurencyjnymi filozoficznymi reakcjami na teorie Einsteina: z refleksją rozwijaną w obrębie Szkoły Lwowsko-Warszawskiej (Kazimierz Ajdukiewicz) i z wybraną refleksją neoscholastyczną (Feliks Hortyński SJ, Ludwik Wrzoł SJ, Jan Stepa). Na tym tle lepiej można dostrzec specyfikę filozofii analizowanej w tym studium przypadku – najważniejsze uwagi na ten temat zostały zawarte w zakończeniu niniejszej pracy.Niniejszy artykuł ukazuje fakt, że szczególna i ogólna teoria względności Einsteina znacząco wpłynęły na rozwój refleksji filozoficznej w Polsce. Warto również zaznaczyć, że refleksja filozoficzna stała się w opisywanym okresie ważnym uzupełnieniem pracy naukowej przyrodników, co decyduje również o specyfice rozwoju ówczesnej nauki.

We investigate whether photon ring observations in black hole imaging are able to distinguish between the Kerr black hole in general relativity and alternative black holes that deviate from Kerr. Certain aspects of photon rings have been argued to be robust observables in very-long-baseline interferometry black hole observations which carry imprints of the underlying spacetime. The photon ring shape, as well as its Lyapunov exponent (which encodes the narrowing of successive photon subrings), are detailed probes of the underlying geometry; measurements thereof have been argued to provide a strong null test of general relativity and the Kerr metric. However, a more complicated question is whether such observations of the photon ring properties can distinguish between Kerr and alternative black holes. We provide a first answer to this question by calculating photon rings of the Johannsen, Rasheed-Larsen, and Manko-Novikov black holes. We find that large deviations from Kerr and large observer inclinations are needed to obtain measurable differences in the photon ring shape. In other words, the Kerr photon ring shape appears to be the universal shape even for deviating black holes at low inclinations. On the other hand, the Lyapunov exponent shows more marked variations for deviations from the Kerr metric. Our analysis lays out the groundwork to determine deviations from the Kerr spacetime in photon rings that are potentially detectable by future observing missions.

The special theory of relativity (STR) is operationally expanded onto orthogonal accelerations: normal and binormal that complement the instantaneous tangential speed and thus can be structurally extended into operationally complete 4D spacetime without defying the STR. Thus the former classic Lorentz factor, which defines proper time differential can be expanded onto within a trihedron moving in the Frenet frame (T,N,B). Since the tangential speed which was formerly assumed as being always constant, expands onto effective normal and binormal speeds ensuing from the normal and binormal accelerations, the expanded formula conforms to the former Lorentz factor. The obvious though previously overlooked fact that in order to change an initial speed one must apply accelerations (or decelerations, which are reverse accelerations), made the Einstein’s STR incomplete for it did not apply to nongravitational selfpropelled motion. Like a toy car lacking accelerator pedal, the STR could drive nowhere. Yet some scientists were teaching for over 115 years that the incomplete STR is just fine by pretending that gravity should take care of the absent accelerator. But gravity could not drive cars along even surface of earth. Gravity could only pull the car down along with the physics that peddled the nonsense while suppressing attempts at its rectification. The expanded formula neither defies the STR nor the general theory of relativity (GTR) which is just radial theory of gravitation. In fact, the expanded formula complements the STR and thus it supplements the GTR too. The famous Hafele-Keating experiments virtually confirmed the validity of the expanded formula proposed here.

There is a great deal of justified concern about continuity through scientific theory change. Our thesis is that, particularly in physics, such continuity can be appropriately captured at the level of conceptual frameworks (the level above the theories themselves) using conceptual space models. Indeed, we contend that the conceptual spaces of three of our most important physical theories—Classical Mechanics (CM), Special Relativity Theory (SRT), and Quantum Mechanics (QM)—have already been so modelled as phase-spaces. Working with their phase-space formulations, one can trace the conceptual changes and continuities in transitioning from CM to QM, and from CM to SRT. By offering a revised severity-ordering of changes that conceptual frameworks can undergo, we provide reasons to doubt the commonly held view that CM is conceptually closer to SRT than QM.

A few years ago, in my review of Sean Carroll's book in Classical and Quantum Gravity [1], I wrote that while the 1970s was the decade of Weinberg [2] and Misner, Thorne and Wheeler [3], and while the eighties was the decade of Schutz [4] and Wald [5], the 2000s was clearly the decade of Hartle [6] and Carroll [7]. In my opinion, these books continue to stand out in the surprisingly dense crowd of introductory textbooks on general relativity. At the dawn of this new decade I look forward to see what fresh pedagogical insights will be produced next, and who will be revealed as the winners of the 2010s. It is, of course, much too early to tell, but Schutz is back, and he will set the standard just as he did back in 1985.

The general theory of relativity is considered to be Albert Einstein’s masterpiece in theoretical physics. In contrast with special relativity, where scientists like Hendrik Lorentz and Henri Poincaré worked in parallel, motivated by the unsolved physical problems existing at the beginning of the twentieth century (for instance, motion with respect to the æther and the negative result of the Michelson–Morley experiment), there was no such motivation for general relativity. With the exception of an anomaly in the precession of Mercury’s orbit, the Newtonian theory of gravitation did not manifest symptoms of obsolescence.