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.

We review recent developments and results in testing general relativity (GR) at cosmological scales. The subject has witnessed rapid growth during the last two decades with the aim of addressing the question of cosmic acceleration and the dark energy associated with it. However, with the advent of precision cosmology, it has also become a well-motivated endeavor by itself to test gravitational physics at cosmic scales. We overview cosmological probes of gravity, formalisms and parameterizations for testing deviations from GR at cosmological scales, selected modified gravity (MG) theories, gravitational screening mechanisms, and computer codes developed for these tests. We then provide summaries of recent cosmological constraints on MG parameters and selected MG models. We supplement these cosmological constraints with a summary of implications from the recent binary neutron star merger event. Next, we summarize some results on MG parameter forecasts with and without astrophysical systematics that will dominate the uncertainties. The review aims at providing an overall picture of the subject and an entry point to students and researchers interested in joining the field. It can also serve as a quick reference to recent results and constraints on testing gravity at cosmological scales.

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Testing general relativity in cosmology

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L2(R, Cs) from the orthogonal vector-valued wavelet packets.

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

Der speziellen Relativitatstheorie liegt folgendes Postulat zugrunde, welchem auch durch die Galilei-Newtonsche Mechanik Genuge geleistet wird: Wird ein Koordinatensystem K so gewahlt, das in bezug auf dasselbe die physikalischen Gesetze in ihrer einfachsten Form gelten, so gelten dieselben Gesetze auch in Bezug auf jedes andere Koordinatensystem K′, das relativ zu K in gleichformiger Translationsbewegung begriffen ist. Dieses Postulat nennen wir „spezielles Relativitatsprinzip“. Durch das Wort „speziell“ soll angedeutet werden, das das Prinzip auf den Fall beschrankt ist, das K′ eine gleichformige Translationsbewegung gegen K ausfuhrt, das sich aber die Gleichwertigkeit von K′ und K nicht auf den Fall ungleichformiger Bewegung von K′ gegen K erstreckt.

Die Grundlage der allgemeinen Relativitätstheorie

In this paper, for the first time we analyze the structure of the Italian Airport Network (IAN) looking at it as a mathematical graph and investigate its topological properties. We find that it has very remarkable features, being like a scale-free network, since both the degree and the “betweenness centrality” distributions follow a typical power-law known in literature as a Double Pareto Law. From a careful analysis of the data, the Italian Airport Network turns out to have a self-similar structure. In short, it is characterized by a fractal nature, whose typical dimensions can be easily determined from the values of the power-law scaling exponents. Moreover, we show that, according to the period examined, these distributions exhibit a number of interesting features, such as the existence of some “hubs”, i.e. in the graph theory’s jargon, nodes with a very large number of links, and others most probably associated with geographical constraints. Also, we find that the IAN can be classified as a small-world network because the average distance between reachable pairs of airports grows at most as the logarithm of the number of airports. The IAN does not show evidence of “communities” and this result could be the underlying reason behind the smallness of the value of the clustering coefficient, which is related to the probability that two nearest neighbors of a randomly chosen airport are connected.

Topology of the Italian airport network: A scale-free small-world network with a fractal structure?

The structures formation of the Universe appears as if it were a classically self-similar random process at all astrophysical scales. An agreement is demonstrated for the present hypotheses of segregation with a size of astrophysical structures by using a comparison between quantum quantities and astrophysical ones. We present the observed segregated Universe as the result of a fundamental self-similar law, which generalizes the Compton wavelength relation. It appears that the Universe has a memory of its quantum origin as suggested by R. Penrose with respect to quasi-crystal. A more accurate analysis shows that the present theory can be extended from the astrophysical to the nuclear scale by using generalized (stochastically) self-similar random process. This transition is connected to the relevant presence of the electromagnetic and nuclear interactions inside the matter. In this sense, the presented rule is correct from a subatomic scale to an astrophysical one. We discuss the near full agreement at organic cell scale and human scale too. Consequently the Universe, with its structures at all scales (atomic nucleus, organic cell, human, planet, solar system, galaxy, clusters of galaxy, super clusters of galaxy), could have a fundamental quantum reason. In conclusion, we analyze the spatial dimensions of the objects in the Universe as well as space–time dimensions. The result is that it seems we live in an El Naschie’s E-infinity Cantorian space–time; so we must seriously start considering fractal geometry as the geometry of nature, a type of arena where the laws of physics appear at each scale in a self-similar way as advocated long ago by the Swedish school of astrophysics.

Stochastic self-similar and fractal universe

The conformal gravity fit to observed galactic rotation curves requires $\ensuremath{\gamma}g0$. On the other hand, the conventional method for light deflection by galaxies gives a negative contribution to the Schwarzschild value for $\ensuremath{\gamma}g0$, which is contrary to observation. Thus, it is very important that the contribution to bending should in principle be positive, no matter how small its magnitude is. Here we show that the Rindler-Ishak method gives a positive contribution to Schwarzschild deflection for $\ensuremath{\gamma}g0$, as desired. We also obtain the exact local coupling term derived earlier by Sereno. These results indicate that conformal gravity can potentially test well against all astrophysical observations to date.

Correct light deflection in Weyl conformal gravity

In this paper, we will analyze the Fantappie group and its properties in connection with Cantorian space–time. Our attention will be focused on the possibility of extending special relativity. The cosmological consequences of such extension appear relevant, since thanks to the Fantappie group, the model of the Big Bang and that of stationary state become compatible. In particular, if we abandon the idea of the existence of only one time gauge, since we do not see the whole Universe but only a projection, the two models become compatible. In the end we will see the effects of the projective fractal geometry also on the galactic and extra-galactic dynamics.

Fantappiè's group as an extension of special relativity on ε (∞) Cantorian space-time

In this paper we show a segregated universe in terms of Tolman–Bondi equations of pressure-free spherically symmetric systems of particles in general relativity. We demonstrate the agreement between our model of segregation with a size of astrophysical structures by using a comparison between quantum and astrophysical quantities. This is made within the framework of Mohamed El Naschie's e(∞) Cantorian space–time.

Fractal Cantorian structures with spatial pseudo-spherical symmetry for a possible description of the actual segregated universe as a consequence of its primordial fluctuations

In the context of Mohamed El Naschie ϵ(∞) theory, we present a segregated universe in terms of Tolman–Bondi equations of pressure-free spherically symmetric systems of particles We demonstrate the agreement between our model of segregation with a size of astrophysical structures via a comparison between quantum and astrophysical quantities In [Iovane G, Laserra E, Giordano P Fractal Cantorian structures with spatial pseudo-spherical symmetry for a possible description of the actual segregated universe as a consequence of its primordial fluctuations Chaos, Solitons & Fractals 2004;22(3):521] we considered a scale factor a2(r) = 1, here we will also analyze a2(r) ≠ 1

Ettore Laserra

Papers

Stochastic self-similar and fractal universe

Correct light deflection in Weyl conformal gravity

Fantappiè's group as an extension of special relativity on ε (∞) Cantorian space-time

Fractal Cantorian structures with spatial pseudo-spherical symmetry for a possible description of the actual segregated universe as a consequence of its primordial fluctuations

El Naschie ϵ(∞) Cantorian structures with spatial pseudo-spherical symmetry: A possible description of the actual segregated universe