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Highlights of Riemannian Geometry in Physics Brazilian Journal of Physics, Vol Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Novello, Mario; Bittencourt, Eduardo Metric Relativity and the Dynamical Bridge: Highlights of Riemannian Geometry in Physics Brazilian Journal of Physics, vol. 45, núm. 6, 2015, pp. 756-805 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46442560018 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Braz J Phys (2015) 45:756–805 DOI 10.1007/s13538-015-0362-7 GENERAL AND APPLIED PHYSICS Metric Relativity and the Dynamical Bridge: Highlights of Riemannian Geometry in Physics Mario Novello1 · Eduardo Bittencourt2,3 Received: 4 July 2015 / Published online: 30 September 2015 © Sociedade Brasileira de F´ısica 2015 Abstract We present an overview of recent developments mechanics and consists in the principle of Metric Relativ- concerning modifications of the geometry of space-time to ity. The main difference between gravitational interaction describe various physical processes of interactions among and all other forces concerns the universality of gravity classical and quantum configurations. We concentrate in which added to the interpretation of the equivalence princi- two main lines of research: the Metric Relativity and the ple allows all associated geometries—one for each different Dynamical Bridge. We describe the notion of equivalent body in the case of non-gravitational forces—to be uni- (dragged) metric gμν which is responsible to map the fied into a unique Riemannian space-time structure. The path of any accelerated body in Minkowski space-time same geometrical description appears for electromagnetic onto a geodesic motion in such associated g geometry. waves in the optical limit within the context of nonlinear Only recently, the method introduced by Einstein in gen- theories or material medium. Once it is largely discussed eral relativity was used beyond the domain of gravitational in the literature, the so-called analogue models of grav- forces to map arbitrary accelerated bodies submitted to ity, we will dedicate few sections on this emphasizing their non-Newtonian attractions onto geodesics of a modified relation with the new concepts introduced here. Then, we geometry. This process has its roots in the very ancient pass to the description of the Dynamical Bridge formalism idea to treat any dynamical problem in Classical Mechan- which states the dynamic equivalence of nonlinear theories ics as nothing but a problem of static where all forces (driven by arbitrary scalar, spinor or vector fields) that occur acting on a body annihilates themselves including the iner- in Minkowski background to theories described in asso- tial ones. This general procedure, that concerns arbitrary ciated curved geometries generated by each one of these forces—beyond the uses of General Relativity that is lim- fields. We shall see that it is possible to map the dynam- ited only to gravitational processes—is nothing but the ical properties of a theory, say Maxwell electrodynamics relativistic version of the d’Alembert method in classical in Minkowski space-time, into Born-Infeld electrodynam- ics described in a curved space-time the metric of which is defined in terms of the electromagnetic field itself in Eduardo Bittencourt such way that it yields the same dynamics. It is clear that [email protected] when considered in whatever unique geometrical structure, these two theories are not the same; they do not describe Mario Novello [email protected] the same phenomenon. However, we shall see that by a convenient modification of the metric of space-time, an 1 Instituto de Cosmologia Relatividade e Astrofisica equivalence appears that establishes a bridge between these (ICRA/CBPF) Rua Dr. Xavier Sigaud, 150 - Urca, two theories making they represent the same phenomenon. Rio de Janeiro, RJ 22290-180 , Brazil This method was recently used to achieve a successful 2 CAPES Foundation, Ministry of Education of Brazil, Bras´ılia, geometric scalar theory of gravity. At the end, we briefly Brazil review the proposal of geometrization of quantum mechan- ics in the de Broglie-Bohm formulation using an enlarged 3 Physics Department, La Sapienza University of Rome, non-Riemannian (Weyl) structure. I-00185 Rome, Italy Braz J Phys (2015) 45:756–805 757 Keywords Riemannian geometry · Classical field theory · and to follow the geodesics on this curved space-time, as Nonlinear field theory · Quantum mechanics far as gravity is taken into account solely. If a body suffers a particular acceleration of non-gravitational character, this freedom disappears and the body follows a non-geodesic 1 Preliminary Comments path. The purpose of these notes is to describe some new The success of general relativity (GR) led to the general results that intend, in a very particularly and precise way, to acceptance that the introduction of a geometry to describe a extend the idea that it is possible to deal with any kind of physical process yields a unique structure identified to the force through similar lines as suggested by GR. This means space-time. However, this is a rather general mathematical to introduce the idea that modifications of the geometry of procedure, and one can apply the modification of the back- space-time may describe a large variety of physical pro- ground geometry to substitute the acceleration not only by cesses. In other words, to generalize the approach of GR in gravity but by any kind of force; introducing specific and order to represent all kind of acceleration as nothing but a limited geometries for different observers in many distinct modification of the geometry. Along this line, the path of situations seems to be a very useful tool. any body on which an arbitrary force has acted upon, can One could accept that the achievements of a given the- be described as if this body is free, without any interaction, ory may depend on the uniqueness of its representation. in a particular associated metric structure that depends on However, in more recent years, we have learned that the both, the characteristics of the force and the kinematics of possibility of presenting an alternative equivalent descrip- the body. Let us point out that there are obvious distinctions tion of a given phenomenon may be an important theoretical between the case of GR and all other cases. This is due to tool. The presentation of a theory under distinct formula- the universality of gravitational processes. Indeed, one of tions has been of great help in many occasions, as we will the postulates of GR states that there exists one and only one verify with direct examples in these notes. We shall see how geometric structure of space-time as perceived for all that it is possible to modify the background geometry of space- exists. Besides that, the uniqueness of the scenario where time (intrinsic to the description of gravitational interactions all events occurs lies on the hypothesis that its geometry is according to the rules and proposals of GR) to describe the controlled by the gravitational phenomena. This unification effect of forces of distinct and multiple natures. A well- requires the neglect of all other kind of interaction and must known example concerns the propagation of light inside be interpreted as some sort of approximation related to the moving dielectrics. degree of amplitude of its corresponding description. This We shall describe how very different modifications of the beautiful requirement of GR appears in each observation space-time geometry have been used to describe accelerated as specific examples of geometries obtained as a particular paths in Minkowski space-time as geodesics in an associ- solutions of its dynamics. In these notes, we will analyze ated curved Riemannian geometry and even to exhibit the another possibility. equivalence dynamics of various fields in distinct associated The original GR proposal was concerned with the possi- geometries. bility of eliminating the acceleration induced on an arbitrary An important step in the construction of the theory of spe- body A by a gravitational field by a convenient modification cial relativity was the hypothesis that a specific proper time of the metric associated to the space where A is propagating. can be assigned for each body. Distinct observers establish GR shows that all gravitational effects can be equivalently distinct times. This proposal highlights the importance of described in terms of such an universal modification of the an individual body in the description of its kinematics and space-time geometry. stresses the importance of the global aspects of the physical However, from the technical point of view, this is only description in the domain of mechanics. Such an individ- one possibility. Indeed, as we shall review in these notes, ualization was new in the standard program of physics this should not be considered the only possibility, and one that usually is based, in a very broad sense, to develop could well make other conventions to ascribe specific met- formalisms that intended to be universal. rics to different events. According to this point of view, After the success of this procedure, there has been a need to each interaction, a particular modification of the met- to recover the global and unified character of the theoret- ric environment in which a body moves in such a way that ical framework. This has indeed come about through the any kind of force can be eliminated by this interpretation is next big step made by the entrance of a dynamical scheme likely to be associated. After this characterization of its own associated to the universal gravitational interaction, which metric structure, the body is interpreted as realizing a free becomes identified to an attribute of the structure of space- motion.
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