Tzvi Scarr

Spacetime transformations from a uniformly accelerated frame

We use the generalized Fermi?Walker transport to construct a one-parameter family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. We explain the connection between our approach and that of Mashhoon. We show that our solutions of uniformly accelerated motion have constant acceleration in the comoving frame. Assuming the weak hypothesis of locality, we obtain local spacetime transformations from a uniformly accelerated frame K? to an inertial frame K. The spacetime transformations between two uniformly accelerated frames with the same acceleration are Lorentz. We compute the metric at an arbitrary point of a uniformly accelerated frame.

Uniform acceleration in general relativity

We extend de la Fuente and Romero’s (Gen Relativ Gravit 47:33, 2015) defining equation for uniform acceleration in a general curved spacetime from linear acceleration to the full Lorentz covariant uniform acceleration. In a flat spacetime background, we have explicit solutions. We use generalized Fermi-Walker transport to parallel transport the Frenet basis along the trajectory. In flat spacetime, we obtain velocity and acceleration transformations from a uniformly accelerated system to an inertial system. We obtain the time dilation between accelerated clocks. We apply our acceleration transformations to the motion of a charged particle in a constant electromagnetic field and recover the Lorentz-Abraham-Dirac equation.

Topological and Borel compactifications of Polish G-spaces

Abstract We investigate the relationships between topological and Borel G -spaces, where G is a Polish group. We show that every Polish G -space can be topologically and equivariantly embedded into a compact Polish G -space iff G is locally compact. This answers a question of Kechris. It also provides a striking contrast to the recent result of Becker and Kechris which states that every Borel G -space can be Borel-embedded into a compact Polish G -space.

Classification of JBW *-triple factors

In the previous chapters, we have shown that homogeneous balls and bounded symmetric domains can be realized as unit balls of JB*-triples. Let X be a complex Banach space with the JB*-triple product. The JB*-triple X is called atomic if it is spanned by its minimal tripotents. It is known that the second dual X** of X is a JBW*-triple i.e., X** is a JB*-triple whose triple product is w*-continuous. Moreover, X can be isometrically embedded into the atomic part of its second dual. Hence, any JB*-triples can be considered as a subtriple of an atomic JBW *-triple.

The classical bounded symmetric domains

The definition of bounded symmetric domains is geometric in nature and does not seem, at first glance, to be connected to the category of operators on Hilbert spaces. Nevertheless, we have already seen in the previous chapter that the spin factor has certain properties found in operator spaces, such as spectral decomposition, representation by Pauli matrices, and Peirce decomposition. Surprisingly, most BSDs are unit balls of operator spaces. Such domains are calledclassical domains.They provide a familiar setting in which to introduce some of the more abstract concepts connected with BSDs.

The complex spin factor and applications

In this chapter, we will discuss the complex spin factor, a domain of type IV in the Cartan classification. This domain is symmetric with respect to the analytic automorphisms. In fact, in the previous chapter, we used the analyticity of the spin factor on a two-dimensional plane to solve equations of evolution.

Relativity based on symmetry

In this chapter, we will derive the Lorentz transformations without assuming the constancy of the speed of light. We will use only the principle of special relativity and the symmetry associated with it. We will see that this principle allows only Galilean or Lorentz space-time transformations between two inertial systems. In the case of the Lorentz transformations, we obtain the conservation of an interval and a certain speed. From known experiments, this speed iscthe speed of light in a vacuum.

The algebraic structure of homogeneous balls

In this chapter we will present the main ideas of the theory of homogeneous balls and bounded symmetric domains and the algebraic structure associated with them. The domains will be domains in complex Banach spaces and their homogeneity and symmetry will be with respect to analytic (called also holomorphic) maps. Thus we will start with the definition and study of the analytic mappings on Banach spaces. Next we will consider the group of analytic automorphisms Aut a (D) of a bounded domain D and show that the elements of this group are uniquely defined by their value and the value of their derivative at some point.

The real spin domain

In the previous chapter, we used the principle of special relativity to obtain the real bounded symmetric domainD v .This domain is symmetric with respect to the projective automorphisms and is a domain of type I in the Cartan classification of bounded symmetric domains In this chapter, we will discuss another real domain, called the real spin factor, which is a domain of type IV in the Cartan classification. The complex spin factor will be studied in Chapter 3.