Pointless Spaces in General Relativity

The new approach to quantize the gravity based on the notion of differential algebra is suggested. It is shown that the differential geometry of this object can not be described in terms of points. The spatialization procedure giving rise to points by loosing a part of the entire structure is discussed. The counterpart of the traditional objects of differential geometry are studied. FOREWORD In general relativity an event in spacetime is idealized to a point of a fourdimensional manifold. Such idealization is adequate within classical physics, but is unsatisfactory from the operationalistic point of view. In quantum theory the influence of a measuring apparatus on the object being observed can not in principle be removed. We could expect the metric of a quantized theory to be subject to fluctuations, whereas the primary tool to separate individual events is just the metric. Thus a sort of smearing procedure for events is to be imposed into the theory. An essential step in this direction was the idea to build the differential geometry in terms of abstract algebras. Geroch (1972) proposed to generalize the notion of algebra of smooth functions on a manifold to that of Einstein algebra whose elements are not yet functions. This generalization was successful since the entire content of general relativity can be reformulated in such a way that the underlying spacetime manifold is used only once: to define the collection of smooth functions. Although, since the commutative case is considered, the absence of points is, roughly speaking, an illusion. As a matter of fact, a commutative algebra can always be represented by functions on an underlying space. Such representation is, for instance, the Gel’fand construction (for normed algebra) which is the special case of representation of commutative algebra on its spectrum. So, in