Relativistic Hypercomputing: Problems and Prospects from the Physicist’s Point of View

Abstract

The paper presents the main results on hypercomputing based on the use of relativistic effects. Two approaches to the problem are compared – formal-logical and physical. The basis of the physical approach is the study of the metric of curved space-time manifolds on which hypercomputing are realized, obtained either by applying the equivalence principle or by solving Einstein's equations. The properties of Malament-Hogarth spaces arising in these manifolds are discussed. The advantages of the physical approach are shown, which make it possible to verify the possibility of hypercomputing by the example of the problem of calculating the sum of the divergent Dirichlet series for the Riemann zeta function, which requires overcoming the so-called Turing barrier. It is stressed the possibility of using numerical algebras that differ from the field of real numbers, which promises significant progress in the development of modern physical theories first of all in cosmology. The issues of relativistic theory are considered separately. The relativistic solution of the problem of motion with constant acceleration by finding the gravitational potential field of an infinite homogeneous plane is discussed. The solution of this problem by applying the equivalence principle is also discussed. The results are compared with the well-known solutions of V. Fock and R. Tolman.