Quantum Particle in the Gravitational Field

In this chapter, we demonstrate the main theorems of the general theory of relativity, and derive dynamic equations for a quantum particle in a gravitational field. For two curvilinear time-space systems of reference, we obtain the transformed matrices of vectors and tensors. We define contravariant and covariant representations, and the quotient theorem. We consider the curvature of the physical four-dimensional time-space system, in the total system, including a number of extracoordinates, enabling this curvature. For the physical system, as a hypersurface in the total system, we obtain the metric tensor, as a function of the extra-coordinates, which, on this hypersurface, depend on the physical coordinates, as parameters. In this framework, we obtain the Christoffel symbols as functions of the coordinate derivatives of the metric elements. We obtain the covariant derivative, including only physical effects, without the curvature effects described by the ordinary derivative. We show that the covariant derivative of a metric element is null, and derive equations for the matter conservation, harmonic oscillations, and curvilinear forms of the Gauss and Stokes theorems. From the inertial and gravitational forces, we obtain the geodesic equations of a null covariant acceleration, as from the invariance of the timespace interval, we obtain that any vector is perpendicular to its covariant derivative with any coordinate. We define the curvature tensor and derive the symmetry and the Bianci relations. We define the Ricci tensor, and from the Bianci relation for this tensor we obtain Einstein’s law of gravitation. From this law for a gravitational system with spherical symmetry, we obtain the Schwarzschild solution for the metric tensor. We derive the Einstein gravitational law in the presence of matter, and study the dynamics of a quantum particle in a black hole. Outside a black hole we consider a time-like region, with a far region where all the bodies are attracted, and a near region, where the coming bodies are strongly decelerated, to a null velocity at the Schwarzschild boundary, but reaching this boundary only in an infinite time. The internal part of a black hole is defined as a space-like region, where at its formation, the central matter explodes, having the tendency to concentrate at the Schwarzschild boundary, but reaching this boundary also in an infinite time.