The paper by A. Einstein presents the foundation of the general theory of relativity, which is a significant generalization of the special theory of relativity. The special theory, which is based on the principle of relativity and the constancy of the speed of light, does not account for gravitational effects. The general theory extends this to include gravitational fields, leading to a more comprehensive understanding of space and time. Einstein discusses the need for a more general mathematical framework, specifically the "absolute differential calculus," which was developed by Gauss, Riemann, and Christoffel, and later systematized by Ricci and Levi-Civita. This framework allows for the description of curved spacetime, which is essential for the general theory. The paper also addresses the philosophical and physical arguments for extending the principle of relativity. It highlights the limitations of classical mechanics and the special theory in explaining certain phenomena, such as the behavior of distant masses and the nature of gravitational fields. The introduction of the general theory of relativity is motivated by the need to account for these phenomena, particularly the influence of distant masses on the motion of bodies and the curvature of spacetime. Einstein defines the four-dimensional spacetime continuum and introduces the concept of general covariance, which requires that physical laws must be expressed in equations that are valid for all coordinate systems. This principle ensures that the theory is consistent with the principle of relativity and allows for a unified description of gravity and electromagnetism. The paper also delves into the mathematical tools used in the general theory, including tensors and their transformations. It explains how these tools are used to describe the curvature of spacetime and the behavior of objects in gravitational fields. The fundamental tensor \( g_{\mu \nu} \) plays a crucial role in this framework, and its properties are derived and discussed. Finally, the paper derives the equations of geodesic motion, which describe the paths of freely falling particles in curved spacetime. These equations are essential for understanding the behavior of objects in gravitational fields and are the basis for many predictions of the general theory of relativity, such as the bending of light and the Shapiro time delay.The paper by A. Einstein presents the foundation of the general theory of relativity, which is a significant generalization of the special theory of relativity. The special theory, which is based on the principle of relativity and the constancy of the speed of light, does not account for gravitational effects. The general theory extends this to include gravitational fields, leading to a more comprehensive understanding of space and time. Einstein discusses the need for a more general mathematical framework, specifically the "absolute differential calculus," which was developed by Gauss, Riemann, and Christoffel, and later systematized by Ricci and Levi-Civita. This framework allows for the description of curved spacetime, which is essential for the general theory. The paper also addresses the philosophical and physical arguments for extending the principle of relativity. It highlights the limitations of classical mechanics and the special theory in explaining certain phenomena, such as the behavior of distant masses and the nature of gravitational fields. The introduction of the general theory of relativity is motivated by the need to account for these phenomena, particularly the influence of distant masses on the motion of bodies and the curvature of spacetime. Einstein defines the four-dimensional spacetime continuum and introduces the concept of general covariance, which requires that physical laws must be expressed in equations that are valid for all coordinate systems. This principle ensures that the theory is consistent with the principle of relativity and allows for a unified description of gravity and electromagnetism. The paper also delves into the mathematical tools used in the general theory, including tensors and their transformations. It explains how these tools are used to describe the curvature of spacetime and the behavior of objects in gravitational fields. The fundamental tensor \( g_{\mu \nu} \) plays a crucial role in this framework, and its properties are derived and discussed. Finally, the paper derives the equations of geodesic motion, which describe the paths of freely falling particles in curved spacetime. These equations are essential for understanding the behavior of objects in gravitational fields and are the basis for many predictions of the general theory of relativity, such as the bending of light and the Shapiro time delay.