In 1907, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. Riemann’s work was so ahead of his time that it did not receive its proper attention in the mathematical community until Einstein’s formulation of general relativity in 1915. Einstein imagined the spacetime as a geometric object whose curvature is determined by the distribution of energy and matter. Thus gravitational force is no longer a force in the Newtonian sense but a mere manifestation of the curvature of spacetime . The space-time is a manifold and a space-time event is represented by a point . The worldline of the particle is traced by a curve in . There are several basic tenets of general relativity. • Space-time is a semi-Riemannian manifold. • Free particles follow geodesics. • Curvature tells matter how to move, and matter tells space-time how to curve. Intrinsic properties - those that can be measured on the surface without regard to how it is embedded in an ambient space. In Euclidean geometry, the shortest distance between two points can be found using Pythagoras's theorem. What Riemann discovered was a more powerful, general form of Pythagoras's theorem that works on curved surfaces, even when the curvature is in more than two dimensions and varies from one place to another. In this looking-glass world of curved space, the familiar idea of distance is replaced by the broader concept of something called a metric, from the Greek for "measure," while curvature is similarly described by a more elaborate mathematical object. Gauss had found that the curvature in the neighborhood of a point of a specified two-dimensional geometry is given by a single number: the Gaussian curvature. Riemann showed that six numbers are needed to describe the curvature of a three-dimensional space at a given point, and that 20 numbers at each point are required for a four-dimensional geometry: the 20 independent components of the so-called Riemann curvature tensor.
Before of Einstein, Hilbert had submitted a paper the foundations of physics which also contained the correct field equations for gravitation.
General relativity is the geometric theory of gravitation and the current description of gravitation in modern physics. In general relativity, the universe has three dimensions of space and one of time and putting them together we get four dimensional spacetime, which gravity as an emergent effect from the spacetime curvature associated with distributions of energy. The central ideas of general relativity have been neatly summarized by John Archibald Wheeler: "matter tells space how to bend; space tells matter how to move".
Energy cause space-time warp and curve and gravity is the exchange of gravitons. As the photon is packet of electromagnetic energy, gravitons would be considered packets of the gravitational energy or space-time curvature.
One of the key requirements for the formulation of General relativity was the principle of equivalence between inertial mass and gravitational mass.
The difficulty now is that if light can follow curved paths, these curved paths must be the path of least time between two points. The only way that the curved path can be the path of least time and the speed of light remain constant is that if space and time are compressed and stretched respectively.
Of course he wrote this nearly two centuries before the concept of a field, such as the electromagnetic field, emerged in physics: a field that has a material existence in that it contains and carries energy and momentum, and importantly, mediates influences between bodies at a finite speed.
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