# What is General theory of relativity? Equations & Examples

The General Theory of Relativity was published by Albert Einstein in 1915. The General Theory of Relativity is the accepted name published by Albert Einstein in 1915 for the theory of gravity. The force of gravity is a representation of the local geometry of space-time, according to the General Theory of Relativity. While the modern theory is due to Einstein, its roots lie in the axioms of Euclidean geometry and the numerous attempts to prove Euclid’s fifth postulate over the centuries, which says that parallel lines always remain equidistant, and that culminated in the finding by Bolyai and Gauss that this axiom is not necessarily true. The general mathematics of non-Euclidean geometry was developed by Riemann, a disciple of Gauss.

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Gauss demonstrated that there is no reason why the geometry of space should be Euclidean, which means that if a physicist places a mark, and a cartographer remains at a certain distance and its length is measured by triangulation based on Euclidean geometry, then no the same answer is guaranteed if the physicist carries the mark with him and measures its length directly.

The difference between the two measurements could not, of course, be determined in practise for a brand, but there are analogous measures that must specifically detect the non-Euclidean geometry of space-time , for example, the Pound-Rebka experiment (1959) detected the difference in light wavelength from a cobalt source occurring by 22.5 metres against gravity in a room at the Jefferson Physics Laboratory at Harvard.

In relativity, the basic principle is that without first specifying its reference structure, we should not speak about the physical quantities of velocity or acceleration. And the private choice is defined by this reference system. In such a case, all movement is defined and quantified relatively to another matter.

In the special theory of relativity it is assumed that reference systems can be extended indefinitely in all directions in space-time. But in general theory it is recognized that the definition of approximate systems locally and for a finite time is only possible for finite regions of space (similar to how we can draw flat maps of regions of the Earth’s surface but we cannot extend them to cover the surface of all land without distortion).

In the general theory of relativity, Newton’s laws are assumed only in relation to local reference systems. In particular, free particles travel by drawing straight lines in local inertial systems (Lorentz). When those lines are extended, they do not appear as straight, being called a geodesic. So Newton’s first law is replaced by the law of geodetic motion.

We distinguish inertial reference systems, in which bodies maintain uniform movement without the action of or on other bodies, from non-inertial reference systems in which freely moving bodies undergo acceleration derived from the reference system itself. In non-inertial reference systems, force derived from the reference system is perceived, not by the direct influence of another matter. We feel “gravitational” forces when we go in a car and turn a curve as the physical basis of our reference system.

The Coriolis effect and the centrifugal force act similarly when we define rotating matter-based reference systems (just like the Earth or a child spinning). The equivalence principle in general theory states that there are no local experiments that are capable of distinguishing a non-rotational fall in a gravitational field from uniform motion in the absence of a gravitational field. That is, there is no gravity in a free fall reference system.

From this perspective, the gravity observed on the surface of the Earth is the force observed in a reference system defined by the matter on the surface that is not free (it is bound) but is activated downwards by the terrestrial matter, and is analogous to the “gravitational” force felt in a car making a curve.

Mathematically, Einstein modelled space-time by a pseudo-Riemannian variety, and his field equations establish that the curvature of the variety at one point is directly related to the energy tensor at that point; This tensor is a measure of the density of matter and energy.

Curvature tells matter how to move, and reciprocally matter tells space how to curve. The possible field equation is not unique, with other models available without contradicting observation. The general theory of relativity is differentiated from other gravity theories by the simplicity of the coupling of matter and curvature, although its unification with Quantum Mechanics and the replacement of the field equation with a law appropriate to quantum has not yet been resolved.

Few physicists doubt that such a theory, Einstein’s field equation contains a parameter called “cosmological constant”? which was originally introduced by Einstein to allow a static universe.

This effort was unsuccessful for two reasons: the instability of the universe resulting from such theoretical efforts, and Hubble’s observations a decade later confirm that our universe is indeed not static but expanding, So was abandoned, but quite recently, astronomical techniques found that a nonzero value for? it is necessary to explain some observations.

## General theory of relativity Equations

The field equations read as follows:

where *R i k* is the Ricci curvature tensor, R is the Ricci curvature scalar, *g i k* is the metric tensor, is the cosmological constant, *T i k* is the energy tensor, p is pi, **c** is the speed of light in vacuum and **G** is the universal gravitational constant, similar to what happens in Newtonian gravity. *g i k *describes the variety metric and is a 4×4 symmetric tensor, so it has 10 independent components. Given the freedom of choice of the four Spatio-temporal coordinates, the independent equations are reduced to six.

**Recommended books:**

**1. Relativity: The Special and General Theory (Dover Books on Physics)** by Albert Einstein

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2. The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity

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