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Concept
Mathematics of general relativity
Summary
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.
Note: General relativity articles using tensors will use the abstract index notation.
The principle of general covariance was one of the central principles in the development of general relativity. It states that the laws of physics should take the same mathematical form in all reference frames. The term 'general covariance' was used in the early formulation of general relativity, but the principle is now often referred to as 'diffeomorphism covariance'.
Diffeomorphism covariance is not the defining feature of general relativity,[1] and controversies remain regarding its present status in general relativity. However, the invariance property of physical laws implied in the principle, coupled with the fact that the theory is essentially geometrical in character (making use of non-Euclidean geometries), suggested that general relativity be formulated using the language of tensors. This will be discussed further below.
Spacetime and Spacetime topology
Most modern approaches to mathematical general relativity begin with the concept of a manifold. More precisely, the basic physical construct representing gravitationa curved spacetimeis modelled by a four-dimensional, smooth, connected, Lorentzian manifold. Other physical descriptors are represented by various tensors, discussed below.
The rationale for choosing a manifold as the fundamental mathematical structure is to reflect desirable physical properties. For example, in the theory of manifolds, each point is contained in a (by no means unique) coordinate chart, and this chart can be thought of as representing the 'local spacetime' around the observer (represented by the point).
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