Concept
Einstein tensor
Related concepts (16)
Metric tensor (general relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past. In general relativity, the metric tensor plays the role of the gravitational potential in the classical theory of gravitation, although the physical content of the associated equations is entirely different.
Introduction to the mathematics of general relativity
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion.
Scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls.
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor).
Mathematics of general relativity
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.
Ricci-flat manifold
In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a (pseudo-)Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in vacuum with vanishing cosmological constant. In Lorentzian geometry, a number of Ricci-flat metrics are known from works of Karl Schwarzschild, Roy Kerr, and Yvonne Choquet-Bruhat.
Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Let G be a Lie group with Lie algebra , and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a -valued one-form on P). Then the curvature form is the -valued 2-form on P defined by (In another convention, 1/2 does not appear.
Gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenomena, and is measured in newtons per kilogram (N/kg). Equivalently, it is measured in meters per second squared (m/s2). In its original concept, gravity was a force between point masses.
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space.
Exact solutions in general relativity
In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.