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Lecture

Scalar Gravity: Einstein Equations

In course

PHYS-427: Relativity and cosmology I

Introduce the students to general relativity and its classical tests. ...

Description

This lecture covers the concept of scalar gravity, starting with a recap on covariant derivatives and the Levi-Civita connection. It then delves into the Ricci tensor, Weyl tensor, and the Einstein tensor, discussing their symmetries and properties. The principle of equivalence, Einstein Equivalence Principle, and the plan for the day focusing on scalar gravity are also explained. The lecture further explores the generalization of Newtonian gravity equations to scalar gravity, emphasizing the Einstein-Hilbert action and the cosmological constant. Various exercises are presented to demonstrate the application of scalar gravity in different scenarios.

Official source

Ontological neighbourhood

Physics

Relativity: General relativity

Mathematics

Geometry: Differential geometry

Astronomy

Astrophysics: Physical cosmology

Related concepts (32)

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).

Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force.

Gravitational redshift

In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well (seem to) lose energy. This loss of energy corresponds to a decrease in the wave frequency and increase in the wavelength, known more generally as a redshift. The opposite effect, in which photons (seem to) gain energy when travelling into a gravitational well, is known as a gravitational blueshift (a type of blueshift).

Einstein tensor

In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum. The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds.

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