Energy Momentum Tensor: Derivation and Conservation

This lecture covers the derivation of the energy momentum tensor for point particles, focusing on the covariant derivative and conservation. It also explores the impact of adding an electromagnetic field, computing the divergence of the energy momentum tensor, and deriving the expression for the electromagnetic current. Additionally, it delves into the symmetry and static nature of generic metrics in spherically symmetric spacetimes, emphasizing the Schwarzschild metric as a solution. The exercise involves analyzing the Schwarzschild metric, identifying problematic radii, and applying coordinate changes to investigate physical effects.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related lectures (44)

Related concepts (27)

Schwarzschild coordinates

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is adapted to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere.

Electromagnetic field

An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by moving electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical counterpart to the quantized electromagnetic field tensor in quantum electrodynamics (a quantum field theory). The electromagnetic field propagates at the speed of light (in fact, this field can be identified as light) and interacts with charges and currents.

Electromagnetic tensor

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below.

Spherically symmetric spacetime

In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving matter or energy. Because spherically symmetric spacetimes are by definition irrotational, they are not realistic models of black holes in nature. However, their metrics are considerably simpler than those of rotating spacetimes, making them much easier to analyze.

Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun.

Introduce the students to general relativity and its classical tests.

This course is the basic introduction to modern cosmology. It introduces students to the main concepts and formalism of cosmology, the observational status of Hot Big Bang theory and discusses major

Einstein Equations PHYS-427: Relativity and cosmology I

Covers the derivation of Einstein equations and the Schwarzschild metric.

Linear Momentum Conservation and Stress in Continuum

Explores the conservation of linear momentum and stress in a continuum, focusing on governing equations and constitutive laws.

Christoffel Symbols and Gravity Before Einstein PHYS-427: Relativity and cosmology I

Introduces Christoffel symbols and gravity concepts before Einstein, discussing mathematical tensors and the Nobel Prize in Physics.

Special and General Relativity PHYS-427: Relativity and cosmology I

Introduces special and general relativity, Einstein equations, and gravitational dynamics.

Radiation-reaction Force: Lagrangian Formulation PHYS-324: Classical electrodynamics

Covers the radiation-reaction force, conservation of momentum, and correct force expressions.