Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Spherically symmetric spacetime
Summary
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.
Spherically symmetric models are not entirely inappropriate: many of them have Penrose diagrams similar to those of rotating spacetimes, and these typically have qualitative features (such as Cauchy horizons) that are unaffected by rotation. One such application is the study of mass inflation due to counter-moving streams of infalling matter in the interior of a black hole.
A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3) and the orbits of this group are 2-spheres (ordinary 2-dimensional spheres in 3-dimensional Euclidean space). The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations". The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere). Conventionally, the metric on the 2-sphere is written in polar coordinates as
and so the full metric includes a term proportional to this.
Spherical symmetry is a characteristic feature of many solutions of Einstein's field equations of general relativity, especially the Schwarzschild solution and the Reissner–Nordström solution. A spherically symmetric spacetime can be characterised in another way, namely, by using the notion of Killing vector fields, which, in a very precise sense, preserve the metric. The isometries referred to above are actually local flow diffeomorphisms of Killing vector fields and thus generate these vector fields.
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.
Ontological neighbourhood
Related courses (2)
MATH-530: Differential geometry IV - general relativity
This course will serve as a basic introduction to the mathematical theory of general relativity. We will cover topics including the formalism of Lorentzian geometry, the formulation of the initial val
PHYS-427: Relativity and cosmology I
Introduce the students to general relativity and its classical tests.
Related lectures (9)
Deflection of Light: Observations and Methods
Explores the deflection of light by gravity and methods for studying matter distribution.
Energy Momentum Tensor: Derivation and Conservation
Explores the derivation and conservation of the energy momentum tensor for point particles, including the impact of electromagnetic fields and the Schwarzschild metric.
Gravitational Red-Shift and Escape Velocity
Covers the discovery of integrals of motion and the analysis of escape velocity and gravitational red-shift.
Related publications (5)
Related concepts (3)
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries. Physical problems are often investigated and solved by noticing features which have some form of symmetry.
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.
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.