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 GraphSearch.
Concept
Causal structure
Summary
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events.
The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships.
If is a Lorentzian manifold (for metric on manifold ) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types.
A tangent vector is:
timelike if
null or lightlike if
spacelike if
Here we use the metric signature. We say that a tangent vector is non-spacelike if it is null or timelike.
The canonical Lorentzian manifold is Minkowski spacetime, where and is the flat Minkowski metric. The names for the tangent vectors come from the physics of this model. The causal relationships between points in Minkowski spacetime take a particularly simple form because the tangent space is also and hence the tangent vectors may be identified with points in the space. The four-dimensional vector is classified according to the sign of , where is a Cartesian coordinate in 3-dimensional space, is the constant representing the universal speed limit, and is time. The classification of any vector in the space will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the metric.
At each point in the timelike tangent vectors in the point's tangent space can be divided into two classes. To do this we first define an equivalence relation on pairs of timelike tangent vectors.
Official source
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 publications (1)
Related concepts (25)
Globally hyperbolic manifold
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is (t and r being the usual variables of time and radius) which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime.
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events. The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature.
Event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact objects that even light cannot escape. At that time, the Newtonian theory of gravitation and the so-called corpuscular theory of light were dominant. In these theories, if the escape velocity of the gravitational influence of a massive object exceeds the speed of light, then light originating inside or from it can escape temporarily but will return.