Lorentzian manifold

Lorentzian Manifold

A **Lorentzian manifold** is a smooth manifold equipped with a Lorentzian metric, which is a non-degenerate, symmetric bilinear form of signature \((n-1, 1)\) or \((1, n-1)\). This structure is fundamental in the formulation of general relativity, where spacetime is modeled as a four-dimensional Lorentzian manifold.

Definition

A Lorentzian manifold \( (M, g) \) consists of a smooth manifold \( M \) and a Lorentzian metric \( g \). The metric \( g \) is a tensor field of type \((0, 2)\) that assigns to each point \( p \in M \) a bilinear form \( g_p \) on the tangent space \( T_pM \). The signature of \( g \) is \((n-1, 1)\) or \((1, n-1)\), meaning that at each point \( p \), the metric \( g_p \) has one negative and \( n-1 \) positive eigenvalues (or vice versa).

Properties

A key property of Lorentzian manifolds is the distinction between timelike, spacelike, and null vectors. A vector \( v \in T_pM \) is:

**Timelike** if \( g_p(v, v) < 0 \),
**Spacelike** if \( g_p(v, v) > 0 \),
**Null** (or lightlike) if \( g_p(v, v) = 0 \) and \( v \neq 0 \).

These distinctions are crucial in the study of causal structures and the propagation of signals and particles in spacetime.

Causal Structure

The causal structure of a Lorentzian manifold is determined by the light cones at each point. The light cone at a point \( p \) is the set of all null vectors in \( T_pM \). Timelike vectors lie inside the light cone, while spacelike vectors lie outside. This structure allows the definition of causal relationships between events:

**Causal**: An event \( p \) can causally influence an event \( q \) if there exists a future-directed timelike or null curve from \( p \) to \( q \).
**Chronological**: An event \( p \) is chronologically before \( q \) if there exists a future-directed timelike curve from \( p \) to \( q \).

Geodesics

Geodesics in a Lorentzian manifold are curves that locally extremize the spacetime interval. They are solutions to the geodesic equation: \[ \frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{\nu \lambda} \frac{d x^\nu}{d \tau} \frac{d x^\lambda}{d \tau} = 0, \] where \( \Gamma^\mu_{\nu \lambda} \) are the Christoffel symbols of the metric \( g \), and \( \tau \) is an affine parameter.

Timelike geodesics represent the trajectories of freely falling particles, while null geodesics represent the paths of light rays.

Curvature

The curvature of a Lorentzian manifold is described by the Riemann curvature tensor \( R^\rho_{\sigma \mu \nu} \), which encodes how vectors are transported around infinitesimal loops. The Ricci curvature tensor \( R_{\mu \nu} \) and the scalar curvature \( R \) are derived from the Riemann tensor and play significant roles in the Einstein field equations of general relativity: \[ R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}, \] where \( \Lambda \) is the cosmological constant, \( G \) is the gravitational constant, \( c \) is the speed of light, and \( T_{\mu \nu} \) is the stress-energy tensor.

Examples

Minkowski Space

The simplest example of a Lorentzian manifold is Minkowski space, which is the flat spacetime of special relativity. It is a four-dimensional manifold with the metric: \[ ds^2 = -dt^2 + dx^2 + dy^2 + dz^2. \] Minkowski space is globally hyperbolic and has no curvature.

Schwarzschild Solution

The Schwarzschild solution describes the spacetime geometry outside a spherically symmetric, non-rotating massive object. The metric in Schwarzschild coordinates \((t, r, \theta, \phi)\) is: \[ ds^2 = -\left(1 - \frac{2GM}{r}\right) dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2, \] where \( d\Omega^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 \).

Kerr Solution

The Kerr solution generalizes the Schwarzschild solution to rotating black holes. The metric in Boyer-Lindquist coordinates \((t, r, \theta, \phi)\) is: \[ ds^2 = -\left(1 - \frac{2GMr}{\rho^2}\right) dt^2 - \frac{4GMar \sin^2 \theta}{\rho^2} dt \, d\phi + \frac{\rho^2}{\Delta} dr^2 + \rho^2 \, d\theta^2 + \left(r^2 + a^2 + \frac{2GMa^2 r \sin^2 \theta}{\rho^2}\right) \sin^2 \theta \, d\phi^2, \] where \( \rho^2 = r^2 + a^2 \cos^2 \theta \) and \( \Delta = r^2 - 2GMr + a^2 \).

Global Hyperbolicity

A Lorentzian manifold is said to be globally hyperbolic if it admits a Cauchy surface, which is a spacelike hypersurface that every non-spacelike curve intersects exactly once. Global hyperbolicity ensures well-posedness of the initial value problem for the Einstein field equations and is a key condition for the predictability of physical theories.

Penrose Diagrams

Penrose diagrams are a tool used to represent the causal structure of Lorentzian manifolds. These conformal diagrams compactify infinite regions of spacetime, allowing the visualization of causal relationships between different regions. They are particularly useful in the study of black holes and cosmological models.

Applications in Physics

Lorentzian manifolds are central to the formulation of general relativity, where the Einstein field equations describe the dynamics of spacetime geometry in response to matter and energy. They also appear in string theory and other theories of quantum gravity, where higher-dimensional Lorentzian manifolds are considered.

Mathematical Techniques

The study of Lorentzian manifolds involves various mathematical techniques, including differential geometry, topology, and partial differential equations. Tools such as the ADM formalism, spinor fields, and the Newman-Penrose formalism are employed to analyze the properties and behavior of these manifolds.