Singularities in general relativity

The equations of general relativity (see box) rank among the most fundamental equations in all of modern physics. Introduced in 1915 and 1916 by Albert Einstein (1879–1955), they are now known more simply, in the singular, as Einstein's field equation. It presents countless formidable mathematical challenges of great beauty (the hoop conjecture, nonlinear stability of the Kerr family, the BKL conjecture…), including the famous conjectures concerning cosmic censorship (see below): the equation is still far from having revealed all its secrets!

Einstein's field equation ------------------------------

In its most familiar form, Einstein's equation reads:

\text{R}_{\mu, v} - \dfrac{1}{2}\text{R} g_{\mu, v} + \Lambda g_{\mu, v} = \dfrac{8 \pi}{c^4} \text{GT} _{\mu, v}.

It involves R*_μ*_,*_ν, the Ricci tensor (a second-order tensor encoding the deformation of spacetime); R, the scalar curvature (a number measuring the curvature of spacetime, namely the trace of R_μ*_,*_ν); g_μ*_,*_ν, the metric tensor, namely a second-order tensor that assigns a dot product to any two vectors at each point in spacetime (its signature is (+, —, —, —), characteristic of Lorentzian geometry); T_μ*_,*_ν, the energy-momentum tensor, which characterizes the distribution of mass and energy in spacetime; Λ, a fundamental physical constant (the cosmological constant*); G, the gravitational constant ( G = 6.67430 × 10^-¹¹ m³ kg^-¹ s^-² ± 0.00015 × 10^-¹¹ m³ kg^-¹ s^-²); and c, the speed of light in a vacuum (c = 299,792,458 m s^-¹).

The mathematical study of Einstein's field equation is extremely challenging. To begin with, how are we to understand what it means? Tensors are multilinear maps that assign a scalar to vectors and covectors in a vector space. The key point is that Einstein's field equation allows us to determine the metric tensor (*g_μ*_,*_ν, which is essential for measuring lengths and angles in spacetime) from a given distribution of matter and energy (expressed by the energy—momentum tensor T_μ*_,*_ν). The geometry of spacetime (described by g_μ*_,*_ν) is determined locally by its energy—momentum content (given by T_μ*_,*_ν); spacetime tells matter how to move (via R_μ*_,*_ν); matter tells spacetime how to curve (via* R).

References:

Les équations de la physique moderne (The Equations of Modern Physics). Bibliothèque Tangente 71, 2020.

Vecteurs et espaces vectoriels (Vectors and Vector Spaces). Bibliothèque Tangente 65, 2018.

Mathematically, a "spacetime" is a Lorentzian manifold equipped with a metric. The "shortest path" between two points of the manifold, with respect to the metric in question, is called a geodesic. The "curvature" of the spacetime under consideration is encoded and quantified by a curvature tensor (here, R*_μ*_,*_ν). The more "curved" spacetime is, the more rapidly its geodesics diverge or converge. Finally, if g_μ*_,*_ν(V) > 0, the tangent vector V is spacelike; if g_μ*_,*_ν(V) = 0, it is lightlike; and if g_μ*_,*_ν(V) < 0, V is timelike*.