3+1 Formalism and Bases of Numerical Relativity

TL;DR

The notes deliver a comprehensive, geometrically grounded account of the 3+1 formalism for general relativity, detailing how spacetime is foliated into spacelike hypersurfaces and how Einstein’s equations split into evolution equations for the 3-metric and extrinsic curvature together with Hamiltonian and momentum constraints. The framework introduces lapse and shift as coordinate freedom tools that control foliation and spatial coordinates, and recasts the equations as a 3+1 PDE system amenable to Cauchy problem analysis and numerical integration. A central theme is the conformal decomposition, which isolates the true dynamical degrees of freedom of gravity in a conformal class of 3-metrics and a traceless, conformally scaled extrinsic curvature, underpinning popular formulations such as BSSN. The material also surveys the initial data problem, global quantities, and matter couplings, thereby providing a foundational toolkit for modern numerical relativity used in simulating compact binaries and gravitational wave sources.

Abstract

These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, 4. 3+1 decomposition of Einstein equation, 5. 3+1 equations for matter and electromagnetic field, 6. Conformal decomposition, 7. Asymptotic flatness and global quantities, 8. The initial data problem, 9. Choice of foliation and spatial coordinates, 10. Evolution schemes.

Figures (24)

• Figure 1: Embedding $\Phi$ of the 3-dimensional manifold $\hat{\Sigma}$ into the 4-dimensional manifold $\mathcal{M}$, defining the hypersurface $\Sigma = \Phi(\hat{\Sigma})$. The push-forward $\Phi_*\bm{v}$ of a vector $\bm{v}$ tangent to some curve $C$ in $\hat{\Sigma}$ is a vector tangent to $\Phi(C)$ in $\mathcal{M}$.
• Figure 2: Plane $\Sigma$ as a hypersurface of the Euclidean space $\mathbb{R}^3$. Notice that the unit normal vector $\bm{n}$ stays constant along $\Sigma$; this implies that the extrinsic curvature of $\Sigma$ vanishes identically. Besides, the sum of angles of any triangle lying in $\Sigma$ is $\alpha+\beta+\gamma=\pi$, which shows that the intrinsic curvature of $(\Sigma,\bm{\gamma})$ vanishes as well.
• Figure 3: Cylinder $\Sigma$ as a hypersurface of the Euclidean space $\mathbb{R}^3$. Notice that the unit normal vector $\bm{n}$ stays constant when $z$ varies at fixed $\varphi$, whereas its direction changes as $\varphi$ varies at fixed $z$. Consequently the extrinsic curvature of $\Sigma$ vanishes in the $z$ direction, but is non zero in the $\varphi$ direction. Besides, the sum of angles of any triangle lying in $\Sigma$ is $\alpha+\beta+\gamma=\pi$, which shows that the intrinsic curvature of $(\Sigma,\bm{\gamma})$ is identically zero.
• Figure 4: Sphere $\Sigma$ as a hypersurface of the Euclidean space $\mathbb{R}^3$. Notice that the unit normal vector $\bm{n}$ changes its direction when displaced on $\Sigma$. This shows that the extrinsic curvature of $\Sigma$ does not vanish. Moreover all directions being equivalent at the surface of the sphere, $\bm{K}$ is necessarily proportional to the induced metric $\bm{\gamma}$, as found by the explicit calculation leading to Eq. (\ref{['e:hyp:Kab_sphere']}). Besides, the sum of angles of any triangle lying in $\Sigma$ is $\alpha+\beta+\gamma>\pi$, which shows that the intrinsic curvature of $(\Sigma,\bm{\gamma})$ does not vanish either.
• Figure 5: In the Euclidean space $\mathbb{R}^3$, the plane $\Sigma$ is a totally geodesic hypersurface, for the geodesic between two points $A$ and $B$ within $(\Sigma,\bm{\gamma})$ (solid line) coincides with the geodesic in the ambient space (dashed line). On the contrary, for the sphere, the two geodesics are distinct, whatever the position of points $A$ and $B$.