Lorentz Hyperboloid Model

• The Lorentz hyperboloid model is a representation of n-dimensional hyperbolic space as a Riemannian submanifold in (n+1)-dimensional Minkowski space with constant negative curvature.
• It defines geodesic distances and isometries through the Minkowski bilinear form, supporting group actions by O⁺(1,n) and other symmetry groups.
• The model enables explicit construction of fundamental solutions of the Laplace–Beltrami operator and supports series expansions via Fourier and Gegenbauer methods.

The Lorentz (hyperboloid) model is a canonical representation of $n$-dimensional hyperbolic geometry, embedding hyperbolic space as a Riemannian submanifold within $(n+1)$-dimensional Minkowski space. This construction provides a mathematically rigorous framework for analyzing properties of spaces of constant negative curvature and is fundamental to both differential geometry and the theory of Lie groups. The model naturally realizes hyperbolic $n$-space as the upper sheet of a two-sheeted hyperboloid defined by the Minkowski bilinear form, with the isometry group given by an orthogonal group of signature $(1,n)$, $O^+(1,n)$.

1. Minkowski Space and the Hyperboloid Construction

Let $\mathbb{R}^{n+1}$ denote $(n+1)$-dimensional Minkowski space equipped with the bilinear form

$\eta(x, y) = x_0 y_0 - \sum_{i=1}^n x_i y_i,$

where $x = (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1}$. A non-zero vector $x$ is classified as time-like if $\eta(x,x) > 0$, light-like if $\eta(x,x) = 0$, or space-like if $\eta(x,x) < 0$.

The two-sheeted unit hyperboloid is then

$H = \{ x \in \mathbb{R}^{n+1} : \eta(x,x) = 1 \},$

with connected components

$$H^+ = \{ x \in H : x_0 > 0 \}, \quad H^- = \{ x \in H : x_0 < 0 \}.$$

The upper sheet $H^+$ becomes a model of $n$-dimensional hyperbolic space $\mathbb{H}^n$ when endowed with the Riemannian metric induced by $-\eta$: $$\langle v, w \rangle_x = -\eta(v, w), \quad \text{for } v, w \text{ in } T_x(H^+),\; \eta(x,v) = 0.$$ This yields a simply connected, complete Riemannian manifold of constant sectional curvature $-1$ (for unit hyperboloid) or $-1/R^2$ for $H_R^+ = \{ x : \eta(x,x) = R^2, x_0 > 0 \}$ (Foldes, 2010, Cohl et al., 2011).

2. Geodesic Structure and Polar Coordinates

The geodesic distance in the hyperboloid model is determined by the Minkowski inner product: $$d(x, x') = R \cosh^{-1} \left( \frac{\eta(x, x')}{R^2} \right ),$$ so that for $x, x' \in H_R^+$, one defines the radial coordinate $\rho = d(x, x') / R \in [0, \infty)$, and $\cosh \rho = \frac{\eta(x, x')}{R^2}$.

Using geodesic polar coordinates centered at $x'$, a point $x \in H_R^+$ is parametrized as

$$x = (R \cosh r,\, R \sinh r\, \omega), \quad r \ge 0,\, \omega \in S^{n-1},$$

with induced Riemannian metric

$ds^2 = R^2 (dr^2 + \sinh^2 r \, d\Omega_{n-1}^2),$

where $d\Omega_{n-1}^2$ is the round metric on the unit $(n-1)$-sphere (Cohl et al., 2012, Cohl et al., 2011).

3. Isometry Group Characterization

Let $O^+(1,n)$ denote the identity component of the Lorentz group preserving the sign of $x_0$. This group acts transitively and isometrically on $H^+$ with stabilizer $O(n)$. The full isometry group of the Lorentz model of $\mathbb{H}^n$ is $O^+(1,n)$, realizing all orientation- and time-orientation-preserving hyperbolic isometries.

The group of affine transformations preserving all hyperboloids in Minkowski space coincides with the Poincaré group, i.e., every bijection $f: \mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$ preserving $H + v$ for all $v$ is affine of the form $f(x) = Lx + b$, where $L \in O(1,n)$ and $b \in \mathbb{R}^{n+1}$. Importantly, no nontrivial dilation preserves the hyperboloid, in sharp contrast to Alexandrov–Zeeman theorems for the light cone, which allow for dilations in the characterization of the symmetry group (Foldes, 2010).

4. Fundamental Solution of the Laplacian

The Lorentz (hyperboloid) model supports the explicit construction of spherically symmetric fundamental solutions for the Laplace–Beltrami operator. In geodesic polar coordinates on $\mathbb{H}_R^d$, the Laplace–Beltrami operator is

$$\Delta = \frac{1}{R^2} \left[ \frac{\partial^2}{\partial r^2} + (d-1) \coth r \frac{\partial}{\partial r} + \frac{1}{\sinh^2 r} \Delta_{S^{d-1}} \right].$$

The unique (decaying at infinity) Green's function $G(\rho)$ solves

$- \Delta G = \delta_g(x, x'),$

and is given by

$$G(\rho) = \frac{\Gamma(d/2)}{2\pi^{d/2}} I_d(\rho)^{-1}, \quad I_d(\rho) = \int_\rho^\infty \sinh^{1-d} \eta \, d\eta,$$

with $c_0$ determined by matching the Euclidean singularity. Several closed-form representations exist for $I_d(\rho)$, including sums over hyperbolic functions (for even and odd $d$), Gauss hypergeometric functions,

$$I_d(\rho) = \frac{1}{(d-1) \cosh^{d-1}\rho} \, {}_2F_1 \left( \frac{d-1}{2}, \frac{d}{2}; \frac{d+1}{2}; \frac{1}{\cosh^2 \rho} \right ),$$

and in terms of the associated Legendre function of the second kind,

$$I_d(\rho) = \frac{e^{-i\pi(d/2-1)} 2^{d/2-1}}{ \Gamma(d/2) \, \sinh^{d/2-1}\rho} Q_{d/2-1}^{d/2-1}( \cosh \rho ),$$

yielding

$$G(\rho) = \frac{e^{-i\pi(d/2-1)} 2^{d/2-1}}{2\pi^{d/2} \sinh^{d/2-1}\rho } Q_{d/2-1}^{d/2-1}(\cosh \rho).$$

This construction guarantees uniqueness up to the addition of a global harmonic function with the decay requirement $G(\rho) \to 0$ as $\rho \to \infty$ (Cohl et al., 2012, Cohl et al., 2011).

5. Special Expansions and Addition Theorems

Spherical symmetry allows expansions of the Green's function in both azimuthal Fourier series and Gegenbauer (zonal harmonic) series. In two and three dimensions simplified closed forms exist, for example,

$$G_2(x, x') = \frac{1}{2\pi} \ln \coth\frac{p}{2}, \qquad G_3(x, x') = \frac{1}{4\pi R}(\coth p-1),$$

where $p = d(x,x') / R$.

The azimuthal Fourier expansion, employing geodesic polar and azimuthal angles, takes the general form

$$G(x, x') = \sum_{m=0}^\infty \cos( m(\phi - \phi') ) H_m^{(d)}( r, r', \theta, \theta' ),$$

with $H_m^{(d)}$ given by explicit integrals and, in $d=3$, relates directly to toroidal harmonics.

The Gegenbauer expansion exploits the invariance under $O(d,1)$ and the associated Casimir operator, providing a series in spherical harmonics: $$G(x, x') = \sum_{l=0}^\infty w_l(r, r') \sum_K Y_{l,K}(\Omega) Y_{l,K}(\Omega'),$$ with explicit weights involving Legendre and Gegenbauer polynomials. Addition theorems in $d=3$ relate the Fourier and Gegenbauer coefficients, yielding single-sum representations and facilitating computations involving angular dependencies (Cohl et al., 2011).

6. Asymptotics and Correspondence with Euclidean Theory

The singularity structure of $G(\rho)$ as $\rho \to 0$ matches that of the Euclidean Green’s function: $$I_d(\rho) \sim \begin{cases} - \ln \rho, & d=2, \ \frac{1}{d-2} \rho^{2-d}, & d \geq 3, \end{cases}$$ so that

$$G(\rho) \sim \frac{\Gamma(d/2)}{2\pi^{d/2}} \begin{cases} - \ln \rho, & d=2, \ \frac{1}{d-2}\rho^{2-d}, & d \geq 3, \end{cases} \quad (\rho \ll 1).$$

At infinity, both $G(\rho)$ and $I_d(\rho)$ decay exponentially as $e^{-(d-1)\rho}$, ensuring rapid spatial decay at large distances (Cohl et al., 2012).

7. Connections to Group-Theoretic Characterization and Broader Context

The Lorentz (hyperboloid) model provides the geometric realization underpinning the isometry group structure of hyperbolic spaces. Preservation of hyperboloids, contrary to light-cone preservation (Alexandrov–Zeeman theorem), leads precisely to the Poincaré group as the maximal symmetry group, excluding dilations (Foldes, 2010). This structural characterization is essential in mathematical physics, representation theory, and the analysis of partial differential equations in negatively curved spaces.

The model’s compatibility with explicit analytic, algebraic, and group-theoretic structures makes it central to harmonic analysis, special function theory, geometric group theory, and the study of fundamental solutions to geometric PDEs on symmetric spaces (Cohl et al., 2012, Cohl et al., 2011).