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Background:

Let $M$ be a smooth Riemannian manifold of dimension $n$ and scalar curvature $R$ (with respect to the Levi-Civita connection). Let $m \in M$ and let $B$ be the geodesic ball of radius $r$ centered at $m$. That is,

$B = \{ Exp_m(v)\ |\ \ v\in T_mM,\ ||v||< r\ \} $

Then for small $r$, the volume of $B$ is:

\begin{equation} Vol(B) = (constant)\ r^n\ \left[1 - \frac{R}{6(n+2)}r^2 + O(r^4)\ \right] \end{equation}

where the constant depends only on $n$, and $R$ is evaluated at $m$. So $R$ basically tells us the difference between the volume of a small ball in $M$ and the volume of a small ball in Euclidean $\mathbb{R}^n$ (to leading order in the radius $r$).

Questions:

1) Is there any generalization of this to the case of a semi-Riemannian manifold?

2) If not, is there an analogous result for the case of a Lorentzian manifold (that is, a semi-Riemannian manifold whose metric signature has $n-1$ pluses and one minus)?

3) If not, is there some other result that gives a nice way to interpret the scalar curvature $R$ in a semi-Riemannian (or Lorenztian) manifold?

Comment:

The problem I see is this: to define a geodesic ball, we want to start with a ball "of radius $r$" in $T_mM$. But the metric is indefinite, so there is no norm. I looked in Barrett O'Niell's book "Semi-Riemannian Geometry" but did not find the answer there.

Thanks in advance for your help!

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  • $\begingroup$ In the $(k+1)$ hyperbolic space $H^{k+1}$ we have that $\mathrm{vol}(B_r)=\pi^{k/2}\Gamma(k/2)e^{kr}/k!+O(re^{(k-2)r})$. That helps? $\endgroup$
    – emiliocba
    Commented Jan 22, 2012 at 6:02
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    $\begingroup$ @emiliocba I'm afraid I don't see it. A hyperbolic space is an example of a Riemannian manifold, isn't it? The scalar curvature may be negative, but the metric is still positive definite. $\endgroup$
    – marlow
    Commented Jan 22, 2012 at 16:36
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    $\begingroup$ One thing that makes the relevant calculation in the Riemannian case so nice is the fact that a Riemannian ball is a region with finite area which is invariant under all orthogonal transformations, so when calculating its volume one can freely change bases. Unfortunately, there are no such shapes in Minkowski space -- there doesn't even exist a piecewise-continuous, finite-volume, positive Lorentz-invariant measure on Minkowski space. $\endgroup$
    – tcamps
    Commented Jun 22, 2019 at 16:39

1 Answer 1

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Note that existence / uniqueness of the Levi-Civita connection holds for arbitrary signature nondegenerate pseudo-Riemannian metrics. Covariant derivatives, geodesics, parallel transport, and exponential maps are all defined for arbitrary affine connections. (See Chapter 5 of [Lee] for instance.)

Let $(M, g)$ denote a Lorentzian surface and consider some point $p \in M$. The scalar curvature of $M$ at $p$ determines how two unit-speed timelike geodesics with velocity vectors $v$ and $u$ passing through $p$ are spreading out (see picture). Relatedly, it also determines the rate of growth of the area of a "geodesic sector" centered on $p$, and of the length of the curve swept out when varying a given radial geodesic through an angle without changing its "length" (which is analogous to an arc of a geodesic circle). The sign of the curvature has the opposite effect in the Lorentzian case, however (see below).

More precisely: Consider $e_1, e_2 \in T_p M$ satisfying $g_p(e_1, e_1) = 1$, $g_p(e_2, e_2) = -1$, $g_p(e_1, e_2) = 0$. For some fixed $\rho_0$ and $\theta_1 < \theta_2$, $x(\rho, \theta) := \mathrm{exp}_p(\rho(\sinh(\theta)e_1 + \cosh(\theta) e_2))$ parameterizes a geodesic sector as $\rho$ varies in $[0, \rho_0)$ and $\theta$ varies in $(\theta_1, \theta_2)$. A $\theta$-curve $\theta \mapsto x(\rho_0, \theta)$ parameterizes the "geodesic circle" at radius $\rho_0$. And a sector in the tangent plane is shown between the vectors $v$ and $u$ in the following picture.

Blue geodesic sector Geogebra applet

$\frac{\partial}{\partial \rho}$ and $\frac{\partial}{\partial \theta}$ denote coordinate vector fields associated to the domain of $x$. Let \begin{align*} \begin{bmatrix} E & F \\\ F & G \end{bmatrix} := \begin{bmatrix} g\left(x_\ast\frac{\partial}{\partial \rho}, x_\ast\frac{\partial}{\partial \rho}\right) & g\left(x_\ast\frac{\partial}{\partial \rho}, x_\ast\frac{\partial}{\partial \theta}\right) \\\ g\left(x_\ast\frac{\partial}{\partial \theta}, x_\ast\frac{\partial}{\partial \rho}\right) & g\left(x_\ast\frac{\partial}{\partial \theta}, x_\ast\frac{\partial}{\partial \theta}\right) \end{bmatrix}. \end{align*}

You can easily adapt the proof I give here that $E = -1$ (because $\rho$-curves of $x$ are unit-speed time-like geodesics, by definition of the exponential map), $F = 0$ (analog of the Gauss lemma), and $G > 0$ (since $-G = EG - F^2 < 0$, as the metric is Lorentzian). Pages 405-406 of [O'Neill] discuss the sense in which $\sqrt{G}$ is the distance between nearby radial geodesics in the Riemannian analog, and $(\sqrt{G})_\rho$ is the rate at which the geodesics are spreading. The same ideas apply here. On the page I linked I also show $\lim_{\rho \to 0} (\sqrt{G})_\rho = 1$, which can be interpreted as saying the rate of spreading is like that of Minkowski space for some arbitrarily short time. On the page I linked I give a derivation of the following relationship between $\sqrt{G}$ and the scalar curvature $K$ (the sign of the right hand side in the following is the opposite in the Riemannian case, because the sectional curvature formula involves the determinant of the metric) $$(\sqrt{G})_{\rho\rho} = K\sqrt{G},$$ which [O'Neill, Chapter 8, 3.3 Theorem] refers to as the Jacobi equation (though that name is used for a related but different/more general equation in [Lee]). (This may be missing a factor of $\frac{1}{2}$ in the righthand-side term, which I will have to check later.) The derivation I give holds for Lorentzian surfaces because it only depends on properties of objects defined in terms of the connection. As O'Neill writes, this equation says that the rate of spreading of geodesics depends on the curvature, with the rate increasing with $\rho$ when the curvature is negative and decreasing when the curvature is positive. Since $\sqrt{G}(0, \theta) = 0$ and $(\sqrt{G})_\rho(0, \theta) = 1$ (by [O'Neill], or as shown in my link), it also follows that $\sqrt{G}$ depends only on $\rho$. So one can express the following Taylor series expression for $\sqrt{G}$ centered at $\rho = 0$, $$\sqrt{G}(\rho, \theta) = \rho + \frac{K(p)}{6}\rho^3 + r(\rho, \theta) \rho^3,$$ where $r = r(\rho, \theta)$ is some smooth function for which $\lim_{\rho \to 0} r(\rho, \theta) = 0.$ Then we can finally use this to calculate the length $L_{\rho_0, \theta_0}$ of an arc of angle $\theta_0$ of a radius $\rho_0$ "geodesic ball" (as in the picture above): $$L_{\rho_0, \theta_0} = \int_0^{\theta_0} \sqrt{G}(\rho_0, \theta) d \theta = \theta_0\rho_0 + \theta_0\frac{K(p)}{6}\rho_0^3 + \rho_0^3 \int_0^{\theta_0} r(\rho_0, \theta) d\theta.$$ This means if curvature is negative, then the length is less than in the flat case, and if the curvature is positive, then the length is greater. The observations related to area are similar.

(I can copy the arguments from my linked page to here if that is desirable, though it would significantly increase the length of this post.)

If your time-like geodesics represent the path of free-falling particles, then in the case of flat curvature this says they should move as if they have no relative acceleration. The decrease of the rate of spreading in the negative curvature case signifies that they accelerate towards one-another, and the increase of the rate of spreading in the positive curvature case signifies that they accelerate away from one-another. I learned of this interpretation from [Beem Ehrlich] and it is also corroborated in this Physics SE post.

Also, the paper Some Semi-Riemannian Volume Comparison Theorems by Ehrlich and Sánchez seems to develop all of this in higher generality (and probably better).

Lee, John M., Introduction to Riemannian manifolds, Graduate Texts in Mathematics 176. Cham: Springer (ISBN 978-3-319-91754-2/hbk; 978-3-319-91755-9/ebook). xiii, 437 p. (2018). ZBL1409.53001.

O’Neill, Barrett, Elementary differential geometry., Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-088735-4/hbk). xii, 503 p. (2006). ZBL1208.53003.

Beem, John K.; Ehrlich, Paul E., Global Lorentzian geometry, Pure and Applied Mathematics, 67. New York, Basel: Marcel Dekker, Inc. VI, 460 p. SFr. 126.00 (1981). ZBL0462.53001.

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