How does electromagnetism work in a negatively curved universe?

Let's say that there was a negatively curved universe (in particular $\Omega < 1$).

I assume that means the universe would be like Hyperbolic space.

In our universe, electromagnetism obeys an inverse-square law. As it just so happens, the surface area of a sphere in our universe are proportional to the the radius of the sphere.

In hyperbolic space, the surface area of a sphere grows exponentially as its radius increases. Does that mean electromagnetism obeys an exponential decay law, or what? (As a follow up question, is the same true of other inverse square laws, (such a gravity)?)

1 Answer

The fall-off of field intensity is measured in physics by the van Vleck determinant :

$$\Delta_\gamma (x,y) = (-1)^d \frac{\det (\nabla_\mu^x \nabla_\nu^y \sigma_\gamma(x,y))}{\sqrt{g(x)g(y)}}$$

Which is defined for geodesics $\gamma$ between the points $x$ and $y$, with $\sigma_\gamma$ the geodetic interval between the two. The van Vleck determinant describes the expansion of the geodesic flow in spacetime, with in particular for ultrastatic spacetimes

$$ds^2 = -dt^2 + g_{ij}(x^i) dx^i dx^j$$

the flux of a field at $y$ with source at $x$ is described by

$$\| \vec J \| = \frac{\Delta_\gamma(\vec x, \vec y)}{s^{d-1}}$$

with $s$ the spacelike distance between the two points and $\Delta$ the van Vleck determinant of the spacelike hypersurface. In the weak field limit, the van Vleck determinant is

$$\Delta_\gamma(x, y) \approx 1 + \frac 16 R_{ab} t^a t^b s^2_\gamma(x, y)$$

with $t$ the tangent of $\gamma$. Hence, for hyperbolic space, with the spacelike hypersurface

$$R_{ab} = \frac{d-1}{\alpha} g_{\mu\nu}$$

then we have the approximation

$$\| \vec J \| = \frac{1}{s^{d-1}} + \frac{d-1}{6\alpha} \frac{|t|}{s^{d-3}}$$

So globally the flux drop-off will depend on the distance in hyperbolic space, for instance in the hyperboloid model

$$s_\gamma(x,y) = \operatorname{arcosh}(x^0 y^0 - \sum x^i y^i)$$

For more details on the van Vleck determinant you can check "Relativity : the general theory" by Synge, or the paper by Visser on the topic.

• So it would asymptotically decay exponentially with distance, right? – PyRulez Feb 9 '18 at 16:07