3
$\begingroup$

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)?)

$\endgroup$
2
$\begingroup$

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

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

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

\begin{equation} ds^2 = -dt^2 + g_{ij}(x^i) dx^i dx^j \end{equation}

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

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

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

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

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

\begin{equation} R_{ab} = \frac{d-1}{\alpha} g_{\mu\nu} \end{equation}

then we have the approximation

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

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

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

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.