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## The Einstein-Hilbert action over S4 hypersurfaces

The classical Einstein-Hilbert action is defined over a Lorentzian spacetime.  The base stays fixed there and one varies the metric to find critical points.  The interpretation is that the solution metric is the gravitational field.  The S4 picture would be the same Einstein-Hilbert action but over hypersurfaces of $S^4(1/h)$ or more properly with time added over hypersurfaces of type $M \times \mathbf{R}$ over $S^4(1/h)\times\mathbf{R}$ with a Lorentzian metric.  The major issues are likely clearer in the case of fixed time.

Consider the action $S_{EH}(M) = \int_M R d\mu_M$ where $M$ is a hypersurface of $S^4(1/h)$.  We want to understand the critical points of this action by analyzing normal perturbations from a fixed hypersurface.  A very useful paper by Gerhard Huisken and Alexander Polden provides us with the tools to accomplish this: HuiskenGeometricEvolutionEquationsHypersurfaces.  Consider the hypersurface evolution function from a fixed base hypersurface via a parametrized map $F_t: M\rightarrow S^4(1/h)$ of isometric immersions.  Given some function $f$ on $M$ and consider the evolution equation $\frac{\partial}{\partial t} F = - f \nu$ where $\nu$ is the normal to the hypersurface $F_o(M)$.  When $f$ is thought of as a small compactly supported perturbation of $M_0$, then the evolution equation is giving us the derivative of the the movement along hypersurfaces.  In this situation the following formulae hold for various geometric quantities: the metric is $g_{ij}$, the second fundamental form $h_{ij}$, the mean curvature $H$ and quantities with a bar on top refer to the ambient manifold in this case of constant sectional curvature $h^2$ and the 0-th coordinate in the ambient manifold is normal to the hypersurface, $|A|^2$ is the squared norm of the second fundamental form.

(i)  $\frac{\partial}{\partial t} g_{ij} = 2 f h_{ij}$

(ii)  $\frac{\partial}{\partial t} (d\mu) = f H d\mu$

(iii)  $\frac{\partial}{\partial t} \nu = -\nabla f$

(iv) $\frac{\partial}{\partial t} h_{ij} =-\nabla_i \nabla_j f + f(h_{ik}h^k_j - \bar{R}_{0i0j}$

(v) $\frac{\partial}{\partial t} H = -\Delta f - f( |A|^2+\bar{Ric}(\nu,\nu) )$

The parameter $t$ does not represent for us time but rather a formal parameter to use for calculus of variations.  We are interested in the hypersurfaces for which $d/dt S_{EH} = 0$.  So

$0 = d/dt \int_M R d\mu = \int_M dR/dt d\mu + \int_M R d/dt(d\mu)$

We can read off the second integrand as $f H R$ from (ii) above.  As for the first term, we have to use the hypersurface scalar curvature formula which amounts to:

$R = \bar{R} - 2 \bar{R}_{00} + H^2 - |A|^2$

and since the ambient manifold has constant curvature, the derivative we have

$dR/dt = 2 H dH/dt - d/dt |A|^2$.

At this point the critical point equation is
$0 = H( 2 dH/dt + fR) + d/dt |A|^2$

We can now read off the first term from (v) above and thus far we get:

$0 = H( -2\Delta f + f(R - |A|^2 + \bar{Ric}(\nu,\nu)) + d/dt |A|^2$

In order to understand this equation a bit better, we note that if $H = 0$ at $t=0$, then the equation says that $|A|^2$ does not change.  The next step is to expand the last term.  So

$|A|^2 = g^{li}g^{kj} h_{lm}h_{ij}$

One calculated $d/dt g^{li}$ using by differentiating $g^{ab}g_{bc} = \delta^a_c$ and then one uses (i) and (iv) above.  More work is needed to understand how a similarity to classical Einstein gravitational equation arises from the critical point equation.