Two sphere - a spacelike hypersurface Is the surface of a 2-sphere spacelike? What are the corresponding tangent vectors to the surface of a 2-sphere?
This question arises from the point of a trapped surface. In Schwarzschild spacetime the surfaces inside the event horizon are trapped and they are all 2-spheres. Does this mean that trapped surfaces are always 2-spheres? Are there trapped surfaces which are not 2-spheres but other spacelike hypersurfaces?
 A: The Schwarzschild metric in radial coordinates $(t, r, \theta, \phi)$ is given by
$ds^2 = -(1 - 2M/r) dt^2 + (1 - 2M/r)^{-1} dr^2 + r^2 d\theta^2 +r^2 \sin^2 \theta d\phi^2$
where $M$ is the black hole mass.
A hypersurface is defined by a constant function $f$, in case of a 2-sphere $r = constant$. A vector normal to a hypersurface is described by $\xi^\mu = g^{\mu \nu} \nabla_\nu f$, where $\nabla_\mu$ is the covariant derivative (reducing to the partial derivative when applied to a scalar) and $g^{\mu \nu}$ is the inverse metric tensor. As for a 2-sphere, we have
$\xi_\mu = (0, 1, 0, 0)$ dual vector
$\xi^\mu = (0, (1 - 2M/r), 0, 0)$ vector
The norm squared is $\xi^\mu \xi_\mu = (1 - 2M/r)$, which is positive (space-like) when $r > 2M$, that is outside the event horizon, zero (null) when $r = 2M$, that is on the event horizon and negative (time-like) when $r < 2M$, that is inside the event horizon.
As a hypersurface is time-like if the nornal is space-like, null if the normal is null and space-like if the normal is time-like, we have
A 2-sphere is time-like when $r > 2M$, outside the event horizon
A 2-sphere is null when $r = 2M$, on the event horizon
A 2-sphere is space-like when $r < 2M$, inside the event horizon  
The tangent vectors to the 2-sphere are
$\zeta^t = (1, 0, 0, 0)$
$\zeta^\theta = (0, 0, 1, 0)$
$\zeta^\phi = (0, 0, 0, 1)$  
Trapped surfaces not necessarily are 2-spheres, that depends on the metric. For instance trapped surfaces in Kerr metric (rotating black hole) do not exibit a radial symmetry.
