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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?

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  • $\begingroup$ If someone says "2-sphere" to me in the context of GR, the connotation is that they're talking about something that's topologically a 2-sphere, and nothing more specific than that. I'm not even sure that there's a useful metrical definition of a 2-sphere in a Riemannian or semi-Riemannian space that plays the same role as the Euclidean notion of a sphere. If there is such a definition, it's not obvious to me whether such spheres exist in an arbitrary Riemannian or semi-Riemannian space. $\endgroup$ – Ben Crowell Apr 11 '18 at 15:36
  • $\begingroup$ Re the definition of spacelike, timelike, and null surfaces, I have a discussion of how to do this in a coordinate-independent way in my SR book, section 7.6: lightandmatter.com/sr . $\endgroup$ – Ben Crowell Apr 11 '18 at 15:39
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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.

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  • $\begingroup$ How did you calculate the tangent vectors to a 2-sphere? Also, are all trapped surfaces spacelike? $\endgroup$ – Khushal Apr 11 '18 at 15:18
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    $\begingroup$ Given the way in which the OP asked the question, I think it may not be obvious to them that your definitions here are coordinate-independent, even though you give concrete examples in Schwarzschild coordinates. $\endgroup$ – Ben Crowell Apr 11 '18 at 15:41
  • $\begingroup$ As for the tangent vectors, it is easy to check that the scalar product of each tangent vector with the normal vector is zero. You compose them with the dual vector expression of the normal vector. $\endgroup$ – Michele Grosso Apr 11 '18 at 16:37

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