Amrit: I am not going to make arguments about the ontology of space or time. Spacetime is the field of general relativity. Fields are not really measured directly. The electric and magnetic fields in electromagnetism are inferred by how they induce the motion of currents or pondermotive effects on matter. Vic Stenger makes a big issue out of particles as the only thing which exists, while fields, wave functions, space, time and so forth are only models. Saying that x^4 = ct is just a conversion factor, which takes one measure of a coordinate (time with a clock) and converts it to a length as measured by the proper interval. Everything is moving the speed of light --- even at rest in a reference frame. This could also be argued to be "just a model." Yet, frankly I choose not to engage issues of ontology in physics as having primary importance. Physics is not in my opinion in the business of determining whether fields or wave functions and so forth are actually real or not. Quantum field theory in curved spacetime suggests that particles are frame dependent and that fields are more fundamental, which messes up ideas of particle ontology.
Georgina: Singularities are examples of geodesic incompleteness (b*-incompleteness etc), where geodesics or path come to an end at some region or point on a Riemannian manifold where curvature diverges. There are indeed differences between timelike and spacelike singularities. The Schwarzschild metric is
ds^2 = A(cdt)^2 - (1/A)dr^2 angular parts
with A = 1 - 2GM/rc^2. Now we can look at a region very close to the event horizon r = 2GM/c^2 = 2m. So we have that
A = (r - 2m)/r ~= (r - 2m)/2m, 1/A = r/(r - 2m) ~= 2m/(r - 2m)
We can write this with a further simplification with R = r - 2m and R/2m = ρ. This means the metric near the event horizon looks like
ds^2 = ρdt^2 - (1/ρ)dr^2 ...
which is the Rindler metric. As an exercise, convert dr to coordinates in ρ, set dt = 0 for a spatial surface and compute the proper distance, which will be of the form
du = dρ/sqrt{ρ}.
Now integrate this and then insert into the total metric. The equation is a hyperbola. So the proper time (interval) of an observer with a constant proper distance from the horizon defines a hyperbola. For the metric change with r < 2m the sign of the line element changes and there are spatial hyperbola with constant proper interval from r = 0, where at r = 0 there is a spacelike singularity. The region with spatial hyperbola define the trapping region of the black hole
Timelike singularities are a little stranger, but these have geodesic incompleteness within a timelike region where the above hyperbola which converge to the singularity are timelike. Generally these are problematic, for it means that causal information can end or emerge from them into the timelike region of the universe.
Lawrence B. Crowell
