Question on Singularities

[Jaepheth]

My question is, do objects that fall into a singularity or black hole ever reach the center?

My reasoning is, that a particle falling towards the center would be passing through an increasingly powerful gravity field, and the stronger gravity would slow time down, so the closer it gets to the center, the slower it's moving to an outside observer. Would the center then be the asymptote that the particle approaches, but never reaches?

[Kuroneko]

My question is, do objects that fall into a singularity or black hole ever reach the center?

Absolutely. In fact, by playing with the Schwarzschild metric, one can derive that for radial infall (dθ=dφ=0), freefall is completely Newtonian in proper time in the Schwarzschild radial coordinate r. Actually, it is possible to say much more than this, with some neat relations between the coordinates--for example, if the freefall is not constrained in the radial direction, the specific angular momentum is also exactly Newtonian in proper time, and this quantity is conserved (it is not conserved relative to a non-freefalling observer).

My reasoning is, that a particle falling towards the center would be passing through an increasingly powerful gravity field, and the stronger gravity would slow time down, so the closer it gets to the center, the slower it's moving to an outside observer. Would the center then be the asymptote that the particle approaches, but never reaches?

Here, your reasoning pertains to something completely different. Whether or not the particle reaches the singularity is a question about the particle, and should be answered in its proper time--relative to which, the answer is an unambiguous yes, since it experiences Newtonian freefall relative to its own proper time, as measured by the Schwarzschild radial coordinate r. I emphasize this last bit because although this is what an idealized stationary observer at infinity would measure as the object's distance from the singularity, it is not what the infalling object would observe. As the object approaches the event horizon, it is accelerated relative to an observer hovering near the horizon, and the relative speed approaches the speed of light (this should not be suprising; the event horizon has an escape velocity of the speed of light, and infalling trajectories are identical to escape ones, just time-reversals of one another). Ordinary Lorentz-contraction kicks in, and a small distance in the r-coordinate may be a very large one in the object's reference frame. (Why not the other way around? Because the stationary objects are the ones that are accellerated in GTR, and freefalling ones are inertial.)

The slowing-down effect you reference is relative to an external observer, and leads to what may be construed as a kind of paradox: does the object reach the singularity? According to it, yes. Is the object ever observerd to reach the singularity? According to anyone that stays away from the event horizon, no, never; instead, it is observed to hover near the event horizon, ever-slower, and ever-fainter. (If one inspects the Schwarzschild metric, one will see that the time coordinate is singular at r = 2GM/c^2, which is a finite distance from the singularity, so the object is observered to be asymptotic to this distance rather than the singularity.)

[Zoink]

What I read:

The asymptote occurs on the event horizon. You don't see the person enter because they are essentially frozen in time on the event horizon. They fall into the black hole in some imaginary time in the future from your perspective.

From the perspective of the person falling in, you pass the event horizon and fall towards your death at the singularity, if you survive that long.

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What I don't understand is that if you pile in tons of matter the event horizon will increase, so I would think its possible to see someone 'disappear' into the event horizon in some cases?!?

With Hawking's radiation, the black hole will eventually evaporate. Wouldn't you observe them being cooked by the Hawking's radiation. You can't both be cooked and fall in at the same time...

[Kuroneko]

The asymptote occurs on the event horizon. You don't see the person enter because they are essentially frozen in time on the event horizon. They fall into the black hole in some imaginary time in the future from your perspective.

No. In 'your' perspective, the object is never observerd to cross the event horizon.

From the perspective of the person falling in, you pass the event horizon and fall towards your death at the singularity, if you survive that long.

Yes.

What I don't understand is that if you pile in tons of matter the event horizon will increase, so I would think its possible to see someone 'disappear' into the event horizon in some cases?!?

No. If the event horizon increases, as it must, what will be observed is that the hovering images of the objects that approached the singularity are actually pushed outward.

With Hawking's radiation, the black hole will eventually evaporate. Wouldn't you observe them being cooked by the Hawking's radiation. You can't both be cooked and fall in at the same time...

Er, no. Hawking radiation, like Unruh radiation, is an effect of acceleration. The freefalling objects are inertial in GTR, hence they will experience no Hawking radiation (well, not quite; if the tidal force near the horizon is large enough, the object will not be in perfect freefall). Something else to consider: if the emission of light is quantized into individual photons, there will be a last photon emitted as the object crosses the event horizon in its own frame, so, quantum-mechanically, one can't observe the images of the objects hovering there forever, even if one has pseudomagic amplification device to compensate for the ever-increasing redshift.

[Braedley]

My head hurts...