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It is often stated that a distant observer will observe a freefalling object as experiencing an infinite journey to the event horizon, but that a traveler in the local frame will experience a finite-time journey to a singularity.

Is this however true? During the inward journey to a black hole (non-rotating for simplicity) the traveler will accelerate towards the speed of light, while travelling through an increasing gravitational field. At some point the velocity based local length contraction (special relativity) will be overtaken by gravitational length contraction(general relativity), after which point space will continue expanding as observed locally. This point is when the local uniform length contraction (correlating to a locally observed space expansion) from an external frame point of view has exceeded the speed of light. The traveler will then be experiencing standstill in a singularity-free space undergoing an accelerating expansion. After some (local) time, this space will have many characteristics that are similar to the expanding universe in which it is embedded.

Update: In this view, there is even from an external observer point of view no point or ring singularity, but rather the singularity is an ellipsoid, which is inside the event horizon and can be viewed as a compressed boundary, lying a (perhaps infinitesimally) small distance inside the event horizon. In the local reference frame, the proper distance from the event horizon to the sphere singularity is infinite, and a local traveler will experience the "shell" of space between the event horizon and the compressed boundary as an infinite, singularity-free space, which is expanding uniformly in all directions. The traveler cannot observe the singularity, which indeed does not manifest as a singularity in the local reference frame, but rather appears indirectly as the distant edges of a vast, finite, and expanding universe, which at its boundaries is expanding faster than the speed of light, thereby appearing infinite. In this universe, with progression of time, matter will condense into stars. Some of the stars eventually create black holes, wherein new universes emerge.

See also: A Universe without expansion, C. Wetterich, trefoil.math.ucdavis.edu/1303.6878

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  • $\begingroup$ The calculation of the path an infalling observer follows is straightforward and you'll find it in any introduction to GR. The observer hits the singularity in a finite (as measured by the observer) time. $\endgroup$ – John Rennie Dec 25 '13 at 12:38
  • $\begingroup$ Yes, free fall to the singularity in the Schwarzschild metric can easily be shown to have a finite proper time duration. The question assumes GR does not hold generally, which I think is hardly a stretch, all things considered. The question was originally titled "Does space expand locally without restriction in freefall to elusive black hole singularity", meaning while the singularity appears to be real for some observers, it never manifests for our traveler. $\endgroup$ – Halfdan Faber Dec 31 '13 at 1:18
  • $\begingroup$ Removed reference to singularity in title question. $\endgroup$ – Halfdan Faber Dec 31 '13 at 1:59
  • $\begingroup$ I think it's a big stretch to say GR does not hold generally. Yes, we assume some form of quantum gravity will modify GR at the Planck scale, but any claim that GR doesn't hold above this scale is going to have to be backed up by some extraordinary evidence to be taken seriously. $\endgroup$ – John Rennie Dec 31 '13 at 7:01
  • $\begingroup$ Agree. Does not hold generally = does not hold universally. I.e. does not hold in a neighborhood of singularities in GR... $\endgroup$ – Halfdan Faber Jan 1 '14 at 5:25
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Does matter shrink uniformly without restriction in freefall to central region of black hole?

No.

It is often stated that a distant observer will observe a freefalling object as experiencing an infinite journey to the event horizon, but that a traveller in the local frame will experience a finite-time journey to a singularity.

That's what people say. But think about it. Imagine the infalling traveller started falling into the black hole a year ago. His finite proper time hasn't happened yet. If he started falling a thousand years, his finite proper time still hasn't happened yet. Ditto if he started falling a million years ago, or a billion. That finite proper time is a fairy tale I'm afraid.

Is this however true?

No.

During the inward journey to a black hole (non-rotating for simplicity) the traveller will accelerate towards the speed of light, while travelling through an increasing gravitational field.

And meanwhile the "coordinate" speed of light is decreasing. That's the speed of light at some location as measured from this location. See John Rennie talking about that here. Also take a look at Friedwardt Winterberg's Gamma-Ray Bursters and Lorentzian Relativity. He says an infalling body gets destroyed on the way in.

At some point the velocity based local length contraction (special relativity) will be overtaken by gravitational length contraction(general relativity), after which point space will continue expanding as observed locally.

Space doesn't change when you move through it. You and your measurements change when your motion changes. Or see the Shapiro delay, which occurs because "the speed of a light wave depends on the strength of the gravitational potential along its path". Hence your measurements change. But space isn't actually expanding.

This point is when the local uniform length contraction (correlating to a locally observed space expansion) from an external frame point of view has exceeded the speed of light. The traveller will then be experiencing standstill in a singularity-free space undergoing an accelerating expansion.

No, sorry, that's not right. Space just isn't expanding. Instead the traveller is subjected to an increasing gravitational time dilation. He sees external events progressing at an ever-faster rate.

After some (local) time, this space will have many characteristics that are similar to the expanding universe in which it is embedded.

Again sorry, but this just isn't right. A gravitational field is a place where space is "neither homogeneous nor isotropic", this being modelled as curved spacetime. It isn't a place where space is expanding.

In this view, there is even from an external observer point of view no point or ring singularity, but rather the singularity is an ellipsoid, which is inside the event horizon and can be viewed as a compressed boundary, lying a (perhaps infinitesimally) small distance inside the event horizon. In the local reference frame, the proper distance from the event horizon to the sphere singularity is infinite, and a local traveller will experience the "shell" of space between the event horizon and the compressed boundary as an infinite, singularity-free space, which is expanding uniformly in all directions. The traveller cannot observe the singularity, which indeed does not manifest as a singularity in the local reference frame, but rather appears indirectly as the distant edges of a vast, finite, and expanding universe, which at its boundaries is expanding faster than the speed of light, thereby appearing infinite. In this universe, with progression of time, matter will condense into stars. Some of the stars eventually create black holes, wherein new universes emerge.

The universe is akin to an "inside out" black hole where you swap space for time. Moving away from a black hole across space can be likened to the universe expanding over time. But a black hole isn't as similar to the universe as your description. Sorry.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Manishearth Jan 11 '16 at 10:06

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