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## Main Question or Discussion Point

I am in the process of trying to learn about some of the details that support general relativity. Therefore, my question is a genuine inquiry and not intended as oblique speculation. At the moment, I only have a basic understanding of general relativity in terms of all the apparent complexity of tensors and differential geometry; therefore I am initially focusing on the Schwarzschild metric, as it seems to encompass most of the basic principles. However, it also seems to lead towards a number of anomalies that I have not been able to fully resolve, which I was hoping a member of this forum might help me clarify. As a cross-referenced, I have also raised another thread regarding `

Anyway, back to the question at hand, starting with the Schwarzschild metric. I prefer not to use geometric units so that I have keep track of the `

[tex] c^2 d\tau = c^2\left(1-Rs/r\right)dt^2-\left(1-Rs/r\right)^{-1}dr^2 - r^2d\theta-r^2sin^2\theta d\phi^2[/tex]

By only considering free-fall radial paths and circular equatorial orbits this expression can be simplified to:

[tex] c^2 d\tau = c^2\left(1-Rs/r\right)dt^2-\left(1-Rs/r\right)^{-1}dr^2-r^2 d\phi^2[/tex]

Many standard texts then proceed to solve this equation by dividing through by [tex][dt^2][/tex] or [tex][d\tau^2][/tex] to resolve the implications of general relativity from the perspective of either a distant observer [dt] or an onboard observer [tex][d\tau][/tex]. In the context of a free-falling observer, an anomaly appears to arise because the Schwarzschild metric suggests that the distant observer will `

To establish some physical comparison between these two frames of reference, i.e. distant observer [tex][dt=A][/tex] and free-falling observer [tex][d\tau=B][/tex], we might consider that our two observers (A) and (B) are twins. Overall, there seems to be little argument that the twin free-falling into a black hole is usually on a one-way trip, i.e. at some point there will be no escaping the physical singularity hidden behind the event horizon. However, what if we assume that the our free-falling twin does not initially plunge into the black hole, but only goes as close as possible before returning to (A).

As far as I can see, theory and consensus seems to suggest that twin (B) will indeed be younger than twin (A). As such, the time dilation that results from the increased gravity and velocity is a real effect and not just an aberration of the mathematics. However, this line of thought appears to lead to the conclusion that time will freeze at the event horizon. In contrast, solutions of both the Schwarzschild metric and the Gullstrand-Painleve variant for [tex]dr/d\tau[/tex] suggest that:

[tex] dr/d\tau = -c\sqrt{\left(\frac{Rs}{r}\right)}[/tex]

It appears this equation can even be solved by integration:

[tex] d\tau = -\left(\frac{1}{c}\right)\int\left( \frac{Rs}{r}\right)^{-1/2} dr[/tex]

[tex]d\tau = -\left(\frac{1}{c}\right)\left[ \frac {2/3Rs} {\left(Rs/r\right)^{1.5}}\right]^{0}_{Rs} = \frac{2}{3}*\frac{Rs}{c}[/tex]

If the integration is right, it suggests that it would take about 22us to travel from the event horizon to the central singularity of a small black hole with a mass [M] of about 4 solar masses. On the other hand, this would increase to ~6hours for the black hole said to exist at the centre of our galaxy. However, all this said, I am still left with the nagging question:

As a side issue, it would seem that relativity is restricted to describing a black hole purely in terms of mass and gravity without really considering the quantum implications of the underlying nature of matter. The following links extend this discussion beyond the guidelines of this forum; therefore they are only referenced for interest and not discussion:

http://www.physorg.com/news101560368.html

http://arstechnica.com/news.ars/post/20070622-apotential-solution-to-the-black-hole-information-loss-paradox.html

http://www.sciencenews.org/articles/20040925/bob9.asp

However, I would be interested in any other references, which might expand on this aspect of modern research.

*Effective Potential*` that is unresolved, at least, in my mind.Anyway, back to the question at hand, starting with the Schwarzschild metric. I prefer not to use geometric units so that I have keep track of the `

*real*` units, e.g. [tex]Rs=2GM/c^2[/tex] which reduces to [2M] in geometric units. Therefore, the following form includes the speed of light [c].[tex] c^2 d\tau = c^2\left(1-Rs/r\right)dt^2-\left(1-Rs/r\right)^{-1}dr^2 - r^2d\theta-r^2sin^2\theta d\phi^2[/tex]

By only considering free-fall radial paths and circular equatorial orbits this expression can be simplified to:

[tex] c^2 d\tau = c^2\left(1-Rs/r\right)dt^2-\left(1-Rs/r\right)^{-1}dr^2-r^2 d\phi^2[/tex]

Many standard texts then proceed to solve this equation by dividing through by [tex][dt^2][/tex] or [tex][d\tau^2][/tex] to resolve the implications of general relativity from the perspective of either a distant observer [dt] or an onboard observer [tex][d\tau][/tex]. In the context of a free-falling observer, an anomaly appears to arise because the Schwarzschild metric suggests that the distant observer will `

*see*` time [dt] stop for the free-falling observer at the event horizon [Rs]. However, the same equation suggests that the perspective for the free-falling observer is very different. At this point, some texts highlight the difference between a physical singularity and a coordinate singularity and make reference to specific variations of the Schwarzschild metric, e.g. Gullstrand-Painleve or Eddington-Finkelstein, which are said to avoid the coordinate singularity associated with the standard metric when [r=Rs]. So my expectation was that the Gullstrand-Painlevé solution would resolve many of the apparent anomalies arising from the Schwarzschild metric. However, on review, I am not sure that this variant simply avoids the coordinate singularity rather than resolving it. This is not a statement of fact, simply my understanding so far.To establish some physical comparison between these two frames of reference, i.e. distant observer [tex][dt=A][/tex] and free-falling observer [tex][d\tau=B][/tex], we might consider that our two observers (A) and (B) are twins. Overall, there seems to be little argument that the twin free-falling into a black hole is usually on a one-way trip, i.e. at some point there will be no escaping the physical singularity hidden behind the event horizon. However, what if we assume that the our free-falling twin does not initially plunge into the black hole, but only goes as close as possible before returning to (A).

*Will the free-falling twin be younger?*As far as I can see, theory and consensus seems to suggest that twin (B) will indeed be younger than twin (A). As such, the time dilation that results from the increased gravity and velocity is a real effect and not just an aberration of the mathematics. However, this line of thought appears to lead to the conclusion that time will freeze at the event horizon. In contrast, solutions of both the Schwarzschild metric and the Gullstrand-Painleve variant for [tex]dr/d\tau[/tex] suggest that:

[tex] dr/d\tau = -c\sqrt{\left(\frac{Rs}{r}\right)}[/tex]

It appears this equation can even be solved by integration:

[tex] d\tau = -\left(\frac{1}{c}\right)\int\left( \frac{Rs}{r}\right)^{-1/2} dr[/tex]

[tex]d\tau = -\left(\frac{1}{c}\right)\left[ \frac {2/3Rs} {\left(Rs/r\right)^{1.5}}\right]^{0}_{Rs} = \frac{2}{3}*\frac{Rs}{c}[/tex]

If the integration is right, it suggests that it would take about 22us to travel from the event horizon to the central singularity of a small black hole with a mass [M] of about 4 solar masses. On the other hand, this would increase to ~6hours for the black hole said to exist at the centre of our galaxy. However, all this said, I am still left with the nagging question:

*How old is twin (A) when twin (B) hit the black hole?*As a side issue, it would seem that relativity is restricted to describing a black hole purely in terms of mass and gravity without really considering the quantum implications of the underlying nature of matter. The following links extend this discussion beyond the guidelines of this forum; therefore they are only referenced for interest and not discussion:

http://www.physorg.com/news101560368.html

http://arstechnica.com/news.ars/post/20070622-apotential-solution-to-the-black-hole-information-loss-paradox.html

http://www.sciencenews.org/articles/20040925/bob9.asp

However, I would be interested in any other references, which might expand on this aspect of modern research.