Hi,

I have been playing around with some figures in a spreadsheet linked to the Schwarzschild metric and the equivalent solution in terms of

As a very broad generalisation, it appears that the Gullstrand-Painlevé solution to the normal Schwarzschild metric essentially replaces [dt] with the perception of time in the free-falling frame [dtc]. The objective of this solution is to avoid the ‘

If we assume that a series of stationary shell {B} observers are positioned along the free-falling radial path, they would be able to resolve their relative time with respect to some remote distant observer {A} in flat spacetime. Time in each {B} frame would slow with respect to {A} due to gravity only. However, as {C} passes each {B} observer on-route towards the black hole horizon at [Rs]; {B} would observe an additional slowing of time in the {C} frame due to its relative and increasing velocity. As such, it should be possible to prove that the effects of time dilation, as described, are real.

My second question relates to the velocity of the free-falling observer and the speed of light after crossing the event horizon. A quadratic solution of Gullstrand-Painlevé equation with respect to [dtc] arrives at the classical free-fall velocity:

[1] [tex]v = -c \sqrt { \frac {Rs}{r} } [/tex]

Using the same approach, but setting the variable [dtau=0], gives up the following velocity of a photon in the {C} frame:

[2] [tex]v = -c \sqrt { \frac {Rs}{r} } \pm c [/tex]

As a footnote to the last question, the following equation also appears to suggest that time taken to fall from the event horizon [Rs] to the central singularity [r=0] must involve a velocity [v] in excess of [c]?

[3] [tex]dtc = \frac {2Rs}{3c}[/tex]

Therefore, I would appreciate any further insights, clarifications or corrections to any of the assumptions above. Thanks

I have been playing around with some figures in a spreadsheet linked to the Schwarzschild metric and the equivalent solution in terms of

*http://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates" [Broken]*. This spreadsheet is only considering the specific case of a free-falling observer {C} such that the relativistic effects of velocity and gravity are both proportional to the coordinate radius [r]. However, there are some aspects that I am not sure I understand.As a very broad generalisation, it appears that the Gullstrand-Painlevé solution to the normal Schwarzschild metric essentially replaces [dt] with the perception of time in the free-falling frame [dtc]. The objective of this solution is to avoid the ‘

*coordinate singularity’*at [r=Rs], which occurs with the Schwarzschild metric. While this approach does seem to lead to a more consistent solution in the sense that the free-falling time and velocity remain continuous through the event horizon at [r=Rs], it not clear to me how some of the more fundamental issues are resolved.If we assume that a series of stationary shell {B} observers are positioned along the free-falling radial path, they would be able to resolve their relative time with respect to some remote distant observer {A} in flat spacetime. Time in each {B} frame would slow with respect to {A} due to gravity only. However, as {C} passes each {B} observer on-route towards the black hole horizon at [Rs]; {B} would observe an additional slowing of time in the {C} frame due to its relative and increasing velocity. As such, it should be possible to prove that the effects of time dilation, as described, are real.

*So my first question is that while the free-falling observer may appear to sail through the event horizon, ignoring tidal effects at this point, how do you relate the time in {C} back to {A}?*My second question relates to the velocity of the free-falling observer and the speed of light after crossing the event horizon. A quadratic solution of Gullstrand-Painlevé equation with respect to [dtc] arrives at the classical free-fall velocity:

[1] [tex]v = -c \sqrt { \frac {Rs}{r} } [/tex]

Using the same approach, but setting the variable [dtau=0], gives up the following velocity of a photon in the {C} frame:

[2] [tex]v = -c \sqrt { \frac {Rs}{r} } \pm c [/tex]

*While this appears to preserve the velocity of light [c] with respect to the free-falling observer {C} is there a suggestion that this process somehow exceeds the normal speed of light?*As a footnote to the last question, the following equation also appears to suggest that time taken to fall from the event horizon [Rs] to the central singularity [r=0] must involve a velocity [v] in excess of [c]?

[3] [tex]dtc = \frac {2Rs}{3c}[/tex]

Therefore, I would appreciate any further insights, clarifications or corrections to any of the assumptions above. Thanks

Last edited by a moderator: