Timeline for Intuitively, why do attempts to delay hitting a black hole singularity cause you to reach it faster?
Current License: CC BY-SA 4.0
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| Oct 5, 2018 at 16:59 | comment | added | Colin MacLaurin | The quantity $dr/d\tau$ is not called "time-dilation". You might call it the "$r$-component of the 4-velocity", assuming coordinates including Schwarzschild $r$ are being used. In curved spacetime, time-dilation is well-defined for two observers at the same event (same place and time), but not defined between separated observers. One exception is with a "stationary" spacetime, where the time symmetry lets you compare a "gravitational time-dilation" between any two observers at rest. But spacetime is not stationary inside the Schwwarzschild horizon. | |
| Sep 6, 2018 at 4:14 | comment | added | safesphere | The centrifugal force reversal happens outside the horizon: adsbit.harvard.edu/full/1990MNRAS.245..720A - There are no forces in the radial direction inside, because it is the direction in time. Your analysis suggests that time inside depends on energy, but the total time inside is constant for all bodies and is $r=2M$. The different part is their proper time. The ratio of time to proper time $\dfrac{dr}{d\tau}$ refers to time dilation. While the gravitational potential depends on the time dilation, the dilation simply follows directly from the metric with no need to refer to energy. | |
| Sep 4, 2018 at 21:31 | vote | accept | tparker | ||
| Sep 4, 2018 at 20:52 | comment | added | Colin MacLaurin | indeed you do... I miscounted minus signs! | |
| Sep 4, 2018 at 20:40 | comment | added | tparker | @ColinMacLaurin Yes, I use the usual sign convention for $E$ in my answer, not Lewis and Kwan's convention. My $E$ is indeed positive for a geodesic that extends outside the horizon - I took out the incorrect statement that it's negative. | |
| Sep 4, 2018 at 20:38 | history | edited | tparker | CC BY-SA 4.0 |
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| Sep 4, 2018 at 20:31 | comment | added | Colin MacLaurin | the usual convention is to define E as positive (for all observers in the exterior region r>2M), the Lewis & Kwan paper notwithstanding. Also it helps to give E a name: the "Killing energy (per mass)", also interpreted as the "energy (per mass) measured at infinity" | |
| Sep 4, 2018 at 2:52 | vote | accept | tparker | ||
| Sep 4, 2018 at 2:52 | |||||
| Sep 4, 2018 at 2:51 | history | answered | tparker | CC BY-SA 4.0 |