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Zanket
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This is an analysis involving general relativity (GR). Let a spherically symmetric, uncharged, nonrotating star (not a black hole—that comes later) have an escape velocity = infinitesimally less than c at its surface. Then its r-coordinate radius (also known as the reduced circumference and hereafter just R) is infinitesimally above its Schwarzschild radius. Let an observer having negligible mass free-fall from rest at infinity to traverse the star via a tunnel drilled diametrally through it. Escape velocity is the velocity needed by an object to rise freely from a given altitude to rest at infinity. An object’s fall reverses its rise, so an object free-falling from rest at infinity passes each altitude at the escape velocity for that altitude. Then the observer enters the tunnel at infinitesimally less than c, accelerates to reach a maximum velocity still infinitesimally less than c at the star’s center, and then decelerates to exit the tunnel at infinitesimally less than c, where these velocities are all directly measurable.
The equivalence principle tells us that special relativity (SR) applies in each infinitesimal region of spacetime that the observer passes. Then the Lorentz factor of SR tells us that, because the observer passes each and every infinitesimal region of spacetime within the star at infinitesimally close to c, the observer passes each and every infinitesimal region of spacetime within the star in an infinitesimal proper time (time elapsed on the observer’s watch). The sum of any number of infinitesimal times is an infinitesimal time, so the observer traverses the whole star in an infinitesimal proper time, regardless of the magnitude of R. Were the R of the star even a million light years, the observer would still traverse it in an infinitesimal proper time.
Now consider a black hole. A GR equation for the proper time of the same observer (that is, an observer having free-fallen from rest at infinity) to fall from the black hole’s event horizon to its central singularity is proportionate to the black hole’s mass (equation provide upon request). The more mass, the more proper time elapses for the observer. The R of a black hole is proportionate to its mass. So the greater the R of the black hole, the more proper time elapses for the observer.
Here is the riddle: How do you explain that the observer could take millions of proper years to fall from the black hole’s event horizon to its central singularity, for a black hole with a certain R, whereas the observer could traverse the whole of a star having the same R in an infinitesimal proper time? After all, the only relevant difference between these objects is that the black hole has an infinitesimally greater mass.
(Don’t ask me; I don’t have a mainstream answer. I don’t see how it can be a “time/space coordinates are reversed below the event horizon” thing—that doesn’t seem to explain how an infinitesimal time for the star becomes x number of years when a gram of material is added to the star to make it a black hole.)
The equivalence principle tells us that special relativity (SR) applies in each infinitesimal region of spacetime that the observer passes. Then the Lorentz factor of SR tells us that, because the observer passes each and every infinitesimal region of spacetime within the star at infinitesimally close to c, the observer passes each and every infinitesimal region of spacetime within the star in an infinitesimal proper time (time elapsed on the observer’s watch). The sum of any number of infinitesimal times is an infinitesimal time, so the observer traverses the whole star in an infinitesimal proper time, regardless of the magnitude of R. Were the R of the star even a million light years, the observer would still traverse it in an infinitesimal proper time.
Now consider a black hole. A GR equation for the proper time of the same observer (that is, an observer having free-fallen from rest at infinity) to fall from the black hole’s event horizon to its central singularity is proportionate to the black hole’s mass (equation provide upon request). The more mass, the more proper time elapses for the observer. The R of a black hole is proportionate to its mass. So the greater the R of the black hole, the more proper time elapses for the observer.
Here is the riddle: How do you explain that the observer could take millions of proper years to fall from the black hole’s event horizon to its central singularity, for a black hole with a certain R, whereas the observer could traverse the whole of a star having the same R in an infinitesimal proper time? After all, the only relevant difference between these objects is that the black hole has an infinitesimally greater mass.
(Don’t ask me; I don’t have a mainstream answer. I don’t see how it can be a “time/space coordinates are reversed below the event horizon” thing—that doesn’t seem to explain how an infinitesimal time for the star becomes x number of years when a gram of material is added to the star to make it a black hole.)