How Does Gravitational Time Dilation Affect Light Emitted from a Black Hole?

[Naty1]

Based on recommendations here, I'm reading Kip Thorne's BLACK HOLES AND TIME WARPS which was somewhat uninteresting during the first 100 pages or so but then he redeems himself...here's some excerpts I found insightful and clearly stated (pgs 130-133) After noting that time moves more slowly at the surface of a star as viewed by a stationary distant observer he says:

One consequence of time warpage is the gravitational redshift of light emitted from a star's surface. Since the light's frequency is governed by the flow of time at the place where the light is emitted, light emerging from atoms on a star's surface will have a lower frequency when it reaches the Earth than light emitted by the same kind of atoms in interstellar space. The frequency will be lowered by precisely the same amount as the flow of time is slowed...for a star whose actual circumference was the same as it's critical one (as predicted by Schwarzschild geometry) the flow of time at the star's surface is infinitely dilated; time does not flow at all...in modern language, the stars surface, with it's critical circumference is precisely at the horizon of a black hole...this infinite shift in wavelength, in effect, removes all the light's energy; and the light, thereupon, ceases to exist!

So the overall explanation, I think, ties together a number of concepts...I'm wondering how others might explain the boldface phrase...sounds on one hand like time warps energy! Or is it more correct to view this as only an analogy, with the real explanation being gravitational potential is behind the shift (both in time and wavelength)?

 

[rrogers]

I don't understand your problem. "Energy" and time both transform the same under Lorentz transforms. Remember energy is not power. You can reasonably (but carefully) interpret the frequency shift either as time dilation or potential loss in terms of classical physics; but you really should treat the GR theory on it's own terms as a geometric theory that we make up labels (energy and time) to think with. Labels are used to provide context for our mind, but can be misleading.

Ray

 

[stevebd1]

Hi Naty1

Personally I see the energy from the redshifted light trying to escape the BH, in some way, being transferred to light falling into the black hole (which in turn is infinitely blueshifted) so the first law of thermodynamics applies at least up to the EH of the BH. It also explains why the invariant curvature scalar remains flat at the event horizon (the energy of the light falling in plus the energy of light trying to escape is equivalent to the isotopic energy of the light in flat space). It's supposedly a different kettle of fish at the Cauchy horizon where all light trapped by the event horizon, blueshifted (ingoing) and redshifted (outgoing), is pulled towards the inner horizon and all frequencies (including flat & finite) are infinitely blueshifted, causing the invariant curvature scalar to supposedly diverge.

 

[rrogers]

Sorry about the term "Lorentz transform". I meant the generalized transform between tangent spaces defined by parallel transport. The point being that time and energy are the same component positions; timelike.

Ray

 

[DrGreg]

Naty1

Energy is a frame dependent concept. Even in Newtonian mechanics, an object can have kinetic energy in one frame and none in another.

Here, the frame we are using is one of an observer hovering outside the event horizon. In this frame any photon rising from the black hole is gaining potential energy and losing frequency-energy i.e. red-shifting as it rises.

If we "run time backwards" to work out where the photon has come from, as it nears the horizon its frequency-energy increases to infinity, which is impossible; so photons cannot escape the event horizon. Note that the frame of the hovering observer stops at the event horizon and can't be used to measure anything (e.g. photon energy) at or inside the event horizon.

I would say that, at the event horizon, it's not really that the light "ceases to exist"; it's the observer's frame that ceases to exist, so you can't measure energy in that frame (but you could in other frames).