- #1
nutgeb
- 294
- 1
We all know that nothing can travel faster than the speed of light in a local frame. We also know that distant galaxies in our observable universe have recession 'velocities' relative to us which are far in excess of the speed of light, c, despite the fact that the velocity in their local frames never exceeds c. This apparent contradiction is most often resolved by adopting the paradigm that distant galaxies are not in a local reference frame and are not actually moving away from us; instead the 'expanding space' between us causes them to become progressively more distant. As I've mentioned in other threads, in the 'expanding space paradigm' empty (Lambda=0) vacuum doesn't act like some kind of self-reproducing expansion force that pushes objects apart. Instead, the hypersurface substrate of geometry simply expands over time, resulting in increasing separation between 'stationary' comoving objects.
However, there is another well known situation in which a particle (including a photon) has a coordinate velocity faster than c, and the 'expanding space' paradigm is sometimes but not usually invoked to explain it.
That situation is believed to occur inside the event horizon of a black hole. Professors Taylor & Wheeler give a very accessible description of this situation in their textbook 'Exploring Black Holes'. If you don't have that book, some of the Taylor & Wheeler description is paraphrased in the Wikipedia Article on http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates" . Taylor & Wheeler make a three-part argument in favor of faster than light travel:
1. A particle gravitationally accelerated by the BH from rest at an infinite distance plunges radially through the horizon at a coordinate velocity of exactly c, as calculated by a faraway observer (which is the BH's escape velocity at that radius). Since the in-falling particle continues to accelerate gravitationally after it passes the horizon and moves closer to the center, its coordinate velocity must exceed c.
2. They plot the equation:
[tex] \frac{ dr } { dt_{rain }} = - \left( \frac{2M}{r} \right) ^{1/2} [/tex]
where dr is the Schwarzschild r-coordinate and dtrain is their name for the proper time lapse in the frame of a particle plunging radially at escape velocity. Their plot shows that the coordinate velocity increases without limit above c as the particle approaches the center.
3. They show that the elapsed proper time of the in-falling particle from the horizon to the center represents an average speed greater than c.
On the other hand, they emphasize that in its own local rest frame, the in-falling particle never observes light traveling faster than c, and measures in-falling light to pass by it at exactly c. (Note that light inside the event horizon can never travel outward.)
You might think that there isn't much time for an in-falling particle to conduct measurements inside the horizon before the particle is torn apart by growing tidal forces. But the larger the mass of a BH, the more travel time and distance there is inside the horizon before tidal forces become significant. The authors use the example of a "20-year BH" of 1011 solar masses, for which 20 years is required in an in-falling particle's proper time for it to pass from the horizon to the center, even at superluminal coordinate velocities.
The authors also show that a flash of light emitted radially inward from the in-falling particle inside the horizon will have a coordinate velocity faster than the particle's, and therefore even more in excess of c. And they show that a flash radially outward from the in-falling particle will approach the center with a coordinate velocity that can be less than c.
. . . . . . . . . .
So naturally I'm thinking that the 'kinematic' paradigm for homogeneous, expanding, matter-only, FRW space can be analogized to a BH situation. If so, then superluminal recession velocities might also be explainable without resorting to the 'expanding space' paradigm (not that I have a problem with that paradigm), and without introducing any new physics. (Of course we still have to assume that the superluminal recession velocities of the original cosmic expansion resulted from some exotic physics (Big Bang, inflation, whatever) in the initial conditions.) No one knows for sure what happens inside a BH event horizon, or what combination of effects is possible.
In an FRW model, the Hubble Sphere is defined as the distance at which the coordinate velocities of galaxies become superluminal. Apparently recession velocity always exceeds c outside the Hubble Sphere, but never exceeds c inside it.
The Hubble Sphere in a spatially flat FRW model also happens to be the smallest cosmic sphere containing exactly the total mass of matter (matter parameter) which exceeds the BH threshold, in the sense that 2M > r. That's easy to see intuitively, because any radius in a spatially flat FRW model changes at exactly the Newtonian escape velocity of its mass parameter; and the escape velocity at a BH horizon is exactly c. So we see that a spatially flat, homogeneous cosmos always has BH characteristics outside the Hubble Sphere, and never has BH characteristics inside the Hubble Sphere. This is an interesting correlation. It seems entirely logical that galaxies beyond our Hubble Sphere must have coordinate recession velocities > c; otherwise they would all be inevitably collapse into a true BH.
I’ll refer to these Hubble Spheres as cosmic BH’s, even though they have some unique characteristics (such as the fact that they can expand, and the fact that they do not have BH characteristics inside the Hubble Sphere).
[to be continued in another post]
However, there is another well known situation in which a particle (including a photon) has a coordinate velocity faster than c, and the 'expanding space' paradigm is sometimes but not usually invoked to explain it.
That situation is believed to occur inside the event horizon of a black hole. Professors Taylor & Wheeler give a very accessible description of this situation in their textbook 'Exploring Black Holes'. If you don't have that book, some of the Taylor & Wheeler description is paraphrased in the Wikipedia Article on http://en.wikipedia.org/wiki/Gullstrand-Painlev%C3%A9_coordinates" . Taylor & Wheeler make a three-part argument in favor of faster than light travel:
1. A particle gravitationally accelerated by the BH from rest at an infinite distance plunges radially through the horizon at a coordinate velocity of exactly c, as calculated by a faraway observer (which is the BH's escape velocity at that radius). Since the in-falling particle continues to accelerate gravitationally after it passes the horizon and moves closer to the center, its coordinate velocity must exceed c.
2. They plot the equation:
[tex] \frac{ dr } { dt_{rain }} = - \left( \frac{2M}{r} \right) ^{1/2} [/tex]
where dr is the Schwarzschild r-coordinate and dtrain is their name for the proper time lapse in the frame of a particle plunging radially at escape velocity. Their plot shows that the coordinate velocity increases without limit above c as the particle approaches the center.
3. They show that the elapsed proper time of the in-falling particle from the horizon to the center represents an average speed greater than c.
On the other hand, they emphasize that in its own local rest frame, the in-falling particle never observes light traveling faster than c, and measures in-falling light to pass by it at exactly c. (Note that light inside the event horizon can never travel outward.)
You might think that there isn't much time for an in-falling particle to conduct measurements inside the horizon before the particle is torn apart by growing tidal forces. But the larger the mass of a BH, the more travel time and distance there is inside the horizon before tidal forces become significant. The authors use the example of a "20-year BH" of 1011 solar masses, for which 20 years is required in an in-falling particle's proper time for it to pass from the horizon to the center, even at superluminal coordinate velocities.
The authors also show that a flash of light emitted radially inward from the in-falling particle inside the horizon will have a coordinate velocity faster than the particle's, and therefore even more in excess of c. And they show that a flash radially outward from the in-falling particle will approach the center with a coordinate velocity that can be less than c.
. . . . . . . . . .
So naturally I'm thinking that the 'kinematic' paradigm for homogeneous, expanding, matter-only, FRW space can be analogized to a BH situation. If so, then superluminal recession velocities might also be explainable without resorting to the 'expanding space' paradigm (not that I have a problem with that paradigm), and without introducing any new physics. (Of course we still have to assume that the superluminal recession velocities of the original cosmic expansion resulted from some exotic physics (Big Bang, inflation, whatever) in the initial conditions.) No one knows for sure what happens inside a BH event horizon, or what combination of effects is possible.
In an FRW model, the Hubble Sphere is defined as the distance at which the coordinate velocities of galaxies become superluminal. Apparently recession velocity always exceeds c outside the Hubble Sphere, but never exceeds c inside it.
The Hubble Sphere in a spatially flat FRW model also happens to be the smallest cosmic sphere containing exactly the total mass of matter (matter parameter) which exceeds the BH threshold, in the sense that 2M > r. That's easy to see intuitively, because any radius in a spatially flat FRW model changes at exactly the Newtonian escape velocity of its mass parameter; and the escape velocity at a BH horizon is exactly c. So we see that a spatially flat, homogeneous cosmos always has BH characteristics outside the Hubble Sphere, and never has BH characteristics inside the Hubble Sphere. This is an interesting correlation. It seems entirely logical that galaxies beyond our Hubble Sphere must have coordinate recession velocities > c; otherwise they would all be inevitably collapse into a true BH.
I’ll refer to these Hubble Spheres as cosmic BH’s, even though they have some unique characteristics (such as the fact that they can expand, and the fact that they do not have BH characteristics inside the Hubble Sphere).
[to be continued in another post]
Last edited by a moderator: