How can Block Holes form in finite time?

[skeptic2]

An observer falling in and looking backwards still sees the stars because as you have confirmed, the photons following your trajectory still overtake you.

If an object falls towards a black hole from infinity, at large distances from the black hole its infalling velocity is equal in magnitude to the escape velocity at its location, is it not? If its infalling velocity is less than c at the event horizon, what factor has reduced the acceleration of the object?

If you were falling at c at the event horizon, you would still be able to see the stars behind you. Normally you would think that traveling at c, the light from the stars behind you would be red shifted to zero but the light from those stars is also falling towards the black hole and has been blue shifted by the same amount that your velocity would have red shifted it.

 

[Trenton]

skeptic2,

There might actually be a discrepancy. If we consider the case of particles accelerated in an accelerator, the velocity could never get close to c because the particle just gets heavier and so requires more energy per unit increase in velocity. If we then tried to extract energy from the particle we would find (except for losses in the equiptment) that it had as much energy as was put in.

When matter falls into a black hole though, increases in mass due to velocity are matched by extra gravitational attraction. This will not allow acceleration to c (in my view) because the falling process occurs over finite space and time.

But it makes escape impossible for matter because no matter how fast it's initial upward velocity, it will have to climb against a force that matches it. In this sense and only in this sense, can it be said that the escape velocity is c - and then this would be hopelessly misleading.

It is far better to say escape for matter is impossible because for all values of energy (an infinite scale), the gravity will always just overwhelm.

Saying the escape velocity is c is a nonsense in two ways. It suggest that if something could reach c (which would make it more massive by an infinite factor that the entire observable universe). And it leads to the notion that light cannot escape (this would only be true if time stands still at Rs which I think is wrong)

The idea that time stops at Rs where the gravitational field is finite seems to me to be wrong. The field equations do have two solutions for singularities, one at Rs and one at R=0. I think (much to the horror of others) that singularity means a region of stopped time and infinite density etc which CAN ONLY occur at R=0. I question if such a thing can actually form but I consider singularity at Rs to be invalid solutions. After all, if matter cannot reach c with finite energy how can time stop in a finite gravitational field?

 

[JesseM]

Trenton, the problem is that you try to import ideas from Newtonian gravity directly into GR and it doesn't work, GR is a very different theory. For example:

But it makes escape impossible for matter because no matter how fast it's initial upward velocity, it will have to climb against a force that matches it. In this sense and only in this sense, can it be said that the escape velocity is c - and then this would be hopelessly misleading.

There is no such thing as a gravitational "force" in general relativity! Gravitational effects are due to the fact that matter curves spacetime, and free-falling objects follow "geodesic" paths through this curved spacetime. Some intros to the basic concepts of GR, including curved spacetime and geodesics, can be found in this thread.

Saying the escape velocity is c is a nonsense in two ways. It suggest that if something could reach c (which would make it more massive by an infinite factor that the entire observable universe).

No, I've already told you it doesn't suggest that. In Newtonian physics it's true that an object falling in from infinity will have a velocity equal to the escape velocity when it hits the surface of a planet (or star or whatever), as seen in the inertial frame where the planet is at rest. But in GR escape velocity doesn't have this meaning! For one thing there is no possibility of an "inertial frame" in a large region of curved spacetime, you have an infinite variety of equally valid non-inertial frames and they all measure "velocity" differently (they also don't necessarily say that light itself has a constant velocity). As I explained to you, in a very small patch of spacetime you can have a "local inertial frame" where an object in free-fall is at rest, and where the laws of physics are like those of SR (including the idea that light moves at c), thanks to the equivalence principle. But in any local inertial frame at the horizon, it will be the horizon itself that's moving outward at c, not the free-falling object.

And it leads to the notion that light cannot escape (this would only be true if time stands still at Rs which I think is wrong)

There is no need for time to stand still at the horizon, you can pick coordinate systems where the time dilation experienced by clocks falling through the horizon is finite, and they cross the horizon at a finite coordinate time, like Eddington-Finkelstein coordinates (if you're weirded out by coordinate systems where the horizon is moving outward, you may like this one better since it does have the horizon at a fixed radial coordinate). In these coordinates, the fact that light can't escape from a black hole is explained by the fact that the future light cones of events closer to the horizon become increasingly tilted, until at the horizon the entire future light cone is inside or on the horizon, as illustrated in this diagram from the textbook Gravitation:

[PLAIN]http://www.valdostamuseum.org/hamsmith/DFblackIn.gif

The idea that time stops at Rs where the gravitational field is finite seems to me to be wrong.

Please note that the question of whether "time stops" has no coordinate-independent meaning, it simply depends on how you choose to define your coordinate system. As I mentioned earlier, one can define a coordinate system where "time stops" at any arbitrary boundary like the center of your room, in the sense that in this coordinate system the ratio of your clock time to coordinate time would approach zero as you approached the center, so the event of your crossing the boundary of the center would not be assigned any finite time-coordinate. Please read up on "diffeomorphism invariance" in this article if you have trouble with the idea that non-inertial coordinate systems can be defined in absolutely any way we want them to.

 

[Trenton]

As regard to co-ordinates, I accept a lack of fluency in transforming these at will. The co-ordinates I use by default are spherical and have the center of the black hole rather than any point on the event horizon as the origin. I can't add the time progressions to the coordinates at the moment because all the ones inside Rs seem not to progress according to the formula. This seems to have horrible implications for frame dragging.

I am fully aware that in making this statement I will get a barrage of complaints that I just don't get GR and to a certain extent I don't. I deny the charge of being stuck on Newton. I might be prepared to admit being stuck with SR. I my defence I would say I only did a module in SR. There wasn't one on GR on the course I did!

However, I think the GR concepts are simple enough but the language used to describe them are too easy to be taken out of context. In just the same way as one can choose any co-ordinate regime you want you can also choose how something is described. You can translate between co-ordinate systems and you can translate between terms. In fact doing so, highlights exactly where GR's advantage over SR and SR's advantage in Newtonian mechanics lie - and it serves to check concepts have been appropriately or inapproapriately mapped between the three different regimes.

Take the 'curvature of spacetime' and the geodesic. Light is said to travel in a straight line but it has to follow a geodesic so it describes a curved path. Such is the language of GR but this is surely not the only valid way of explaining it even if it might be the most convient method of applying the math. It is not invalid to say light falls under gravity even in so thinking, one might have trouble deriving the location of such as the photon sphere.

Another GR term that has been brought to my attention is that there is no such thing as the force of gravity - only the curvature of spacetime due to the presence of mass. This distinction is quite correct of course but why is it correct? Probably only because time dilation is a function of gravitational potential and not of field strength as might be tempted to think if the word 'force' were to leave one's lips instead of the word 'curvature'. Or is there a more fundamental reason for the distiction?

One of the things I have studied is NLP (neuro-linguistic programing). This makes very clear why it is important to explain the terms of a superior model in terms of an inferior one - it is to easy to think one has a complete theory in one's mind when in fact key tennets have been learned by rote or are the results of unsound conceptual leaps.

I want to make it absoutely clear I am NOT suggesting that anyone in this forum is making that mistake but I have in the past managed to tie up in knots, more than one public speaker at meetings on astronomy. This was not my intention as all I wanted was to understand certain assertions. In one example the speaker asserted that at the exact point where the radius of a nuetron star becomes equal to Rs (having been hit by one last cosmic ray) - the whole entity collapses to a point. This did not seem to me to be proven and since the equation of state and the degeneracy limits of neutonium is not fully known, I asked if the rusult could simply be a cloaked neutron star. He started out thinking he could answer the question but quickly descended into incoherence with several members of the audiance baying for blood (none of whom understood more than SR). It was a shame because it ruined what had otherwise been a well executed and absolutely exhillirating presentation.

All that said, is there anyting wrong with the idea that (except where curvature is such that time has stopped), it is impossible for matter to effect displacement at the same rate as a photon? (in any set of co-ordinates)

Likewise does matter having realized some gravitational potential, accuire mass by virtue of velocity (relative to the center of the black hole if I must)? And if so, could this be said to account for any descrepancy between velocity and escape velocity along the infalling trajectory. Or similarly, could degrading orbits be explained in these terms just as they can in GR speak? Or is there no deviation between velocity and escape velocity along the trajectory? And is matter falling into a black hole somehow bereft of mass aquisition?

In a similar vane, what happens in erergy terms to light escaping very dense objects? If a neutron star emits say 1 million tonnes of light per second (energy equivelence) but only a proportion of this escapes due to the red shift, I am guessing the net loss of mass to the star is only what escapes? What are the machanisums involved here?

And in reverse, of infalling CBR microwaves blue shifted to far gamma rays by black holes, what would the rate of mass aquisition be?

And what of the interior of the black hole, of R < Rs. Does matter and light move to the center at the same rate (of c)?

And finally (for today), how does the accepted model of a black hole fit with the rather tempting idea that the observable universe is all contained within the interior of a very large black hole?

 

[Trenton]

Having read up on this some more, it is clear that matter cannot follow lightlike paths into a black hole. They have to follow timelike paths which might be asymtopes of lightlike paths but are not actual lightlike paths. This directly equates to matter failing to quite reach the speed of light.

The only real question remaining is why the horizon is said to move outward (at c or any other speed). I can't see any reason why this is the case. If time stands still at the horizon so will light and so will the horizon - but does it? If so isn't time inside the horizon going backwards since the gravitational potential is much higher the further in you go? From what I have read this is not allowed.