Black Hole field relativity

[SamBridge]

When mass is added to a black hole, by the consequences of mass being added, the shape of the event horizon changes with a small bulge then eventually levels off to form a perfect sphere again as the mass approaches the singularity inside the black hole. However, from outside a black hole, theoretically time is stopped at the event horizon. So, a problem I have is how do we measure the mass of a black hole changing when theoretically we would never measure mass entering a black hole due to the effects of time dilation? How could the change in it's gravitational field be measured by an outside observer as soon as a piece of matter entered the black hole from its own frame of reference?

What's the solution to this problem? Something to do with the relativity of simultaneity?

Edited January 16, 2013 by SamBridge

[elfmotat]

When people say "time stops according to an outside observer" what they are referring to is the fact the 00 component of the metric in Schwarzschild coordinates is zero at the event horizon. This is merely an artifact of the particular coordinate system, and is called a "coordinate singularity." The fact that Schwarzschild coordinates can't describe matter crossing the event horizon is physically meaningless. If you do the calculation in a coordinate system which covers the entire spacetime, such as Kruskal coordinates, you find that in-falling matter crosses the event horizon in finite proper time for an outside observer. Now, all of this was assuming the in-falling matter was small enough so that it didn't significantly affect the space-time in the first place.

 

If the matter falling into the BH is significant enough to perturb the metric, what you really need is a new solution to the Einstein Field Equations which describes the space-time in the scenario. I.e. the typical vacuum BH solutions (Schwarzschild, Kerr, etc.) are no longer valid anyway, so now it really doesn't make sense to apply Schwarschild coordinate time to the scenario.

Edited January 17, 2013 by elfmotat

[SamBridge]

When people say "time stops according to an outside observer" what they are referring to is the fact the 00 component of the metric in Schwarzschild coordinates is zero at the event horizon. This is merely an artifact of the particular coordinate system, and is called a "coordinate singularity." The fact that Schwarzschild coordinates can't describe matter crossing the event horizon is physically meaningless. If you do the calculation in a coordinate system which covers the entire spacetime, such as Kruskal coordinates, you find that in-falling matter crosses the event horizon in finite proper time for an outside observer. Now, all of this was assuming the in-falling matter was small enough so that it didn't significantly affect the space-time in the first place.

 

If the matter falling into the BH is significant enough to perturb the metric, what you really need is a new solution to the Einstein Field Equations which describes the space-time in the scenario. I.e. the typical vacuum BH solutions (Schwarzschild, Kerr, etc.) are no longer valid anyway, so now it really doesn't make sense to apply Schwarschild coordinate time to the scenario.

But there's a problem with it being physically meaningless. As a photon approaches a black hole, it's wavelength extends to infinity, and it's frequency goes to 0. This is a physical phenomena. Is there some way you could describe the flow rate of time in different 3 dimensional locations as a 3-D derivative model of a 4 dimensional manifold? Visuals would help more.

You don't apply the Schwarchild coordinates, but the change in the system of significant mass I'm guessing would be measured at the speed of light, which is the same from all frames of reference, but for matter actually falling in, you can't just ignore time dilation, time slows down, therefore you should see that the velocity of an observer slows down because his own frame is counting units of time slower compared to your frame, but you say that it doesn't matter that it should theoretically come to a stop, I don't think it makes sense either, but I can't see a clear solution from what you're saying.

Edited January 17, 2013 by SamBridge

[elfmotat]

But there's a problem with it being physically meaningless. As a photon approaches a black hole, it's wavelength extends to infinity, and it's frequency goes to 0. This is a physical phenomena.

 

You need to be careful when you're talking about what's "observed" in relativity. "Observe" is not at all synonymous will "seen." What you can actually see with your eyes can drastically differ from what we know is "actually" happening. We can calculate that in-falling matter will cross the event horizon in finite proper time. What we see is light reflected from said matter until it is red-shifted enough to be undetectable. But if one believes GR is an accurate description of gravity at macroscopic scales, then one necessarily believes that the matter indeed crossed the event horizon. Science itself undoubtedly gets a bit fuzzy here, since we are extrapolating our model past the point where it makes predictions we can test - namely that objects cross the event horizon in finite time. We can't ever know for sure that this is true, because an object that has already crossed the horizon is forever causally disconnected with the rest of the universe.

 

 

Is there some way you could describe the flow rate of time in different 3 dimensional locations as a 3-D derivative model of a 4 dimensional manifold? Visuals would help more.

 

The problem with your question is that it begs another question: "flow rate of time" according to who? An observer "hovering" a short distance outside the event horizon? An observer free-falling toward the BH? An observer an infinite distance away from the BH?

 

The second two are easy: The free-falling observer measures no change in the "rate of time" as he falls toward (and eventually through) the event horizon. The observer at infinity measures in-falling matter to experience asymptotically zero time as it approaches the horizon.

 

The first is harder. I'll do the calculation later when I have time.

 

 

You don't apply the Schwarchild coordinates, but the change in the system of significant mass I'm guessing would be measured at the speed of light, which is the same from all frames of reference,

 

Undoubtedly changes in the metric would propagate at c. I'd imagine the event horizon would move outward at c, perhaps even "before" the in-falling matter has crossed it (but don't quote me on that).

 

 

but for matter actually falling in, you can't just ignore time dilation, time slows down,

 

Okay, according to an outside observer.

 

 

therefore you should see that the velocity of an observer slows down because his own frame is counting units of time slower compared to your frame, but you say that it doesn't matter that it should theoretically come to a stop, I don't think it makes sense either, but I can't see a clear solution from what you're saying.

 

I'm not saying "it doesn't matter that it should theoretically come to a stop," I'm saying that it doesn't theoretically come to a stop. The main point of my post, however, was that the Schwarzschild solution is no longer valid when in-falling matter is significant enough to perturb the metric. As for what the correct solution would look like for a massive body falling towards a black hole, I don't really have any idea. I imagine somebody's done research into the topic before, though I don't know of the papers myself.

[SamBridge]

The problem with your question is that it begs another question: "flow rate of time" according to who? An observer "hovering" a short distance outside the event horizon? An observer free-falling toward the BH? An observer an infinite distance away from the BH?

 

I was assuming from the frame of reference of someone outside the black hole, since that's really the only way you could see the entire model of it.

The second two are easy: The free-falling observer measures no change in the "rate of time" as he falls toward (and eventually through) the event horizon. The observer at infinity measures in-falling matter to experience asymptotically zero time as it approaches the horizon.

 

Wait, that doesn't sound right. A Free-falling observer should measure time is changing, it's only to an outside observer that they would not measure the free-faller's clock as changing, because if that happened the free-faller couldn't travel distance over time and therefore would never reach the singularity.

Okay, according to an outside observer.

I'm not saying "it doesn't matter that it should theoretically come to a stop," I'm saying that it doesn't theoretically come to a stop. The main point of my post, however, was that the Schwarzschild solution is no longer valid when in-falling matter is significant enough to perturb the metric. As for what the correct solution would look like for a massive body falling towards a black hole, I don't really have any idea. I imagine somebody's done research into the topic before, though I don't know of the papers myself.

It doesn't come to a stop....from what frame of reference? I can't use the Scharzschild metric to calculate the gravitational time dilation you say, if we assume the black hole isn't rotating, how else are you suppose to be able to confirm what you're saying? What other equation is there for calculation the time dilation caused by a black hole?

There has to be some solution to the problem, we know black holes get bigger, that means matter has the cross the event horizon, unless perhaps being at the boundary of the event horizon is equivalent to being within the black hole...?

[imatfaal]

If you want to consider the event horizon you cannot use the Schwarzchild coordinate system. It has a mathematical singularity at the event horizon, ie a divide by zero - this is a mathematical/practical problem not the realisation of a physical circumstance.

 

There are other coordinate systems that you can use to investigate the physics at or near the event horizon - Elfmotet mentioned the Kruskal-Szekeres system and there is Eddington-Finkelstein coordinate system as well.

 

This is from the Edddington-Finkelstein page

 

One advantage of this coordinate system is that it shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and not a true physical singularity.

 

These are just tools for looking at the physics of a black hole. The Schwarzchild coordinates work great a distance off - and are simple and have primacy; but the range is limited, because of the maths involved they cannot give a sensible answer at the even horizon. To look at the physics of the vent horizon we just use different tools.

[SamBridge]

If you want to consider the event horizon you cannot use the Schwarzchild coordinate system. It has a mathematical singularity at the event horizon, ie a divide by zero - this is a mathematical/practical problem not the realisation of a physical circumstance.

 

There are other coordinate systems that you can use to investigate the physics at or near the event horizon - Elfmotet mentioned the Kruskal-Szekeres system and there is Eddington-Finkelstein coordinate system as well.

 

This is from the Edddington-Finkelstein page

 

 

These are just tools for looking at the physics of a black hole. The Schwarzchild coordinates work great a distance off - and are simple and have primacy; but the range is limited, because of the maths involved they cannot give a sensible answer at the even horizon. To look at the physics of the vent horizon we just use different tools.

Hmm, I can see the asymtote at the boundary, I'll have to look into it.

[MigL]

Elfmotat is absolutely correct with all his assertions about black holes, although I have to question his hint of causality violation ( if that is what he meant ), that the event horizon may expand even before mass-energy has actually crossed it. My thinking has been that as the mass-energy initiates crossing the event horizon, the horizon 'swells' outward to totally envelope it, at that point. This 'bulge' is then radiated away as a gravity wave to return the horizon to spherical ( for a simple Swartzchild black hole ) as per Wheeler's 'no hair' theorem..

 

Swartzchild co-ordinates can be used to describe simple, non-rotating, non-charged black holes as long as you keep in mind the limitations. The singularity at the horizon is mathematical, not physical

Edited January 22, 2013 by MigL