What is the relationship between black holes and time dilation?

[pervect]

Nice reformulation! - it's less abstract and more physical.

I'll take this opportunity to provide what I think is a simpler derivation of the original formula.

First, a very quick review of 4-velocities (I know George and Chris know all this, but I'm not sure of how much our other readers know).

If we have coordinates T, r, $\theta$, and $\phi$, we can write out the components of the 4-velocity as

$$u^a = \left[ \frac{dT}{d\tau}, \frac{dr}{d\tau}, \frac{d\theta}{d\tau}, \frac{d\phi}{d\tau} \right]$$

using geometric units where c=1. [tex]\tau[/itex] should be interpreted as the proper time of the observer, i.e. we can parameterize the wordline of the observer by four functions $$T(\tau),r(\tau),\theta(\tau),\phi(\tau)$$ where $\tau$ represents the proper time or 'age' of the observer, and the above set of derivatives of the coordinates with respect to the proper time is the 4-velocity.

It should also be apparent that
$$g_{ab} u^a u^b$$ has a value of plus or minus 1, +1 if we have a +--- metric signature, -1 if we have a -+++ metric signature.

Now let's consider how to find the magnitude of the relative 3-velocity of two observers if one observer has a 4-velocity of $u^a$ and the other has a 4-velocity of $v^b$.

The product $$\vec{u} \cdot \vec{v} = g_{ab} u^a v^b$$ is going to be independent of the choice of frame or coordinate system. It will determine the angle between the two 4-vectors on a space-time diagram, and hence it will determine the magnitude of the 3-velocity.

By considering the case in flat space-time where u = (1,0,0,0) and v = ( $1/\sqrt{1-\beta^2},\beta/\sqrt{1-\beta^2},0,0)$, we can see that the magnitude of the 3-velocity is just

$$\beta = \sqrt{1 - \left(\frac{1}{\vec{u}\cdot\vec{v}}\right)^2}}$$

To apply this formula to the original problem, we need only the 4-velocity of a stationary observer at radius r, and the 4-velocity of an infalling observer with energy E at radius r.

In Schwarzschild coordinates, using a +--- metric, the 4-velocity of a stationary observer will just be

$$\left[ \frac{1}{\sqrt{1-2M/r}},0,0,0 \right]$$

This results from the normalization requirement that $g_{00} u^a u^b = 1$

We only need the first component, $v^0$ of the 4-velocity of an infalling observer to compute the dot product.

http://www.fourmilab.ch/gravitation/orbits/, or the argument I presented about finding conserved quantites from Killing vectors, gives the value of this quantity.

$$v^0 = \frac{E}{1-2M/r}$$

Therfore the dot product $\vec{u} \cdot \vec{v}$ is just

$$g_{00} \left( \frac{1}{\sqrt{1-2M/r}} \right) \left( \frac{E}{1-2M/r} \right) = \frac{E}{\sqrt{1-2M/r}}$$

since $g_{00}$ = 1-2M/r using the +--- sign convention.

So the relative velocity is just

$$\beta = \sqrt{1 - \frac{1-2M/r}{E^2}}$$

We can also re-write this in the form suggested by George by solving for the energy E in terms of the maximum height.

 

[thothicabob]

I've been following this thread with interest. Tho the math is beyond me for the most part, a lot of questions have been cleared - or so I think. If it's not considered impolite, I'd like to drop a couple of my own to check my understanding, albeit well after this thread apparently ends, as it seems the discussion is the 'right type' for them. I'm hoping Chris H. or pervect will have time to answer them:

The questions are "Observer" type questions...

1. To an observer looking "out" from within the black hole (inside the event horizon), how would the outisde universe appear to him? I am thinking one possibility is that it would appear "smeared" over the boundary of the event horizon, but that could be due to a faulty understanding on my part. Further, would 'time' outide appear to have been 'infinitely' accelerated, which would (could) add to that "smearing" effect?.

2. As a black hole gains mass, as I understand it, the diameter of the sphere bounded by its event horizon increases, at least from the point of view of an outside Observer. How would this appear to an Observer within the event horizon? Would his "universe" (for lack of a better term) appear to be expanding? Would he notice it at all?

Thanks in advance - I'll go look for my Hefty(tm) brand protective garment in the meantime... )

 

[pervect]

I haven't worked out in detail how the outside universe appears to someone falling into a black hole. One thing you'll have to specify is the velocity of the infalling observer - he won't be stationary, no stationary observers exist inside the event horizon. Equivalently, you could specify his energy or the height from which he fell.

I have worked out the redshift such an observer sees for radially infalling rays, and while it depends on the energy parameter of the observer, it is finite.

While I haven't worked this out personally, there's a reliable website that discusses this issue, which includes some movies of obsevers falling into a Schwarzschild black hole as well.

See http://casa.colorado.edu/~ajsh/schw.shtml

I believe he also discusses the issue of redshift vs angle.

Sort of as an aside, a real physical black hole probably won't have an interior Schwarzschild geometry, especially if the black hole is rotating as is very likely.

 

[thothicabob]

Thanks, Pervect. Your answer and that page you suggested were helpful, and they also allow me to better formulate my questions in other ways.

Unfortunately, it also lends fuel to one idea I've wanted to be able to 'debunk or confirm' (for myself) that I'll need to work out somehow (to wit: that for all practical purposes, the Universe itself can be thought of as a black hole, and we exist inside it). I know it's been treated here before, so I'll not ask you to do so again - I'll just re-read the stuff, and also do some more pokin' around and try to find some quality grokkage time. The thing is, it seems that, even after all this time and all we've learned, there's still no conclusive answer that's not dependent on "how you look at it" or what initial assumptions you accept, and I've not yet been able to resolve some "logical" questions that I had hoped would at least be satisfactory enough for "me".

On the one hand, it would be a neat and tidy package if it were the case, in some respects. On the other, it opens other cans of worms that serve mainly ony to push some basic questions backwards, away, or calls them "silly" (or meaningless, as you prefer), and not really answer them.

Thanks again!


And yes, I know I really should get my butt back to school and stuff some "real math" into my head, but in a life-context, I'm a afraid I've already passed my own personal event-horizon for that sort of thing, so the best I can do is to try to reason things best I can, and - when necessary and possible - ask people who: a) know better, and; 2) are willing to take the time to answer questions seriously. In that sense, I appreciate the time you (and others here like you) take out for people like me (and others here like me). I doubt I'm alone in that, either.