# Approaching a Black Hole

1. May 10, 2005

### NoClue

:surprised Something that I have been wondering about...
If space traveller A moves directly toward a giant black hole, into an ever-increasing spacetime compression (while spacetime decompresses as traveller A compresses), would the perceived remaining distance to the black hole decrease at an ever slowing rate, making the trip to the black hole seem endless (Kinda like the camera trick from horror movies, where the victim is running down a hallway but isn't getting any closer to the end)? Also, from an stationary observer, would traveller A seem to acquire a spiral path to the black hole, due to spacetime twisting (due to the rotation of the BH), even though traveller A perceives he is moving directly toward it? Also, could it be possible that black holes are actually extremely large (in size, possibly larger than entire galaxies) but distort our perception of themselves by pulling in the space around themselves to appear to be a singularity? I am just curious.

2. May 10, 2005

### pervect

Staff Emeritus
A traveller will reach the event horizon of the black hole in a finite amount of time by his clocks ("proper time"). There won't be any infinite delay. I'm not sure, though, how the traveller will measure the distance to the black hole - the usual method of using a radar set won't work, there won't be any return signals - so I can't answer as to how he will perceive distance vs time. I could work out (at least in principle) at what time by his watch (proper time) the traveller would pass a station-keeping buoy with a given Schwarzschild R coordinate (such devices could hold station only for R>Rs), but I'm not sure if that would answer your question, and it'd be a fair amount of work.

Actually it's not that much work, and I've done it already. It's t(tau) that's a messy, more complicated expression. (I've got an expression for that, too, but I'm less confident in it's correctness).

*fixed sign error*
r(tau) is just c*(-tau)^(2/3) for tau<0. When tau=0, r=0, and the object is at the singularity (not the event horizon).

c=(2/3)*6^(1/3)*m^(2/3) according to my notes. The object is at the event horizon when r=2m.

This has all been done in geometric units, where c=1. Tau is the proper time as measured by the infalling observer's clock. The above expression is for a fall "from infinity" with zero initial velocity.

Last edited: May 10, 2005
3. May 10, 2005

### NoClue

Okay. I get the concepts, but not the math. I wish I understood that. I need to study more.

What about the path distortion? The infalling traveller would be travelling in a "straight" path toward the singularity, but the path would be relativistically "twisted" around the singularity (along the path of the singularity's rotation) if observed from a distance, correct? Would the traveller perceive the distortion, or would he always appear to be heading right toward the singularity? I am trying to unify my thoughts about how intense gravity affects orbiting or infalling bodies. The same things seem to be happening at various scales, and I am just trying to understand what happens. It would seem to me that spiral galaxies, such as the Milky Way are simply super-massive black holes with billions of infalling stars. We are circling the drain, so to speak. I am wondering if the actual size (meaning undistorted spacetime, which doesn't really exist) of the galaxy is much larger, perhaps by an astonishing factor, due to the progressive compression of spacetime nearer to the center of mass. Due to our "decompressed" state (relative to the center) are we underestimating the vastness of things. The MW is about 100,000 LY in diameter (widely accepted). Is that estimate taking into account the distortion of spacetime nearer the center? It would seem to me that the black hole at the center would be "storing" a great deal of the gravitational diameter of the galaxy inside a pocket of "inside-out space" that makes the galaxy only appear to be 100,000 LY across. It's "gravitationally-effective" diamter may actually be more like 300,000 LY, and only 100,000 LY of it is outside the singularity. Am I just babbling nonsense, or does this make sense? Like I said before, I haven't taken the time to learn the supporting math. :grumpy:

4. May 10, 2005

### Zanket

Your questions are kinda babbling, but I can still mostly tell what you're asking. You might benefit from a virtual trip to a black hole. For learning purposes I'd focus on a Schwarzschild black hole, which is a no-frills black hole that is nonrotating, spherically symmetric, and uncharged. This is the one most books cover. And there are tons of web sites to google for that indirectly answer your questions. To answer one of them, the radius of the Milky Way is 50,000 ly measured as if any black hole at the center did not affect the measurement. This type of radius is called an r-coordinate.

If you're really interested in this subject, a good book to start with is Black Holes: A Traveler's Guide; for the layman with easy math.

5. May 10, 2005

### pervect

Staff Emeritus
Well, I had to fix a sign error in the math - but I don't see how much clearer I can make it than expressing the Schwarzschild coordinate 'R' as a function of proper time 'tau'. It's a fairly simple power-law formula, distance being proportional to the 2/3 power of the proper time for the "fall from infinity with zero velocity".

Proper time is the time measured by a clock carrried along with the observer, and is a lot better behaved than the Schwarzschild coordiante time which "blows up" near the event horizon. 'R' is still somewhat badly behaved near the event horizon too, but it turns out that R(tau) is a reasonably well behaved function (still, dr/dtau = c for the example I gave, which is mildly pathological.)

I do think that some of the previous posters recommendations for resources are good ones - the "virtual trip into a black hole" is good and for books I'd recommend also Taylor's "Black Holes and Time Warps: Einsteins Outrageous Legacy".

I did come to realize that there is one reasonable approach to defining how one would get a radar signal back from the event horizon, of a BH, though. Whille you can't actually have an object hover at the horizon of the hole, you can get one arbitrarily close. When you take the limit of an object arbitrarly close to the horizon, the behavior of radar signals becomes very simple - no matter what time tau you emit the signal, you receive it at at the exact instance that you cross the event horizon. (Neat, eh?). This should allow you to compute some sort of apparent velocity and apparent distance from the horizon as a function of proper time, though I didn't go through the detailed calcuations (also have run into some car trouble, that will be sucking up some of my time).