# Is the space-like distance to a black holes event horizon infinite?

1. Oct 14, 2013

### mrspeedybob

From a distant frame of reference a falling object never reaches the event horizon due to time dilation. If I drop a meter stick into a black hole lengthwise I should see both ends of the stick getting asymptotically closer and closer but never reaching the horizon, thus the stick should appear to get shorter from my frame of reference. Assuming that the black hole is large enough that tidal forces are negligible then the stick should not experience anything abnormal, therefore I conclude that the shortening of the stick is a relativistic length contraction type of phenomena. This same logic should apply equally well to a stick of a kilometer or a light-year. If I can drop a stick of arbitrarily long length toward a black hole and never see the far end reach the event horizon can I not conclude that the distance from myself to the event horizon is longer then any arbitrarily long stick?

2. Oct 14, 2013

### MrAnchovy

No, you said yourself that the the apparent shortening of the stick is due to relativistic effects, why would you imagine that you can ignore these effects and use the apparent length as a reliable ruler?

3. Oct 14, 2013

### Staff: Mentor

This is not incorrect, exactly, but it can be misleading, since it's tempting to draw inferences from it such as the ones you draw, which are incorrect.

Does "lengthwise" mean oriented radially (i.e., vertically)? I'll assume it does in what follows, since otherwise the problem is much less interesting.

No, it won't. The apparent "time dilation" that a distant observer sees when watching an object fall into a black hole doesn't work the same as time dilation due to relative velocity. That's one of the main reasons I don't think applying the term "time dilation" in the gravity case is a good idea.

Correct.

This is one of those incorrect inferences that the term "time dilation" tempts you to draw.

Even if the logic were correct, it wouldn't apply to a stick of arbitrarily long length, because at some point the stick's length would be comparable to the size of the black hole, and tidal forces would certainly not be negligible in that case. (The possibility of non-negligible tidal forces also has to be taken into account in the correct logic, which I'll describe in a moment.)

No, and that's not just because of what I said just now about tidal forces becoming non-negligible when the stick is large enough. The key here is to carefully define what we mean by "distances" in the radial direction, and to recognize that what the distant observer sees is distorted by the curvature of spacetime, so it's not a good way to judge what is actually happening to the stick, as MrAnchovy pointed out.

To properly judge radial distances, the distant observer has to correct for the distortion in what he sees. He does this by using the radial metric coefficient, $g_{rr}$, which is $1 / (1 - 2M / r)$, where $M$ is the mass of the black hole. At any given instant of the distant observer's time, the stick extends from $r_1$ to $r_2$ (where we stipulate that $r_1$ is the smaller of the two), so its proper length is given by

$$L = \int_{r_1}^{r_2} \sqrt{g_{rr}} dr$$

If you work out this integral, you will see that it is finite for any values of $r_1$ and $r_2$ greater than $2M$; in fact, it will be finite even in the limit as $r_1 \rightarrow 2M$. That indicates that the distance from any $r > 2M$ to the horizon at $r = 2M$ is finite. (This is still true when the distance is large enough that tidal forces are not negligible; but in that case the range of $r$ coordinates occupied by the stick will be increased significantly as it falls once the tidal forces get strong enough, so its proper length will increase.)

4. Oct 14, 2013

### phinds

In addition to what has already been said, think about the simple logic of what you have said and consider a stick that extends out twice as far from the black hole as you are from the black hole. How would you conclude that you are closer to the black hole than the length of the stick when one end of it is clearly farther from the black hole than you are?

5. Oct 17, 2013

### stevebd1

You might find the following equation of interest-

Source- http://www.mathpages.com/rr/s7-03/7-03.htm

6. Oct 18, 2013

### pervect

Staff Emeritus
I think this question has come up before? The answer is still "no", the distance isn't infinite.

Note that the size of a stick as measured from infinity doesn't have anything to do with the distance that an observer on the stick would compute and/or measure as the distance to the event horizon.

To see this, consider the barn-pole paradox of special relativity (henceforth, SR). From the viewpoint of the barn, the moving pole shrinks. From the viewpoint of the pole, the barn does not grow - the barn shrinks.

Thus one can't conclude from the fact that the pole shrinks that the barn grows. It doesn't follow.

Thus one way to see that the above argument doesn't work is to study SR enough to understand the barn-pole paradox.

Unfortunately, without understanding some SR, the most one can hope to communicate is that the proposed answer (the distance to the event horizion is infinite) is wrong. Hopefully this much will be communicated if the OP studies the responses of science advisors and/or mentors to the several posts on this topic.