Crossing the Horizon on a Street of Houses: Last House Passed

[Zanket]

General relativity tells us that an infinite number of houses can be built at fixed altitudes in the one meter (say) above the horizon, as measured in Euclidian geometry. Let this infinite number of houses exist. Let the street numbers of the houses increment with decreasing altitude. For someone crossing the horizon at the end of this street of houses, what is the street number of the last house passed?

 

[PeteSF]

Insufficient data to provide a definite answer: "the last house passed" is not well defined.

How does this relate to black holes?

 

[JesseM]

General relativity tells us that an infinite number of houses can be built at fixed altitudes in the one meter (say) above the horizon, as measured in Euclidian geometry.

What do you mean by "as measured in Euclidean geometry"? In a continuous classical world it would also be possible to build an infinite number of ever-thinner, ever-more-closely-packed houses in any given meter of flat space--what role does the event horizon play in this problem?

 

[PeteSF]

Oops... I skipped through too fast. For some reason, I missed the GR reference and was considering an abitrary Euclidian space...

Anyway, the core idea of my last post remains, I think... it is not possible to uniquely identify "the last house" in such a situation.

 

[Zanket]

What do you mean by "as measured in Euclidean geometry"?

This means, “measured as if spacetime were flat,” and not curved by gravity. It’s easier to talk about circumferential measurements, which are not affected by gravity. Radially, between the circumference of the horizon and a circumference infinitesimally longer, or let’s say one meter longer, an infinite number of houses can fit, so says general relativity. The directly measured radial distance between these circumferences is infinite.

In a continuous classical world it would also be possible to build an infinite number of ever-thinner, ever-more-closely-packed houses in any given meter of flat space--what role does the event horizon play in this problem?

You can assume that these houses are all the same proper size; e.g. they can all fit people within. Just above the horizon is where the last house is. At and below the horizon, no house can remain at a fixed altitude.

 

[Zanket]

Anyway, the core idea of my last post remains, I think... it is not possible to uniquely identify "the last house" in such a situation.

On the way to the horizon, houses are passed. Upon reaching the horizon, there are no more houses. Then there must have been a last house, uniquely identifiable.

 

[JesseM]

In order for all the houses to maintain a constant distance from the horizon, it will have to expend large amounts of energy--would the energy density of the ever-more-tightly packed houses (in some coordinate system at least, I don't know which system would be the relevant one) cause the houses to form a black hole themselves, or the horizon of the original black hole to expand to include them? If so, maybe this thought-experiment wouldn't be possible even in principle...

 

[JesseM]

On the way to the horizon, houses are passed. Upon reaching the horizon, there are no more houses. Then there must have been a last house, uniquely identifiable.

Do you think there is a uniquely identifiable smallest positive fraction of the form 1/n, where n is a positive integer? After all, as you approach zero when moving along the real number line in the negative direction, fractions of this form are passed, but upon reaching zero there are no more such fractions.

 

[Zanket]

In order for all the houses to maintain a constant distance from the horizon, it will have to expend large amounts of energy--would the energy density of the ever-more-tightly packed houses (in some coordinate system at least, I don't know which system would be the relevant one) cause the houses to form a black hole themselves, or the horizon of the original black hole to expand to include them?

Such details can be ignored in a thought experiment like this. These are not ever-more-tightly packed houses, at least not in their own frames. They can all have the same proper sized lots.

Do you think there is a uniquely identifiable smallest positive fraction of the form 1/n, where n is a positive integer? After all, as you approach zero when moving along the real number line in the negative direction, fractions of this form are passed, but upon reaching zero there are no more such fractions.

True, but it doesn’t answer the riddle. It doesn’t explain, for example, how there is not a uniquely identifiable last house. Have you ever passed a row of houses, and upon coming to the end of the row, found that there was no last house in the row?

 

[JesseM]

Such details can be ignored in a thought experiment like this. These are not ever-more-tightly packed houses, at least not in their own frames. They can all have the same proper sized lots.

But in terms of the formation of a new black hole, does GR say that the fact that the distance to their nearest neighbors in their own frames is the same for each house is enough to insure that no black hole will form? Isn't the fact that houses closer and closer to the horizon will have to thrust with greater and greater energy in their own frames, with the thrust energy approaching infinity as you approach the horizon, also relevant to answering the question of whether any of the houses form a black hole (or cause the horizon of the existing black hole to expand to swallow them)?

True, but it doesn’t answer the riddle. It doesn’t explain, for example, how there is not a uniquely identifiable last house. Have you ever passed a row of houses, and upon coming to the end of the row, found that there was no last house in the row?

Well, even if this thought-experiment is possible in principle, won't a freely-falling observer see the houses more and more tightly packed as he approaches the horizon, rather than an equal distance apart? If so, he is experiencing the same thing you would experience if you walked along a number line towards zero with each number of the form 1/n marked off--at some point the markings would become too close to distinguish.

 

[PeteSF]

On the way to the horizon, houses are passed. Upon reaching the horizon, there are no more houses. Then there must have been a last house, uniquely identifiable.

That's certainly intuitive, but is it true? Unituitive things happen when you're dealing with infinities, which is why Zeno is remembered so fondly.

 

[Zanket]

But in terms of the formation of a new black hole, does GR say...

Such details put the cart before the horse, so they can be ignored.

Well, even if this thought-experiment is possible in principle, won't a freely-falling observer see the houses more and more tightly packed as he approaches the horizon, rather than an equal distance apart? If so, he is experiencing the same thing you would experience if you walked along a number line towards zero with each number of the form 1/n marked off--at some point the markings would become too close to distinguish.

That is indeed the way one of my books explains how an observer free-falling across a horizon can travel an infinite distance (an infinite number of houses) in a finite time. But it still does not answer the riddle. I pass an infinite amount of infinitesimally-sized stuff with every step I take. But that doesn’t explain how, when I come to the end of a row of houses, there is no last house.

 

[PeteSF]

That is indeed the way one of my books explains how an observer free-falling across a horizon can travel an infinite distance (an infinite number of houses) in a finite time. But it still does not answer the riddle. I pass an infinite amount of infinitesimally-sized stuff with every step I take. But that doesn’t explain how, when I come to the end of a row of houses, there is no last house.

Sure it does. If there are infinity houses, there is no uniquely identifiable last house, just like there is no uniquely identifiable last number. What's left to explain?

 

[Zanket]

That's certainly intuitive, but is it true? Unituitive things happen when you're dealing with infinities, which is why Zeno is remembered so fondly.

Infinities can be used to mask absurdity. The solution to Zeno’s Paradox is logical.

 

[Zanket]

Sure it does. If there are infinity houses, there is no uniquely identifiable last house, just like there is no uniquely identifiable last number. What's left to explain?

What is left to explain is how the last house can be passed without it being uniquely identifiable. Houses are passed, and then there are no more houses. There must have been a last house. (But now we’re going in circles.)

 

[JesseM]

Such details put the cart before the horse, so they can be ignored.

What do you mean "put the cart before the horse"? Do you agree that houses closer and closer to the horizon will have to be thrusting with greater and greater G-forces to maintain a constant height? And wouldn't it be true that if a given mass thrusted with greater and greater G-force in flat spacetime, eventually the energy density would be high enough to form a black hole? If that's correct, then it seems the situation you describe is physically impossible, even in principle.

That is indeed the way one of my books explains how an observer free-falling across a horizon can travel an infinite distance (an infinite number of houses) in a finite time. But it still does not answer the riddle. I pass an infinite amount of infinitesimally-sized stuff with every step I take. But that doesn’t explain how, when I come to the end of a row of houses, there is no last house.

OK, but the black hole doesn't seem to be relevant to this particular "paradox". Do you agree that the exact same problem occurs in my thought experiment where you are walking towards 0 along a number line where every number of the form 1/n has been marked? I would suggest looking into some real analysis, particularly the concept of open sets vs. closed sets.

 

[pervect]

The main resolution of this "paradox", which I think needs a little work as well, is that it is not possible to "walk" across the event horizon of a black hole in any physically meaningful sense. An observer falling into a black hole will always be traveling at 'c', the speed of light, when he crosses the event horizon.

The reason I think the paradox needs work is that the coordinate system where one could hypothetically build an infinite number of houses is not even approximately Euclidean or flat. It's a highly non-physical geometry. One of the properties of this coordiante system is that the proper acceleration of gravity will tend towards infinity. Infinite proper accelerations can hardly be called a "flat" or "euclidean" geometry.

This is actually not a coordinate system I've seen discussed much - I'm more or less taking your word for it that it's possible to build an infinite number of houses in it. Where did this riddle come from originally? It seems like it might be related to Milne cosmology.

Anyway, these infinite proper accelerations are why it's impossible for an observer to cross the event horizon at any speed other than 'c', BTW. If you pick an observer with a finite proper acceleration, you will find that his velocity when he crosses the event horzion is equal to 'c'.

 

[Zanket]

What do you mean "put the cart before the horse"?

Sorry for the delay in replying…

The theory allows for an infinity of houses along a segment of a radius above the horizon. We can discuss the logical ramifications of that before accounting for the mass of the houses. If we say that discussion is precluded by the mass issue, then IMO that is putting the cart before the horse; i.e. putting things in the wrong logical order.

A similar example in relativity discussions is the infinitely rigid rod. Such a rod cannot exist in principle, but you’ll see thought experiments using it.

Do you agree that houses closer and closer to the horizon will have to be thrusting with greater and greater G-forces to maintain a constant height? And wouldn't it be true that if a given mass thrusted with greater and greater G-force in flat spacetime, eventually the energy density would be high enough to form a black hole? If that's correct, then it seems the situation you describe is physically impossible, even in principle.

Agreed.

OK, but the black hole doesn't seem to be relevant to this particular "paradox". Do you agree that the exact same problem occurs in my thought experiment where you are walking towards 0 along a number line where every number of the form 1/n has been marked? I would suggest looking into some real analysis, particularly the concept of open sets vs. closed sets.

I agree that it’s a similar situation. And I agree that there simply won’t be an infinite number of houses. Not because of the mass issue per se, but that’s one way to enforce the limitation. Even though there is room for an infinity of houses, there will be a finite number of house always, hence a last house with a uniquely identifiable number.

Thanks for the link.

 

[Zanket]

This is actually not a coordinate system I've seen discussed much - I'm more or less taking your word for it that it's possible to build an infinite number of houses in it. Where did this riddle come from originally?

I just thought of it off the cuff. I think it is impossible to have an infinite number of houses, but might be worth imagining.

Anyway, these infinite proper accelerations are why it's impossible for an observer to cross the event horizon at any speed other than 'c', BTW. If you pick an observer with a finite proper acceleration, you will find that his velocity when he crosses the event horzion is equal to 'c'.

That raises another riddle: If everyone crosses at c, then if you and I are crossing independently but beside each other, what happens when I accelerate to pass you? It seems that we must stay in lockstep since we both must cross at exactly c. Our ships could be in a light-year-across region of almost perfectly flat spacetime for a large enough black hole. Our ships could each be a light-hour long, so it would take an hour on our clocks to completely cross. It would be strange if I accelerated and didn't make any headway relative to you. And if I did make headway then I'd be crossing at a different velocity than you and we could not both be crossing at c.

 

[Mortimer]

If everyone crosses at c...

From the perspective of which observer??
An observer at infinity does not measure the falling object cross at c. At the event horizon he measures the velocity V(r) to be zero (assuming zero starting velocity at infinity):

V(r)=\left(1-\frac{2MG}{rc^2}\right)\sqrt{\frac{2MG}{r}}

If you put that in a graph it looks like this:
http://www.rfjvanlinden171.freeler.nl/blackhole2.jpg

 

[pervect]

The Scwarzschild coordinate system just isn't well behaved at r=rs. The problem, though, is not in the physics, it's just a bad coordiante system. It's like trying to use lattitude and longitude at the north pole of the Earth.

An observer falling into the event horizon will not notice anything unusual, if he accelerates an object towards the BH it will move faster relative to him, if he applies the accleration in reverse, it will fall slower.

 

[pervect]

From the perspective of which observer??
An observer at infinity does not measure the falling object cross at c. At the event horizon he measures the velocity V(r) to be zero (assuming zero starting velocity at infinity):

V(r)=\left(1-\frac{2MG}{rc^2}\right)\sqrt{\frac{2MG}{r}}

If you put that in a graph it looks like this:
http://www.rfjvanlinden171.freeler.nl/blackhole2.jpg

I think I'm the one who actually said the velocity would be 'c' going into the event horizon. I remembered reading this, but I'm not sure where I read this, so I feel obligated to work it out to make sure I am getting the right answer for the specific question at hand.

Your calculation is correct for dr/dt, which is the velocity in Schwarzschild coordinates. But what we want to do is to find the velocity not in Schwarzschild coordinates, but in the Schwarzschild basis. Another way of saying this is that we want to translate to a Locally lorentzian frame where g_00 = g_rr = 1, and find the velocity in that frame, where the local basis vectors are parallel to the Schwarzschild vectors, but orthonormal.

If the observer falls straight in from infinity, in geometric units

dr/dtau = sqrt(2*m/r)
dt/dtau = 1/(1-2*m/r)

which gives your results for dr/dt of zero when we take the ratio
dr/dt = (dr/dtau) / (dt/dtau)

i.e. dr/dt = sqrt(2*m/r)*(1-2*m/r)

But what we want is to transform to a local coordinate system with unity coefficients for the metric. Thus

r' = L^r{}_{r'} * r
t' = L^t{}_{t'} * t

To make the transformed metric coefficients unity in magnitude we need

<br /> L^r{}_{r} = \sqrt{1-2m/r}<br />
<br /> L^t{}_{t&#039;} = 1/\sqrt{1-2m/r}<br />

The desired result will be dr'/dt', which will be (dr/dt) * (dr'/dr) / (dt'/dt)

This means your result gets multiplied by (1-2m/r) in geometric units)

or in other words v = sqrt(2*m/r), which is equal to '1' (the speed of light in geometric units) at the event horizon (r=2m).

I have to take back part of what I said, though, it does look like v is not always equal to 'c' depending on the value of E, the energy of the infalling observer.

If we put in the factor E, I get

dr'/dt' = sqrt(E) at r=2M, where E is the normalized energy of the infalling body, equal to 1 when the observer falls straight in from infinity (v=0 at infinity).

Or to summarize - the velocity of an observer crossing the event horizon is equal to 'c' in the Schwarzschild basis, but only when he falls in from infinity with v=0 at infinity.

 

[pervect]

OK, so now that we've been around the barn, it seems like the original question needs to be addressed.

There may well still be some confusion about coordinates, but let's look at the question of whether or not an infinite number of houses can fit in betweeen r=2m and r=4m in the Schwarzschild basis that I was using.

We have dr' = sqrt(1-2*m/r)dr

Let's let m = 1/2 for simplicity, making the schwarzschild radius r=1, then we have

dr' = sqrt(1-1/r)

Now, this obviously diverges as r->1, but does the integral diverge?

I'm getting a negative answer to this, my favorite integrator gives

<br /> \sqrt {r-1}\sqrt {r}-1/2\,\ln \left( -1+2\,r+2\,\sqrt {r-1}\sqrt {r}<br /> \right) <br />

as the intergal of sqrt(1-1/r) dr, and this gives a finite answer for the integral between r=1 and r=2.

 

[Mortimer]

This in interesting, although you could wonder if it physically means anything. The transformation to unity coefficients transforms to a local approximation of a flat space, am I right (aka geodesic coordinates?)? Can this be done unpunished in the neigborhood of 2m where curvature is extreme and approximations have large errors? You'd have to restrict yourself to an extremely small area. Your integration interval is not particularly small.
I am rather new at GR by the way.

 

[pervect]

I know there isn't any problem with the transformation to the locally flat coordinates of an infalling observer near the event horizon of a black hole - (because there isn't any actual physical singularity at the event horizon)

But of course these aren't those coordinates, these are slightly different.

This difference means that these coordinates are not geodesic coordinates (Gaussian normal coordinates), because the body being used as a reference is not following a geodesic, it's accelerating.

These coordintes (the Schwarzschild basis coordiantes that I was using) still require an infinite proper acceleration for an observer to hold station at a constant value of the Schwarzschild radius (measuring the acceleration is measured in the locally flat coordinates I was just talking about, the Schwarzschild basis coordinates). So there may be some issues left, these coordinates are not really as well behaved as some other choices (i.e they are not well behaved like Kruskal coordinates). They will be at least as well behaved as Schwarzschild coordinates, though.

Certainly these coordinates will have meaning at any point that's not exactly equal on the event horizon, so I'd say that if the limit when you go exactly to the event horizon is still finite, there shouldn't be a problem.

 

[Zanket]

An observer falling into the event horizon will not notice anything unusual, if he accelerates an object towards the BH it will move faster relative to him, if he applies the accleration in reverse, it will fall slower.

How can it do that while the observer is crossing the horizon, when the horizon must be crossed at exactly c, no more no less?

 

[Zanket]

From the perspective of which observer??

From the perspective of the observer crossing the horizon.

 

[Zanket]

I think I'm the one who actually said the velocity would be 'c' going into the event horizon. I remembered reading this, but I'm not sure where I read this, so I feel obligated to work it out to make sure I am getting the right answer for the specific question at hand.

You can know it's valid because an observer hovering infinitesimally above the horizon will measure a velocity of the in-falling observer to be c in the limit (that is, at the horizon, the limiting case for a hovering observer). It can't be more than c anywhere above the horizon, or else the fixed observer would measure that directly and find special relativity to be violated. And it can't be less than c in the limit, or else the in-falling observer would not be forced to cross the horizon.

The book Exploring Black Holes has the calculations for three scenarios: free-falling from rest at infinity, free-falling from rest at a finite altitude, and being hurled toward the hole from a finite altitude at any initial velocity. In all these cases the observer crosses the horizon at a proper velocity of c.

 

[Mortimer]

Thanks Pervect. That gives me some food for thought. I did a quick research on the various coordinate systems used in black hole discussions and noticed that there are plenty (in ICT someone once said that the good thing of standards is that you can choose from so many...).
A question out of curiosity. Obviously many of these coordinate systems assume 4D. Do 5D coordinate systems bring any new options (I know there are speculations on black holes being 5D manifolds)? I could imagine for instance that whatever becomes infinite or zero in 4D vectors is in fact a result of the vector being rotated into a fifth dimension (i.e. in 4D we only see some kind of projection).