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Premature Optimization is a Prerequisite for Success

Archive for January 2011

Misconceptions about Event Horizons

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I’m still up at 1am because someone on the Internet is wrong. Someone has posted on Reddit a short sci-fi essay about black holes, and many people have taken it for an accurate description: http://www.reddit.com/r/askscience/comments/f1lgu/what_would_happen_if_the_event_horizons_of_two/c1cuiyw?context=2

I am not going to address every statement in that post. Anyone who has seen space-time diagrams and causal diagrams of black holes will be able to see the most striking inaccuracies. I’ll just say a few words about horizons and singularities which may help to clear some things out.

First of all, nothing special happens as you cross the event horizon. There is no discontinuity in your observations. Moreover, it is fundamentally impossible for you to determine the event when/where you cross it.

So is it true that you’re doomed once you’ve crossed the horizon? Yes. But if you stay away from black holes you are still doomed. All world lines end sooner or later (or maybe infinitely later) at the End of the Universe. The question is, how many ticks of your clock it will take. Take the photons for example. Poor things are already dead the same instance they are born, by their clock.

The black hole singularity is another kind of the End of the Universe. It can be reached sooner rather than later not just by the photons, but also by subluminal travelers. For a nearby observer, the singularity is not a point in space. It is a moment in time. So when is it? It is precisely at the End of the Universe.

So what is the event horizon? It is a surface in space-time that separates the events that are doomed to reach the singularity from all other events. What shape is the horizon? It is not useful to think of it as a surface in space. It is shaped exactly as a light cone: a null (light-like) surface in Minkowski space. So are you really “surrounded by the singularity” once you fall through the horizon? Only in the same sense as you now are “surrounded by tomorrow”. And of course the singularity will not prevent you from seeing the light from the stars. The fact that you are doomed to enter the tomorrow’s day doesn’t prevent you from meeting other doomed people today.

Is there at least anything Lovecraftian about the black hole? Well, on the causal diagram the singularity looks as if someone bit off the time-like infinity to bring the End of the Universe closer to us. But that doesn’t look like Cthulhu’s bite. Looks more like Stephen King’s Langoliers to me.

That’s about it with regards to the causal issues. If you are not familiar with causal diagrams, please do yourself a favor: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html

A few words on the gravitational issues. What about the light orbit, isn’t that a distinctive feature of the horizon? No, the light cannot orbit the blackhole at the horizon. It can orbit the blackhole at 1.5 Schwarzschild radii. Below that distance there are no closed orbits. What about the “infinite tidal forces that tear apart the fabric of space itself”? Isn’t that a distinctive feature of the horizon? No. Supermassive blackholes may have very small tidal forces at the horizon.

If you want an accurate picture of what happens as you fall through the horizon, read this: http://casa.colorado.edu/~ajsh/singularity.html

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Written by bbzippo

01/24/2011 at 9:17 am

Posted in Uncategorized

Coloring of an infinite map: Proof by compactness

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https://bbzippo.wordpress.com/2011/01/18/coloring-of-an-infinite-map/

https://bbzippo.wordpress.com/2011/01/20/coloring-of-an-infinite-map-inductive-solution/

Assume that any finite map on the plane can be colored using 4 colors. Prove that any map on the plane can be colored using 4 colors.

Proof

Let’s number the regions: r_1, r_2, …. Let’s number the colors: 0, 1, 2, 3.

For any n there exists a coloring of {r_1, r_2, …, r_n}. Let’s encode this coloring by a real number x_n with the decimal expansion 0.c1c2…cn where each digit is the number of the color of the corresponding region.

So we have an infinite sequence {x_n} of real numbers from [0, 1/3]. By compactness, it has at least one limit point X. Easy to see that X encodes a proper coloring of the whole map.

Notice that here (unlike in the inductive case) we didn’t bother to build a self-extending tower from ground up. The limit point is guaranteed to be such a tower. If it wasn’t, it wouldn’t be possible to find a proper coloring arbitrarily close to it.

CONTINUE READING:

https://bbzippo.wordpress.com/2011/02/04/back-to-compactness/

Written by bbzippo

01/21/2011 at 6:14 am

Posted in math

Coloring of an infinite map: inductive solution

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https://bbzippo.wordpress.com/2011/01/18/coloring-of-an-infinite-map/

Assume that any finite map on the plane can be colored using 4 colors. Prove that any map on the plane can be colored using 4 colors.

Proof

Definitions:

Let’s call a proper coloring of a set S extendable if for any finite set E it can be extended to S U E.

If a coloring is not extendable then there exists a finite set D (a “dead end” set) such that our coloring can not be extended to S U D.

Inductive Step Lemma:

Let C be an extendable coloring of a set S. Let r be a single region. There exists an extendable extension of C to S U {r}.

Proof of the Lemma:

C has at most 4 extensions to S U {r}. Assume none of them is extendable. So each of them has a dead end set on which it can’t be extended. The union of those dead ends would be a dead end for C, so C would not be extendable. QED.

“Any finite map can be properly colored” basically means that the coloring of the empty set is extendable.

Let’s number our regions: r_1, r_2, …. Let R_n be the set of the first n regions. The coloring of the empty set extends to R_1. And by the lemma, it extends to all R_n. The union of all these extensions gives us a coloring of the whole map.

CONTINUE READING:

https://bbzippo.wordpress.com/2011/01/21/coloring-of-an-infinite-map-proof-by-compactness/

Written by bbzippo

01/20/2011 at 3:14 am

Posted in math

Coloring of an infinite map

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Assume that any finite map on the plane can be colored using 4 colors. Prove that any map on the plane can be colored using 4 colors.

Hints:

  • An infinite map is properly colored if every finite subset of it is properly colored.
  • The set of regions on the map is countable.

I think that this problem is very instructive and it touches some important foundational concepts. So rather than just presenting a solution, I’d like to try and reconstruct the thought process and demonstrate the emergence of those concepts.

First, let’s notice that those hints that I gave may both lead and mislead us. For example, one may be tempted to come up with non-solutions like these:

  1. Non-solution: Since (any finite subset can be properly colored) and (if any finite subset is properly colored then the whole map is properly colored), it immediately follows that the whole map can be properly colored.
  2. Non-solution: Since the set of regions is countable (r_1, r_2, …), and for any n we can color the set {r_1, …, r_n}, it immediately follows that we can color the whole map.

In the first non-solution the fallacy is that the fact that we can find a proper coloring for each finite subset doesn’t mean we can find one for all of them. This in fact is what we are trying to prove. Just like the fact that a function is continuous at every point doesn’t guarantee that it is uniformly continuous. If we can find a delta for the given epsilon at each point, we don’t  necessarily have one “blanket” delta. It is only guaranteed to exist on a compact domain… So we are trying to prove some sort of compactness property of planar maps with respect to coloring. Really, it is compactness that allows us to reduce properties of infinite sets to properties of their finite subsets.

The second non-solution is an attempt to apply induction, but it neglects to demonstrate the inductive step. If we could show how to extend a proper coloring of {r_1, …, r_n} to a proper coloring of {r_1, …, r_n+1} then we would have a recipe for coloring each region that would be the “blanket” coloring that would satisfy all finite subsets.

In the next post I will give an inductive proof. Then I will present a much simpler proof that demonstrates the compactness of the map with respect to coloring by using compactness of some other space.

Then (if I still have inspiration) I’d like to show how the Axiom of Choice allows to “compactify” uncountable sets. And then I’m planning to mention the mother of all compactness: the compactness in Model Theory, and its implications. And did I mention that I have no clue about all this stuff 🙂 I am not sharing knowledge or wisdom, but the excitement of exploration 🙂 Cabbages and kings, you know, and men and cats and cops (anagram of compactness)

CONTINUE READING:

https://bbzippo.wordpress.com/2011/01/20/coloring-of-an-infinite-map-inductive-solution/

https://bbzippo.wordpress.com/2011/01/21/coloring-of-an-infinite-map-proof-by-compactness/

Written by bbzippo

01/18/2011 at 7:14 am

Posted in math

The Rule of 72 revised

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Wanna double your wealth? Invest it at r% rate and wait for 72/r years.

This is the well-known Rule of 72. Why does it work and what is 72 anyway?

72 because for r =0.08, r*log(2)/log(1+r) = 0.720… And 72 = 2^3*3^2 is a well-divisible number 🙂

And it works for other values of r because x/log(1+x) is almost perfectly linear between 0 and 1: http://www.wolframalpha.com/input/?i=x%2Fln%281%2Bx%29+from+0+to+1

But is r = 0.08 (8% rate) a safe assumption these days? 😉

Short-term treasuries rate is 0.007 today… let’s see 0.007*log(2)/log(1.007) is approximately 0.7 (I love the 3 sevens in this formula), so the revised risk-free rule is the rule of 70:

Wanna double your wealth risk free? Invest it at r% rate and wait for 70/r years.

At the high-risk rate you’ll have to wait for 72/8 = 9 years, and at the low-risk rate: 70/0.7 = 100 years.

And the general rule is: if you want to increase your wealth by the factor f, invest at the rate r and wait for log(f)/log(1+r) years.

Disclaimer: this should not be relied on as investment advice, and I shall not be liable for any direct, indirect, exemplary, compensatory, punitive, special or consequential damages, costs, expenses or losses 🙂 🙂 🙂

 

Written by bbzippo

01/06/2011 at 3:45 am

Posted in fun, math