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Q: How hard would it be to make a list of pruducts of primes that could beat public key encryption?

Saturday, March 27th, 2010

The complete question was: I’m assuming almost anyone with sufficient computing power could generate big prime numbers (if these are not already published somewhere). Would making a table of all of the products of these prime numbers be so difficult? Published public keys could then be compared to this list and the prime factors would be known immediately with a simple database look-up. Wouldn’t this make public key cryptography useless?

Mathematician: Let’s say we are working with all prime numbers that are about d digits long or shorter. The largest such number is x=10d – 1. The number of prime numbers less than this (applying the prime number theorem) is about \frac{x}{\ln{(x)}} which is approximately equal to 0.43\frac{10^d}{d}. Making a table of the product of all of these primes would mean storing about \frac{x}{\ln{(x)}} squared numbers, which is about

.18\frac{10^{2d}}{d^2}

numbers. The average size of one of these numbers is about

.5 x 10d

Hence, on average one of these products will be larger than

.25 x 102d

which takes about

6.64 d

bits to store. The total number of bits needed will be about the number of bits per number times the number of numbers. The number of numbers though is about .18\frac{10^{2d}}{d^2} giving a total number of bits of about

(6.64 d )(.18 \frac{10^{2d}}{d^2}) = 1.2 \frac{10^{2d}}{d}

converting this to gigabytes, we have about

1.12 \frac{10^{(2d-9)}}{d} gigabytes.

If we set d=100 (corresponding to storing all products of prime numbers with no more than 100 digits each) that requires about

1.12 x 10189

which is a ridiculously huge number… far, far bigger than the entire size of the internet. For comparison, the Earth only has about 1050 atoms.

If we set d=20 (corresponding to storing all products of prime numbers with no more than 20 digits each) that requires about

5.6 x 1029
which is still much bigger than the size of the internet!

Basically, there are far too many large prime numbers to store them all, even if we restrict ourselves only to numbers of 20 digits in length.

Physicist: This isn’t an answer, just a clarification of what the question was about (for those of you who aren’t hip to the Comp Sci scene), followed by something I think is worth knowing.

Public key encryption is based on a “trap door algorithm”. Basically, you use a mathematical process that is easy to do in one direction and nie-impossible to do in the other direction, unless you know the “secret”. RSA encryption uses multiplication as it’s mathematical process. (There’s a bit more to it, and if anyone wants to know more let us know.)

Multiplication is easy: 170,363 x 83,701,993 = ?

Factoring is hard (secret = knowing the factors): ? x ? = 14,259,722,633,459

Cryptographers like prime numbers because by using only two big primes you make sure that your factors are large and hard to find. If you have b factors, then, for a given product N, the average size of each factor is about N^{\frac{1}{b}} (smaller and easier to find for larger b).

If you’re ever bored and want to find a big prime you can use Fermat’s Little Theorem: if x^{P-1} mod \, P = 1 (for 1

So for example: P = 7, and x= 5 (pick x at random). 56 mod 7 = (52)3 mod 7 = (25)3 mod 7 = (21+4)3 mod 7 = (4)3 mod 7 = 64 mod 7 = 63+1 mod 7 = 1. 7 is almost definitely prime!

To be more sure you can check a couple different x’s. False positives are very rare using this test.

Another example: P = 12, x=3. 311 mod 12 = 38+3 mod 12 = ((32)2)233 mod 12 = (92)227 mod 12 = (81)2(24+3) mod 12 = (72+9)23 mod 12 = (9)23 mod 12 = 81 x 3 mod 12 = (72+9)3 mod 12 = 9 x 3 mod 12 = 27 mod 12 = 3 ≠ 1. 12 is not prime!

Fermat’s Little Theorem is a nearly 100% accurate test, but if you’re some kind of mathematician or something you would want to use the algorithm found in “Primes is in P” (by Agrawal, Kayal, Saxena). This doesn’t have much to do with the original question, but I think it’s worth knowing.

Posted in -- By the Mathematician, -- By the Physicist, Math, Number Theory | No Comments »

Q: What are complex numbers used for?

Wednesday, March 24th, 2010

Physicist: If you’ve ever had to do square roots you’ve probably come up against the problem of taking the square root of a negative number.  If you restrict your attention only to real numbers (0,1, -17, \pi, √2, …, any number you can think of), then there’s no way to take the square root of a negative.  But this makes taking square root, cube roots, and so on, a pain.

You have to remember: 1) The square root, forth root, sixth root, … of a positive number has two answers (positive and negative).  2) The square root, forth root, sixth root, … of a negative number has no answers.  3) The cube root, fifth root, seventh root, … of any number has one answer (with the same sign as the original number).  These are random, frustrating, difficult-to-remember rules.  Mathematicians had to deal with these all the time before the 1700′s.

Then Euler happened.  He was looking at a similar problem; finding the roots of polynomials.  Similar, because the question “x=\sqrt{-1}, what is x?”, is exactly the same as “x2+1=0, what is x?”.  He was annoyed that most polynomials have roots that don’t exist, so he said; “Sure…  But, what if the root did exist.  Like… in an imaginary sense…”.

That’s paraphrasing, but it’s pretty accurate.  He just declared that there’s a new number called “i” with the property that “i2 = -1″.

So, to actually answer: Complex numbers make it easy to take roots, and using complex numbers, all polynomials with terms up to xN have N roots.

Using only real numbers, the cube root of 1 is 1, and only 1. Using complex numbers you can see that the other two roots exist, they just happen to be off of the real line. In this picture the arrow to the right is "1", and the real line is the horizontal line. The other two arrows are the other two roots. The roots of a number are always equally spaced like this.

These two properties help complex numbers “complete” the real numbers.  That is to say; you don’t need to create “super complex numbers” to fix any problems with complex numbers.  One of the first “problems” that people ask about is; “Sure, \sqrt{-1} = \pm i, but what’s \sqrt{i}?”  Well, \sqrt{i} = \pm \left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right).  No problems!

Also, if you’ve ever had to do anything with trigonometry you’ve probably come up against:

\cos{(A+B)} = \cos{(A)} \cos{(B)} - \sin{(A)}\sin{(B)} \sin{(A+B)}=\sin{(A)}\cos{(B)}+\sin{(B)}\cos{(A)}

Which looks horrible, is horrible, and is horrible to deal with.  Here’s the same thing (both equations) using complex numbers:

e^{i(A+B)} = e^{iA}e^{iB}, which is just a direct application of Euler’s equation: e^{i \theta} = \cos{\theta} + i \sin{\theta}.  Almost any time that you have to do lots of summations or multiplications involving trig function, it’s best to bust out some complex numbers.

In the same vein, electrical engineers use “phasors” (phasor, not phaser) to talk about sinusoidal current (like what comes out of the wall).  Again, complex numbers = easy!

If you’ve gotten stuck behind a nasty integral in calculus (and if you haven’t, ignore this), you’ll find that many of them are surprisingly hard using real numbers, but baby simple using complex numbers.  For example: \int_0^\infty \frac{\sin{x}}{x} \, dx, \int_{-\infty}^\infty \frac{1}{1+x^2} \, dx, \int_0^\pi \frac{1}{2 \cos{x} + 1} \, dx, \int_{-\infty}^\infty \frac{p(x)}{q(x)} \, dx (where p and q are polynomials).

Some fields simply can’t be approached without complex numbers, particularly quantum mechanics.  In QM the probability of something happening is the square of the magnitude (absolute value) of the “probability amplitude”, which is complex-valued.  So, if the probability amplitude is \frac{i}{\sqrt{2}}, then the probability is \left| \frac{i}{\sqrt{2}} \right|^2 = \left( \frac{1}{\sqrt{2}} \right)^2 = \frac{1}{2}.  There’s really no way around this.  In fact, the Schrödinger equation, which is arguably the backbone of QM, has an “i” staring you right in the face.  Here’s the equation for a single particle

i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t)

where i is in the first term, and “\Psi” is the probability amplitude of the particle’s location.  That’s a bit much, but the point is; you can’t get rid of the complex numbers and do QM with just real numbers.

The moral of the story is that “complex numbers” are misnamed.  While they are intimidating at first, they make things simple all over the place.  Notably: trigonometry, anything with waves (electricity, light, sound), finding roots, streamlining math (often, but not always), and in quantum mechanics.

Posted in -- By the Physicist, Equations, Math, Quantum Theory | 2 Comments »

Q: Can one truly create something from nothing? If matter formed from energy (as in the Big Bang expansion), where did the energy come from?

Monday, March 15th, 2010

Physicist: That right there is one of the great unsolved questions.  Every experiment that’s ever been done (on this subject) verifies the conservation of mass and energy.  While the amount of mass or the amount of energy may change (they can be interchanged), the sum of the two is absolutely invariant.

This naturally leads to the question above.  There are plenty of theories bouncing around, but without a couple more big bangs to do tests on, it’s unlikely we’ll ever know for sure.  We’ll be more sure.  As we learn, many of the theories will be ruled out, but we’ll probably never be for sure sure.  Here are some examples that almost certainly won’t pan out:

Spectacular Uncertainty: Energy and time cannot both be exactly known.  Over any time scale there is always a little energy error, but the larger the time scale the smaller the energy error.  Generally, this takes the form of one or two extra particles that last effectively no time.  For example; gluons (the carrier of the nuclear strong force) usually don’t even exist long enough to cross an atomic nucleus at the speed of light.  But maybe, just maybe, the unimaginable amount of matter in the universe is some kind of amazing quantum clerical error.  You could tie in the anthropic principle (if there weren’t lots of matter there would be no one around to see it, so since we can see it…) if you want, but still.  That shouldn’t explain there being any more matter than the absolute minimum amount needed to have observers.

It is what it is: Maybe mass/energy conservation only works for T>0, but doesn’t mean bupkis for T=0.  In other words, there’s an unexplainable asterisk on the law.

God/Goddess/Gods/Higher Power: Sure.

All this has happened before, and all this will happen again: Maybe the big bang wasn’t the beginning, but in fact the universe just goes through expansion, collapse, and re-expansion cycles.  This has the advantage of explaining the big bang, and also eliminates the question about conservation of mass/energy, but it does leave lots of other questions.  Maybe worse questions.  Also, the universe gives every sign of wanting to expand forever, making it less likely that a previous iteration would have collapsed.

Bubbles: It could be that our universe “bubbled” off of an even larger universe, that had plenty of mass and energy to spare.  This theory don’t answer the question, but it does push it back far enough that it’s hopeless to try and answer rigorously.

Keep in mind that none of these theories are technically scientific.  Scientific knowledge is nothing more than what we can learn from observation, inference, and experiment.  Beyond inference, we’ve got very little to work with as far as the big bang goes.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 6 Comments »

Q: Why does wind make you colder, but re-entry makes you hotter?

Saturday, March 13th, 2010

The original question was:

Why is it that, when you are outdoors and the atmosphere is moving past you at a moderate rate (wind), you get colder; but when the atmosphere is moving past a space shuttle during re-entry, it gets hotter?

Things falling from space get crazy hot. Unless they're designed specifically to survive the onslaught of air, they tend to aerosolize. That doesn't seem to be the case on the ground.

Physicist: The air right on a surface stays effectively fixed to that surface, an effect that the fluid dynamicists call “the no-slip boundary condition”.  You can imagine this like a deck of cards being spread out on a table, the bottom card barely moves, the card above that moves a little more and so on.  By the time you get to the top card there’s plenty of movement, but the first couple of cards would seem to be effectively “stuck”.

The top card moves a lot, the bottom card barely moves, and the distance each card moves with respect to its neighbors is about the same.

We experience this as a “bubble” of air around our bodies.  The air right next to our skin is almost stationary, while the air a foot or two away barely notices us at all.  The greater the air speed, the more shearing the air near our bodies is, and the thinner the layer of roughly-stationary-air.  The boundary layer (since it’s near us) is warmer than the surrounding air, and acts as insulation.  So more wind means less insulation, which makes you colder.

Unless you live somewhere terrible like Florida or Morocco.  Then the wind just blows away the air that was keeping you cool.  Terrible places to live.  If you want to know what Florida is like, get a dozen humidifiers and hairdryers, and lock yourself in a tiny, air-tight room.

When you wave a piece of cardboard, or something, through the air the resistance you feel is (mostly) a build up of pressure on one side, and a drop of pressure on the back.  Increasing the pressure on a region of air compacts it and causes it to heat up.  When you walk down the street your front should actually be a little warmer than your back, but if you really think you can feel it, then you’re probably walking with the Sun in your face.

The space shuttle does the exact same thing.  It’s just better at compressing the air in front of it, since it’s moving at about Mach 23.  The rush of air moving past the shuttle does cool it off, but the effect is completely overwhelmed by the effect of the compression.  The effect is most pronounced above the speed of sound (Mach 1).  Below Mach 1 air will just get out of the way rather than compress much, but above Mach 1 you can basically “sneak up” on the air.  It doesn’t know you’re coming until it gets hit.

Posted in -- By the Physicist, Physics | 1 Comment »

Q: Are explosions more or less powerful in space?

Friday, March 12th, 2010

Physicist: Brace yourself for a mess of guesses:

I’ve heard (anecdotally, and on Mythbusters) that explosions in water are more dangerous that than explosions in the air.  I don’t know if that has more to do with the specific properties of the medium, or merely with the presence of the medium itself, but (with a reasonable guess) it seems that having material around should help transmit the energy.

For an explosion in space, the only thing that carries the energy to you is light (which you would have gotten anyway) and material from the explosive itself.  So the explosion itself should do less damage, while the shrapnel should actually be more dangerous (in air it would get slowed down, and even somewhat cushioned on impact).

Also, in a medium energy tends to travel at specific speeds (the speed of sound generally) which means that most of the energy will hit you all at once (Shock waves can travel faster than the speed of sound, but they burn up a lot of energy doing it)  An explosion in space will cause material to fly out with a broad distribution of velocities, so you’ll experience the explosion over a more drawn-out time.

That’s all guess work.  If anyone knows anything for certain, please post a comment.

Interesting aside: The famous EMP (electro-magnetic pulse) that follows soon after an atmospheric nuclear detonation would be absent in space (deep space anyway).  The bulk of the pulse is actually generated by the explosion moving the ionosphere (it moves… the ionosphere).

Deep seated pet peev: The kick-ass billowing explosions of Vin-Diesel movies will also be missing from explosions in space.  Instead, the explosion would appear as a “star burst” of material all flying outward in straight lines.

A billowing explosion and a star burst explosion.

Posted in -- By the Physicist, Physics | 2 Comments »

Q: What is infinity? (A brief introduction to infinite sets, infinite limits, and infinite numbers)

Thursday, March 11th, 2010

Mathematician: To mathematicians, infinity is not a single entity, but rather a label given to a variety of related mathematical objects. What unites these “infinities” is that they are all, in some sense, larger than anything that can be obtained or enumerated in the real physical world. Below, I will discuss a few of the infinities that crop up most frequently. One common feature they share is that our intuition about how things should behave often break down in these cases and the math requires some subtlety. Hang onto your hat.

1. Infinite Sets

Whereas sets like {1,2,3} and {dog, cat, apple, bat} clearly are finite in size (with size 3 and 4 respectively), it is natural to say that the set of all integers and the set of real numbers have infinite size. What is particularly interesting though is that while both infinite sets, the integers have an infinite size that’s smaller (in a precise sense) than that of the real numbers.

To observe this, we begin by noting that we can relabel the elements of {dog, cat, apple, bat} to be {1,2,3,4} by assigning dog=1, cat=2, apple=3, bat=4, which does not alter the size of our set (since the size of a set is independent of the names given to its items). However, {1,2,3} is obviously a subset of {1,2,3,4}, so {1,2,3} cannot be larger in size than {1,2,3,4}, and therefore {1,2,3} cannot be larger than {dog, cat, apple, bat} since this set was constructed from {1,2,3,4} just by renaming elements. More generally, if one (finite) set is larger than another, then we can always relabel the larger set so that the smaller one becomes a subset of it. Let’s now assume that this property continues to hold for infinite sets (or, if you like, we can use this very natural property as part of the foundation for the definition of the sizes of infinite sets).

Now, we apply this reasoning about relabeling to the real numbers and integers. First, we observe that the integers are a subset of the real numbers, and hence cannot have size larger than the real numbers. On the other hand though (and this is subtle requiring proof, which can be found here and here) it is impossible to relabel the integers in such a way that the real numbers become a subset of them. Hence, the real numbers are indeed larger than the integers in some important sense.

It may seem obvious that the size of the set of real numbers is in some sense a larger infinity than that of the size of the integers. What may come as a greater surprise however, is that the set of integers {…, -2, -1, 0, 1, 2, …}, the set of positive integers {1, 2, 3, 4, …}, and the set of all rational numbers {p/q where p and q are integers and q > 0} are all infinite but have exactly the same infinite size. The reason is simply because relabelings of these sets exist that make them all into the same set. For example, note that the assignment 1=0, 2=1, 3=-1, 4=2, 5=-2, 6=3, 7=-3, etc. will turn the positive integers into the set of all integers.

As it turns out, there is an infinite “chain” of infinities that measure the size of sets, including \aleph_{0}, the size of the integers, \aleph_{1}, the size of the real numbers, \aleph_{2}, the size of the set of all functions from the real numbers to binary values, and so on. In fact, for every set of size \aleph_{n} one can form a set of size \aleph_{n+1} by taking the set of all subsets of the original set. A disturbing questions with an even more disturbing answer can then be posed: “does there exist a set whose size is greater than \aleph_{0} but less than \aleph_{1}?” Bizarrely, this question turns out to be independent from the standard axioms of mathematics. That leaves us with just three options:

(a) Accept the fact that this mathematical question in unanswerable or “outside of math”.

(b) Reject the existence of sets that are larger than the integers and smaller than the real numbers which would be confirming what is known as the Continuum Hypothesis and amounts to adding a new axiom to math.

(c) Accept the existence of sets with this in between size, which implies adding the existence of such sets as a new mathematical axiom.

2. Limits

Infinities often arise when using “limits”, mathematical constructions which provide a rigorous backbone for calculus. When we have a function f(x) and write

\lim_{x \rightarrow \infty} f(x)

what we mean is the value (if one exists) that the function f(x) approaches as x gets larger and larger. So, for example, we have

\lim_{x \rightarrow \infty} \frac{1}{x} = 0

since by making x large enough, we can make \frac{1}{x} as close to 0 as we like. We would also write that

\lim_{x \rightarrow \infty} x^{2} = \infty

since as x gets larger and larger, x^{2} grows bigger without end (for any real number r there exists an x large enough so that x^{2} exceeds r) . We note though that there is a “rate” at which x^{2} approaches infinity. To see this, we can consider taking limits of the ratio of x^{2} to other functions which also grow without bound as x grows. If these ratios “go to infinity”, then x^{2} goes to infinity faster than these other functions. For example, we have:

\lim_{x \rightarrow \infty} \frac{x^{2}}{x} = \infty

and

\lim_{x \rightarrow \infty} \frac{x^{2}}{\log(x)} = \infty

whereas

\lim_{x \rightarrow \infty} \frac{x^{2}}{x^{2}-1} = 1

so x^{2} goes to infinity faster than x and \log(x) but at the same rate as x^{2}-1.

3. Algebra

Yet another way to think about infinity, is to introduce it as a special “number” with certain properties. For example, we can define \infty so that it satisfies (for all real numbers x):

\infty > x

\infty + x = \infty

\infty \times x = \infty when x > 0

\frac{x}{\infty} = 0

Similar rules could be used to define -\infty.  Some tricky cases arise though for which no sensible definition of \infty seems possible. For example

\infty \times 0 = ?

\frac{\infty}{\infty} = ?

\infty - \infty = ?

Defining each of these latter statements is problematic. However, as long as we never need to multiply infinity by zero, or divide infinities, or do any other “undefinable” operations in whatever context we happen to be working, we can introduce \infty as if it were just a special new type of number.

4. Topology

Topology is the study of topological spaces (which are sort of like a generalized notion of surfaces) and the properties they have that are independent of angles and distances. If two surfaces can be made the same through stretching or pulling without requiring any cutting or gluing (more technically, if they can be mapped onto each other by a continuous function with a continuous inverse) then they are considered identical from a topological perspective. Hence, a disc and rectangular surface are topologically equivalent, as are a rubber band shaped surface and a disc with one hole punched in it (imagine stretching the hole until you get a band like object). You may begin to see why it has been said that a topologist is a person who can’t tell a teacup from a doughnut.

When topologists work with the real number line (i.e. the set of real numbers together with the usual notion of distance which induces topological structure), they sometimes introduce a “point at infinity”. This point, denoted \infty can be thought of as the point that you would always be heading towards if you started at 0 and traveled in either direction at any speed for as long as you liked. Strangely, when this infinite point is added to the real number line, it makes it topologically equivalent to a circle (think about the two ends of the number line both joining up to this single infinite point, which closes a loop of sorts). This same procedure can also be carried out for the plane (which is the two dimensional surface consisting of points (x,y) where x and y are any real numbers). By adding a point at infinity we compactify the plane, turning it into something topologically equivalent to a sphere (imagine, if you can, the edges of the infinite plane being folded up until they all join together at a single infinity point).

How to get a circle from a line in four easy steps.

In some sense these “points at infinity” that are introduced are not special in any way (they behave just like all other points from a topological perspective). However, if measures of distance are thrown back into the mix, it seems fair to say that these points at infinity are infinitely far away from all the others.

Conclusion

Ultimately, the question “what is infinity?” is not one that really can be answered, as it assumes that infinite things have a unique identity. It makes more sense to ask “how do infinite things arise in mathematics”, and the answer is that they arise in many, very important ways.

Posted in -- By the Mathematician, Math, Philosophical | No Comments »

Q: Are there physical limits in the universe other than the speed of light?

Wednesday, March 10th, 2010

Physicist: Hells yeah.

Fastest fast: This is worth commenting on since you often hear “nothing can travel faster then light”, but the justification is almost always missing.  The universe seems to be pretty happy thinking of the speed of light as being the same to everybody first (Maxwell’s Laws give you the speed of light, but Maxwell’s laws are the same to everybody so the speed of light is the same to everybody), and as a speed limit second.  Since you always see light moving at the same speed, then no matter how much you speed up, it will always pass you by.  So catching up to it isn’t an option, and everyone will always see you traveling slower than the speed of light.

Densest dense: The harder you compress something, the denser it becomes.  Normally this is reflected in the distance between atoms shrinking.  However, if the pressure is great enough, the atoms will find that it’s easier to have their electrons merge with their protons which then turn into neutrons (and also spit out neutrinos, but whatever).  Without battling electron shells, the once mostly-empty atoms can be packed nucleus-to-nucleus.  Pressures and densities this high only seem to show up in neutron stars (guess where the name comes from).  By way of comparison, here are some densities (in kilograms per liter): Air = 0.0012, People = 1, the Sun = 1.4, Iron = 7.8, Gold = 19.3, Neutron Star = 500,000,000,000,000.

You can also cheat a little.  If a neutron star has a mass of more than about 5 Suns it will collapse into a blackhole, which is technically more dense.

Coldest cold: You might have guessed: zero.  Specifically 0°K = -273°C = -460°F.  However, this is more of an “asymptotic limit” and can never quite be reached.  An object with a temperature of absolute zero will have no atomic movement (heat) whatsoever, but that’s not possible.  One way of thinking about it is in terms of the Heisenberg uncertainty principle which, in a paraphrased nutshell, states: “You can’t have both a perfectly certain position and a perfectly certain momentum”.  Where \Delta x and \Delta p are the position and momentum errors respectively, the uncertainty principle can be written: \Delta x \Delta p \ge \frac{\hbar}{2}.

So if you’ve got a substance and you have any idea where it is (\Delta x < \infty), then you can’t be sure that the momentum is zero, and the object will always have at least a little atomic movement.  Most people who have heard of Heisenberg’s uncertainty principle are under the impression that it’s a limit on how well we can know about an object.  In fact, it’s far better to think of it as a description of how well the universe can know about an object.

Despite the difficulties imposed by the uncertainty principle, we can still get things crazy cold.  The world record for lowest temperature now stands at 0.0000000001°K = 0.1 nK.

Hottest hot: There are actually two limits here, depending on how you phrase the question.  The first is the theoretical upper limit, which depends on which theory you’re working with, but is often quoted around 1030 °K.  These limits have to do with “the graininess” of space, and how much energy can be forced into a particular region.

The second kind of limit is more practical.  As a gas is heated its atoms move faster and faster.  When they collide they bounce of each other and often create photons (light), which generally just go on to push other atoms around.  However, as the temperature approaches about 4 billion °C, the atoms of the gas will often have enough energy to create electron/positron pairs (“E=mc2“, where “E” is the kinetic energy of the gas atoms, and “m” is the total mass of the electron/positron pair).  Normally these newly created particles will almost immediately find other electrons and positrons and annihilate, creating light.  But sometimes they’ll create neutrinos instead of light.  Neutrinos are “weakly interacting” (which is science speak for “goes through walls, no problem”), so the energy used to create them just flies into space, never to be seen again (or just about never).  This has the effect that a gas with a temperature above around 4,000,000,000°C will cool off on its own (without seeming to radiate any energy).  For comparison, the core temperature of the Sun is about 15.7 million °C.

The Sudbury neutrino detector: 40 feet across, and among the more evil looking things ever built. Image stolen without remorse from "http://zuserver2.star.ucl.ac.uk/~idh/apod/ap990623.html"

This is sometimes important during stellar collapse.  If a star needs to have a core temperature above the cut-off point to hold itself up, then it’s not going to hold itself up.

Smallest small: Again, for “uncertainty principle type reasons” it doesn’t make sense to talk about objects or events smaller than the Planck scale, which is about 10-35m.  So far, nobody can think of anything in the universe, at any scale, that would really care, or be able to tell the difference between two points separated by 10-35m.

Emptiest empty: One version of the Heisenberg uncertainty principle can be written “\Delta E \Delta t \ge \frac{\hbar}{2}“, which means that the time and energy of something can’t both be perfectly well known (not even by the universe, the quantities themselves are uncertain).  If you apply this principle to empty space you’ll notice that over short enough time scales there will be measurable, non-zero energy, and over really short time scales you’ll find particles popping in and out of existence.  These particles are called “virtual particles”, and this phenomena is sometimes described as a “particle foam”.

So even with a perfect vacuum, you’ll still have crap around.  This crap is often called the “vacuum energy” or “zero point energy”.

One of the few examples of a device that can harness the vacuum energy of the universe to charge your chrystals or whatever. This illustration of "pyramid power" stolen from "http://www.merlinsrealm.com/pyramid-power.htm"

Sadly, harvesting the vacuum energy is physically impossible (it would violate the uncertainty principle).  The vacuum energy amounts to about 10-13J/m3, or about “the energy a baseball has falling off a table per volume of Lake Superior“.

Posted in -- By the Physicist, Astronomy, Physics, Quantum Theory, Relativity | 3 Comments »

Q: Is it of any coincidence that mathematics is able to describe physical reality – given that both are inventions of the human mind?

Friday, March 5th, 2010

Physicist: There’s a lot of math that doesn’t describe physical reality at all, and even some (few) mathematicians who feel that
“applicability” is just another word for “impurity”.  The ability of math to describe reality is just a consequence of the fact that reality is nice and consistent.

The fact that the math we use (addition, subtraction, geometry, calculus, whathaveyou) works is no coincidence at all.  Mathematics literally evolves in the sense that, if something doesn’t work, then people will ignore it.  So if you have a theory that \pi = 7, great, but no one will use it because it’s patently, provably false.  It doesn’t describe reality (in this case the reality that the ratio of the circumference to the diameter of a circle is \pi), so it goes the way of the Woolly Mammoth.

π=7

I assume that this question is about perceived reality (colors only exist in the brain, whereas in reality there is no “blueness” or “redness”), and not physical reality.  The fact that we can only describe (mathematically and otherwise) the reality we perceive does guide the direction of mathematical research, and as we perceive more we find that the field of math expands accordingly.  For example; number theory wasn’t much more than a hobby before digital communication and RSA encryption, and differential geometry was mostly a nuisance (and anal-retentive over-generalization) until general relativity cropped up.  Now these are both thriving fields of research (in computer science and physics, respectively).

However, just because something works in your head has absolutely no bearing on whether or not it will work in reality (which you would expect if the physical world were created by our minds).  Very good, very reasonable ideas get shot down by experiment every day, and we are constantly surprised.

Philosopher: If we assume the external world exists (independent of our minds), Math’s correspondence to reality is no more coincidental than the correspondence to reality of theories stated in any other language.  This isn’t dependent on the existence of mathematical objects, and it’s not dependent on Mathematical truths existing independently of humans (though I think they do).  If we assume the external world is merely an “invention of the human mind”, then the correspondence of Math to the world is even less coincidental, since the same thing is the author of both.

Posted in -- By the Physicist, Evolution, Math, Paranoia, Philosophical | 1 Comment »

Q: If you were to break down an average human body into its individual atoms, and then laid the atoms out in a single straight line, how far would it stretch?

Wednesday, March 3rd, 2010

Physicist: Atoms are a little “fuzzy”, so their exact size is a little tricky to define.  So taking their size in terms of bond length, and looking at the most common elements in the human body (by mass: 65% oxygen, 18% carbon, and 10% hydrogen), you’ll find that 1kg of person will stretch about 7 trillion km.  So an average (80kg) human would extend about 550 trillion km, or about 14 billion loops around the equator, or 1.4 billion trips to the moon, or about 58 light years.

So you can fit a rich man through the eye of a needle, but be sure to coil him up after you string him out.  Otherwise the process will take at least 58 years.

Posted in -- By the Physicist, Biology, Brain Teaser | No Comments »

Q: What’s it like when you travel at the speed of light?

Wednesday, March 3rd, 2010

Physicist: From a classical (Newtonian) view point this is a completely solid question.  However, in the context of special relativity the question itself is (unfortunately) non-sense.  For many practical purposes, the speed of light (hereafter I’ll call it “C”) is “infinitely fast”.  If you define infinitely fast as the speed you’ll be going if you accelerate forever, then C is exactly that.

Normally when you want to figure out “the behavior at infinity” you can “take a limit”.  For example; the limit as x goes to infinity of 1/x is 0.  This statement just means that as x gets bigger and bigger 1/x gets closer and closer to zero.  So by looking at the behavior at larger and larger finite values you can talk about what happens at infinity.  C, on the other hand, is fundamentally different from all other speeds.

At a basic level, speed is just distance traveled over time taken (as in “miles per hour”).  Due to the laws of special relativity, movement affects both the relative distances and relative time between two reference frames.

As a quick aside, a “reference frame” is just the set of all things that are moving at the same speed or, equivalently, are stationary with respect to each other.  So if you’re traveling down the highway you’re in the same frame as all the other cars around you (if everyone’s going the same speed), while the repair teams and clean-up crews on the shoulder are in a different reference frame.

It may seem silly to say it, but no matter how fast you move you still see things passing by, and it still takes at least a little time to get where you’re going.  At C however, the distance to your destination is always zero due to length contraction, while the time it takes to get there is also zero due to time dilation.  If you were to calculate your own speed you would say v= \frac{d}{t} = \frac{0}{0} = ?, which makes no damn sense.  I mean, what is that?

The universe: As seen by something traveling slower than C, and something traveling at C.

Also, consider this: at any other speed you can speed up or slow down, but at C you genuinely don’t have time to step on the brakes or the gas.  Literally, “time” and “distance” are phenomena that only make sense if you’re talking about them at speeds slower than C.  Stuff in the universe is divided into two categories: “massive” and “massless”.  Massive objects (anything with mass) always travel slower than C, while massless things must travel at C.

All that being said, you can wave your hands and talk about what life is like for a photon, that can’t exist at sub-light speeds (after all, what speed would you expect light to move at?).  When a photon is generated it immediately takes off at C, and never slows down until it runs into something.  Photons never experience time or distance.  As far as they’re concerned they are emitted and absorbed at the same place and time.  Many of the radio photons hitting you right now (about a third of them), have been traveling for around 15 billion years, but they think that the beginning of the universe just happened (or would, if they could think).

Posted in -- By the Physicist, Physics, Relativity | 1 Comment »

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