November « 2009 « Ask a Mathematician / Ask a Physicist

Archive for November, 2009

Q: Will CERN create a black hole?

Thursday, November 26th, 2009

Physicist: Unfortunately, ultra small black holes straddle the line between quantum mechanics and general relativity, which makes it difficult to make useful predictions. The answer is: a solid maybe. If your conCERN is that the black hole thus created will destroy the world, you can relax.

First, the hole will be so small that you couldn’t force-feed it an electron.

Second, the TeV (the energy an electron has after being pushed through 1,000,000,000,000 volts) collisions that CERN is aiming for happen in nature. Statistically, they should happen in the upper atmosphere somewhere in the world a couple times a day. The record is held by the OhMyGod particle detected in 1991, which had an energy of around 300 million TeV. So if we can form tiny black holes, then nature’s already beaten us to the punch. The evidence of these ridiculously high energy collisions come in the form of a “shotgun” wave of intense radiation, that only affect small areas on the ground a couple hundred yards across. You’ve been hit by several of these events in your lifetime, and been none the wiser.

Do to the effects of Hawking radiation (which makes small black holes fizzle out and disappear), the scientists at CERN are working on methods to detect the secondary effects of a fresh black hole evaporating, instead of detecting the hole itself. They don’t expect it to last long enough to get from the collision point all the way to the detectors, which are only inches away.

CERN: The last thing James Bond will ever see.

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

Q: What’s the highest population growth rate that the Earth can support?

Tuesday, November 24th, 2009

Physicist: Zero.

Populations tend to grow exponentially, which is why the growth rate is defined as R in P = Ae^Rt, where A is the population at time t=0, and P is the population at any other time t. If the average growth rate is greater than zero, then the population will grow exponentially forever, which is sadly impossible. Here’s why:

If the population could grow forever, then eventually the total mass of Humans would be greater than the mass of the Earth, which makes no damn sense. What were we eating?

The population could grow forever if we found a way to colonize other star systems. However, even with speed-of-light ships we could only colonize something like $\frac{4}{3} \pi (Ct)^3$ planets in time t, since we can’t travel faster than light (this is the volume of a sphere that expands at the speed of light, C). The population density in the colonized area of the universe would then look like $\frac{\textrm{Population}}{\textrm{Volume}} = \frac{3A e^{Rt}}{4 \pi (Ct)^3} \propto \frac{e^{Rt}}{t^3}$ , where “ $\propto$ ” means “proportional to”. You’ll notice that if R is greater than 0, then as time increases the population density goes up forever, which makes no damn sense. If you don’t notice, then just graph it.

What this ultimately means is that the average, over all time, of the number of children that a person has is 2 or less. No way around it.

We’ve had a lot of visits from reddit.com with a little confusion over the line “populations tend to grow exponentially” (my bad). What I should have written is “populations tend to grow exponentially under the assumption that they have not yet begun to saturate the available resources”, but I figured that might be pushing the discussion. The less that resources are available, the more the population will tend to level off. Both the exponential growth and leveling off can be modeled using various scalings of the logistic function. Try graphing: $\frac{1}{1+e^{-t}}$ , and $e^t$ . You’ll find that they line up almost exactly until near $t=0$ , where resources begin to dwindle.

Posted in -- By the Physicist, Biology, Math | 3 Comments »

Q: What is time?

Monday, November 23rd, 2009

Physicist: All philosophy aside, the best answer is due to Einstein, who said (after lots of thought) that “Time is what clocks measure”.

The term “spacetime” gets bandied about a lot. And you may have been lead to (correctly) suspect that time and space are closely related, and even a little interchangeable. Time is still not exactly like another spacial dimension (like the usual 3). Which is why physicists will tend to say that we live in “3+1 dimensions”, for 3 spacial dimensions, and 1 time dimension. If someone says that time is “the 4th dimension”, keep in mind that there’s an asterisk attached.

If you’re trying to measure a distance $D$ between two events $x_1$ and $x_2$ in 1 dimension you’ll find that $D = \sqrt{(x_1-x_2)^2}$ , which is just another way of writing “the absolute value of $x_1 - x_2$ “. In 2 dimensions $D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$ , which is the Pythagorean theorem. And in 3 dimensions $D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$ . Notice a pattern? The kick-ass thing about each of these equations is that no matter how you shift or spin around the coordinates, D will always stay the same, which is exactly what you’d expect. If you measure a yard stick at any angle, anywhere in the world, you’ll always measure the same length. “D” is the only equation that makes that happen.

In 3+1 dimensions (the standard spacetime) $L = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 - C^2(t_1 - t_2)^2}$ . Here the x’s, y’s, and z’s are the coordinates in space of event 1 and event 2, the t’s are the time coordinates of those events, and C is the speed of light. “L” is the measure of distance in spacetime, but to keep everything straight it’s called the “spacetime interval”. You may have noticed that sometimes when you use this equation that you’ll be taking the square root of a negative number. Don’t worry about it. The power of the Interval, L, is that, like the D above, L remains unchanged by changing location, rotating, and even movement (holy crap), and it is the only equation that does all that. So hidden away in all that wordiness is the big difference between space and time:

The time term in L has a minus sign. As far as physics goes, that’s about it. All of the horror and weirdness of special relativity (all the time dilation, length contraction, ruined simultaneity, all of it) can be tracked back to that minus sign. If you wanted to, you could define time to be the direction in spacetime that has a minus in front of it in L. I don’t think many people do that (not many grounded people at least), but they could if they really wanted.

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

Q: What is “Dark Matter”?

Saturday, November 21st, 2009

Physicist: The most common answers are: “heavy neutrinos” and “I don’t know, something else?”. Neutrinos are, for lack of a better word, “ghost” particles, which makes them a decent candidate.

The original dark matter theories came from observations of nearby galaxies. By using the Doppler effect we can detect how fast stars are moving toward or away from us.

The Doppler Effect: Stars moving toward us look bluer, stars moving away look redder.

By graphing the speed of stars vs. their distance from the core of their galaxy you can determine how matter is distributed in that galaxy. This technique has revealed two interesting things: galaxies have about ten times more mass than you’d expect (from adding up stars and dust), and that even flat galaxies (like ours) have most of their mass arranged in a ball shape. The first result implies immediately that there’s matter out there that we can’t see. The second result (spherical distribution) implies that this matter is weakly- or non-interacting. Here’s why that makes sense:

All the particles in a cloud of matter (dust or gas or whatever) more or less orbit the center of the cloud. If those particles can interact, they will collide with each other and accrete. This means the cloud will tend to flatten into a disk, as well as collapse in on itself as it loses energy. This is a very common image for anyone who’s spent time looking at space pictures.

Saturn, Uranus, the Sombrero Galaxy, and the Andromeda Galaxy

A non-interacting cloud has no reason to do anything other than sit there being round. So, there’s the source of the theory. This “stuff” is so non-interacting, that light doesn’t even bounce off of it, so it’s invisible. But it creates gravity so it must have lots of matter. Hence the name, “Dark Matter”. Another, less popular name is “WIMPs” (Weakly Interacting Massive Particles). Many physicists are uneasy about dark matter because, by definition, it’s very difficult to detect. But the computer simulations of galaxies forming under the influence of Dark Matter tend to give results more consistent with the galaxies we can see in the sky today.

There are some alternative theories to dark matter, such as Mo.N.D. (Modified Newtonian Dynamics, which claims that gravity has slightly different rules at very low intensity), but they tend to be even less popular than dark matter.

Posted in -- By the Physicist, Astronomy, Physics | No Comments »

Q: Why do heavy objects bend space and what is it they are bending?

Tuesday, November 17th, 2009

Physicist: Scientists can generally answer “medium complicated” questions. The really hard ones are too damn hard, and the really simple ones are often “just the way things are”. Or they may be really, really difficult questions in disguise. You’ve found one of the fundamental questions that no one seems to have a good answer for. We also don’t know why the mass, $M$ , we use when talking about inertia (as in $F = MA$ ) is the same as the mass we use when talking about gravity (as in $F = \frac{GMm}{R^2}$ ). The best theory I’ve ever heard is that matter is actually made of bunched up “knots” of space-time, and even that isn’t a terribly good a theory.

The second half of your question we do have an answer for, it’s just a little mind bending and hard to picture. What heavy objects are bending is space itself. The way you detect space (follow me here) is with rulers or stretched out strings, or stuff like that. Anything that measures distance. Near heavy objects the distance between points is greater than you would expect, and you can use this fact to detect and measure the stretching of space.

The usual trick when trying to picture higher dimensional space is to knock off a couple of dimensions, and picture that instead. 3 dimensions is too complicated, and 1 is stupid, so here’s 2!

A circle and its diameter in flat space. Here D=6.5'', C=20.4'', and 20.4/6.5=3.13 ≈π.

A circle and its diameter in curved space. Here D=7.2'', C=20.4'', and 20.4/7.2=2.83<π

Take the circumference, $C$ , and the diameter, $D$ . You’ll notice that on paper (flat space) $\frac{C}{D} = \pi$ . Which makes sense, since that’s the definition of $\pi$ . However, on the balloon (curved space) $\frac{C}{D} < \pi$ , since the diameter is longer. This effect is huge (infinite) for things like black holes, but for an object like the Earth the effect is tiny (ignore the hell out of it). The stretching of space caused by the Earth’s gravity (well… that is Earth’s gravity) increases the diameter of the Earth by about 18mm.

That is, the diameter you get by measuring the equator and dividing by $\pi$ ( $D = \frac{C}{\pi}$ ) is the “flat space diameter”, the diameter you get by physically dropping a rope through the planet is the true, “curved space diameter”, and the difference between these two is 18mm.

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

Q: Why does math work so well at modeling the world around us?

Monday, November 16th, 2009

Mathematician: One important thing to realize about mathematics is that it was primarily created for practical purposes. For example, numbers were likely used in the beginning to count possessions, multiplication for trade, and geometry to measure plots of land (or some similar purposes). Mathematicians and scientists use math to model the world by constructing mathematical objects that capture important properties of physical things (while ignoring those properties that are not relevant for the investigation). Hence, it isn’t as though math just happens to work well for analyzing the world we live in, rather, it was specifically designed for that purpose. If our original mathematical objects had failed to capture important properties of real objects, they surely would have been discarded and replaced with ones that would be more useful. To give one example, if the operation of addition did not so closely model so many physical phenomena (e.g. if I have two objects in one group and I combine them with three objects in another group, then my new group has five objects, which is mimicked by 2+3=5) then it might not be considered a basic mathematical operation like it is today.

Once the basic objects of math were introduced (for their practical uses), it was then possible for people to generalize these objects, find connections between them, and prove theorems about them. For example, once we have integers (for counting) we can ask the question whether there is any largest integer. Once we have addition, we can ask the question whether a + (b + c) = (a + b) + c. Once we have division, we can introduce the idea of prime numbers. Once we have exponents and real numbers, we can introduce polynomials, and attempts to find the roots of polynomials will inevitably lead to the introduction of imaginary numbers. Hence, from the basic useful mathematical objects, a whole complicated structure follows which contains many new ideas relating to or emanating from the original ones.

Long after most of the basic objects of math were created, attempts were made to axiomatize the subject (i.e. provide a small set of basic axioms from which the rest of math can be derived), but math was not developed from these axioms. Quite to the contrary, these axioms were developed from the already existing useful mathematical system, and hence the axioms somehow inherently have built into them the usefulness of the entire mathematical structure. By altering these axioms mathematicians can (and have) developed different versions of mathematics. One thing that is special about the version of mathematics that we are used to is that it allows for creating a staggering variety of useful models. When the basic axioms are fundamentally altered, this is not necessarily the case.

A more difficult question than why math works so well at modeling the world, is the question of why math that is developed for one purpose (or, sometimes no purpose at all except theoretical interest) ends up being so useful for other purposes, but this is a subject that deserves a post of its own.

Posted in -- By the Mathematician, Math, Philosophical | 5 Comments »

Q: Why is it that when you multiply a positive number with a negative number you get a negative number?

Sunday, November 15th, 2009

Mathematician: Consider what would happen if when we multiplied a positive number by a negative number we got a positive number. For example, suppose that

3*-5 = 15

rather than the usual

3*5 = 15.

Now, if all the other standard rules of math still applied, then we could write

-15 = -(3*5) = 3*-5 = 15

and hence we would find that -15 and 15 are equal to each other. Depending on how you think about it, this is either a contradiction (if we fundamentally believe that -15 is not the same as 15) or we have destroyed all negative numbers entirely (if we simply interpret this formula to mean that there is no difference between negative and positive numbers). In either case, clearly something has gone screwy.

Perhaps an even simpler way to think about this is to observe that by using the rules of addition and the definition of multiplication, we have for example that

3*(-2) = -2 + -2 + -2 = -6.

More intuitively though, why should we expect that negative numbers multiplying positive ones should lead to negative numbers? Well, consider a financial example. If we denote positive amounts of dollars with positive numbers (i.e. money owned), it makes sense that negative numbers would correspond to money owed (i.e. debt). Then, for example, if one bank has $100 and it merges with a bank that has -$80 then the new, combined bank has a total of $20 in net assets (100-80 = 20). So where does multiplication come into play? Well, let’s now imagine that three banks merge, each of which has -$20. In that case, we should expect the total amount of net assets that the new merged bank has is just three times that of the individual banks, given by the mathematical expression 3*(-20) = -60. Of course, this only works if multiplying the positive number 3 with the negative number -20 leads to a negative number (in this case -60).

Posted in -- By the Mathematician, Math, Philosophical | 1 Comment »

Q: What is the best way to understand relativity theory? Why is it so counter intuitive?

Sunday, November 15th, 2009

Physicist: 1) Pictures and math. 2) Work through and understand the “train struck by lightning” and “barn running pole vaulter” thought experiments, to better understand the relationship between space and time. 3) Pick up a book on modern physics and do a mess of random homework problems. 4) Build something that involves relativity.

As for the second question; This is one of my favorite examples of the Allegory of the Cave. Intuition is based on experience and the mental wiring we have evolved and inherited. Almost everything we interact with day to day is smaller than a mile, slower than 60mph, weighs less than a ton, and takes less than a year. At the scales we’re used to the laws of physics are fairly intuitive. Humankind spent tens of thousands of years thinking that light is instantaneous. And who could blame them? Magellan took a leisurely three years to work his way around the planet (easy to imagine), whereas light can do it in about 1/7 of a second (hard to imagine).

Gamma ( $\gamma$ ) is a term that shows up a lot in special relativity, and describes how much things get messed up by moving fast (time dilation, length contraction, mass increase, …). It’s fairly easy (pre-calculus easy) to show that $\gamma = \frac{1}{ \sqrt{1-v^2/C^2} }$ . Notice from the graph below that relativistic effects are barely noticeable until around $V= 0.5 C$ ( $C$ is the speed of light).

Gamma vs. Velocity. Gamma = 1 is the normal, every day experience.

On this graph the speed of light is at $V=1$ . For comparison, the speed of sound is around $V = 0.000001$ , and the fastest that any human has ever traveled is around $V = 0.00003$ . You can use $\gamma$ to show that Astronauts and Cosmonauts that have spent years in orbit, traveling in excess of 7.5 km/s, still lose less than one second of time total, compared to people on the ground.

When physical laws work actively on scales we’re used to, they tend to become intuitive. The laws of the very big, small, old, hot, cold, fast, … are the same laws, we just experience a special case.

Posted in -- By the Physicist, Philosophical, Physics, Relativity | No Comments »

Q: Will we ever go faster than light?

Saturday, November 14th, 2009

Physicist: Hells no.

There are a lot of conclusions you can draw from the theory of special relativity that rule out FTL (faster than light) travel. From your own point of view you may as well be sitting still (try it), but from the perspective of someone watching you approach the speed of light all kinds of thing get messed up. But all of those bizarre effects can be boiled down to one solid principle, the “Einstein Equivalence Principle”, so I’ll just use that. The EEP states that the laws of physics operate exactly the same no matter where you are or how fast you’re moving. This includes the physics behind electro-magnetism, which dictates the speed of light. So a direct consequence is that the speed of light is the same, no matter where you are or how fast you’re moving.

The best way to tell if something is going faster than something else is to race them and see which wins. So imagine a photon whips by you at the speed of light (hence the name), and you start running (using a rocket, or whatever) to catch up. A minute later you’re going much faster, but the photon is still moving at the speed of light compared to you. In fact, no matter how fast you get going, light will always pass you at the speed of light. So it’s not like the race is even close. As far as the photon is concerned, you may as well not be moving.

So if what you think of as “speed” (miles per hour) gets messed up, then what you mean by “distance” and “time” (miles and hours) must get messed up as well. This is the (unfortunately correct) conclusion that Einstein drew. So, unlike breaking the sound barrier, which was an engineering challenge, but known to be possible, breaking the light barrier is known to be impossible, by everything we know about modern physics.

The most powerful fuel source that humankind is every likely to come across is hydrogen fusion, which can get a ship up to about 11% of light speed. A “Bussard ram scoop” could get you going arbitrarily fast by collecting fuel en route, but there’s very little of it to be had (it wouldn’t be called “space” if it was full), so your acceleration would be very slow. An anti-matter rocket could also get a ship going arbitrarily fast, but there are problems with anti matter: 1) where do you get it?, 2) you can’t ever touch it (it’s the most unstable, dangerous stuff possible), 3) to get up to 90% of light speed half of your ship would have to be matter/anti-matter fuel.

Posted in -- By the Physicist, Physics, Relativity | No Comments »

Q: If you were on a space station, would you be able to tell the difference between centrifugal force and normal gravity?

Tuesday, November 10th, 2009

Physicist: Normal gravity on Earth, $G$ , never changes. However, the acceleration, $C$ , due to centrifugal force on the station is given by $C = \frac{V^2}{R}$ , where $V$ is how fast the station is spinning, and R is the radius of the station. If you’re sitting still or walking slowly on a space station, then you’d probably never notice the difference, but if you run either with or against the direction of spin you can change the value of $V$ (for yourself), and thus change the amount of centrifugal force you experience. Whether or not you notice a difference now depends on how big the station is.

Kick-ass, spinning space station (from "2001")

The acceleration, due to gravity on the surface of the Earth is about $G = 9.8 m/s^2$ , and I would guess that a $5 \%$ change in your weight is generally detectable. Given this $5 \%$ estimate, a running speed of around $4.5 m/s$ (10 mph), and assuming that the station is spinning exactly fast enough to have $C=G$ , then, after some math happens, you’ll find that when the radius of the station is less than around 3.25 km that you can detect the difference. So, you could tell the difference on any ship or station likely to be made in this solar system at least.

If you don’t feel like running back and forth you can try playing catch. You’ll find that when you throw a ball spinward that it curves down, and that when you throw the ball anti-spinward it curves up. In fact, if you can throw the ball as fast as the station spins, then you can make the ball fly at the same height forever (assuming no air resistance). In this case the ball is actually sitting still while the station spins around it.