Q: How did Lord Kelvin come up with the absolute temperature? I mean, how could he say surely that it was 273.15 C below zero?

Posted on May 9, 2010 by The Physicist

Physicist: Lord Kelvin (and others of his ilk) noticed that when you hold the volume of an ideal gas constant you get a nice, linear relationship between pressure and temperature.

Temperature vs. Pressure. PV = nRT works for all ideal gases, independent of the particular gas in question (helium, water vapor, sulfur hexafluoride, ...). Different gases become liquids or solids (and stop exerting pressure) at different temperatures, so they drop off of the "ideal gas graph" at different points.

By the by, an ideal gas is just a gas where you can assume that the particles are bouncing off of each other much harder than they’re trying to stick together (the gas is hot enough that it’s a long way from condensing).  This assumption allows you to use the “billiard ball model” of gas dynamics, which in turn leads to the ideal gas law (PV=nRT), which says that you should expect a nice straight line (like the one above).

Although it’s impossible to cool anything off completely, and despite the fact that all of the gases that Lord Kelvin was working with became liquids when chilled enough, it was still easy to graph temperature vs. pressure (even around room temperature) and then extend the line to find the temperature where the pressure should be zero.  Kelvin figured that this would be a much more natural place for “zero” to be, and he carefully measured it (by extending the line) to be around -273.15°C, which is now 0°K (zero degrees Kelvin).

Using Kelvins instead of Celsius means that you can bust out the ideal gas law without needing to adjust anything.  If you wrote the ideal gas law using Celsius instead, it would be PV=nR(T+275.15), which is ugly.

Posted in -- By the Physicist, Physics | 14 Comments

Q: What do complex numbers really mean or represent?

Posted on May 3, 2010 by The Physicist

Physicist: Nothing really.

Complex numbers are very useful for streamlining a lot of different types of math, generalizing ideas, and “closing” the real numbers.  In quantum theory you’ll find that on the most fundamental level the universe seems to prefer complex numbers to real numbers.

But you can’t use them to count or measure stuff for crap, so most people can live long happy lives without being particularly bothered by complex numbers.  If you can call that living.

Mathematician: There are many ways to view complex numbers, but one of the most intuitive is to think of them as representing points in the plane. Doing so will allow us to interpret basic arithmetic operations like addition and multiplication as performing geometric manipulations.

How does this work? Well, every complex number z can be written as z = x + i y  where x and y are real numbers, and i is the square root of -1. We can then think of z as a point on the (x,y) plane with x being the position along the horizontal (i.e. real) axis, and y the position along the vertical (i.e. imaginary) axis. To understand the operation of adding and multiplying complex numbers though, we need to think about them slightly differently.

The complex number 2+3i represented as a point in the complex plane. Have you ever seen a more boring image in your life?

If we choose, we can treat complex numbers as vectors rather than points. That means that now z = x+i y  represents not the point (x,y) but rather, the directed line segment which extends from (0,0) in the complex plane to (x,y) in the complex plane (unless otherwise stated we will assume here that our vectors are emanating from the origin (0,0)). You can think of a vector as simply representing a magnitude (or length) and a direction. The line segment extending from (1,2) to (-4,5) is the same as the one extending from (-4,5) to (1,2), but the vectors represented would be different because they would be pointing in opposite directions. Vectors are often drawn as arrows.

The complex number 2+3i represented as a vector in the complex plane. Compared to our last image, this is a hoot.

Once we have the vector notion of a complex number, we can think about adding complex numbers as adding vectors. For example, if we have

z_{1} = x_{1} + i y_{1}

and

z_{2} = x_{2} + iy_{2}

then

z_{1} + z_{2} = (x_{1}+x_{2}) + i (y_{1}+y_{2}).

Hence, z_{1} + z_{2} is the vector in the complex plane that extends from (0,0) to (x_{1}+x_{2}, y_{1}+y_{2}). Note that (x_{1}+x_{2}, y_{1}+y_{2}) is the point we get to if we piggy back vectors z1 and z2 and then follow them both. By this I mean that we translate (i.e. move without rotating or stretching) z2 so that its beginning is at the end of z1, and then we follow the path consisting of the two vectors until we get to the (new) end of z2. However, since z_{1} + z_{2} = z_{2} + z_{1}, we can also piggy back z1 at the end of z2 to get the same result. A slightly different view is achieved if we think of our first number z_{1} as being a vector, and our second number z_{2} as being a point. In this case, z_{1} + z_{2} simply corresponds to moving the point z_{2} by the distance and direction represented by the vector z_{1}.

Here are two complex numbers represented as vectors, 2+3i and 1+3i.

When we sum the complex numbers 2+3i and 1+3i we get the complex number 3+6i.

Alright, so addition of complex numbers can be thought of as adding vectors in the complex plane (or moving a point by the distance and direction stored in a vector), but what the heck does multiplication do? Well, to understand multiplication we need yet another geometric way of thinking about complex numbers.

First though, we make a quick (and relevant) aside. For any complex number z, we have by definition that the absolute value |z| of z satisfies

|z| = \sqrt{z \overline z} = \sqrt{(x+iy)(x-iy)} = \sqrt{x^{2} + y^{2}} = \sqrt{(x-0)^{2} + (y-0)^{2}}

which is precisely the formula for the distance between the point (0,0) and the point (x,y). Hence, |z| measures the length of the vector that z represents. Now, according to Euler’s formula, we have that for any real number \theta

e^{i \theta} = cos(\theta) + i sin(\theta).

hence e^{i \theta} is a complex number since it is the sum of a real and imaginary number. In particular, we have

|e^{i \theta}|^{2} = e^{i \theta} \overline{ e^{i \theta} } = e^{i \theta} e^{-i \theta} = e^{i \theta - i \theta} = e^{0} = 1

That means that any complex number representable in this form must have a distance of 1 from the origin. In other words, such numbers represent points on the unit circle, or equivalently, vectors of length 1 extending from the origin. As it turns out, all vectors in the complex plane with length 1 can be represented in this form. But what angle does each of these vectors point at? Well, we compute the angle of the line segment (0,0) to (x,y) with respect to the horizontal axis using arctan(y/x). In our case though, since x=cos(\theta) and y = sin(\theta) this gives:

arctan(y/x) = arctan(sin(\theta)/cos(\theta))

= arctan(tan(\theta)) = \theta.

Hence, the complex number e^{i \theta} represents a vector of length 1 pointed at angle \theta. Similarly, for any non-negative number r, we have that the complex number

r e^{i \theta}

represents a vector of length r pointed at angle \theta . This provides a new way of thinking about a complex number: as a vector specified by its length and angle. What happens when we multiply two such numbers z_{1} = r_{1} e^{i \theta_{1}} and z_{2} = r_{2} e^{i \theta_{2}} together? Well, we have

z_{1} z_{2} = r_{1} e^{i \theta_{1}} r_{2} e^{i \theta_{2}}

= r_{1} r_{2} e^{i (\theta_{1} + \theta_{2})} .

Hence, z_{1} z_{2} is a new complex number representing a vector of length r_{1} r_{2} pointed at angle \theta_{1} + \theta_{2}. Therefore, we can think of the action of z_{1} multiplying z_{2} as causing the vector z_{2} to stretch by a factor of r_{1} and rotate by an angle \theta_{1}.

Hence, to recap, we can view complex numbers geometrically as representing points or vectors in the complex plane. If we do this, then adding complex numbers corresponds to adding together vectors, or equivalently, moving the point that the second complex number represents along the vector that the first complex number represents. On the other hand, we can think of multiplication of complex numbers as corresponding to scaling and rotating the second complex number in the multiplication by the length and angle inherent in the first complex number. Finally, we note that taking the absolute value of a complex number corresponds to measuring the length of the corresponding vector. Therefore, one way to view complex numbers is as a means for converting geometric operations (translation, rotation, scaling) into algebraic operations (adding, multiplication) and back again. As you might imagine, this can be extremely useful!

Posted in -- By the Mathematician, -- By the Physicist, Math, Philosophical | 14 Comments

Q: Is it odd that the universe’s constants are all so perfectly conducive to life?

Posted on April 30, 2010 by The Physicist

Physicist: Maybe.

When written down, most physical laws involve at least one physical constant.  For example, the “G” in gravitational force: F=\frac{Gm_1m_2}{r^2}, or the “h” in the energy of photons: E=hf, or the speed of light: “c”.  There are a couple dozen other (more and more obscure) constants out there.  Changing these constants changes how the universe hangs together, which forces are most important, what chemical elements are possible, how (and if) things interact, whether or not stars exist and for how long, etc.

None of the constants have any reason to be what they are, and not something else.  Sure, G=6.673 × 10-11m3kg-1s-2, but why?  Why not G=2 m3kg-1s-2 or something?

What’s really spooky is that even tiny changes in most of the physical constants tend to make life, and even the universe as we know it, impossible.  Some research (computer simulations mostly) has recently suggested that there are completely different combinations of constants that lead to universes that are unrecognizably strange but still capable of supporting highly complicated systems (and so, possibly life).

Here are my two most fave theories about why the universe is so nice:

If there are plenty of universes: you can use the anthropic principle.  The anthropic principle can be used to justify effects that require an observer.  For example: “you are here” signs are always correct when you’re there to see them, but are always wrong when you’re not.  You can sometimes find people excited about how perfectly the universe (Earth in particular) is suited to Human life, and admittedly the fit is pretty good.  But you can justify that fit using the anthropic principle; if you’re going to find Human life somewhere in the universe, do you really expect to find people on an acid world or some kind of lava monster world?

Mall directories: how do they always know where you are?

To get all the constants just right you’d need a huge (probably infinite) number of universes.  Turns out that picking a number truly at random is tricky.  If you have enough universes, then many of them will have the right balance to allow life, no matter how unlikely it is.  Now everyone capable of asking the question will find themselves asking it in a universe perfectly suited for life, despite the fact that the overwhelming majority of universes are completely inhospitable.

The idea that there are effectively (or actually) an infinite number of universes is not new to physics.  All quantum mechanical systems (which is everything, really) exist in every possible state simultaneously.  However, all of these states use the same physics, so the jump to every possible universe with every possible set of physical laws is a pretty big jump (“Powellian” even).  But, sadly, we don’t know just a hell of a lot about universe creation.  It might be completely reasonable for new universes to have different physical constants.  We’ll have to create a few trillion to know for sure.

If there’s one universe: you don’t get to use the anthropic principle.  But how about this:  When you create a batch of particles (using accelerators mostly) the stuff that flies out is in the highest entropy it can manage.  You’ll see lots of different kinds particles, in different states, flying in different directions, and all of it is random.  Low predictability = high entropy.  The exact results of particle creation are impossible to predict.

What if the creation of the universe followed the same rule?  That a universe with its constants tuned for very high entropy is more likely to be created.  The tendency of things to have high entropy is independent of the constants involved, so it’s not completely unreasonable to apply a rule like this, merely very unreasonable.  Looking around at the universe today, it would seem to be set up to maximize its own entropy.  Chemicals of amazing complexity are possible, there are almost a hundred natural elements (which requires crazy balance), there are dozens of different types of exotic particles that blink in and out everywhere all the time.  Very unpredictable, very high entropy.  By contrast, if you kept everything the same, but changed the mass of the proton from 1.673 × 10-27 kg to 1.675 × 10-27 kg (0.2% increase) you’d find that all the universe’s protons would turn into neutrons in short order.  No more chemicals (or even elements) of any kind.  Very predictable, very low entropy.

It may be that the universe is the way it is just to be as complicated as possible.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 18 Comments

Q: How/when will the world end?

Posted on April 26, 2010 by The Physicist

Physicist: To answer this question definitively would require the destruction of at least a couple dozen other worlds.  But failing that, guesswork:

The little things (people): In the short term (less than several million years) the biggest threat the Earth faces is people.  We’ve already got the Holocene mass extinction going for us, now we’ve just got to step up our game and go for broke.  Hey, Coalition of the Willing!  I think I saw North Korea stealing your cup cakes.  Also, it’s too cold in the winter.  Couldn’t we burn another teraton of coal or something?

Boring, Regular Extinction: If we (Homo sapiens) follow the same fate as all of our predecessors and cousins (homo habilis, rudolfensis, georgicus, ergaster, erectus, cepranensis, antecessor, heidelbergensis, rhodesiensis, and neanderthalensis, for example), then it’s very likely that we’ll be extinct within the next 100,000 to 1,000,000 years.  Statistically speaking anyway.

Total carbon re uptake: Over very large time scales the sun is getting brighter (along the lines of about 10% per billion years).  Astrophysicist Brownlee and paleontologist Peter Ward have written a book espousing the idea that this gradual brightening will cause the Earth to heat up and the natural chemical processes that absorb CO2 from the air (and lock it away in sediment) will speed up.

“Main sequence stars”, which include the sun, are surprisingly stable for a very long time. They do change a little, increasing their brightness by about 10% every billion years.

They figure that inside of 500 million years there won’t be enough CO2 in the atmosphere to support plant life, and that would be the end of complex life.  Although life has been around for at least 3.5 billion years, the interesting stuff (animals) have only been around for about 500 million years.  So if Brownlee and Ward are right, we’re only about halfway done (not nearing the end).

It may seem strange to talk about the loss of CO2 being the end of the world when we so often talk about the dangers of too much CO2.  The difference is in the time scales.  The spike of CO2 we worry about today is on the scale of centuries, while the long term absorption of CO2 is on a time scale a million times larger (unnoticeable in the short term).

Dynamo shutdown: The Earth’s magnetic field is the result of iron rich (electrically conductive) stuff flowing around in the Earth’s core.  The currents are driven by radioactive heating which causes convection, specifically the decay of radioactive potassium, uranium, and thorium.  The half-lives of these materials are 1.25 billion, 4.5 billion, and 14 billion years respectively, so most of the original fuel has already been used up.

The exact nature of magnetic dynamos is not terribly well known, and is still an active area or research.  We don’t know for certain what the minimum energy input is needed to keep the damn thing running.  We do know that it’s certainly possible for a planetary magnetic dynamo to shut down (Mars’ shut down at least a couple billion years ago).  If our dynamo shuts down, then our magnetic field will vanish and (in fairly short order) the atmosphere will be stripped away by solar wind, as happened on Mars.

Never-ending Summer: The increase in the Sun’s output will make it too hot for liquid water on Earth in about 1 billion years.  With the oceans boiled away the pressure everywhere on Earth will be about the same as the pressure on the ocean floor.  The difference between Venus and Earth will be academic.  No matter what else happens before then, this will be the end of life on Earth.

SPF 5,000,000,000,000,000: Somewhere around 5 to 7 billion years from now the Sun will start to run out of fuel.  Ironically this will actually make the core hotter as it collapses in on itself.  The top layers will fluff up and (probably) envelope all the inner planets, including Earth.  For obvious reasons this is called the “red giant” phase of the Sun’s life.  The solar system will eventually settle down with the gas giants still in place, the inner solar system missing, a white dwarf star where the Sun used to be, and the trans-neptunian stuff completely unaffected.

The Sun as we see it today (yellow), and the Sun in its fluffier red giant phase (red).

Lights out: If by “end of the world” you mean “end of the universe”, then a good end of everything is the end of the age of stars.  The universe started out made up of about 75% hydrogen, but today is only about 70% hydrogen.  Stars are almost completely powered by hydrogen fusion, so assuming that the consumption of the universe’s hydrogen stays constant (which isn’t a particularly good assumption), then there will be almost no stars left in 250 to 300 billion years.

The big rip: Not only is the universe expanding, but the speed of that expansion is increasing.  The expansion is a little hard to picture because the expansion isn’t about things moving away from each other in space, it’s about the space in between things actually expanding.  Right now the effect is small enough that it can only be seen on huge, inter-galactic, scales.  But eventually the expansion with be so rapid that the space between the planets and their stars will increase so fast that the planets will be pulled into open space, and not long after than (as in a couple of months or so) the space between atoms will increase so fast that everything will be completely torn apart and atomized.  This is called the “big rip”.  Some estimates put the big rip about 20 billion years out, and some say it won’t happen at all.

Posted in -- By the Physicist, Astronomy, Biology, Paranoia, Physics | 5 Comments

Q: What would happen if an unstoppable force met with an unmovable, impenetrable object?

Posted on April 22, 2010 by The Mathematician

Mathematician: Sometimes, when we don’t use language carefully enough, we can get ourselves into philosophical trouble. For example, consider the following statement:

If a barber shaves all those men (and only those men) who do not shave themselves, does he shave himself?

If the barber shaves himself, then he is shaving a man who shaves himself, which is something that (by definition) he does not do. On the other hand, if the barber does not shave himself, then there is a man who doesn’t shave himself that the barber doesn’t shave, which again contradicts our definition of the barber.

So what is the answer? Well, the question has no answer, because the definition we use for our barber contains within it a logical contradiction. What’s more, it is impossible for such a barber to actually exist in the real world, since the razor burn associated with simultaneously shaving yourself and not shaving yourself is too much for any single person to withstand.

Now, let’s return to the original question:

What would happen if an unstoppable force met with an unmovable, impenetrable object?

Well, let’s suppose that we define an “unstoppable” force to be one that can move absolutely any matter. Furthermore, let’s define an “unmovable” object to be one that cannot be moved by any force. In that case, this question is unanswerable, because like the barber paradox above, it relies on contradictory information. By definition our force can move anything, but then, also by definition, there is an object that the force cannot move. This is a bit like saying “suppose X is true, and not X is true. Then is X true?”. Here  X is the idea that the force can move anything, and not X is the idea that there is at least one object that cannot be moved by the force (which in this case is our unmovable object). Hence, this question has no answer because it relies on assumptions which contradict each other.

Posted in -- By the Mathematician, Brain Teaser, Philosophical | 32 Comments

My bad: Have aliens ever visited Earth?

Posted on April 21, 2010 by The Physicist

Physicist: In the post “Q: Have aliens ever visited Earth?” I said that the maximum velocity that can be obtained by a rocket using fusion is 11% of light speed.  Wrong!

It turns out that (as a commenter had suggested) the maximum velocity of a rocket is actually a function of the exhaust velocity of the rocket (which is about 11% of light speed for fusion), and the ratio of initial to final mass of the rocket (the initial mass includes fuel).  The assumption I had been working with was that a rocket shouldn’t be able to get going any faster than it’s exhaust velocity.

To find the mass ratio in question: \frac{M_i}{M_f} = \left[ \frac{c+V}{c-V} \right]^{\frac{c}{2w}} where Mi is the initial mass, Mf is the final mass, c is the speed of light (2.99 x 108 m/s), w is the exhaust velocity, and V is the final speed of the rocket.  A derivation of this can be found here: www.relativitycalculator.com/images/rocket_equations/AIAA.pdf.

You can also turn this around: V = \frac{\alpha -1}{\alpha +1} c , where \alpha = \left( \frac{M_i}{M_f} \right)^{\frac{2w}{c}}.

A “reasonable” rocket would be something like 99% hydrogen fuel.  Which translates to a final velocity of only about 47% of light speed.  In order to get something to 0.9c (again using fusion) would require a fuel-to-ship ratio of around 700,000.  If you wanted to get the shuttle to 0.9c you’d need about 19 cubic km of liquid hydrogen.  Frustrating, isn’t it?

The space shuttle (lower left) next to the amount of liquid hydrogen it would need to get to 90% of light speed using fusion with perfect efficiency. If it used regular rocket fuel instead, the cube would be really big.

Using normal rocket fuel (w = 4440 m/s) to get to 0.9c, you’d need a fuel to ship ratio of about 1076,985.  So if the space shuttle were to use the engines it has now, it’s fuel tanks would weigh at least as much as about 1076,961 Suns.

Posted in -- By the Physicist, Mistake | 6 Comments