For example, Newton’s law of universal gravitation says that the gravitational force between two objects with masses m₁ and m₂, that are separated by a distance r, is , where G is the “gravitational constant“.  G can be expressed as a lot of different numbers depending on the units used.  For example, in terms of meters, kilograms, and seconds: .

In terms of miles, pounds, and years: .

In terms of furlongs, femptograms, and  fortnights: .

Point is, by changing the units you change the value of G (this has no impact on the physics, just the units of measurement).  So, why not choose units so that G=1, and then ignore it?  The Planck units are set up so that G (the gravitational constant), c (the speed of light), (the reduced Planck constant), and k_B (Boltzmann constant) are all equal to 1.  So for example, “E=mc²” becomes “E=m” (again, this doesn’t change things any more than, say, switching between miles and kilometers does).

The “Planck length” is the unit of length in Planck units, and it’s meters.  Which is small.  I don’t even have a remotely useful way of describing how small that is.  Think of anything at all: that’s way, way, way bigger.  A hydrogen atom is about 10 trillion trillion Planck lengths across (which, in the pantheon of worldly facts, ranks among the most useless).

Physicists primarily use the Planck length to talk about things that are ridiculously tiny.  Specifically; too tiny to matter.  By the time you get to (anywhere near) the Planck length it stops making much sense to talk about the difference between two points in any reasonable situation.  Basically, because of the uncertainty principle, there’s no useful (physically relevant) difference between the positions of things separated by small enough distances, and the Planck length certainly qualifies.  Nothing fundamentally changes at the Planck scale, and there’s nothing special about the physics there, it’s just that there’s no point trying to deal with things that small.  Part of why nobody bothers is that the smallest particle, the electron, is about 10²⁰ times larger (that’s the difference between a single hair and a large galaxy).  Rather than being a specific scale, The Planck scale is just an easy to remember line-in-the-sand (the words “Planck length” are easier to remember than a number).

That all said (and what was said is: don’t worry about the Planck constant because it’s not important), there are some places on the bleeding edge of physics where the Planck length (or distances of approximately that size) do show up.  In particular, it shows up in the “Generalized Uncertainty Principle” (GUP) where it’s inserted basically as a patch to make physics work in some fairly obscure situations (quantum gravity and whatnot).  The GUP implies that at a small enough scale it is literally impossible, in all situations, to make a smaller-scale measurement.  In the right light this makes it look like maybe spacetime is discrete and comes in “smallest units”, and maybe the universe is like the image on a computer screen (made up of pixels).

How bleeding edge is the GUP?  So bleeding edge that there isn’t even a wikipedia article about it.  Like a lot of things in string theory (this is an opinion), these sort of patches may prove to be mistakes.  So, spacetime may come in discrete chunks, but the most we can say is that those chunks (if they exist) are very, very, very, very small.

You’d never notice (at least, the experiments designed to notice haven’t so far).

“Friction” is a blanket term to cover all of the wide variety of effects that make it difficult for one surface to slide past another.

There a some chemical bonds (glue is an extreme example), there are electrical effects (like van der waals), and then there are effects from simple physical barriers.  A pair of rough surfaces will have more friction than a pair of smooth surfaces, because the “peaks” of one surface can fall into the “valleys” of the other, meaning that to keep moving either something needs to break, or the surfaces would need to push apart briefly.

This can be used in hand-wavy arguments for why friction is proportional to the normal force pressing surfaces together.  It’s not terribly intuitive why, but it turns out that the minimum amount of force, F_f, needed to push surfaces past each other (needed to overcome the “friction force”) is proportional to the force, N, pressing those surfaces together.  In fact this is how the coefficient of friction, μ, is defined: F_f = μN.

The force required to push this bump “up hill” is proportional to the normal force.  This is more or less the justification behind where the friction equation comes from.

The rougher the surfaces the more often “hills” will have to push over each other, and the steeper those hills will be.  For most practical purposes friction is caused by the physical roughness of the surfaces involved.  However, even if you make a surface perfectly smooth there’s still some friction.  If that weren’t the case, then very smooth things would feel kinda oily (some do actually).

Sheets of glass tend to be very nearly perfectly smooth (down to the level of molecules), and most of the friction to be found with glass comes from the subtle electrostatic properties of the glass and the surface that’s in contact with it.  But why is that friction force also proportional to the normal force?  Well… everything’s approximately linear over small enough forces/distances/times.  That’s how physics is done!

That may sound like an excuse, but that’s only because it is.

Q: It intuitively feels like the friction force should be directly proportional to the surface area between materials, yet this is never considered in any practical analysis or application.  What’s going on here?

A: The total lack of consideration of surface area is an artifact of the way friction is usually considered.  Greater surface area does mean greater friction, but it also means that the normal force is more spread out, and less force is going through any particular region of the surface.  These effects happen to balance out.

If you have one pillar the total friction is μN. If you have two pillars each supports half of the weight, and thus exert half the normal force, so the total friction is μN/2 + μN/2 = μN.

Pillars are just a cute way of talking about surface area in a controlled way.  The same argument applies to surfaces in general.

Q: If polishing surfaces decreases friction, then why does polishing metal surfaces make them fuse together?

A: Polishing two metal surfaces until they can fuse has to do with giving them both more opportunities to fuse (more of their surfaces can directly contact each other without “peaks and valleys” to deal with), and polishing also helps remove impurities and oxidized material.  For example, if you want to weld two old pieces of iron together you need to get all of the rust off first.  Pure iron can be welded together, but iron oxide (rust) can’t.  Gold is an extreme example of this.  Cleaned and polished gold doesn’t even need to be heated, you can just slap two pieces together and they’ll fuse together.

Inertia welders also need smooth surfaces so that the friction from point to point will be constant (you really don’t want anything to catch suddenly, or everyone nearby is in trouble).  This isn’t important to the question; it’s just that inertia welders are awesome.

Q: Why does friction convert kinetic energy into heat?

A: The very short answer is “entropy”.  Friction involves, at the lowest level, a bunch of atoms interacting and bumping into each other.  Unless that bumping somehow perfectly reverses itself, then one atom will bump into the next, which will bump into the next, which will bump into the next, etc.

And that’s essentially what heat is.  So the movement of one surface over another causes the atoms in each to get knocked about jiggle.  That loss of energy to heat is what causes the surfaces to slow down and stop.

The big difference between regular gas and plasma is that in a plasma a fair fraction of the atoms are ionized.  That is, the gas is so hot, and the atoms are slamming around so hard, that some of the electrons are given enough energy to (temporarily) escape their host atoms.  The most important effect of this is that a plasma gains some electrical properties that a non-ionized gas doesn’t have; it becomes conductive and it responds to electrical and magnetic fields.  In fact, this is a great test for whether or not something is a plasma.

For example, our Sun (or any star) is a miasma of incandescent plasma.  One way to see this is to notice that the solar flares that leap from its surface are directed along the Sun’s (generally twisted up and spotty) magnetic fields.

A solar flare as seen in the x-ray spectrum.  The material of the flare, being a plasma, is affected and directed by the Sun’s magnetic field.  Normally this brings it back into the surface (which is for the best).

We also see the conductance of plasma in “toys” like a Jacob’s Ladder.  Spark gaps have the weird property that the higher the current, the more ionized the air in the gap, and the lower the resistance (more plasma = more conductive).  There are even scary machines built using this principle.  Basically, in order for a material to be conductive there need to be charges in it that are free to move around.  In metals those charges are shared by atoms; electrons can move from one atom to the next.  But in a plasma the material itself is free charges.  Conductive almost by definition.

A Jacob’s Ladder.  The electricity has an easier time flowing through the long thread of highly-conductive plasma than it does flowing through the tiny gap of poorly-conducting air.

As it happens, fire passes all these tests with flying colors.  Fire is a genuine plasma.  Maybe not the best plasma, or the most ionized plasma, but it does alright.

The free charges inside of the flame are pushed and pulled by the electric field between these plates, and as those charged particles move they drag the rest of the flame with them.

Even small and relatively cool fires, like candle flames, respond strongly to electric fields and are even pretty conductive.  There’s a beautiful video here that demonstrates this a lot better than this post does.

The candle picture is from here, and the Jacob’s ladder picture is from here.

If you have a matrix, , then the determinant is , where is a rearrangement of the numbers 1 through n, and is the “signature” or “parity” of that arrangement.  The signature is (-1)^k, where k is the number of times that pairs of numbers in have to be switched to get to .

For example, if , then

Turns out (and this is the answer to the question) that the determinant of a matrix can be thought of as the volume of the parallelepiped created by the vectors that are columns of that matrix.  In the last example, these vectors are , , and .

The parallelepiped created by the vectors a, b, and c.

Say the volume of the parallelepiped created by is given by .  Here come some properties:

1) , if any pair of the vectors are the same, because that corresponds to the parallelepiped being flat.

2) , which is just a fancy math way of saying that doubling the length of any of the sides doubles the volume.  This also means that the determinant is linear (in each column).

3) , which means “linear”.  This works the same for all of the vectors in .

Check this out!  By using these properties we can see that switching two vectors in the determinant swaps the sign.

4) , so switching two of the vectors flips the sign.  This is true for any pair of vectors in D.  Another way to think about this property is to say that when you exchange two directions you turn the parallelepiped inside-out.

Finally, if , , … , then

5) , because a 1 by 1 by 1 by … box has a volume of 1.

Also notice that, for example,

Finally, with all of that math in place,

Doing the same thing to the second part of D,

The same thing can be done to all of the vectors in D.  But rather than writing n different summations we can write, , where every term in runs from 1 to n.

When the that are left in D are the same, then D=0.  This means that the only non-zero terms left in the summation are rearrangements, where the elements of are each a number from 1 to n, with no repeats.

All but one of the will be in a weird order.  Switching the order in D can flip sign, and this sign is given by the signature, .  So, , where , where k is the number of times that the e’s have to be switched to get to .

So,

Which is exactly the definition of the determinant!  The other uses for the determinant, from finding eigenvectors and eigenvalues, to determining if a set of vectors are linearly independent or not, to handling the coordinates in complicated integrals, all come from defining the determinant as the volume of the parallelepiped created from the columns of the matrix.  It’s just not always exactly obvious how.

For example: The determinant of the matrix is the same as the area of this parallelogram, by definition.

The parallelepiped (in this case a 2-d parallelogram) created by (2,1) and (3,5).

Using the tricks defined in the post:

Or, using the usual determinant-finding-technique, .

They show up when you describe a black hole using some weird coordinates, so they’re essentially just a non-real mathematical artifact.  However, white holes are a cute idea so they show up a lot in sci-fi.  White holes are a mathematical abstraction that necessarily exist in the infinite past.  That is to say, if you follow the mathematical model that physicists use, you’ll never have a situation where a white hole exists at the same time as anything else.  Its existence happens infinitely long ago.

Spacetime gets seriously messed up near and inside of a black hole.  To make the math easier, and to help make the situation easier to picture, the Kruskal-Szekeres coordinate system was created.

In this (very unintuitive diagram) straight lines through the center are lines of constant time, with the future roughly up.  The event horizon of the black hole is also the infinite future (from an outside perspective it takes forever to fall all the way into a black hole).  That should make very little sense, but keep in mind: black holes and weird spacetime go together like Colonial Williamsburg and a lingering sense of disappointment.  The black hole’s interior is the upper triangle, the entire universe is the right triangular region and the white hole is the lower region.

The boundary of this lower region is in the infinite past.  That is; in this goofy mathematical idealization of a static and eternal black hole, a white hole shows up automatically in the infinite past.  One of the issues here is that black holes need to form at some point (in the finite past).

Taking this model completely seriously and assuming that it implies that white holes are real is a little like saying “imagine an infinite robot-godzilla”, and then worrying about where it came from.  It’s an abstraction used to think about other things.  Physicists love themselves some math, but the love is tempered by the understanding that writing down an equation doesn’t make things real.

Physicists love themselves some math, but (almost always) recognize the scope and limitations of their own equations.

For example, we can talk about the location “North 97°, East 40°”, but that doesn’t make it exist (North 90° is the north pole, the farthest north you can get by definition).

Sci-fi is about the only place you’ll hear people talking about white holes.  Whites holes are the opposite of black holes: they spit out matter and energy, they’re impossible to enter, they’re very bright, that sort of thing.  In fiction “the opposite of…” is a great way to get weird new ideas (e.g., Bizarro Superman).

The Einstein picture was created here.

In order to move from one place to another always takes a little time, no matter how fast you’re traveling.  But “time slows down close to the speed of light”, and indeed at the speed of light no time passes at all.  So how can light get from one place to another?  The short, unenlightening, somewhat irked answer is: look who’s asking.

Time genuinely doesn’t pass from the “perspective” of a photon but, like everything in relativity, the situation isn’t as simple as photons “being in stasis” until they get where they’re going.  Whenever there’s a “time effect” there’s a “distance effect” as well, and in this case we find that infinite time dilation (no time for photons) goes hand in hand with infinite length contraction (there’s no distance to the destination).

At the speed of light there’s no time to cover any distance, but there’s also no distance to cover.  Left: regular, sub-light-speed movement.  Right: “movement” at light speed.

The name “relativity” (as in “theory of…”) comes from the central tenet of relativity, that time, distance, velocity, even the order of events (sometimes) are relative.  This takes a few moments of consideration; but when you say that something’s moving, what you really mean is that it’s moving with respect to you.

Everything has its own “coordinate frame”.  Your coordinate frame is how you define where things are.  If you’re on a train, plane, rickshaw, or whatever, and you have something on the seat next to you, you’d say that (in your coordinate frame) that object is stationary.  In your own coordinate frame you’re never moving at all.

How zen is that?

Everything is stationary from its own perspective.  Movement is something other things do.  When you describe the movement of those other things it’s always in terms of your notion of space and time coordinates.

The last coordinate to consider is time, which is just whatever your clock reads.  One of the very big things that came out of Einstein’s original paper on special relativity is that not only will different perspectives disagree on where things are, and how fast they’re moving, different perspectives will also disagree on what time things happen and even how fast time is passing (following some very fixed rules).

When an object moves past you, you define its velocity by looking at how much of your distance it covers, according to your clock, and this (finally) is the answer to the question.  The movement of a photon (or anything else) is defined entirely from the point of view of anything other than the photon.

One of the terribly clever things about relativity is that we can not only talk about how fast other things are moving through our notion of space, but also “how fast” they’re moving through our notion of time (how fast is their clock ticking compared to mine).

The meditating monk picture is from here.