Moving along in our countdown to Newton's birthday, we start to deal with equations that Sir Isaac never would've seen, because they deal with more abstract quantities than he worked with. The first and in some ways most important of these is energy:
This is the full and correct expression for the energy of a particle with mass m moving at speed v. The notion of energy traces back to Newton's contemporary and rival Gottfried Wilhelm Leibniz, but this particular equation involves the same square-root factor as Saturday's definition of momentum. That tells you for sure that this particular equation comes from Einstein's theory of special relativity.
So, why is this important?
For starters, energy is important because while it's less tangible than momentum, it gives us another way to understand the motion of objects, as a flow of energy between different systems and subsystems. While you can solve absolutely any problem involving the motion of objects using Newton's second law, the solutions are frequently somewhat cumbersome, and may involve lengthy numerical calculations. In many cases where working with force and momentum directly is tedious and unpleasant, you can get an answer more quickly and easily by thinking about it in terms of energy.
This particular form doesn't give you much to work with in that sense, though, being the energy of a single particle not interacting with anything else. This still tells us something important about the way the universe works, though, particularly when the speed v is zero.
If you let v be zero, then the stuff under the square root is just equal to 1, whose square root is 1. That means, the energy of a single particle that isn't moving is not zero, but its mass multiplied by the speed of light squared (which, incidentally, is a very big number). In other words, this gives you the world's most famous equation, E=mc2.
But what does this mean? How can something that isn't moving contain vast amounts of energy, a quantity typically associated with motion? The short and simple answer is that something that isn't moving through space is nevertheless moving through time, advancing into the future at one second per second. It's that temporal motion that's responsible for the rest energy of Einstein's most famous equation, giving any object with mass a large amount of energy, even when it's standing still.
When the object is moving, the total energy given by this equation is greater than when it's standing still, and that excess energy is the kinetic energy you may have encountered in some past physics class. It may not look like the classical formula (which traces back to Leibniz), but at low energy the classical value is an excellent approximation. And as the speed approaches the speed of light, the energy of a moving particle increases dramatically, eventually becoming many times greater than the original rest energy (which, again, is huge).
This energy, in turn, can be turned into new mass, under the proper circumstances, such as a collision between protons in a giant particle accelerator. That's how we know about all the different types of quarks and leptons and bosons, and how physicists hope to find the Higgs boson, and a whole host of other particles that might explain dark matter and other mysterious properties of the universe.
But that's getting a little ahead of the story. For today, just take a moment to appreciate the energy of a single particle, and tomorrow, we'll continue our countdown with the next equation of the season, and the next piece of the story of physics.
Is there a reason why you chose E=gamma*m*c^2 instead of E^2 = p^2*v^2 + (m*c^2)^2?
Gamma is already present in your definition of p from Saturday, and you can still do the "when v=0, E=m*c^2" bit.
When I was first learning Newtonian mechanics, I was vaguely disturbed by the fact that kinetic energy, unlike momentum, had this strange nonlinear dependence on velocity, such that it behaved oddly under transformations between different inertial frames. Somehow the conservation laws all still worked out in any frame, but it didn't seem like they ought to.
Special relativity actually made the situation much more elegant, to my mind. Suddenly energy and momentum had parallel forms (which you can see in these blog posts), and could be expressed as parts of the same four-dimensional vector, and you could see that the lopsided situation in Newton's world just came from taking different terms in a power series in the low-velocity limit.
Can you elaborate on why a particle moving at close to the speed of light has so much energy and momentum? If I'm reading these equations correctly, then as v tends to the speed of light then v^2/c^2 tends to 1, so the denominator tends towards a very small number. When you divide the numerator by this very small number you get a very large number. I had always thought that a particle nearing the speed of light has a lot of momentum just because the speed of light is very fast, but now there is this additional term that gives very fast objects even more energy and momentum!
I went with this form because of the parallelism between this version and the definition of momentum. Also, I think that momentum is just foreign enough to people to seem unnatural appearing in a formula like this. It ends up being sort of recursive, because you have to go look up the definition of momentum before you can do anything with the definition of energy. This way, everything is expressed in terms of the simple, familiar quantities mass and velocity.
JohnE: That strange factor written as gamma shows up all over the place in special relativity. It's the same factor that appears in the formula for time dilation. And, in fact, energy and momentum vary with a change between differently moving reference frames in exactly the same way as time and position (which is another way that special relativity is more symmetrical and elegant than Newtonian mechanics).
So one way to think of it is that it has to do with the four-dimensional geometry of space-time in special relativity. Energy and momentum together make a four-dimensional vector, and time and position make another four-dimensional vector, and these transform under changes of the velocity of a frame of reference in an analogous way.
I note the use of plain "m" in the formula, when formerly it was common to use "rest mass" m0 to show with Î³. I realize, many have recently argued to keep mass as a constant, and that practice caught on. Yet there are problems. First, the symmetry between matter and energy is spoiled. In the old tradition, I can just say "a given amount of mass-energy" that I find moving past me. I don't have to worry about referring differently to "energy" and "rest energy" versus "mass." The new tradition is galling, I need to imagine that for example the compression energy per se in a spring is "more" at higher velocity, whereas pretend that the spring mass itself is "a constant."
Second, the idea of a definite constant at all is flawed. What if I have a hot gas, spinning wheels etc. inside a box - I can measure its "mass" at rest and get a figure - but that "constant" already takes into account the velocity altered "effective masses" of the constituents!
Which leads to third: the effective "mass" even at rest, *is* the relativistic mass because of the above considerations. Again, consider a flywheel: its effective inertia includes the relativistic mass from rotation, but at rest it is "an object" - should it be treated as an entire thing with effective rest mass, or should we have to break down each little piece of the spinning rim, etc? With "energy" as a given, no worry. Change back, people!
Neil: It becomes more elegant again if you consider that the mass (times c^2) is the magnitude of a four-vector. To get the mass of a composite system, you have to do vector addition. The mass of the system is not the sum of the masses of the components, any more than the length of a vector sum in the plane is the sum of the lengths of the added vectors. You have to use the parallelogram rule. And this is independent of the rest frame, which is nice. (The term "rest mass" gives the impression that it's only a relevant quantity when the object is at rest, which isn't true.)
In particle physics, particularly, it's more useful to define m in Chad's way. Particle theorists use c=1, so reserving m for the relativistic mass just gives us two redundant symbols, E=m, and they'd have to write m0 all over the place in their formulae just to preserve that redundancy. Instead, for the "relativistic mass" they just use E itself.
And formerly before that it was called m, just as it has been since since the 1950s if you look at research articles rather than popular books. Subscripts are for the particle in question (proton, electron, up quark, cannonball) rather than different species of mass. Further, it is incorrect to mix up energy and mass, because each has its own role in the stress-energy tensor of General Relativity.
In addition to what Matt wrote @7 about what is constant in different coordinate systems, you should realize that the internal energy of your box (of hot gas) is no different than the mass of a proton (of quarks and gluons). Both are invariant, and reliably calculated in special relativity just as they formerly were in classical physics. Both can be changed by adding energy to the system (hotter gas, excited states of the proton).