I’ve been thinking about how to make a powerful weapon based on a relatively weak light transmitter which can be tuned in frequency. This could be the basis of what, in Star Trek, they call photon torpedos. The physics principle behind this kind of weapon is called “dispersion.” We commonly talk about oil “dispersing” on the surface of water, or pollen dispersing in the wind, but in physics we use this word to mean something very special and also very interesting.
Try this thought experiment. Take everyone on Earth (7 billion) and have them all stand side by side on a long straight platform perpendicular to and orbiting Earth’s equator. Have everyone face in the direction of the orbital velocity, using a local coordinate system where all people are on the x-axis at z=0. Give each person a baseball, and have them throw their baseball directly forward (in the direction of increasing z coordinate) at the same time. The baseballs leave the hands of every person (z = 0) at exactly the same moment (t=0). We won’t worry about the x or y spatial coordinates, only how far the balls travel in the z direction as time goes on. For the moment, we shall not be concerned about the rebound of the platform due to the ejection of balls. Perhaps there is a small rocket which cancels the momentum of the expelled baseballs, keeping the platform on its original course.
Some balls are thrown faster than others, depending on each person’s ability. At t=0, all the balls are clumped together at z=0. Later at time t=t1, the fast balls have moved farther from the platform than slow balls. The balls are now spread out, or dispersed, along the z-axis. With increasing time, dispersion increases: the distance between the fastest ball and the slowest ball increases with time. Thanks to Kepler’s laws, each ball will return to its starting point after executing one orbit of the Earth, provided the balls don’t hit the Earth (too slow) or escape Earth’s gravity (too fast). If we register the time when the balls arrive, we will typically find a small number of fast balls (thrown by major league pitchers) will arrive first. This is followed by a large “hump” or high density of balls arriving later with time corresponding to the speed of an “average” thrower, since average humans are more numerous than those with special skills. Finally, a small number of balls come dribbling in at the end, thrown by unusually weak throwers such as small children. Plotting the number of balls returning versus the time it takes to circle the Earth, we will find a bell-shaped curve. This bell curve looks a little like the envelope of a “wave packet,” if you have heard of that expression in quantum mechanics.
Now turn the classical baseball experiment on its head. After the first experiment, the experimenter knows the time required for each person’s ball to make one orbit. Starting with the slowest thrower (longest orbital time), this person throws first. The ball comes back to the platform after known time T. Then, we ask the second slowest thrower to throw next, at just the right moment so that the second ball arrives back at the platform at time T. Carrying on with the third slowest, who throws next at the appropriate moment, we continue through all 7 million people until we reach the fastest thrower, who waits until the very last moment such that her ball arrives back at the platform again at the time T.
For the sake of argument, we use our rocket to steer the platform such that when the baseballs return after one orbit, they strike the platform from behind (but don’t hit any people). If just one baseball hits the platform, we don’t expect much of an impact. Even if 7 million baseballs arrive one by one, over a period of a year, each impact is small, so the people on the platform may feel a rumble but not even enough to make them fall down, since the impact of each ball is absorbed separately.
But in our second thought experiment, all of the 7 million balls strike the platform at exactly the same time. The instantaneous Force (transfer of momentum) is huge. Not only are people likely to fall down, but the platform itself may be obliterated by the impact. This might seem somewhat surprising, and it arises from the fact that the “impact” or force felt by the platform depends on both the the amount of momentum that is transferred from the balls and the time period over which momentum is transferred. Newton’s law of action and reaction explains this concisely:
Reactive Force on Platform = (change in momentum caused by balls) / (period of impact), or
F = dp / dt
where F = force, p = momentum, and t = time.
How does this discussion lead to a powerful weapon? Suppose we build a machine that can throw one baseball at a time at a certain speed, v. We can build a destructive weapon merely by preparing 7 billion copies of this machine and causing them to throw at exactly the same moment. Since all balls have the same speed, they arrive at their destination in a giant clump, obliterating the target. But this has two problems: 1) 7 billion are a lot of machines (expensive) and 2) the instantaneous power required to trigger all machines at the same time is extraordinary, and possibly so large that Earth technology cannot feasibly produce so much energy over such a short time.
So we build a different design with only one machine that is capable of throwing one ball after another with a small time delay Dt, but each one having a different speed. The machine starts by throwing slow balls, and increases the ball speed uniformly in proportion to
(V_slowest_ball) ( ball num in throwing order) (time between throws), or
V0 n Dt
where V0 is the speed of the slowest ball, n refers to the nth ball thrown. With this choice, we ensure that every baseball arrives at the same moment, transferring large momentum in a small time and obliterating the target.
What have we gained? 1) Instead of building 7 billion machines, we built only one that is slightly more complicated (cheap). 2) Over any time period ndt, only the energy required to throw one ball is required. This is a dramatically smaller power level, which is extended over a long period of time. In total, about the same amount of energy is required for either of the above weapons, but the latter is astronomically cheaper and more energetically feasible.
Don’t get me wrong, I’m not a big fan of weapons. But I can’t help myself describing this particular use of physical “dispersion” since it is so fascinating.
This blog is already much too long, so we’ll very quickly skip to the quantum case, in which situation we make photon torpedos.
As described in an earlier blog, the space between stars (interstellar medium) is filled with an extremely thin gas, mostly hydrogen, with approximate 1% of hydrogen atoms being ionized into free electrons and protons, called plasma. When light travels through plasma, it picks up a tiny bit of the properties of matter: photons combine with electron motions into quasiparticles that look almost like photons but have a teeny tiny bit of rest mass. This rest mass depends only on the plasma density and not on the photon energy or frequency. These quasiparticle photons, like any massive particle, suffer dispersion. Even though all “pure” photons in vacuum travel with the same speed, c = speed of light, quasiparticle photons with rest mass can travel with any speed v, where (0 <= v < c), just like any other massive particle. This is what makes photon torpedos possible.
Set up a light generator, call it an idealized tunable laser that emits radiation into a region of the interstellar medium (ISM). Low frequency light, like radio waves, travel more slowly through the ISM because they carry less total energy, hence less kinetic energy as compared with their tiny rest mass. Optical light waves travel faster, since their kinetic energy >> rest mass. X-rays, and then gamma-rays travel even faster. As a reality check we note that astronomers can ignore the slow-down even in the optical frequency range since it is small. But the slow down is never zero, even for gamma rays.
Now we perform exactly the same process with light that we did with baseballs. We begin by emitting low-energy (low frequency) radio waves. These waves can travel much less the speed of light since their total energy is not much larger than their quasiparticle rest-mass.** A little later, the laser is tuned to a higher frequency with corresponding higher speed for photon travel. Later, higher and higher frequency waves are emitted. We adjust the time of emission of the different frequencies such that they all arrive at the target at exactly the same time, packing an astounding punch. The result is in perfect analogy to the baseball experiment.
** In a typical region of the interstellar medium, the quasiparticle photon rest mass is 4e-18 eV. Oops! Did I just quote the mass in units of energy? Shame on my lazy physics habits. That should say 7e-54 kg. Despite being a small number, it is easily measured in astronomical observations.
Using only a single laser transmitter and by transmitting different frequencies at specific times, we can use a single machine to simulate the “impact” of a large number of identical machines shooting the same frequency at the same time. Also, the amount of power emitted by the laser is relatively small but carries on for a relatively long period of time. By using “dispersion” to our advantage, we cause all of that energy to arrive at the target in a short moment, packing a giant whallop far beyond the capability of a single-burst from the laser at one frequency.
So that is one way to make a photon torpedo. Or you can use electrons instead, or neutral H or He atoms, or even baseballs. All these weapons are always based on the same principle of dispersion, which is a common feature of every object that has rest mass. Which is everything.
I hope this stimulates some entertaining thoughts about dispersion.