Normally, I limit my writing to topics that I myself have experienced or initiated. This ensures that what you read in Reflections is new, and isn’t just something cut and pasted from the web.
However, I sometimes run across things that I find really interesting and are probably not common knowledge, but which might be of interest to fellow Lowbrows. What follows are a few vignettes that are not original, but which I hope you find interesting.
One of the things I like best about light is that ordinary, familiar things about it turn out to be Quantum Mechanical. Trying to understand Pauli spin matrices? Just look at the polarization of light from the standpoint of Jones matrices. Feynman’s sum-over-histories approach to quantum theory? Look at double-slit light diffraction experiments. Wave-particle duality? Just look at the behavior of light waves (or photons).
Light was first described as waves in a field by James Clerk Maxwell. Calling light a wave explained diffraction, and varying the frequency of that wave explained colors. But when light detectors became sensitive enough to detect very small quantities of light, they always detected individual particles. Even today, you can do experiments which prove that light is a wave, and experiments which prove light is a particle. You just can’t do them at the same time.
The problem seems to be that light is really neither a particle nor a wave. It is a bit of quantum stuff, called a qubit. And the manner in which it appears to us depends on the tools we use to measure it.
I think of light as being a bit like music. We can produce music from plucked strings on violins, each string vibrating and making waves in the field of air that add to form a song, or we can produce music from CD disks which consist solely of discrete bits of dots and dashes, which, when their fast and slow impulses are added together in the air, produce exactly the same song. We depend upon our imperfect tools, because our present senses are inadequate to see the Universe as it really is, and because of that sad fact, for now, we see through a glass, darkly.
Einstein says that matter causes space to curve, and the curved space tells matter how to move. He also said that anything in free fall (a falling rock, a baseball headed for the outfield fence, that buttered toast headed for the floor) is traveling in a straight line through curved space-time. So why do the curvatures of the tracks of the paths of a rock and a bullet and the baseball and the toast look so different? I mean, the radius of curvature of the track of a bullet as it flies over a field is, like, maybe a mile, and the path radius of a flipped coin is just a few inches. Is space-time curved differently for these two objects?
The answer is that space-time is curved equally for all of these objects, and their paths all look exactly alike and have the same curvature, but our view of space-time is highly distorted because our senses leave out the time part.
Let’s say we took movies (one billion frames per second—this is a thought experiment) of the falling toast and of a rifle bullet shot horizontally across a field (viewed along the path), and projected them onto a screen at a 1:1 ratio with the real world. You might notice after playing the movies that both the toast and the bullet hit the ground at the same time after release, but that’s just an aside here. Their actual paths through space still look very different.
Next, we’ll get few billion sticks and put LED’s on the top of the sticks. Then, we’ll push two sticks into the ground in front of the screen and position them so the two LEDs mark the positions of the bread and the bullet in the first frame on the screen. Then, we’ll move the screen back a foot, advance to the next frame, and repeat the process with two more LED sticks.
We do this a few billion times and notice something strange, as we look back on the long line of LED sticks. The pairs of LEDs representing the positions of the bread and the bullet are tracking each other perfectly. If we were to go back and tie a wire between all of the bread LEDs and tie another wire between all of the bullet LEDs, they would describe exactly the same path through space, and the two wires would have exactly the same curvatures.
The act of moving our screen one foot for every billionth of a second allows us to sweep through space at the speed of light, and to “see” the time dimension as a space dimension. That is the part that is missing from our ordinary perceptions of space-time.
So, what is the local curvature of space-time, represented by our wires?
The toast (and the bullet) fell three feet vertically through space, which took about 0.432 seconds. The distance they traveled horizontally on our screen paths (through the time dimension of space-time) was ct, or 186,000mi/sec x 5,280ft/mi x 0.432sec = 424,258,560 light-feet. We thus have the sagitta (3 ft) and half-diameter HD (424 million light-feet) of a parabola, and using the telescope maker’s formula for the sagitta of a parabola: sagitta = HD2/2RC, we find the local radius of curvature RC of space-time to be 3.0 x 1016 feet, or about 0.967 light-years. Close to one light year on the surface of the Earth, as it turns out.
Astronomers have been looking for the Missing Mass in the Universe at least since the 1930’s, when Fritz Zwicky noticed that the arms of spiral galaxies were spinning a lot faster than they would if they were just orbiting the visible mass of the galaxy. There are a couple theories now about what that invisible stuff might be, which include Dark Matter and Dark Energy (the latter of which causes the expansion of the universe to accelerate. Those Science Fiction writers who predicted we’d discover anti-gravity by the year 2000 were right. We call it Dark Energy, but it could also be called anti-gravity, since it looks like matter repelling matter. It is unclear whether it works like gravity in reverse or as a uniform pressure, though), but we are concerned here with the unexpected presence of mass, which seems to come and go with the seasons.
Mercury has this problem. With each turn around the sun, when it swings in close to the sun, it seems to get heavier. Thus, it speeds up in its orbit, and the orbit’s perihelion (the closest point to the sun) advances a bit. When it moves away from the sun in its orbit, it acts as if it were lighter. Newton predicted a lot, but he didn’t predict this.
Einstein said E=mc2, which we all know, but he also said m=E/c2. (He said the latter first, but it was the former that caught on.) But this formula only refers to a body’s rest mass. The formula for the total energy of a moving mass is: E = mc2 /[1-(v2/c2)0.5].
Thus, Mercury’s mass consists not just of its rest mass, but also of the mass that comes from its added kinetic energy as it speeds up in its orbit.
I’ve tried to explain to my wife that my added seasonal mass comes from kinetic energy as I approach the hors d’oeuvres table with a Martini in hand, and she just insists that I’ll lose that weight if I stay farther from the center of the powerful snack attraction field.
And I thought she didn’t know anything about relativity.