DNAunion

DNAunion: I said I would post something about a light clock if I had time. The following is from my newest set of personal notes (these being non-mathematical - i.e., conceptual - notes dealing with physics). They are an early work in progress.

Special Relativity
Einstein’s theory of special relativity produces a Universe most of us would find counterintuitive. Loosely speaking, some examples include: it is possible for one identical twin to end up being many years younger than the other; events one observer sees as being simultaneous are not simultaneous for some other observers; velocities do not add together in simple 20mph + 20mph = 40mph fashion; time is divided not only into the present, the past, and the future, but also into something called the elsewhere; observer A’s clock can be running slower than observer B’s clock and at the same time observer B’s clock can be running slower than observer A’s, without there being any contradiction; and so on.

Yet all of the above-mentioned counterintuitive conclusions follow from the simple idea of special relativity...the laws of physics are the same for all (observers in) uniformly moving reference frames. Let’s look briefly at what this tells us about the speed of light and then about measures of time.

Maxwell’s Equations
Magnetism and electricity – which had been thought to be two completely separate phenomena –
were unified into electromagnetism by James Clerk Maxwell in the 1800s. One consequence of his unifying equations was that there should be electromagnetic waves and these waves must travel at approximately 186,000 mi/s (or about 300,000 km/s). This result – which matched the already known speed of light, c – fell naturally out of Maxwell’s equations: he did not need to manipulate his work to force this result. No counterintuitive conclusions followed yet because it was thought that while the laws of Newtonian mechanics (i.e., the laws of motion, gravity, etc.) were the same for all observers, the laws governing electromagnetism were not.

Speed of Light is a Constant (c)
But Einstein changed all of that when, in 1905, he published his theory of special relativity. Then all of the laws of physics – including the laws of electromagnetism – were the same for all (observers in) uniformly moving reference frames. Since Maxwell’s equations are among the laws of physics and they mandate that electromagnetic waves travel at approximately 186,000 mi/s (in a vacuum), then all uniformly moving reference frames must measure the same speed of light (and other electromagnetic waves), even reference frames that are moving relative to each other! An example might shed some light on this.

Suppose there are two observers, A and B, in two separate uniformly moving reference frames and a pulse of light, emitted from a star, is traveling towards them. Let us suppose that observer A is at rest here on the Earth. Since scientists have already measured the speed of light from here on Earth to be approximately 186,000 mi/s, we know that A will obtain that result, regardless of how B is moving. Let’s now look at things from B’s perspective (reference frame). First, suppose that B is in his car speeding down the highway into the oncoming light pulse at 100 mph. According to Einstein’s theory of special relativity, B will not measure the speed of that light to be 186,000 mi/s + 100 mi/hr, but rather simply 186,000 mi/s (the same speed as that measured by observer A). Furthermore, if observer B were flying in a jet airplane into the oncoming light pulse at 600 mi/hr then he/she would not measure its speed to be 186,000 mi/s + 600 mi/hr, but rather, again, simply 186,000 mi/s (the same as observer A). In fact, if B were rocketing into the oncoming light pulse at half the speed of light, he/she would still measure the speed of that light to be 186,000 mi/s: and not 1.5 c. And finally, even if observer B were rocketing away from the light pulse at half the speed of light, he/she would still measure its speed to be 186,000 mi/s: and not 0.5 c. Unlike the speed of sound and another wave that is propagated only through a medium, not only is the speed of light independent of the motion of its source, it is also independent of the speed of its observers. Thus the speed of light (in a vacuum) is a constant: the same everywhere and for everyone in uniform motion (this is why it came to be symbolized by c).

Light Clock
So what does this invariance of the speed of light do to time? The best way to describe it is to use a thought experiment involving a “light clock”. What is a light clock? Imagine a clear boxlike device consisting of a light emitter fixed to the inside bottom and a mirror attached to the top directly above the emitter. This hypothetical clock keeps time by firing a photon up to the mirror, which reflects it straight back down; and the exact instant that photon strikes the emitter, another is shot out up towards the mirror. These repeating events occur at regular intervals since each cycle – consisting of one “tick” (photon shot up to mirror) and one “tock” (light reflected to the emitter) - would take exactly the same amount of time.

Now let us again imagine observers A and B in two separate reference frames. A is, once again, at rest (here, with respect to the clock) and B is in motion (relative to the clock). We are interested in the events that comprise a single cycle of the light clock: one firing-reflection-return. Observer A sees the photon shoot directly up, reflect off the mirror, and exactly retrace in the opposite direction its upward path. Therefore, the distance the light travels according to observer A’s perspective is simply twice the height of the box. But observer B is moving, let’s say to the left. So the light clock is in motion to the right relative to him/her so he/she sees something different. The photon that is emitted still hits the mirror dead on (it must, since both observers are examining the exact same set of events), but, since the whole device has shifted to the right between the time that the photon is emitted and the time that it strikes the mirror, the mirror has moved to the right slightly. Consequently, observer B does not see the photon travel directly upwards, but rather upwards at a right slant (needed to hit the repositioned mirror). The photon is then reflected and still hits the emitter square on (again, the two observers are witnessing the exact same set of events so they must agree on this), but the whole device has moved rightward again in the time between the photon’s reflection and its return. Thus observer B sees the downward trajectory also slant off to the right. All of this means that observer A and observer B see two different versions of the same exact set of events due to their motions relative to one another. The length of the path the photon travels from observer A’s perspective is simply twice the height of the box, but for observer B, the total length is twice the height plus some additional distance in the right horizontal direction. Therefore, the photon travels a greater total distance in B’s frame of reference frame than it does in A’s. What does this mean?

If this above light clock thought experiment had been performed before Einstein’s special theory of relativity, the explanation for the two observers measuring different distances would have simply been that they measure different speeds for light (electromagnetic waves). In other words, by definition, speed is equal to the distance traveled divided by the amount of time that elapsed (think what a speed of 50 mi/hr means). Since the time that elapses for the same set of events must be equal, and the distances differ, then the speeds must differ. The explanation would have been that simple.

Time Dilation
But Einstein strictly forbade that, stating that the speed of light is a constant, even for observers in motion relative to one another. Now, if the speed of the light pulse is the same for both observers, yet the distances for each differ, then the amount of time that elapsed for observers A and B, who were both in uniform motion, moving relative to one another, must be different, even though they both witnessed the very same set of events! Yes, despite what our everyday experience tells us, time does not pass at an absolute, fixed rate. Two observers in relative motion to each other will measure different rates of the passage of time: if their relative speed is great enough, one person’s second could be the other person’s minute!

A common cram-it-all-into-one-bite-size-nugget-so-that-it-is-easy-to-remember expression used to convey this complex subject is, “moving clocks run slow”. Though helpful as a memory device, besides other problems, this shorthand version (and the light clock example) implies that this effect is restricted to light clocks and mechanical or digital clocks/watches. But that is not so. This phenomenon of time variance relates to all processes, including physiological (heart beat, metabolism, mental processes, etc.) and physical (decay of radioactive isotopes, etc.). It is time itself that is different for those two observers. Time dilation, as it is called, is not mere fantasy: it has been experimentally verified by multiple experiments.

Time-Traveling Twin Paradox
Time dilation explains why two twins can end up being years apart in age (as alluded to previously). Suppose that while one twin remains here on the Earth, the other rockets off at near the speed of light to a star several light years away, turns around, and then returns home. Since the “moving” twin’s “clock” would have been “running slow”, when the pair meet again after the trip, the space-traveling sibling could be up to many years younger than his/her twin.

DNAunion

DNAunion: I just felt that I needed to better support a claim I made earlier (all I did originally was mention the book it could be found in).

[ August 08, 2002: Message edited by: DNAunion ]</p>

Demosthenes

Techincally the fourth dimension is the next direction perpendicular to the three dimensions. But since Einstein has chosen time ot be the fourth dimension, the next spatial dimension is called the fifth dimension. Though with the current vogue of multiple dimensions in the physical theories, we could knock time all the way to the 11th dimension and adjust the numbering of the dimensions accordingly without changing anything about the theories. It's just a matter of arbitrary naming.

ohwilleke

Imagine a cube. Imagine colors from red to blue representing the temperature of different spots within the cube. You have now visualized a four dimensional system.

Now imagine a videotape that shows the colors inside the cube changing over time (sort of like on radar weather on the evening news). Now you have visualized a five dimensional system.

Now imagine that instead of just little red through blue dots, you are seeing little colored arrows throughout the cube of different lengths representing wind speed and direction and temperature at that point. As before this is on video and changes. You are now visualizing a nine dimensional system. (Each frame represents a point in the time dimension, each arrow is at a three dimensional point in space, each point in space has (1) temperature, (2) wind speed, and three dimensions of wind direction).

Now imagine the same cube full of colored arrows of different length on video. Make some colors darker to represent polluted areas, and some colors lighter to represent unpolluted areas. You are now visualizing a ten dimensional system and it wasn't even that hard.

Now, that you can visualize a system in ten dimensions, imagine a little legend at the bottom of the page. This legend contains a conversion chart that says "a difference in color from red to indigo equals five miles which equals twelve seconds. Directional arrows are shown at a 1000:1 scale. Differences in hue of a factor of two equal three miles." Now you have a system with ten dimensions that can be treated as a ten dimensional mathematical space.

(For sticklers: Recognize that this is simply a heuristic example to show one way of visualizing a ten dimensional space. Obviously, the "weather" example used is not really a ten dimensional space, but such a display could fairly represent a real ten dimensional space in a way that a mere human could understand it).

[ August 09, 2002: Message edited by: ohwilleke ]</p>