On Relativity

Einstein’s Theory of Relativity gets you thinking about the nature of space and time. This model of reality got rolling in 1887 when Michelson and Morley observed that the speed of light was invariant. It doesn’t matter in which direction you measure it. We know that any laboratory located on the surface of the earth is rotating around the earth’s axis, which is orbiting the sun, which is traveling through space at several miles per second. If we measure the speed of light in the direction of our motion, or in the opposite direction, or cross-wise to it, it doesn’t make any difference—we always observe the speed of light to be 186,282 miles per second. This implies that light is not propagated in some kind of medium like sound in air, or like waves on water. Its speed doesn’t depend on its source, either. Light from a moving flashlight travels at the same speed as light from a stationary flashlight, no matter who makes the measurement or what the relative speed is between the flashlight and the observer. This is how the Theory of Relativity got its name, all frames of reference appear the same within themselves. There is no clue from within that relates to anything absolute. The only absolute is the speed of light itself.

Now, these facts have implications when worked into a model. Einstein worked them into the model we call the Theory of Relativity. Predictions can be made from this model and tested with observations. The theory predicts a mass increase, a time slow down, and a length contraction. The theory also predicts that matter can only approach, but never quite reach, the speed of light. And, it predicts a relationship between matter and energy, namely the famous equation: Energy equals mass times the square of the speed of light. Another equation is less famous. It is called the Lorentz Transformation. It gives the exact amount of increase or decrease in observed phenomena as a function of the relative speed between an object and an observer.

The Lorentz Transform can be visualized by thinking of a right triangle and remembering that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the speed of light. It remains constant. Only the other two sides will be allowed to vary. One of these sides represents the relative speed between two different frames of reference. It can go from zero to the speed of light. In other words, its length may vary from zero up to the length of the hypotenuse. The third side is determined by the fact that it must be at right angles to the second side and connect between it and the hypotenuse.

At this point, I could produce a diagram showing a right triangle, with hypotenuse and sides all labeled. Or, you could draw one for yourself. Or, you could visualize one in your mind’s eye. I’ve decided to delete the diagram I drew in an earlier draft, and restrict “Numinations” to ideas that I can transmit by words alone. So, your imagination (or scratch paper) are important to both of us to make this process work.

Let’s label the hypotenuse, C, and let it be one unit in length. It represents the speed of light. Its length doesn’t vary. We’ll label the second side, S. It represents the relative speed between two frames of reference. Its length can vary from zero to one. The third side, we’ll label T, to represent the Lorentz Transform. The transform varies over the same range as the speed, but oppositely. In other words, as S goes from zero to one, T goes from one to zero. The angle between S and T is always 90 degrees. The angle between S or T and C varies as S (the relative speed) increases from zero to C (the speed of light). The Lorentz Transform happens to be the sine of the angle between S and C (or the cosine of the angle between T and C).

Let’s run through an example to see how the Lorentz Transform is applied. Suppose an object is traveling straight across (exactly transverse to) our field of vision at half the speed of light. The Theory of Relativity predicts that the passage of time “dilates” when the object’s frame of reference is compared to ours. It also predicts a length contraction in the object’s direction of motion. In other words, if the object is known to be ten feet wide, it will appear to be less than that. But, what about this “time dilation” thing? A simple experiment demonstrates what is meant. Let’s say the object is emitting green light at a frequency of 566 THz (that’s pronounced Terahertz, and means “trillion cycles per second”). Time dilation simply means that fewer cycles of light are emitted from an object in one second of the observer’s time. Fewer cycles means a lower frequency and a different color. This directionality is important. Observed length must be increased to get subjective length. Subjective length must be decreased to get observed length. This is similar for all the values that change: Time intervals, frequencies, mass, and distance in the direction of travel. You need to multiply quantities to be decreased by the Transform, or divide it into quantities that increase. The Lorentz Transform multiplied by the frequency emitted at the source gives us the frequency we would observe. Draw a right triangle. Label the hypotenuse as one unit, representing the speed of light. Label one of the other two sides as 0.5 units, because the object has a relative speed of half the speed of light. Now, the square of the third side, plus the square of 0.5 should add up to the square of the hypotenuse (which has a value of one). The third side, representing the transform, needs to be 0.866 units (0.5 squared is 0.25 and 0.866 squared is 0.75, and these add up to one). Thus, if green light with a frequency of 566 THz were emitted, red light with a frequency of 490 THz would be observed (566 times 0.866 is 490). Likewise, the object would appear to be 8.66 feet wide (since we know it to be ten feet wide at rest).

Scientists have tested the predictions of this model, and so far it has held true. For example, atomic particles have certain fixed rates of decay. When an unstable particle travels at a significant fraction of the speed of light, it takes longer to decay. This time extension is in the exact ratio predicted by the model. But, the theory says that speed is relative. If that’s true, why wouldn’t our time appear slower from the particle’s point of view? Doesn’t this create a paradox?

The theory also says that nothing can travel faster than the speed of light. Consider two events. Event 1 is that I leave earth at almost the speed of light (about 99.999999% of it) heading for the nearest star. Event 2 is that I arrive at the nearest star (four light-years away). Because my time slows down when I travel fast enough, events 1 and 2 could appear to me to be only 10 seconds apart. To you the interval is still a full four years. The catch is that distance in my direction of travel also shrinks as I approach the speed of light. It doesn’t appear to me that I have gone faster than the speed of light, the distance simply becomes much less. My destination now appears (to me) to be only 10 light seconds away. Distance and time shrink and stretch between frames of reference whose relative motion approaches that of the speed of light.

These are the very non-intuitive conclusions of the Special Theory of Relativity. Space and time, and matter and energy, are literally warped by traveling at very high speeds. The General Theory of Relativity says that these same phenomena are likewise warped by the presence of mass. That is, the presence of mass is accompanied by a gravitational field, and a gravitational field is nothing more than a warp in space with effects similar to those produced by travel at very high speeds. Scientists have gone on to conjecture that the same kind of bending of space produced by gravity could also apply over the extent of the entire universe. Just as the surface of the earth is bent back upon itself to form a sphere, the three dimensional “surface” of the universe could be bent back upon itself to form a hypersphere (this would be a four dimensional object standing in relation to a three dimensional sphere, as a sphere stands in relation to a circle). A sphere has a finite surface area, but it’s unbounded—it has no edge. The universe could also be unbounded, but as a hypersphere, the universe would have a finite volume.

Lots of models can and have been built upon these conjectures and much has been confirmed by the data. We’ll look into more of this subject next time. Meanwhile, keep “Numinating!”