4 Dec 2009

Simplifications of Einstein's Special Relativity Theory, Minkowski's Diagrams, and Spacetime. Based on Mook & Vargish, Inside Relativity

[I am not a physicist. Perhaps consult Mook and Vargish's book Inside Relativity. I found it very comprehendible.]

Simplifications of Einstein's Special Relativity Theory, Minkowski's Diagrams, and Spacetime

Delo E. Mook & Thomas Vargish

Inside Relativity

In the third chapter of Duration and Simultaneity, Bergson refers to Minkowski's schema. We will follow parts of Delo Mook's & Thomas Vargish's Inside Relativity. This will allow us to grasp basic principles of relativity and the way they are represented on the Minkowski diagrams.

3.3 The Problem of Defining Time

Einstein will define time in an operational sense. How do we measure time? We use clocks. Their mechanisms are supposed to provide a steady motion as with a second-hand. This steadily moving indicator moves over evenly spaced markings. We measure a time as a simultaneity. We note where the clock-hand is when an event happens. This is the simultaneity between the event and the clock's hand. We then use the spatial measure on the clock to give us a measure of the duration between such event-clock simultaneities. Einstein writes:

' We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, "That train arrives here at 7 o'clock," I mean something like this: "The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events." (Einstein The Principle of Relativity 39, qtd. in Mook & Vargish 56bc). '

Now Einstein will discuss how we know that two events are simultaneous.

There is no difficulty when our clocks are right near the event. But consider if there is a light bulb on a train track. The person right next to the bulb will be able to clock the correct time the bulb lights. But if that person walks-out substantial distance to position A, then the light will take longer to arrive, and hence the simultaneity between the light-flash and the clock-hand will be later in time. Similarly observer B, who is even further out, will have an even later time-reading for the flash.
Light Track
Hence "Different observers will record different times for the same event" (58a).

3.4 Einstein Defines a Common Time System

So Einstein wanted a "procedure that would permit scientists and others to specify and report a single, unambiguous time for an event" (58d).

Imagine Peter and Paul standing far along a railroad track. Both have a clock.
So we now have "two unambiguous, but independent, definitions of time:" Peter-time and Paul-time. Einstein will now show that there is not a common system of time that unites them and that allows everyone to synchronize together.

Next, Mary stands exactly between Peter and Paul.

Both Peter and Paul have a light-bulb. In front of Mary are angled mirrors.
The mirrors allow her to see the light-bulb flashes from Peter and Paul without having to turn her head.

When noon strikes on both Peter's and Paul's clocks, they then flash their lights. If Mary sees them simultaneously, then both Peter's and Paul's clocks are synchronized. But if one is off, Mary can tell that person how much to set his clock forward or backward so to synchronize with the other person.

We could also place synchronized clocks all along the track using such a method.
This would seem to give us a common system of time. With such a supposed system, Einstein then defines the time of an event, even for events that are distant from the observer: "the time given simultaneously with the event by one of the synchronized clocks that happens to be located at exactly the same point in space as the event" (61d).

The common system will not tell us what time an observer actually sees the event. That is determined by the distance of the observer from the event.

3.5 The Problem with Moving Observers

We will now suppose that there is a moving observer on the railroad track. Sara rides a rail-car that moves at a constant speed. And she holds a clock. She will test our network of clocks along the track that are synchronized to common time. So she first synchronizes her clock with the common time clocks.

Peter and Paul still are on opposite sides. Each operates a flash bulb. Sarah has a mirror device that lets her see both flashes with one glance. And Mary stands in the center.

Peter and Paul flash their lights at noon. So that means Mary sees the light shortly after. Also at noon, Sara's train car passes Mary. So first noon happens, and at that time Mary sees Sara. Sara passes right away. Moments after Sarah has passed Mary down the track, Mary sees both flashes at once. Of course Mary sees both flashes simultaneously. But Sarah is further down the track at the moment Mary sees the simultaneity. So Sarah sees Paul's light first.


3.7 The Relativity of Time Measurements

We will now imagine again that Sarah is in a moving train car. She has a light-clock device. It is like a vertical column. On the bottom-inside is a lamp. At the top-inside is a mirror. Light shines from the bottom, bounces off the top, and returns again to the bottom. The outer encasing is glass, so we can see inside.

The length is L. So the whole trip is 2L. C is the speed of light. And T is the time for the full round trip. Recall the speed formula:

Hence for the time of the whole trip, C = 2L/T.

But Mary stands stationary on the ground, while Sara's train-car rolls-by. So for Mary, the light path is longer than 2L, because its journey is diagonal.

Note that because Mary is stationary, she does not perceive simultaneities the same. [The light will return to the bottom of the light-clock. For Sara that will happen at a certain moment. But because she is already further down the track, it will require time for that light to arrive back to Mary. So if they were all synchronized to common time, Mary would perceive the return of the light as happening at a time that comes after the moment that Sarah records it.] We will call the time for the total trip, as seen from Mary's stationary perspective, Tg. And we will say that the speed of Sarah's car is V. Recall the distance formula:

We want now to find the distance that the light traveled, from Mary's perspective. So we will use the pythagorean theorem.

Of course, the hypotenuse is longer than L. So recall that speed is distance over time. For stationary Mary, the light went a greater distance. So let's say both Sarah and Mary will calculate the speed of light just based on the time and distance they perceived it to travel. And let's say that they are also Newtonian. So they go by Sarah's timing for how long the light's round-trip took. For, one event should not last two amounts of times. But if this is so, when Mary calculates the speed of light, she will find that it had to travel a longer distance in the same amount of time. So her calculation would show that light traveled faster. The authors write that for a Newtonian physicist, this is fine. For, Sarah is moving one speed and Mary (on the moving earth) is traveling another speed. That would be like Mary seeing Sarah on a moving boat walk from one end to the next. Sarah would seem to herself to go one speed. But to Mary on the dock, Sarah would go her walking speed plus the boat speed.

Einstein saw things differently. In the first case, Mary and Sarah would not be able to synchronize to the same time, which we saw in this experiment:

Also, Einstein postulated (and it was empirically demonstrated) that the speed of light remains the same no matter how the light source moves with respect to the observer (77c). Instead, Mary and Sarah do not have agreeing systems of time. If we see things Einstein's way, we can make the calculations accord with each other, when we say that light appears to have the same speed in both cases. So now we are supposing that light went the same speed. And we are also supposing that the distance the light traveled for stationary Mary is longer than the distance the light traveled for Sarah. The way we rectify the equations is by saying that from Mary's perspective, Sarah's clock ticks more slowly. So for each tick of Mary's clock, Sarah's tick takes a little bit longer, but all this is from Mary's perspective. For Mary, there was a "time dilation" in Sarah's moving system.

3.11 Space, Time, and Spacetime

Hermann Minkowski in 1908 says of space and time that "space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (Minkowski "Space and Time" reprinted in Lorentz et al., The Principles of Relativity p. 75, qtd in Mook and Vargish p.86d). However, the authors warn us not to think he means that "space and time are somehow similar things, or even manifestations of the same essence. [...] Space and time are very different, measured in very different ways and sensed differently by us" (87a). Minkowski developed a visual way to depict the correlation between space and time. He has us add the forth dimension of time to the three dimensions of space. But this does not mean there is only one spacetime in physical reality. Rather, they are mathematically correlated using Minkowski's schemes. "What Minkowski developed was a mathematical (not physical) unification of these four numbers designating dimensions. Instead of thinking of three spatial dimensions and one time dimension, Mankowski proposed thinking of the world as a grid of four-dimensional spacetime" (88b).

Recall the experiment where Peter was far from a light shining down a railroad track, and Paul was even further.

Light Track
We are only here concerned with the light traveling down a straight line. So in this case we need only correlate one spatial dimension with the time dimension.

Let's suppose that after one second, the light-bulb flashes. It reaches Peter at 4 seconds, and Paul at 7 seconds. And we will say that the distances will be measured beginning from the left-most rail-tie marker. So the light bulb we will say is at 1 kilometer. Peter is at 4 kilometers. And Paul is 7 kilometers away (this distances and times are provisional; they do not reflect the actual speed of light). We would construct our Minkowski diagram this way:

More sophisticated versions of the Minkowski diagrams will allow us to depict the distortion of time and space for objects in motion. [Perhaps later we will explore them in more detail.]

Mook, Delo E. & Thomas Vargish. Inside Relativity. Princeton: Princeton University Press, 1987.

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