The Plane of Existence

Reference frames are three copies of the number line drawn on a surface at angles called, though they really aren't, “right angles.” Einstein invites us to imagine reference frames in space. But we cannot use them to measure the distance from here to there, for reference frames do not allow measurement. They allow only calculations, a procedure performed by writing symbols. The number line, often drawn freehand, is only roughly straight, and the number-divisions marked upon it are only roughly accurately placed freehand to illustrate that no measurements should be made upon them. It is an illustration that justifies certain calculations. To imagine the number line in actual space is to describe an experiment relieved of the obligation to measure.

 

Or is it possible to have a number line upon which measurements can be taken? We are invited to imagine the lines in space shooting out, supplying three numbers that together locate every point. But how might one install a reference frame in space? We must assign one point, zero, to something in space? But what is in space? Lamps, people. Our zero must be something, a smudge, a mark, a recognizable something, perhaps only recognizable by feel, but, as usual, recognizable by more than one person. It must be something each can locate, turn away from, turn back to, locate again and agree upon the position of. Otherwise how could we use it in a procedure others can copy?

 

But we can only identify one such mark with our reference frame. For any two such identifiable points can move in relation to one another, but the points on the number line cannot. All other points must be located by specifying their distance from the one identifiable point in space, the zero point. That is, all points, except the zero point, can only enter into calculations, not measurements.

 

Mathematical calculations offer no difficulties further than naming the answer, for to name them is to do them. All you have to do is write something down. The difficulties in doing a corresponding measurement are either theoretical, if an impossible procure is called for, or practical, if the procedure is standard but still cannot be done for some reason particular to this token of the procedure. Too big, too far away, and so on is a way of saying that something that seemed to be a token of a type isn't. You can't weigh the earth though it has mass. That is a practical problem.

 

Einstein, in his discussion of special relativity, invites us to imagine two reference frames, one moving in relation to the other. Since the motion involved only the x-axis, the y and z axis are superfluous and can be ignored.

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We provide each X-axis with numbers to designate the location for every point on the axis at each moment in time. Every point at every moment has two names. If x is the name of a point in the S reference frame, the point's other name, in S', is x'= x-vt. The procedure described by this equation is simply translating the name of a point from one frame to another by performing an arithmetic calculation. The equals sign indicates that the two expressions are the two different names for the same point.

 

The point consists of nothing more than these two names, for after another period and the movement of one of the reference frames, the two names (numbers) will no longer coincide. The relativity of motion makes it impossible to tell which of the frames moved and which, if any, of the two original names still name the point. So these are names of nothing but themselves, in fact, numbers. They are names of names and also numbers that can enter into calculations. But if being in space means usable in measurements of distance, these points are not in space.

 

Einstein often illustrated his thought experiments with images of trains used as reference frames. Imagine a train moving along an embankment. On this train is a flat car. At the front end of the car is a screen, and at the other end, a man with a flashlight. The train, traveling at nearly the speed of light, is the moving reference frame, the embankment the stationary one with its own observer. The man flashes the light at the screen. What does each observer see?

 

The observer on the embankment, if he could see light travel at the speed of light, would see the beam slowly gain on the train. Since the train is going at nearly the speed of light, the light is going only slightly faster. The train will be well down the track by the time the light hits the screen. The observer on the train, oblivious of the train's movement, also sees the light travel at the speed of light. From his perspective it has simply gone from one end of the flat car to the other, a much shorter distance than that measured by the stationary observer.

 

Because both observers see light travel at the speed of light, and the light, from the two perspectives, has traveled two very different distances, it will have taken different elapsed times for the two observers of the same event. Since the observer on the bank sees the path the light traveled as much longer, the event will have taken more time for him. Also, he must divide that path into many units of measurement, say the meter, for them to add up to that distance. That path is the same path the moving observer sees. If, for example, we mark the point on the embankment where and when the beam completes half the journey, and make a mark on the train at that same moment, it will also be halfway to the screen. We can make such marks anywhere along the light's journey. the marks on the train will be much closer together.  

 

That is what we do if we measure a moving length with a stationary meter stick. If the moving observer measures a moving distance, and the stationary observer measures “the same distance,” the moving measurement will be smaller than the stationary measurement. The moving meter stick is shorter than the stationary one. However, motion is relative. Distances on the embankment will be smaller when measured with a meter stick on the train.

 

We can do the calculation in a thought experiment, but can we do the measurement in the real experiment? To measure a distance with a meter stick we place it alongside the length to be measured, and line up the zero in the meter stick with one end of the length. Then we observe which mark on the meter stick lines up with the other end of the length. While we do this we must be careful that the meter stick doesn't move relative to the length to be measured. Not moving the meter stick while making the measurement is a procedure essential to the measurement, something we must do if the measurement is to be accurate (repeatable). But this is just what we must not do if we are to measure the moving length with the stationary meter stick. For that would mean tht the formerly stationary meter stick is moving.  Other methods of measurement, such as using photography, are attempts to stop motion without stopping it. Is this a theoretical or merely a practical problem? 

 

If instead of the embankment we imagine another train going equally fast but in the other direction, each observer would see the other as moving and himself as still. Thus each would measure the moving meter in the other train as shorter than his own. Should we say that each of these meter sticks is shorter than the other or that each observer sees something the other can't that is still somehow “real.”