"Have you guessed the riddle yet?" the Hatter said, turning to Alice again.
"No, I give it up," Alice replied: "what's the answer?"
"I haven't the slightest idea," said the Hatter.
"Nor I,"said the March Hare.
Alice sighed wearily. "I think you might do something
better with the time," she said, "than wasting it in
asking riddles that have no answers."
A hallmark of 20th century physics is the operational definition. In the past the connection with physics and metaphysics was much murkier. The concept of the operational definition has helped clarify the boundaries.
An operational definition is simply a way of defining a physical quantity or quality in terms of an operation performed. Instead of saying length is "the amount of distance between two points" a modern physicist would give a procedure (an operation) for measuring the distance between two points. ("lay metersticks end to end and count how many" or "send a light signal from point A to point B and reflect it back to A again; then multiply the elapsed time for the signal by 1/2 c.") The result of such a measurement would be the DEFINITION of the length.
The need for the concept of an operational definition became apparent when considering the "ether" of 19th century mechanics and electro-dynamics.
In the famous "Principia" Isaac Newton stated.
"Absolute space, in its own nature, without relation to anything external, remains always similar and immovable."
Newton said all other frames that were moving at constant velocity relative to this frame (and hence to each other) acted just as this "absolute" frame did. He gave no way of determining that one's frame was the absolute one or not, however. Thus his definition was useful in deriving his theory but was not an operational definition.
In fact even before Newton, Galileo noted that it was impossible to
tell which of two frames, with one moving at constant velocity relative
to the other, was "really" at rest. He stated:
"Shut yourself up with some friend in the main cabin below decks
on some large ship, and have with there some flies, butterflies,
and other small flying animals. Have a large bowl of water with
some fish in it; hang up a bottle that empties drop by drop into
a wide vessel beneath it. With the ship standing still, observe
carefully how the little animals fly with equal speed to all sides
of the cabin. The fish swim indifferently in all directions; the
drops fall into the vessel beneath; and, in throwing something to
your friend, you need throw it no more strongly in one direction
than another, the distances being equal; jumping with your feet
together, you pass equal spaces in every direction. When you
have observed these things carefully (though there is no doubt
that when the ship is standing still everything must happen in
this way), have the ship proceed with any speed you like, so long
as the motion is uniform and not fluctuating this way and that.
You will discover not the least change in all the effects named,
nor could you tell from any of them whether the ship was moving
or standing still."
Newton needed an absolute space with which to compare acceleration since you can feel the effects of acceleration. Just try to take a swig of your coffee mug while riding in a car just as the driver steps on the gas.
But the problem is that there is no experiment that can be performed which distinguishes absolute space from relative space. Thus the concept cannot be tied to experiment.
Or can it?
The development of Maxwell's beautiful electromagnetic theory in the 1860's predicted electromagnetic waves which behaved exactly like light waves. Thus it was learned that light was an electro-magnetic wave. The equations also predicted the speed of these light/electromagnetic waves.
Well, waves are waves in something. The ocean waves are traveling disturbances of the ocean surface. Sound waves are traveling disturbances of the air pressure. Traveling waves on a rope are traveling disturbances of the rope's position. Thus light/EM waves must be traveling disturbances of something. But sunlight traveled across the vast vacuum of space from the sun to the earth. Whatever that something was must be something we have not yet detected. That something was called the "ether" It was considered to be at absolute rest and it was assumed to have various other qualities that accounted for the behavior of light. It was realized that by measuring the speed of light in various directions our motion through the ether could be detected since the ether was what light moved relative to. Thus absolute space could be found.
The experiment (The famous Michaelson/Morely experiment) was done, and redone and redone again and again. The result was always the same: No motion through any ether was ever detected.
The world's physicists used these results to add on a few qualities to the ether. They assumed that the movement through the ether "compressed" the apparatus in the direction of the motion so that the "no movement" reading always resulted.
It was interesting. The Ether had been defined to be impossible to detect by any means. "If it was impossible to detect," one might ask, "how did they know it existed?"
It was against this background of theories that were fundamentally untestable that Einstein and others stated that any experimental science such as physics must have its fundamental entities defined in terms of some experiment. To avoid this was to invent ghosts.
Since physics is a way of predicting and explaining the observations that one will encounter under certain circumstances, it should be somehow based on observation and experiment. Any concept that could not, in principle, be connected with experiment was declared to be not physics but metaphysics.
Space and time in special relativity are defined to be what is measured. There is no "real" length different from what is measured. There is no "real" time besides that which is measured.
To ask what the "real" length of something is independent of any measurement is to pose Mad Hatter's riddles.
In physics what you measure is what you get.
Next: Einstein's Postulates
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