Invariance appears to mean just the opposite: definiteness, absence of relativity. In the theory of relativity, it refers to areas of agreement, those aspects of a phenomenon, and even more important, those laws of phenomena, that are the same for different observers. Instead of relativity and invariance, we could just as well have used two more familiar words, subjectivity and objectivity, except perhaps that these words are too familiar. The subjectivity of relativity is a definite kind of physical subjectivity, not referring merely to differences in human perception. And the objectivity of relativity is not the philosophers objective reality. It is rather objectivity by definition, an agreement among observers to accept as real the common aspects of their measurements, and to assign significance to laws that are the same for different observers.
The theory of relativity has, surprisingly, added both more relativity, or subjectivity, and more invariance, or objectivity, to science (see Essay R5). Einstein showed that a number of quantities previously thought to be invariant are in fact relative, most notable of these being time. But at the same time he showed how to extract from the increased relativity of observation new invariant quantities. More important, he raised to the level of a fundamental postulate of science the principle that, despite the relativity of the raw observations of phenomena, the laws governing these phenomena must be invariant. From the increased subjectivity of observation that relativity brought to the world came a new and deeper view of the objectivity of physical laws.
The ideas of relativity and invariance exist in ordinary Newtonian mechanics, and can be illustrated with simple mechanical experiments. Consider a child in a uniformly moving train who releases a ball and allows it to fall straight down. From the childs point of view it starts from rest, and falls vertically downward with uniform acceleration. Someone watching from outside the train would have a different view. The actual path of the ball in space, the outside observer would say, is a parabola. The ball had a forward motion when released, and fell like a projectile through a parabolic arc. The outside observer would concede, however, that the child was right in assigning to the ball a uniform downward acceleration, and indeed both would agree on the magnitude of the acceleration. We may summarize the areas of agreement and disagreement between the two observers in a short table. Bear in mind that this table refers to Newtonian mechanics. The new mechanics of Einstein will require that it be altered.
|Laws of motion|
There is obvious disagreement about position, since there is disagreement about the shape of the trajectory. There is also obvious disagreement about velocity, since the outside observer considers the ball to have some horizontal component of velocity, whereas the inside observer considers the ball to have only vertical velocity. There is also possible disagreement about coordinates, something that is not necessarily connected with the relative motion of the observers. Any two observers are always free to choose different coordinate systems to which to refer their measurements.
Despite the disagreements, there remains a large area of agreement between the observers. Both would measure the same acceleration. This may not be immediately obvious. It comes about because there is no relative acceleration between the two observers. Since each has zero acceleration with respect to the other, both agree about the acceleration of moving objects. Mass and time are assumed in mechanics to be definite invariant quantities, and the theory of mechanics would be considerably upset if it were not so, as indeed it is upset by the theory of relativity. Force would be measured to be the same by both observers, since experimentally train travelers do not change their weight. Most important of all, if the train traveler, who had better now be a scientist and not a child, supplements the experiment of dropping a ball with many other experiments, he or she would arrive at Newtons laws, exactly those that the outside observer already knows to be valid at rest outside the train.
The fact that the observers moving relative to each other agree about the laws of motion is known as Galilean invariance. It means that the Earth is neither a better nor a worse laboratory than the train, and that the traveling scientist has as much right as the earthbound scientist to claim to be at rest with the other moving. This invariance eliminates any single preferred frame of reference for mechanics, and thereby eliminates absolute motion. The essence of this invariance principle, as its name implies, was already appreciated by Galileo. He used it to support the Copernican view that the Earth moves around the Sun. Like passengers on a moving ship, he argued, who cannot directly perceive motion on a smooth sea, we cannot detect the motion of our vehicle Earth transporting us through space.
The implications of the fact that the same laws of motion hold in the train and outside it can be appreciated by considering possible types of motion in the two frames of reference. The train defines the travelers coordinate system, or frame of reference; the Earth defines the frame of reference of the fixed observer. Although the traveling observer and the fixed observer disagree about the trajectory of the dropped ball—one says it fell straight down, the other that it followed a parabolic arc—each concedes that what the other claims to see is a possible motion according to Newtons laws. The outside observer could easily duplicate the straight vertical motion that the train traveler saw, and the traveler could without difficulty cause the ball to follow a parabolic arc with respect to the railroad car, if he or she launched it with a horizontal component of velocity. Invariance of the laws of motion means that both observers agree about the possible paths of moving particles, but not necessarily about the motion of a particular object that they are both observing.
Galilean invariance is in fact well known, at least qualitatively, to everyone. We all know that in a uniformly moving elevator, or in an airplane in smooth air, there is no sensation of motion. We feel neither heavier nor lighter; there are no unexpected physiological sensations. If we drop an object, it falls to the floor. Everything seems normal and that normalcy means only that the same laws of mechanics apply to us and our surroundings in the elevator or in the airplane as at rest on Earth.