Finally, in the twentieth century, angular momentum joined momentum and energy as one of the preeminent mechanical concepts.1 There are several reasons for its present status. One reason is its conservation; another is its quantization; a third is its relation to a simple symmetry of empty space.
As momentum conservation is related to, and indeed can be founded upon, the homogeneity of space (the indistinguishability of one point in space from another—see Essay M5), angular-momentum conservation is similarly tied to the isotropy of space (the indistinguishability of one direction from another). An isolated object at rest in space is not expected to be self-accelerating in some direction, for that would imply an inhomogeneity of space. Nor is it expected to set itself spontaneously into rotation, for that would imply an anisotropy of space. The absence of spontaneous rotation requires the absence of any net internal torque, which in turn implies that the angular momentum of an isolated system is conserved. The bland sameness of space is at the root of both momentum conservation and angular-momentum conservation.
Let me expand on this idea. Consider a wheel that is uniform except for a weight placed at one point of its periphery. If suspended by its axle near the earth, initially at rest, it will begin to rotate “spontaneously” if the weight starts at any point other than the lowest point. There is nothing surprising about this. An external force, the force of gravity, acts on the wheel, and it produces an external torque. (Another external force, supporting the wheel at its axle, contributes no torque with respect to the axle.) Because of the external torque, the wheel’s angular momentum changes. If initially at rest, it begins to rotate. For the wheel, angular momentum is not conserved. Another way to describe this situation is to say that in the neighborhood of the Earth there exists a preferred direction, the vertical direction of the earth’s gravitational force. The preferred spoke—the one connecting the hub to the weight on the rim—moves in such a way as to align itself with the preferred direction or to oscillate equally about the preferred direction.
In empty space, on the other hand, far from the Earth or other external influences, there should be no preferred direction and no spontaneous rotation. The same wheel, placed at rest in an ideally remote location, should remain at rest. The “should” in these sentences is based on the fundamental postulate of the isotropy of space. If space possesses the indistinguishability of direction called isotropy, no isolated object should spontaneously begin to rotate. This is the first key step in the argument. The second is to note that no rotation means no torque. Torques, if any, within the isolated object must cancel exactly if there is no rotational tendency. Having reached the conclusion that total internal torque must equal zero, we may, for the final step in the argument, allow our isolated object to be rotating initially instead of being stationary. Since it experiences neither external torque nor internal torque, its angular momentum remains constant.
This argument, intended only to be indicative of the existence of a link between the isotropy of space and the conservation of angular momentum, is less rigorous than it may appear. It provides a tight logical link only for rigid objects such as the wheel just discussed. For looser systems whose parts are in relative motion, the connection between spatial isotropy and angular-momentum conservation is more subtle (but just as real). Then the absence of any preferred direction leads only to the conclusions that some rotational property should be conserved. Angular momentum is defined in just such a way that it is the conserved quantity. It is not hard to think of other rotational quantities—angular velocity, for instance—that are not conserved.
Angular-momentum conservation has not been put to the test over domains of space larger than the solar system. It remains a question for the future, and a most intriguing question, whether this conservation law will fail in the galactic and intergalactic domains. If it does, scientists will have learned that space in the large is not perfectly isotropic, a discovery that would have important bearing on the structure of the universe as a whole, and on the question of whether the universe is finite or infinite.
1 All three of these concepts have required new definitions in the modern theories, definitions that encompass the old.