So much is being spoken and written in both a highly technical and a popular vein about relativity that the theoretical astronomer and navigator is forced to take account of stock at increasingly frequent intervals, lest he find himself caught with an array of discarded concepts on his mental shelves.
When relativity is mentioned, we think of Dr. Einstein and associate the word with a theory that as often as not bears his name. We should keep in mind, however, that the law of relativity is as old as the world. In its simplest form, it may be stated this way: “There are always two parties to every observation—the observed and the observer.”
The basic formulas employed in the positional problems of the solar system are derived from what are ordinarily referred to as Newtonian concepts. The laws of Newton serve us well, except for the orbit of Mercury and the orbit of the moon. The Newtonian world is tridimensional. The Einsteinian continuum is poly-dimensional. There are certainly many practical considerations that make it desirable for the positional astronomer and for the navigator to cling to the tridimensional concept. If perchance these practical considerations can be supported by acceptable philosophical and mathematical reasoning, we can still dwell with confidence in a physical cosmos governed by understandable laws of cause and effect, especially if our reasoning responds also to experimental fact. The theory of uncertainty to which many physicists are for the moment receptive can have no appeal for the navigator on soundings or off.
In the light of our present knowledge of the physical universe, it is not difficult for us to accept the thought that it takes force time to act. It is essential that we accept this as a fact, if we accept the results of experiments in the electromagnetic field of which Newton could have had no possible knowledge. Just as Newton, however, did not accord to light a sufficient time to travel, so also he accorded to gravity no time whatever to act. In other words, the Newtonian mathematics is predicated on the assumption that a center of force may be responsible for instantaneous action at a distance.
For as long as men accepted that the world was round or spherical, they knew by observation that falling bodies directed themselves toward the earth’s center. Yet Newton was the first man who stated in a definitive way the law of gravitation. Observations for centuries have indicated that the orbits of the planets about the sun are rotating ellipses. Hence there can scarcely be a question but that the equation of their orbit must be of the form:
1/p = u = a + b cos g?
which is the polar equation of a rotating ellipse,
where p is the radius vector;
u is its reciprocal
a is the reciprocal of the semi-latus rectum
— b is the eccentricity divided by the semi-latus rectum
? is the angle about the center of mass from the initial aphelion point to the radius vector
g is the factor in the argument of the cosine whose departure from unity indicates the extent of the rotating motion of the apses
A consideration of the Maxwell electromagnet c theory, the mathematics of which in relation to the forces here dealt with parallel the Einstein mathematics, gives us warrant for accepting that these forces travel with the speed of light. In consequence, we must have an aberration of force.
A comparison of this expression of gravitational force acting between celestial bodies with the relativistic deductions applied to Kepler’s laws, as given in Arnold Sommerfeld’s Atomic Structure and Spectral Lines (Eng. trans. Methuen & Co., 1923), page 467, will serve to indicate that the Einstein mathematics approximates this more exact and definitive result. Perhaps a better comparison is made by considering the vis viva integral and the law as given in Eddington’s Mathematical Theory of Relativity (Camb. Univ. Press, 1924), Eq. 38-8.
It is at this point that the astronomer and the navigator should note a feature that has apparently been overlooked in a too rigid acceptance of Newton’s concepts. The elements of the earth’s motion are known with presumably greater accuracy from the standpoint of astronomical observations than those of any other body. The orbital elements of the moon, our nearest celestial neighbor, should be known with practically equal accuracy. Why the wide discrepancy? It is believed that the explanation is found in the following considerations.
The elements of the earth’s motion are employed in such a way as to satisfy the law expressed by the formula,
Fp = k2/p2
If the law governing planetary motions is correctly expressed by this formula, it follows clearly that an employment of the constants which are derived from the observed motion of any one of the bodies should equally satisfy each of the others in their respective motions. This is not borne out by astronomical observations.
From a consideration of the equation, we see that for a body moving in a circular orbit about the central force the law expressed by formula (1) above is fulfilled exactly. The earth’s orbit is made to fit it, although the earth’s orbit is not circular. Planets with orbits approaching in their elements the orbit of the earth will show little departure from this law, but an orbit such as that of Mercury will show a considerable discrepancy.
The discrepancy will disappear, if formula (2), taking into account the considerations set forth above, is employed. In the case of the moon, the still wider discrepancy will presumably also be brought into such limits as will vindicate this Newtonian approach to a solution of the moon’s orbit in contradistinction to an Einsteinian approach.
After all, the physical universe is the same universe today as it was in the days of Newton or even Euclid. The physical laws governing it do not change because Riemann devises a new geometry as acceptable mathematically as the geometry of Euclid. In other words, mathematical knowledge of the physical laws governing our universe or all of the physical universes we may conceive to exist does not alter the action of these laws. Such mathematical knowledge, no matter on what postulates it may initially be predicated, affords only a means of indicating events in the physical continuum that will happen or that have happened in the unrecorded past.
The mathematics pertaining to any physical law is merely a statement or series of statements in symbols which appeal to the human mind as a logical presentation of the law. Contrariwise the verbal expression of the same law is a logical exposition in words of how some definitive mathematical formula works. That is to say, speaking mathematically, if we are given results as an hypothesis, we should be able to trace back by logical mathematical concepts to the causes; or given the causes in a mathematical setting, we should be able to determine the results mathematically. For instance, if we obtain the elements of an orbit by observation and express that orbit mathematically, it should be possible and we know it is possible, accepting the law of cause and effect, to express mathematically the forces which are acting to produce this orbit. Correspondingly, if observations can determine for us sufficient data concerning the force of gravitation, then it should be entirely possible to express the resultant orbital paths of bodies affected by this force in appropriate mathematical formulas. Investigations in the electromagnetic field reveal the technique of ascertaining such forces and their character. We are stating, in short, that mathematical theories ought properly to be reversible, because we believe the law of cause and effect is an absolute law.
The theory of relativity, in its present stage of development, is in many respects a one-way theory, affording to physical quantities no definitive meaning. Until the theory passes through this stage, the astronomer is on safe ground, if he holds to a mathematics that enables him to predict logically the ephemerides of the future rather than to geometrize uncertainly the physics of space.
Note. By a comparison of the basic formula here submitted with that part of the theory of relativity which has a physical significance, the abstract constant n is found to have a value of minus six.