The Quest of the Absolute

Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. The Quest of the Absolute

THE QUEST OF THE ABSOLUTE

Modern Developments in Theoretical Physics, and the Climax Supplied by Einstein

BY DR. FRANCIS D. MURNAGHAN,
JOHNS HOPKINS UNIVERSITY
BALTIMORE

We shall discuss the more important aspects of the theory popularly known as the “Einstein Theory of Gravitation” and shall try to show clearly that this theory is a natural outcome of ideas long held by physicists in general. These ideas are:

(a) The impossibility of “action at a distance;” in other words we find an instinctive repugnance to admit that one body can affect another, remote from it, instantaneously and without the existence of an intervening medium.

(b) The independence of natural, i.e., physical, laws of their mathematical mode of expression. Thus, when an equation is written down as the expression of a physical law it must be satisfied, no matter what units we choose in order to measure the quantities occurring in the equation. As our physics teacher used to say “the expression of the law must have in every term the same dimensions.” [277]More than this the choice of the quantities used to express the law—if there be a choice open—must have no effect on its correctness. As we were told—“all physical laws are capable of expression as relations between vectors or else as relations between magnitudes of the same dimensions.” We shall hope to make this clearer in its proper place in the essay, as its obvious generalization is Einstein’s cardinal principle of relativity.

The measurements which an experimental physicist makes are always the expression of a coincidence of two points in space at the same time. If we ask such an experimenter what he means by a point in space he tells us that, for him, the term has no meaning until he has a material body with reference to which he can locate the point by measurements; in general it requires three measurements and he expresses this by saying that space has three dimensions. He measures his distance, as a rule, parallel to three mutually perpendicular lines fixed in the material body—a Cartesian reference-frame so-called. So that a “point in space” is equivalent to a given material reference-frame and three numbers or coordinates. If, for any reason, we prefer to use a new material reference-frame the coordinates or measurements will change and, if we know the relative positions of the two material reference-frames, there is a definite relation between the two sets of three coordinates which is termed a transformation of coordinates. But which particular material reference-frame shall we use? The first choice would, we think, be that attached to the earth. But, even yet, we are in doubt as there are numberless Cartesian [278]frameworks attached to the earth (as to any material body) and it is here that our idea (b) begins to function. We say it must be immaterial which of these Cartesian frames we use. In each frame a vector has three components and when we change from one frame to another the components change in such a way that if two vectors have their three components equal in one framework they will be equal in any other attached to the same material system. So our idea (b), which says that our physical equations must be vector equations, is equivalent to saying that the choice of the framework attached to any given material body can have no effect on the mode of expression of a natural law.

Shall we carry over our idea (b) to answer the next question: “To which material body shall we attach our framework?” To this question Newton gave one answer and Einstein another. We shall first consider Newton’s position and then we may hope to see clearly where the new theory diverges from the classical or Newtonian mechanics. Newton’s answer was that there is a particular material frame with reference to which the laws of mechanics have a remarkably simple form commonly known as “Newton’s laws of motion” and so it is preferable to use this framework which is called an absolute frame.

We have tacitly supposed that the space in which we make our measurements is that made familiar to us by the study of Euclid’s elements. The characteristic property of this space is that stated by the theorem of Pythagoras that the distance between two points is found by extracting the square root of the sum of the squares of the differences of the Cartesian coordinates of the two points. Mathematicians have long recognized the possibility of other types of space and Einstein has followed their lead. He abandons the empiricist method and when asked what he means by a point in space replies that to him a point in space is equivalent to four numbers how obtained it is unnecessary to know a priori; in certain special cases they may be the three Cartesian coordinates of the experimenter (measured with reference to a definite material framework) together with the time. Accordingly he says his space is of four dimensions. Between any two [281]“points” we may insert a sequence of sets of four numbers, varying continuously from the first set to the second, thus forming what we call a curve joining the two points. Now we define the “length” of this curve in a manner which involves all the points on it and stipulate that this length has a physical reality, i.e., according to our idea (b) its value is independent of the particular choice of coordinates we make in describing the space. Among all the joining curves there will be one with the property of having the smallest length; this is called a geodesic and corresponds to the straight line in Euclidean space. We must now, for lack of an a priori description of the actual significance of our coordinates, extend the idea of vector so that we may speak of the components of a vector no matter what our coordinates may actually signify. In this way are introduced what are known as tensors; if two tensors are equal, i.e., have all their components equal, in any one set of coordinates they are equal in any other and the fundamental demand of the new physics is that all physical equations which are not merely the expression of equality of magnitudes must state the equality of tensors. In this way no one system of coordinates is privileged above any other and the laws of physics are expressed in a form independent of the actual coordinates chosen; they are written, as we may say, in an absolute form.

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