General Principle of Relativity
General Principle of Relativity
The beginning part of Einstein's 1916 paper "Die Grundlage der allgemeinen Relativitaetstheorie" (Ann. der Phys.) is a "must" for any students of spacetime philosophy. It is already quoted many times (e.g. M. Friedman 1983, 205-207), and a Japanese translation is also available (『アインシュタイン選集2』、59-114). Section 2 of this paper begins as follows (quotations from Friedman 1983, and the whole English translation is in Perrett & Jeffrey 1923; a new translation is in The Collected Papers of Albert Einstein, Princeton Univ. Press):
In classical mechanics, and no less in the special theory of relativity, there is an inherent epistemological defect which was, perhaps for the first time, clearly pointed out by Ernst Mach. We will elucidate it by the following example:
This example may be reproduced by the following figure.
Now suppose one is a sphere and the other is an ellipsoid of revolution.
Thereupon we put the question---What is the reason for this difference in the two bodies? No answer can be admitted as epistemologically satisfactory, unless the reason given is an observable fact of experience. The law of causality has not the significance of a statement as to the world of experience, except when observable facts ultimately appear as causes and effects.
With this "verificationist" principle in hand, Einstein argues as follows:
Newtonian mechanics does not give a satisfactory answer to this question. It pronounces as follows: The laws of mechanics apply to the space R1, in respect to which the body S1 is at rest, but not to the space R2, in respect to which the body S2 is at rest. But the privileged space R1 of Galileo thus introduced, is merely factitious cause, and not a thing that can be observed. It is therefore clear that Newton's mechanics does not really satisfy the requirement of causality in the case under consideration, but only apparently does so, since it makes the factitious cause R1 responsible for the observable difference in the bodies S1 and S2.
Einstein tried to solve this epistemological problem by his general principle of relativity. But Friedman points out that we have to distinguish two starategies: (1) one is a simple extension of relativity principle to any kind of motion, not merely to all inertial motions, and (2) the other is Machianization, which tries to explain the difference between S1 and S2 by introducing the relative motion of the two to some third physical object (or class of physical objects). But (1) is hopeless in that it leads to physically false consequences, so that we have only (2) left. Thus Einstein continues:
The cause must therefore lie outside this system. We have to take it that the general laws of motion, which in particular determine the shapes of S1 and S2, must be such that the mechanical behaviour of S1 and S2 is partly conditioned, in quite essential respects, by distant masses which we have not included in the system under consideration. These distant masses and their motions relative to S1 and S2 must then be regarded as the seat of the causes (which must be susceptible to observation) of the different behaviour of our two bodies S1 and S2.
As is clear to any readers of Mach, this is nothing but what Mach suggested in his criticisms of Newton. However, from this suggestion, Einstein moves to an extension of the principle of relativity: "The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."
Now, Friedman clearly points out that Einstein's next step of identification of this "extended relativity" with the "general covariance" (section 3) was decisive for producing grave confusions and mistakes in later writers.
The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant).
This is general covariance, but it is much weaker than the proposed "extended relativity", a generalized physical equivalence of reference frames (see Friedman 1983, 208).
Moreover, as is already pointed out by Sklar, many people do not even think that General Relativity theory is a Machian theory (however, this depends on how one understands the "Machian" character; some people, like Julian Barbour, regard the theory as Machian).
Added March 23, 2001
The same passages are also quoted and criticized by John Earman as follows:
Behind this attempted extension lies the stroy of a triple confusion. First, in searching for the field equations of GTR, Einstein was guided by the idea that the distribution of matter should determine the metric; the difficulties inherent in this idea will be discussed ... . Second, for a period extending from 1913 into 1915, Einstein convinced himself that matter couldn't determine the metric through generally covariant field equations, and in response he proposed to abandon the requirement of general covariance. After he realized his mistake and reembraced general covariance, he was led to the brink of another confusion: general covariance, he thought, entailed a generalized principle of relativity by which all frames of reference are equivalent; hence the title of section 2 of his 1916 paper and the appellation of "general theory of relativity" for his new theory of gravitation.
None of this detracts from Einstein's monumental achievement, but it should serve as a cautionary tale for philosophers of science who seek to draw wisdom from the philosophical pronouncements of scientists, even the greatest scientists. (World Enough and Space-Time, 1989, 105)
The last comment should be kept in mind, especially because some scientists, issuing their own philosophical views, often claim in effect "we know our field better than anyone else; therefore our philosophical comments should be accepted as having the same authority".
Added November 30, 2001
Despite the preceding addition, I am not sure whether most students can understand the distinction between "generl covariance" and "general relativity" in the sense Einstein intended. So let me add quotations from Friedman (1983), which is one of the first to emphasize the significance of the distinction. Referring to the last quoation from Einstein (above), Friedman continues:
The notions of "sameness of form" and covariance correspond to the notions of physical equivalence and relativity only in the context of flat space-time theories in which there exists a priviledged class of inertial coordinate systems. Such theories possess a standard formulation ... that takes the same form in all inertial systems and whose covariance group is just the group of all transformations from one inertial frame to another. All inertial frames are indistinguishable, and so the indistinguishability group = the covariance group of the standard formulation .... But in nonflat space-time theories like general relativity these "nice" connections between indistinguishability and covariance break down. Instead, we have no priviledged subclass of inertial frames and no standard formulation ... holding in all and only the inertial frame. The only "standard formulation" available is the generally covariant formulation holding in all coordinate systems, and this fact does not imply a generalized equivalence of reference frames. (Friedman 1983, 208)
Thus, notice that we can give a generally covariant formulation for the Euclidean geometry, for the Newtonian mechanics, etc. Galilean relativity tells us that we cannot distinguish one inertial frame from another. And in the Newtonian mechanics, its metric is fixed once for all, the same through all "dynamically possible" worlds (distances and time intervals are the same among them); and the same for the Minkowski metric in special relativity (however, here intervals are invariants), and the Lorentz invariance holds for all inertial frames. We cannot find any symmetries of this sort in general relativity, except for local Lorentzian (inertial) frames.
Looking back on the development of relativity from our present point of view, we can see that there are three distinct notions that have been inadvertently conflated: symmetry, indistinguishability, and covariance. The symmetry group of a space-time theory characterizes the objects of that theory: it tells us which objects are absolute and which dynamical, and the size of the symmetry group is inversely proportional to the number of absolute objects. The indistinguishability group of a space-time theory characterizes the laws of that theory: it determines which reference frames (states of motion) are distinguishable (by a "mechanical experiment") relative to those laws, and in well-behaved theories the indistinguishability group is contained in the symmetry group (indistinguishable models are identical). Covariance, on the other hand, is really a property of formulations of space-time theories rather than space-time theories themselves: it characterizes systems of differential equations ... representing the intrinsic laws of a space-time theory relative to some particular coordinatization .... The covariance group of such a formulation reflects the range of coordinate systems in which that particular system of equations holds good.
In pre-general-relativistic physics these three distinct notions happen to coincide. In well-behaved theories with inertial coordinate systems (flat space-time) the symmetry group = the indistinguishability group = covariance group of the standard formulation (with respect to a subclass of inertial coordinate systems). Thus, for example, in classical electrodynamics the symmetry group = ... = the indistinguishability group = covariance group of the standard formulation .... In relativistic electrodynamics, on the other hand, the symmetry group = the Lorentz group = indistinguishability group = the covariance group of the standard formulation ....
In general relativity, however, our three notions are not interchangeable. As we have seen, we cannot interpret the general principle of relativity as an indistinguishability requirement, for the indistinguishability group of the general theory is just the restricted group of transformations from one local inertial frame to another. Nor can we interpret it as a covariance requirement, for the general theory has no standard formulation in the usual sense, and the covariance group of the theory is the same as the covariance group of every other space-time theory. Hence, in neither of these interpretations is the general principle of relativity any kind of generalization of the sspecial principle of relativity. As Anderson was the first to realize, the only way to interpret the general principle as such is to make it a symmetry requirement. That is, we interpret the general principle of relativity s the requirement that the symmetry group of our theory include all differentiable transformations: in effect, that it be just the group M. This mrequirement means that our theory can have no absolute objects, for the only geometrical objects invariant under all differentiable transformations are constant-valued scalars. (Friedman 1983, 212-4).
Thus, we have to be very careful, not to be misled by the similarity of names, "special relativity" and "general relativity"; in a nutshell, there is no "generalized principle of relativity" as Einstein once intended. Instead, we have a nice theory of gravity, with no priviledged frame of reference, except for local Lorentzian systems.
References
Friedman, Michael (1983) Foundations of Space-Time Theories, Princeton University Press, 1983.
Perrett, W. and Jeffrey, G. B. (1923) The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity, Dover, 1923.
『アインシュタイン選集2』（湯川秀樹監修、内山龍雄訳編）、共立出版、1970。
Last modified March 28, 2003. (c) Soshichi Uchii
