Then again.

" There are many instances of later accounts misrepresenting Einstein’s ideas.

None is as universal and complete as the later treatments of Einstein’s principle of equivalence. In his 'Meaning of Relativity', Einstein gives a statement of the principle typical of all his writing. K is an inertial system in special relativity and K‘ a system of coordinates uniformly accelerating with respect to K. Having noted ’that free masses in K are accelerated‘ just as if a gravitational field were present and K‘ unaccelerated’, Einstein (1922a. p57-8) then writes:

". . . there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that K is ‘at rest’ and a gravitational field is present we can consider as equivalent to the conception that only K is an ‘allowable’ system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of coordinates, K and K, we call the ‘principle of equivalence’; . . . [it] signifies an extension of the principle of relativity to co-ordinate systems which are in non-uniform motion relatively to each other. In fact. through this conception we arrive at the unity of nature of inertia and gravitation."

Einstein, however, is nearly universally understood as urging a rather different principle, which I shall call the ‘infinitesimal principle of equivalence’. A canonical formulation is given in Pauli (1921,pl45): "For every infinitely small world region (i.e. a world region which is so small that the space and time-variation of gravity can be neglected in it) there always exists a coordinate system K

_{0}(X

_{1} X

_{2} X

_{3} X

_{4}) in which gravitation has no influence either on the motion of particles or any other physical process."

The key idea here is that in adopting a sufficiently small region of spacetime, an arbitrary gravitational field becomes homogeneous and can be transformed away by a suitable choice of coordinate system. This principle exists in many variant forms. Sometimes it is strengthened to require that when the gravitational field is transformed away, we recover special relativity locally (for example, Misner et al., 1973, ~386).

With somewhat different qualifications, Pauli’s infinitesimal principle correspond to Dicke’s ‘strong equivalence principle (Roll et al., 1964, p444). Dicke’s ‘weak equivalence principle’, however, requires only the uniqueness of gravitational acceleration, which amounts to requiring that the trajectories of free fall of suitably idealized bodies are independent of their constitutions. Unlike most other writers, Pauli (1921, p 145) acknowledged that his infinitesimal version of the principle of equivalence differed from Einstein’s, suggesting that, where Einstein’s principle applied only to homogeneous gravitational fields, Pauli’s version was for the ‘general case’. However the differences ran far deeper than Pauli allowed and pertain to quite fundamental questions of the role of the principle of equivalence in general relativity. These differences can be summarized in three essential aspects of the principle which remained fixed throughout Einstein’s writings on general relativity, from the earliest moments in 1907 to his final years in the 1950.

..Einstein’s principle of equivalence was a covariance principle..

Special relativity required the complete physical equivalence of all inertial coordinate systems; for Einstein, general relativity required the complete equivalence of all coordinate systems.

Einstein’s principle of equivalence required the complete equivalence of a set of coordinate systems of intermediate size: inertial coordinate systems plus uniformly accelerated coordinate systems. That is, the principle sanctioned the extension of the covariance of special relativity beyond Lorentz covariance but not as far as general covariance. Thus, for Einstein, the principle of equivalence was a relativity principle intermediate in range between the principle of relativity of special relativity and of general relativity. The point is so important for our concerns here that it is helpful to have it in Einstein’s own words of (1950, p347):

"This is the gist of the principle of equivalence: In order to account for the equality of inert and gravitational mass within the theory it is necessary to admit non-linear transformations of the four coordinates. That is, the group of Lorentz transformations and hence the set of ‘permissible’ coordinate systems has to be extended."

Or, more succinctly, in an article devoted to explicating precisely what he intended with his principle of equivalence, Einstein (1916a, p641) wrote in emphasized text: "The requirement of general covariance of equations embraces that of the principle of equivalence as a quite special case."

The function of the alternative, infinitesimal principle of equivalence is to stipulate that a spacetime of general relativity with an arbitrary gravitational field is in some sense locally, that is, in infinitesimal regions, like the spacetime of special relativity. Einstein objected in correspondence with Schlick to the latter’s use of this idea, pointing out to Schlick that the sense in which special relativity holds locally must be so weak that accelerated and unaccelerated particles cannot be distinguished. For details, see Norton (1985, section 9).)

As a covariance principle, Einstein’s version of the principle served no such function. Therefore it was invariably restricted in the following related ways: Einstein’s principle of equivalence was applied only in special relativity to what we now would call Minkowski spacetimes. That is, the inertial coordinate system K of Einstein’s formulation of the principle is not some kind of free fall coordinate system of general relativity. It is simply an inertial coordinate system of special relativity. Thus the coordinate systems K and K’ are both coordinate systems of a Minkowski spacetime. Because of this, we would now be inclined to picture the entire principle as operating within special relativity.

This seems not to have been Einstein’s view. He seems to have regarded special relativity supplemented with the principle of equivalence as having more physical content than special relativity alone. The supplemented theory had a wider covariance and it dealt with a new phenomenon, homogeneous gravitational fields. Einstein’s principle of equivalence was not a prescription for transforming away arbitrary gravitational fields; it was just a recipe for creating a special type of gravitational field. Einstein’s principle of equivalence gave a recipe for creating a homogeneous gravitational field by transforming to a uniformly accelerated coordinate system.

The infinitesimal principle gives a recipe for transforming away an arbitrary gravitational field: one first homogenizes it by considering an infinitesimal region of spacetime and then transforms it away by the reverse transformation of Einstein’s principle. Einstein repeatedly insisted that his principle of equivalence did not allow one to transform away an arbitrary gravitational field, but only gravitational fields of a quite special type, those produced by acceleration of the coordinate system. (Einstein devotes a paragraph of near page length to this point (1916a. pp640-I). See Norton (1985, section2) "

From 'General covariance and the foundations of general relativity: eight decades of dispute, by J. D. Norton.' And if you didn't have a headache before I hope you'll find it now

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Scanning this text puts in 'spaces' between words, so bear with it.