* Investigations like the one just made, which begin from general concepts, can serve only to ensure that this work is not hindered by too restricted concepts, and that progress in comprehending the connection of things is not obstructed by traditional prejudices.
Riemann, 1854

* After eschewing the comfortable but naive principle of inertia as a suitable foundation for physics, Einstein concluded that "in the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial coordinates can be directly measured by the unit measuring rod, or differences in the time coordinate by a standard
clock...this requirement ... takes away from space and time the last remnant of physical objectivity". It seems that we're completely at sea, unable to even begin to formulate a definite solution, and lacking any definite system of reference for defining even the most rudimentary quantities. It's not obvious how a viable physical theory could emerge from such an austere level of abstraction.

These difficulties no doubt explain why Einstein's route to the field equations in the years 1907
to 1915 was so convoluted, with so much confusion and backtracking. One of the principles that heuristically guided his search was what he called the principle of general covariance.

* It strikes many people as ironic that Einstein found the principle of general covariance to be so compelling, because, strictly speaking, it's possible to express almost any physical law, including Newton's laws, in generally covariant form (i.e., as tensor equations). This was not clear when
Einstein first developed general relativity, but it was pointed out in one of the very first published
critiques of Einstein's 1916 paper, and immediately acknowledged by Einstein. It's worth remembering that the generally covariant formalism had been developed only in 1901 by Ricci and LeviCivita, and the first real use of it in physics was Einstein's formulation of general relativity. This historical accident made it natural for people (including Einstein, at first) to imagine that general relativity is distinguished from other theories by its general covariance,
whereas in fact general covariance was only a new mathematical formalism, and does not connote a distinguishing physical attribute. For this reason, some people have been tempted to conclude that the requirement of general covariance is actually vacuous. In reply to this criticism, Einstein clarified the real meaning (for him) of this principle, pointing out that its heuristic value
arises when combined with the idea that the laws of physics should not only be expressible as
tensor equations, but should be expressible as simple tensor equations. In 1918 he wrote "Of two
theoretical systems which agree with experience, that one is to be preferred which from the point
of view of the absolute differential calculus is the simplest and most transparent". This is still a
bit vague, but it seems that the quality which Einstein had in mind was closely related to the
Machian idea that the expression of the dynamical laws of a theory should be symmetrical up to
arbitrary continuous transformations of the spacetime coordinates. Of course, the presence of any
particle of matter with a definite state of motion automatically breaks the symmetry, but a
particle of matter is a dynamical object of the theory. The general principle that Einstein had in
mind was that only dynamical objects could be allowed to introduce asymmetries. This leads naturally to the conclusion that the coefficients of the spacetime metric itself must be dynamical elements of the theory, i.e., must be acted upon. With this Einstein believed he had addressed what he regarded as the strongest of Mach's criticisms of Newtonian spacetime, namely, the fact that Newton's space acted on objects but was never acted upon by objects.