Mario Castagnino and Rafael Ferraro explain the differentiation between curved space-time, and the measurement of it by a clock. Whilst the physical parameters recorded by the clock are recorded as natural time, there is said to exist an infinite number of such parameters, all of which are defined as observer-dependent.
It is well known that there is a small confusion between a physical observer’s system and a geometrical coordinate system (or chart) in several papers. Of course they are two different concepts, e.g., in classical physics, an observer’s system is a rigid frame and a clock, where we can use all kinds of charts, for instance, Cartesian or polar coordinates. In curved space-time we cannot use a rigid frame and the natural generalization of the observer’s system will be a timelike fluid of observers, each one endowed with a clock, i.e., a set of timelike paths, each one with a different parameter, the “time” measured by the clock. This time is not necessarily the proper time; it is only an arbitrary continuous function of space-time. Of course we can describe this fluid of observers with any chart we like. We will find that physics is, in fact, observer dependent, but it is of course, chart independent. We shall restrict ourselves to irrotational fluid; thus we can define a set of orthogonal timelike hypersurfaces to the fluid paths, and we can define a parameter T, on each surface, such that the equations T = const would define the orthogonal hypersurfaces. We shall call this parameter a “natural time.” Of course, there exists an infinite set of natural times. We can pass from one to another via a continuous function T -> T’ = T'( T). We shall see that physics is independent of the natural time we use; it is only dependent on the chosen observer’s fluid. Of course, in general, natural time is different from proper time. We can label each fluid world line by three real parameters x1, x2 , x3 ,* and we can call x0 to the natural time T induced by the fluid of observers. Then x0, x1, x2, x3 is a chart and every event of space-time has its coordinates x0, x1, x2, x3 – namely, the space coordinates of the fluid world line, where the event happens, plus the natural time measured by the clock of this world line when the event happens. We shall call this chart an adapted chart (Castagnino and Ferraro 1988, 52-53).
Castagnino, Mario, and Ferraro, Rafael. 1988. “Toward a complete theory for unconventional vacua,” In Claudio Teitelboim (ed.) Quantum mechanics of fundamental systems 1, 51-62. New York: Springer Science+Business Media.