What is time relative to?

1. Absoluteness as having an intrinsic metrics, i.e., independence from measurements and instruments of measure.

2. Absoluteness as homogeneity, i.e., independence from any reference frame.

3. Absoluteness as immutability, i.e., independence from any physical interaction.

The meaning (1) is what we have discussed till now and should be what Newton had sustained with the first two definitions of the Scholium (§§ I-II). One of the results that the English physicist probably wanted to obtain with that conception of absoluteness would be that the concept of absolute space and time does not derive from the one of relative space and time. This conclusion is connected to a general interpretation of the role of those notions in the wider framework of the Newtonian physics.

Now, the ontological characterization of space and time as abstract mathematical objects seems to correspond with an epistemological char- acterization of them as axioms. So, Dorato supposes – following [DiS02]

and [Ryn95] – that, with that account of absolute space and time, New- ton has no need to ground those notion with empirical arguments, e.g., with the famous bucket’s argument10_{. Moreover, contrary to DiSalle, who}

sustains that the meaning of absolute space and time is explained by the Newtonian laws of motion, Dorato insists on the axiomatic value of those notions. According to him, the definitions of space and time are presup- posed by the laws of motion, and not vice versa (see for details [ADLZ05], pp. 27-33).

Measure is the key-concept in order to understand these conclusions. Indeed, the whole story revolves around why a measure is required and what is required to have a measure. Measure, in fact, should be necessary to have an empirical verification of physical theories, and then also for the laws of motion. On the other hand, according to Newton we cannot measure – by definition – absolute space and time: what we measure is always relative space or relative time. So, one may conclude that we have no empirical proof of the existence of absolute space or time, or, rather, we cannot have in principle such a proof.

Nevertheless, another one (maybe a reader of Hartmann’s) can object that absolute space and time are not a possible object of measurement, but a condition of possibility of measure in general. Thus, with Dorato’s not- substantival interpretation of Newton’s concepts of absolute space and time, and with the help of Hartmann’s concept of category, it is possible to conclude that absolute space and time as mathematical notions, and pre- cisely as geometrical notions, are necessary to define – at least – their own measure, but also motion and its laws11_.

The opposition between Dorato and DiSalle is similar to the one be- tween the (strong) dynamical approach and the geometrical approach, which we shall discuss in the next chapters, talking about relativity and QG. Also in that case the axiomatic conception of spatiotemporal notions will return as fundamental. In an important paper, which I shall analyze in the next chapter, we can read something like that.

10_{A particular interpretation of this argument – partially in line with Dorato’s approach}

(i.e., the arguments presupposes that the geometrical notions are axiom for the dynamical laws) – is in [Har50], ch. 7 d.

11_{Dorato, more precisely, assumes – in agreement with DiSalle – that the first Newto-}

nian law of motion allow at most to make more accurate measurement of the temporal magnitude, and then to approximate to the ideal limit of the absolute time. However, one cannot do the same for space. For the second law of motion permits only to derive the difference of velocities (accelerations and rotations), neither the absolute velocity nor the absolute position. Against DiSalle, thus, Dorato concludes that the notions of absolute space and – symmetrically – of absolute time are not defined by the laws of motion, but presupposed by the latter (see [ADLZ05], pp. 32-34).

[T]he notion of velocity, for instance, presupposes a reference to prim- itive geometrical notions such as length (or distance) and time, so that the linkage between the equations of motion and the geometry is already made, by construction, at the level of the kinematics.

[HH13], p. 359

The concepts of motion and measure are also involved in the mean- ing (2) of absoluteness. In fact, according to Newton, absolute space and time are independent of any reference frame in two senses. First, they are thought of as metrically homogeneous: a spatial distance or a tempo- ral interval are the same in every reference frame, in the sense that their measures do not depend on the relative velocities of the reference frame. This homogeneity will become untenable in the contest of STR, in which two observers in different reference frames (with a different state of mo- tion) obtain different measurements of the same interval or distance. The reason, as well known, is that the velocity of light (in the void) is what is "absolute" or independent from the state of motion of any reference frame, or better invariant under (the Lorentz) transformations from one reference frame to another.

Second, absolute space can be considered as a sort of immobile ref- erence frame. Although this account has become the ground of the tra- ditional interpretation of the concept of absolute space as a substance (a container of all the material bodies) in Newton, it can appear weird if com- pared with the previous sense of space as abstract notion. But the account seems less unpalatable, if one considers Dorato’s identification of abstrac- tion with ideal limit, and the presence of another spatial notion among the definitions of the Scholium, i.e., the one of "place" (locus), from which the concept of motion derives.

III Locus est pars spatii quam corpus occupat, estque pro ratione spatii vel absolutus vel relativus. [. . .]

IV Motus absolutus est translatio corporis de loco absoluto in locum absolutum, relativus de relativo in relativum.

[New09], Def., Schol., § III-IV

Connecting these two definitions, and taking in consideration the con- cept of absolute simultaneity, surely present in Newton’s system, Earman derives the modern concept of reference frame, which can be associated to both absolute and relative space if interpreted as "instantaneous space",

i.e., a hyperplane of absolute simultaneity in the spacetime system deriv- able from Newton’s theory (see [Ear89], p. 9-10). Thus, the immobility of the absolute reference frame depends on the definition of the relative space, which has been defined as a measure of the absolute one, "or rather a certain mobile dimension, which is defined by our senses in accordance with the position of space to bodies" ([New62], Scholium, § II). If the lat- ter is mobile, being connected to bodies, which are generally subject to forces (as Dorato notes, Newton was aware that there could be no refer- ence frame at rest in the universe), the former is supposed to be immobile, being not connected to any bodies12_.

The concept of absolute simultaneity is rejected by STR, so even this sense of absoluteness for space (or slices of spacetime) is untenable in that context. I will not discuss whether it is possible to maintain this form of absoluteness, if space and time are interpreted as an abstractions or a conditions of possibility (Hartmann believed such a thing; see [Har50], ch. 18).

The General Theory of Relativity, on the other hand, rejected the no- tion (3) of absoluteness. For relativistic spacetime cannot be immutable. Its metrics changes because of its variable curvature, which depends on the distribution of energy and matter in the universe. As known, such an insight would be impossible for Newtonian physics, the latter having not the necessary geometrical basis (the general notion of not-Euclidean geometrical structures, and then, in particular, the concept of Riemannian manifold) even to imagine this conclusion.

We shall examine in the next chapters the relation between geometry and dynamical interactions (gravitation), and the sense of this relativity of spacetime with respect to matter/energy.

4.2.2.1.3 Substance = Field? For the moment it’s important to notice