From time to spacetime in relativity

Block view, as well as some eliminativist conception of time, has been historically connected to Einstein’s relativity and the diffused relational- ist interpretation of its content (based on the fact that in STR no reference frame in absolute rest is postulated and every velocity is relative to a ref- erence frame, except the velocity of light, of course). In this context, a first stage in the direction of a reduction of time to other concepts has been cer- tainly the geometric formulation of STR by Minkowski (see [Min08]), in which time appeared as the fourth dimension of a four-dimensional sys- tem, namely, spacetime. So begins (in a classical english translation) the famous Minkowski’s Raum und Zeit1:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

[Min52], p. 75

In STR time and space lose their mutual independence, in order to maintain the relativity principle, i.e., that every physical law must be in- variant in every inertial frame. And the law in question is the invariance of the value of the velocity of light (in the void) c. Indeed, this invariance in different reference frames with different relative velocities is guaranteed by Lorentz transformations, in which, unlike in Galilean transformations, the time coordinate t is not independent of the spatial position and the condition of motion of the coordinate system. Thus, even the measures of temporal intervals, but also of spatial distances, must vary from a ref- erence frame to another, the effect being the so-called time dilatation and segment contraction. In this sense, time and space lose their absoluteness as metrical homogeneity (see ch. 4 above)2_.

Despite this differences from the Euclidean space, used to represent the Newtonian Mechanics, however, Minkowski spacetime remains for STR a background, i.e., its dimensions are parameters according to which the basic elements of the theory (events) receive coordinates. As such, the

1_{In this second formulation Minkowski used the time coordinate without explicit ref-}

erence to its imaginary coefficient.

2_{In Philosophie der Natur Hartmann dedicates a chapter (the 18th) to the philosophi-}

cal problems of (special and general) relativity. He exposes his point of view about the problem at the base of STR in [Har50], ch. 18 a.

dimensions cannot be modified by the dynamical features of the events. In this sense of immutability Minkowski spacetime is still absolute.

While its causal structure has changed, spacetime is still absolute in the sense of being non-dynamical: it acts on matter and fields, but is itself not acted upon; it is only an arena for the dynamics.

[Kie11], p. 663

As well known, this sense of absoluteness is abandoned with the ad- vent of GTR. In this context the curvature (and then the metrical struc- ture) of spacetime regions varies in relation to the distribution, density and pressure of matter/energy (and momentum) in those regions. As synthe- sized by Wheeler in a famous motto, "Spacetime tells matter how to move; matter tells spacetime how to curve" ([FW98], p. 235).

Such an account of that relation, from which no hierarchy emerges be- tween the geometrical side (spacetime or, better, the pseudo-Riemannian manifold with its metric tensor) and the dynamical side (gravitational field or, rather, the stress-energy tensor), is difficult to find in the GTR theorists, and even more in recent theorist3_{. Indeed, if one goes on reading the above}

quoted passage by Kiefer, who is a QG theorist, one finds already a specific interpretation of that relation.

Spacetime becomes dynamical only with the advent of general rel- ativity in 1915. Its geometry is a manifestation of the gravitational field, and gravity is dynamical. There is now a complicated non- linear interaction between gravity and non gravitational fields, as en- coded in the Einstein field equations. The running of a clock depends on its position in spacetime, and the clock acts back on spacetime due to its mass. This back action is very natural because ’it is contrary to the scientific mode of understanding to postulate a thing that acts, but which cannot be acted upon’ [[Ein22], ch. 3].

[Kie11], p. 664

All the attention is on gravity and its "interaction" with the "non gra- vitational fields". This interaction is felt as the actually urgent problem in

3_{Wheeler himself was not always so "equilibrate". Indeed, he thought – and affirmed}

– that at the Planck scale "ordinary ideas of length would disappear. Ordinary ideas of time would evaporate" ([FW98], p. 247). Moreover, the famous equation that is generally used to "demonstrate" the absence of time at the high energy limit takes its name (in part) from him: the Wheeler-DeWitt equation.

modern physics, while the more philosophical question of the relation be- tween gravity and spacetime geometry seems to be solved affirming that the latter is a "manifestation of the gravitational field". The problem with this way of talking is that saying "manifestation" looks like saying "effect", namely, that something (spacetime) is, as it were, "produced" by some- thing else (gravitational field). Even Wheeler’s motto, on the other hand, seems to talk of a (mutual) causation.

Wheeler’s formulation could mislead. It should not be taken to mean that mass causes spacetime to curve. Einstein’s equations do no more than posit a systematic relationship between spacetime curvature and distribution of matter: certain distributions of matter can only ex- ist with certain spacetime curvatures. There is no "one causing the other" here (or at least, the theory is agnostic on this issue); there is simply a constraint on the way physically possible worlds that con- form to GTR might be.

[Dai10], p. 353

Of course, the question for the theorist that propose such an interpre- tation of GTR is not – explicitly at least – about causation, but the idea of derivability, even in the sense of conceptual derivability, remains. Thus, in GTR and, as we shall see soon, beyond GTR, if one thinks that time, or rather, spacetime is somehow derivable from anything else, then one can also think that time or spacetime is not always or not everywhere present. And the problem is just here, in talking about moments in which time does not exist, or about moments and places in which spacetime does not exist. This is the problem of time and of spacetime. Now we shall examine the latter in the context of QG, while in the next chapter we shall consider the problem of time between GTR and QG.