Spacetime Is Material

Abstract

Space and time are central concepts for understanding our World. They are important ingredients at the core of every scientific theory and subject of intense debate in philosophy. Albert Einstein’s Special and General theories of Relativity showed that space and time blend in a single entity called spacetime. Even after a century of its conception, many questions about the nature of spacetime remain controversial. In this chapter, we analyze the ontological status of spacetime from a realistic and materialistic point of view. We start by outlining the well-known controversy between substantivalism and relationalism and the evolution of the debate with the appearance of General Relativity. We analyze how to interpret spacetime as a physical system and how to model its properties in a background-free theory where spacetime itself is dynamical. We discuss the concept of change, energy, and the ontology of spacetime events. In the last section, we review the mereology of spacetime and its relevance in cosmology.

Appendix: Dictionary of Technical Terms

In this section, we give some useful definitions regarding spacetime and spacetime symmetries to complement the article.

Spacetime model :

A spacetime model $M= (\mathcal {M},\mathbf {g})$ is a real, four-dimensional connected C ^∞ Hausdorff manifold with no boundaries, with a globally defined C ^∞-tensor field g of type (0,2), non-degenerate, and Lorentzian (Wald 1984).

Affine connection :

A covariant derivative in the direction of a vector W, ∇_W, is a derivative operator that transforms tensors into tensors. The operator is completely determined through its application to a basis of spacetime e _a, in the direction of the same basis

$$\displaystyle \begin{aligned} \nabla_a {\mathbf{e}}_b = \omega^{c}_{ab} {\mathbf{e}}_c. \end{aligned} $$

(5.9)

The quantity $\omega ^{c}_{ab}$ is called the affine connection of ∇.

Equation of motion :

Following Giulini (2007), we represent a general equation of motion (EOM) of a given physical theory by:

$$\displaystyle \begin{aligned} E[g,\gamma, \Phi; \Sigma] = 0, \end{aligned} $$

(5.10)

where E is some differential operator, g designates the spacetime metric, γ designates a set of particle models, Φ designates a set of physical fields, and Σ designates some geometrical and non-dynamical structures that must be given. The latter is usually referred to as the absolute structure of the theory and can be referential or non-referential. There is no general consensus about a formal definition of the absolute structures of a given theory. The dynamical fields $\left (g,\gamma , \Phi \right )$ are unknown and meant to be solved, given Σ.

Covariance :

An EOM is covariant under the subgroup G ⊆Diff(M) iff for all f ∈ G:

$$\displaystyle \begin{aligned} E[g,\gamma, \Phi; \Sigma] = 0 \Longleftrightarrow E[f \cdot g,f \cdot \gamma, f \cdot \Phi; f \cdot \Sigma] = 0. \end{aligned} $$

(5.11)

Remark

As stated by Giulini, covariance requires the equation to ‘live on the manifold’; in other words, to refer to well-defined geometrical objects with given transformation laws under f ∈ G. The equation remains valid after the action of f ∈ G, but it is different from the original, since the components of Σ are different.

Invariance :

An EOM is invariant under the subgroup G ⊆Diff(M) iff for all f ∈ G:

$$\displaystyle \begin{aligned} E[g,\gamma, \Phi; \Sigma] = 0 \Longleftrightarrow E[f \cdot g,f \cdot \gamma, f \cdot \Phi; \Sigma] = 0. \end{aligned} $$

(5.12)

Remark

An invariant diffeomorphism f ∈ G keeps the equation identical, since the components of Σ are unchanged. In this way, invariance is much more restrictive than covariance. The invariant group of a given equation is inherited by the invariance group of the absolute structures Σ. Since the transformed equation is identical to the original, from a solution of the dynamical fields we can obtain a whole set of solutions related by invariant diffeomorphisms.

General Covariance :

An EOM is general covariant if it is covariant under the group G = Diff(M).

Remark

The equations of any theory can be written, in principle, in a general covariant way (Rovelli 2004). This process is usually carried out to make the equations of a given theory compatible with GR, but this is not necessarily the case. In fact, a given covariant theory under a specific group of transformations can be made general covariant in many non-equivalent ways. The key assumption in GR “covariantization” is that we assume the metric is the only external geometrical object, and thus, the connection, introduced to define a meaningful notion of change, is the Levi-Civita connections. Adding, for instance, additional tetrad fields or other geometrical objects to construct a more general connection would render a Lorentz covariant equation general covariant but with the wrong limit in GR. Finally, let us define

General Invariance :

An EOM is general invariant if it is invariant under the group G = Diff(M).

Remark: This definition seems to imply that a theory with a general invariant EOM would be background independent. As we pointed out earlier, however, a precise definition of background independence is very subtle and is not equal to general invariance. We shall present the classical definition by Anderson (1967) that is not, however, free of problems (see also Friedman 1973):

Background Independence (Anderson):

A theory is background independent if their dynamical EOM are free of absolute structures.

Remark

From the previous definitions, it follows that the equations of motion of any background independent theory, written in a general covariant way, are general invariant as well. That is the case with GR. There are various constructs in GR, however, that can be considered absolute, e.g. dimensions and signature, volume elements, etc.