On the Ontology of Spacetime: Substantivalism, Relationism, Eternalism, and Emergence

For there neither is nor will be anything else besides what is, since Fate has fettered it to be whole and changeless. Parmenides (Fragment 8. From the translation in Kirk et al. 1983 ).

Abstract

I present a discussion of some issues in the ontology of spacetime. After a characterisation of the controversies among relationists, substantivalists, eternalists, and presentists, I offer a new argument for rejecting presentism, the doctrine that only present objects exist. Then, I outline and defend a form of spacetime realism that I call event substantivalism. I propose an ontological theory for the emergence of spacetime from more basic entities (timeless and spaceless ‘events’). Finally, I argue that a relational theory of pre-geometric entities can give rise to substantival spacetime in such a way that relationism and substantivalism are not necessarily opposed positions, but rather complementary. In an appendix I give axiomatic formulations of my ontological views.

Appendix: Axiomatics

1.1 Basic Definitions

In this appendix I give some basic definitions used in the two axiomatisations that follow.

The Einstein tensor is:

$$\begin{aligned} G_{ab}\equiv R_{ab}-\frac{1}{2}R g_{ab}, \end{aligned}$$

(1)

where $R_{ab}$ is the Ricci tensor formed from second derivatives of the metric and $R\equiv g^{ab}R_{ab}$ is the Ricci scalar. The geodetic equations for a test particle free in the gravitational field are:

$$\begin{aligned} \frac{d^{2}x^{a}}{d\lambda ^{2}}+ \varGamma ^{a}_{bc}\frac{dx^{b}}{d\lambda }\frac{dx^{c}}{d\lambda }=0, \end{aligned}$$

(2)

with $\lambda$ an affine parameter and $\varGamma ^{a}_{bc}$ the affine connection, given by:

$$\begin{aligned} \varGamma ^{a}_{bc}= \frac{1}{2}g^{ad}(\partial _{b}g_{cd} +\partial _{c}g_{bd}-\partial _{d}g_{bc}). \end{aligned}$$

(3)

The affine connection is not a tensor, but can be used to build a tensor that is directly associated with the curvature of spacetime: the Riemann tensor. The form of the Riemann tensor for an affine-connected manifold can be obtained through a coordinate transformation ${x^{a}\rightarrow {\bar{x}^{a}}}$ that makes the affine connection vanish everywhere, i.e.

$$\begin{aligned} \bar{\varGamma }^{a}_{bc}(\bar{x})=0, \;\;\; \forall \; \bar{x},\; a,\;b,\; c. \end{aligned}$$

(4)

The coordinate system ${\bar{x}^{a}}$ exists if

$$\begin{aligned} \varGamma ^{a}_{bd, c}-\varGamma ^{a}_{bc, d} + \varGamma ^{a}_{ec}\,\varGamma ^{e}_{bd} - \varGamma ^{a}_{de}\,\varGamma ^{e}_{bc}=0 \end{aligned}$$

(5)

for the affine connection $\varGamma ^{a}_{bc}({x})$. The left hand side of Eq. (5) is the Riemann tensor:

$$\begin{aligned} R^{a}_{bcd}=\varGamma ^{a}_{bd, c}-\varGamma ^{a}_{bc, d} + \varGamma ^{a}_{ec}\,\varGamma ^{e}_{bd} - \varGamma ^{a}_{de}\,\varGamma ^{e}_{bc}. \end{aligned}$$

(6)

When $R^{a}_{bcd}=0$ the metric is flat, since its derivatives are zero. If

$$\begin{aligned} K=R^{a}_{bcd}R^{bcd}_{a}>0 \end{aligned}$$

the metric has a positive curvature. Sometimes it is said, incorrectly, that the Riemann tensor represents the gravitational field, since it only vanishes in the absence of fields. On the contrary, the affine connection can be set locally to zero by a transformation of coordinates. This fact, however, only reflects the equivalence principle: the gravitational effects can be suppressed in any locally free falling system. In other words, the tangent space to the manifold that represents spacetime is always Minkowskian.

1.2 Axiomatic Ontology of Spacetime

The basic assumption of the ontological theory of spacetime I propose is:

Spacetime is the emergent system of the ontological composition of all events.

Events can be considered as primitives. They are characterised by the axiomatic formulation of the theory. Since composition is not a formal operation but an ontological one, spacetime is neither a concept nor an abstraction, but an emergent entity. What I present here is, then, a substantival⁸ ontological theory of spacetime. As any entity, spacetime can be represented by a concept. The usual representation of spacetime is given by a 4-dimensional real manifold E equipped with a metric field $g_{ab}$:

$$ {\text{ST}}\hat{ = }\left\langle {E,g_{{{\text{ab}}}} } \right\rangle . $$

I insist: spacetime is not a manifold (i.e. a mathematical construct) but the “totality” of all events. A specific model of spacetime requires the specification of the source of the metric field. This is done through another field, called the “energy-momentum” tensor field $T_{ab}$. Hence, a model of spacetime is:

$$\begin{aligned} M_\mathrm{ST}=\left\langle E, g_{ab}, T_{ab}\right\rangle . \end{aligned}$$

The relation between both tensor fields is given by the field equations. The metric field specifies the geometry of spacetime. The energy-momentum field represents the potential of change (i.e. event generation and density) in spacetime.

We can summarise all this through the following axioms. The axioms are divided into syntactic, if they refer to purely formal relations, ontological, if they refer to ontic objects, and semantic, if they refer to the relations of formal concepts with ontological ones. There are no physical axioms at this level.

The basis of primitive symbols⁹ of the theory is:

$$\begin{aligned} B_\mathrm{Ont} =\left\langle {\mathcal {E}}, \; E, \; \left\{ \mathbf{g}\right\} , \;\left\{ \mathbf{T} \right\} , \;\left\{ \mathbf{f} \right\} , \;\varLambda ,\;\kappa \right\rangle . \end{aligned}$$

• P1—Ontological/semantic $\mathcal {E}$ is the collection of all events. Every member e of $\mathcal {E}$ denotes an event.

• P2—Syntactic E is a $C^{\infty }$ differentiable, 4-dimensional, real pseudo-Riemannian manifold.

• P3—Syntactic The metric structure of E is given by a tensor field of rank 2, $g_{ab}$, in such a way that the differential 4-dimensional distance ds between two events is:

  $$\begin{aligned} ds^{2}=g_{ab} dx^{a} dx^{b}. \end{aligned}$$

• P4—Syntactic The tangent space of E at any point is Minkowskian, i.e. its metric is given by a symmetric tensor $\eta _{ab}$ of rank 2 and trace $-2$,

  $$\begin{aligned} \eta _{ab} = \left( \begin{array}{llll} 1 &{}0&{}0&{}0\\ 0&{}-1&{}0&{}0\\ 0&{}0&{}-1&{}0 \\ 0&{}0&{}0&{}-1\\ \end{array}\right) . \end{aligned}$$

• P5—Syntactic The symmetry group of E is the set of all 4-dimensional transformations $\left\{ \mathbf{f} \right\}$ among tangent spaces.

• P6—Syntactic E is also equipped with a set of second rank tensor fields $\left\{ \mathbf{T} \right\}$.

• P7—Semantic The elements of $\left\{ \mathbf{T} \right\}$ represent a measure of the clustering of events.

• P8—Ontological: inner structure The metric of E is determined by a rank 2 tensor field $T_{ab}$ through the following equations:

  $$\begin{aligned} \mathbf{G}-\mathbf{g} \varLambda =\kappa \mathbf{T}, \end{aligned}$$

  (7)

  or

  $$\begin{aligned} G_{ab}-g_{ab}\varLambda =\kappa T_{ab}, \end{aligned}$$

  (8)

  where $G_{ab}$ is the Einstein tensor. Both $\varLambda$ and $\kappa$ are constants.

• P9 —Semantic The elements of E represent physical events.

• P10—Semantic Spacetime is represented by an ordered pair $\left\langle E, \; g_{ab}\right\rangle$:

  $$\begin{aligned} \mathrm{ST}\hat{=}\left\langle E, g_{ab}\right\rangle . \end{aligned}$$

• P11—Semantic A specific model of spacetime is given by:

  $$\begin{aligned} M_{\mathrm{ST}}=\left\langle E, g_{ab}, T_{ab}\right\rangle . \end{aligned}$$

This theory characterises an entity that emerges from the composition of basic, timeless and spaceless events (see below). On the basis of this theory we can formulate a physical theory about how this entity, spacetime, interacts with other systems and the corresponding dynamical laws. Such a theory is General Relativity. The axioms we should add to obtain General Relativity form our ontological theory are:

• A.1—Semantic The tensor field $\mathbf{T}$ represents the energy, momentum, and stress of any physical field defined on E.

• A.2—Physical $\varLambda$ is a constant that represents the energy density of spacetime in the absence of non-gravitational fields. The constant $\kappa$ represents the coupling of the gravitational field with the non-gravitational systems.

• A.3—Semantic $k=-8\pi G c^{-4}$, with G the gravitational constant and c the speed of light in vacuum.

From $\bigwedge _{i=1}^{11} \mathbf{P_i} \; \wedge \; \bigwedge _{i=1}^3 \mathbf{A_i}$, all standard theorems of General Relativity follow (see Bunge 1967; Covarrubias 1993; Romero 2014).

1.3 Towards an Axiomatic Pre-geometry of Spacetime

The ontological, substantival theory of spacetime outlined above characterise an entity, spacetime, that is formed by events. If events are the basic constituents of spacetime, a constructive theory of spacetime can be proposed. In such a theory, spacetime emerges from timeless and spaceless events whereas metric properties and the internal spacetime structure are the result of the transition to large numbers of events that allows to adopt a continuum description. The development of a theory of this class is the major goal of several approaches to quantum gravity. In what follows, I outline a minimum axiomatic system that might be useful as a guiding framework for such an enterprise (see Perez-Bergliaffa et al. (1998) for an alternative relational approach).

The basis of primitive symbols of the theory is:

$$\begin{aligned} B_\mathrm{Pre-Geom} =\left\langle {\mathcal {E}}_\mathrm{B}, \; E_\mathrm{B}, \; P, \; \preceq , \; {\mathcal {W}}, \; l_\mathrm{P}, \; \circ \right\rangle . \end{aligned}$$

Tentative axiomatic basis:

• O1—Ontological/semantic ${\mathcal {E}}_\mathrm{B}$ is the collection of basic events. Every x in ${\mathcal {E}}_\mathrm{B}$ denotes an event.

• O2—Syntactic/semantic There is a set ${E}_\mathrm{B}$ such that every $e \in {E}_\mathrm{B}$ denotes a basic event of ${\mathcal {E}}_\mathrm{B}$.

• O3—Syntactic There is a binary operation $\circ$ from $E_\mathrm{B} \times E_\mathrm{B}$ into a set $E^*$ that composes basic events into complex events (Def. Complex events: processes).

• O4—Syntactic There exists a partially ordered set $P\subset {E}_\mathrm{B}$ (poset) endowed with the ordering relation $\preceq$.

• O5—Syntactic The partial order binary relation $\preceq$ is:

  • Reflexive: For all $x \in P$, $x \preceq x$.

  • Antisymmetric: For all $x,\; y\; \in P$, $x \preceq y \preceq x$ implies $x = y$.

  • Transitive: For all $x,\; y,\; z \in P$, $x \preceq y \preceq z$ implies $x \preceq z$.

  • Locally finite: For all $x,\; z \in P$, card $(\{y \in C | x \preceq y \preceq z\}) < \infty$.

  Here card (A) denotes the cardinality of the set A. Notice that $x \prec y$ if $x \preceq y$ and $x \ne y$.

• O6—Ontological. The elements of $\mathcal {E}_\mathrm{B}$ have an extensive property called energy ${\mathcal {W}}(x): {E}_\mathrm{B} \rightarrow \mathfrak {R}$. The larger ${\mathcal {W}}(x)$, the more numerous are the events that can be linked to x by $\preceq$ in $E^*$.

• O7—Ontological. If Comp$(e)=\{e_1, e_2, \ldots, e_n\}$ then ${\mathcal {W}}(e)={\mathcal {W}}(e_1)+{\mathcal {W}}(e_2)+\cdots+{\mathcal {W}}(e_n)$, where all $e_i$ are basic events.

• O8—Ontological If $E' \subset E_\mathrm{B}$ has n elements, then

  $$\begin{aligned} {\mathcal {W}}(E') =\varSigma _{i=1}^n {\mathcal {W}}(e_i),\;\;\; e_i \in E' \end{aligned}$$

• O9—Syntactic $E_\mathrm{B}$ is embedded in $E^*$ in such a way that $E^*$ preserves the internal structure of $E_\mathrm{B}$ given by the relation of precedence.

• O10—Syntactic The set $E^*$ has a (pseudo) metric structure.

• O11—Syntactic $E^*$ can be extended into a continuous, differentiable pseudo-Riemaniann 4-dimensional manifold E.

• O12—Ontological The energy density is $\rho ={\mathcal {W}}(E')/V$, where V is the volume of a region $E'$ in E. This energy density forms a component of a tensor field on E that is related to the curvature of E by Einstein field equations.

From $\mathbf{O11}$, the continuum approximation is valid in the large number limit of basic events and allows to match the pre-geometric structure with the ontological one. To prove $\mathbf{O11}$ as a theorem from more basic axioms is a major problem of the causal set approach to quantum gravity. I hope to discuss this issue elsewhere.