Eternalism and the Problem of Continuous Space-Time

Abstract

Special and general relativity motivate the “block-universe’’ (eternalist) ontology: all space-time events are equally real, and “past’’ or “future’’ simply index slices through a four-dimensional manifold in which the notion of global simultaneity is ill-defined. Yet the block picture implicitly relies on the individuation of events—an assumption that becomes problematic if space-time is a continuous field with no fundamental grain. We review the mathematical and philosophical obstacles that an infinitely differentiable space-time poses for eternalism, contrast them with prospects in discrete approaches such as causal-set theory, and argue that the very smoothness that underwrites relativity undermines the metaphysical coherence of a fixed, completed block.

Relativity, Simultaneity and Eternalism

Minkowski’s fusion of space and time dissolves absolute simultaneity; observers connected by Lorentz transformations disagree on which distant events are “now.” Eternalism takes this relativity of simultaneity as evidence that all events—past, present, future—must coexist in a single geometric entity.(medium.com) General relativity extends the idea to curved manifolds, apparently reinforcing the block. At first glance, continuous space-time seems tailor-made for such a view: the manifold is a four-dimensional continuum MM with metric gabg_{ab}.

The Individuation Problem in a Continuum

A smooth manifold contains an uncountable infinity of points. But points carry no physical identity—they can be “dragged’’ by active diffeomorphisms without changing observable relations. This is the core of Einstein’s hole argument: if one treats manifold points as ontic, general covariance yields indeterminism; the remedy is to declare only relations (field coincidences) physical.(plato.stanford.edu, plato.stanford.edu) A fixed block requires well-defined events, yet general covariance tells us that the manifold has surplus structure, so the “events’’ populating the block lack intrinsic labels. The universe, on this reading, is more like a web of evolving field correlations than a completed catalogue of point-events.

Infinite Degrees of Freedom and Ontological Overload

A continuum contains infinitely many independent modes. Quantum field theory on such a background must tame ultraviolet divergences via renormalisation; gravitationally, it faces the non-renormalisability of perturbative quantum gravity. The need for cut-offs or coarse-graining suggests that the smooth manifold is an effective, not fundamental, description. If the manifold is merely emergent, the block that rests on it is emergent – and therefore provisional rather than metaphysically fundamental.

Discrete Alternatives as a Rescue Attempt

Discrete programmes—causal-set theory, spin foams, loop-quantum gravity—replace the continuum with fundamentally countable structures. In causal-set theory, the basic elements are elementary “events’’ ordered by causality; the manifold is recovered only statistically at large scales.(link.springer.com, link.springer.com) Such atomism restores unambiguous event identity and can in principle support a block-like ontology, albeit one that is locally finite rather than continuous. Philosophers sympathetic to eternalism increasingly look to these discrete frameworks to secure the reality of all events.(link.springer.com)

Why Smoothness Undermines the Block

Gauge redundancy: Without fundamental labels, continuum points are not physical entities; only relational structures are. A block that treats points as occupants of “places’’ smuggles in non-physical structure.
Indeterminacy of global structure: In a manifold lacking a preferred foliation, no unique slicing into “times’’ exists. The block collapses into an equivalence class of diffeomorphically related pictures, none privileged.
Openness under quantum dynamics: If quantum evolution is unitary yet indeterministic relative to classical outcomes, the future cannot be fixed within the manifold without specifying how superpositions obtain definite values. Collapse mechanisms (if real) are stochastic; path-integral sums weigh not just one classical history but many. A continuous space-time forces us to confront this “open future’’ at every scale.
Holographic hints: Entropy bounds in quantum gravity count degrees of freedom by area, not volume, implying that continuum event-counting over-states reality. Again, discreteness—or at least a finite state count—seems fundamental.

Breaking Diffeomorphism Invariance?

One might try to explicitly break general covariance to recover preferred coordinates that individuate events. But even tiny violations of diffeomorphism invariance produce observable deviations in cosmology and solar-system tests. Recent work shows that such models survive only at the price of fine-tuning and introduce new instabilities.(arxiv.org) The cure looks worse than the disease.

Discussion: From “Block’’ to “Becoming’’

If physical reality is given only by field coincidences and if those coincidences themselves require coarse-graining over fundamentally indeterminate quantum degrees of freedom, then the block universe loses its explanatory traction. Process- or “evolving block’’ views, where the space-time set grows as new relational facts come into existence, gain ground. Interestingly, discrete approaches can marry such becoming with Lorentz-invariant growth laws, sidestepping the simultaneity objection without presupposing a completed continuum.(philarchive.org)

Conclusion

Relativity removes absolute simultaneity, inviting the block-universe metaphor. But the same theory’s commitment to a smooth, label-less manifold undercuts the metaphysical pillars on which eternalism rests. A truly continuous space-time offers no primitive events to “fill’’ the block and no finite pattern of physical facts to fix its content. Unless fundamental discreteness (or an equivalent finite structure) is built in, the block universe is arguably an artefact of idealised mathematics rather than an ontologically robust picture of the cosmos.

References

Key discussions of eternalism and relativity: (medium.com)
General covariance and the hole argument: (plato.stanford.edu, plato.stanford.edu)
Discreteness in causal-set theory: (link.springer.com, link.springer.com)
Philosophical appraisal of discrete blocks: (link.springer.com)
Phenomenology of diffeomorphism-breaking models: (arxiv.org)

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