Ⅶ: Is Spacetime Discrete?

Pick up any physics textbook and you will find spacetime described as a smooth, continuous manifold. At every point, distances and times can take any real value. Between any two points in space, no matter how close, there is always another point. The geometry of the universe, in the picture established by Einstein and developed over a century of general relativity, is infinitely divisible.

This assumption has never been seriously questioned at the scales physics can reach. The smallest distances probed by particle accelerators are around 10⁻¹⁹ metres. General relativity describes those distances and everything larger without any detectable failure. The assumption of continuity works.

But sixteen orders of magnitude below the reach of any accelerator lies the Planck scale — 10⁻³⁵ metres — where, as Article I established, general relativity and quantum mechanics simultaneously break down. Whether spacetime remains smooth and continuous at that scale, or whether it has a granular structure, a smallest possible unit below which the concept of a distance loses meaning, is a question physics cannot currently answer by experiment. It can only approach it theoretically.

The three approaches examined in Group 2 give different answers. Loop quantum gravity derives discrete spectra for area and volume. Asymptotic safety suggests spacetime becomes effectively two-dimensional at the Planck scale. String theory treats spacetime as continuous at the classical level, though its dualities suggest a form of minimum length at the string scale. And a fourth approach, not yet discussed in this series, begins with discreteness as its founding assumption.

This article examines what spacetime discreteness would actually mean, why the question is harder than it first appears, what the leading theories say, what experiments can and cannot tell us, and why the answer matters for everything that follows in this series.

What Continuity Actually Assumes

Before asking whether spacetime is discrete, it is worth being precise about what continuous spacetime actually assumes.

In mathematics, a continuum is a set in which between any two distinct points there are infinitely many others. The real number line is a continuum. Between 0 and 1 there are infinitely many real numbers. Between 0 and 0.000001 there are infinitely many real numbers. This property of infinite divisibility holds at every scale without exception.

General relativity models spacetime as a four-dimensional continuum of this kind. Every event in the universe corresponds to a point in this continuum, specified by four real-valued coordinates. The metric, the mathematical object that encodes distances and times between events, is a smooth function defined at every point. There are no gaps, no minimum intervals, no smallest unit of space or time.

This is not a statement that has ever been confirmed experimentally at the Planck scale. It is an assumption that has been confirmed to work extremely well at every scale physics has so far probed, and that is extended by default to all smaller scales. Whether it continues to hold sixteen orders of magnitude below the current experimental frontier is precisely the question.

Quantum mechanics offers a reason to doubt that it does. Quantum mechanics introduces a fundamental minimum action, the Planck’s constant, and a fundamental minimum of energy for any oscillating system. It is natural to ask whether quantum gravity introduces a fundamental minimum of space. The heuristic argument for this was given in Article I: at the Planck length, a particle’s quantum uncertainty in position is comparable to its Schwarzschild radius. Localising a particle to a region smaller than the Planck length would require so much energy that the region would collapse into a black hole. The concept of a sub-Planckian position measurement is arguably incoherent, not merely difficult.

This heuristic does not prove discreteness. It suggests that the smooth continuum of general relativity cannot be the correct description of spacetime below the Planck scale. What replaces it is the question the rest of this article addresses.

Loop Quantum Gravity: Discreteness Derived

Article IV established the most mathematically precise argument for spacetime discreteness in the current literature. In loop quantum gravity, the geometric observables of area and volume have discrete spectra — they can only take specific values, separated by gaps, in the same way that the energy levels of a hydrogen atom can only take specific values.

This result was derived by Carlo Rovelli and Lee Smolin in 1994 and 1995 from the mathematical structure of the quantum theory. In loop quantum gravity, quantum states of spatial geometry are represented by spin networks — graphs whose edges carry quantum numbers related to the mathematical group SU(2). The area of any surface in space is determined by counting the contributions from each spin network edge that crosses it. Each edge contributes a minimum, discrete quantum of area on the order of the Planck area — the Planck length squared, approximately 10⁻⁷⁰ square metres. The area operator has a discrete spectrum: the area of any surface can only take specific values, and there is a smallest non-zero value it can take.

This is a precise statement. The area of a surface in loop quantum gravity is not a continuous variable that can take any real value. It is a quantum observable whose possible measured values form a discrete set, with a minimum spacing determined by the Planck scale.

The same holds for volume. Each node of a spin network contributes a discrete quantum of volume on the order of the Planck volume. Below this scale, the concept of spatial volume has no meaning within the theory.

What loop quantum gravity does not yet establish is whether this mathematical discreteness corresponds to the physical world. The theory has not proven its classical limit — it has not demonstrated that smooth four-dimensional spacetime correctly emerges from spin networks at large scales. Until that demonstration is achieved, the discrete spectra are predictions of a candidate theory rather than established features of nature. However, they are the most mathematically rigorous predictions of discrete spacetime geometry that currently exist.

Causal Set Theory: Discreteness Assumed

A separate approach to quantum gravity takes a fundamentally different starting point. Rather than deriving discreteness from a quantisation procedure, it assumes discreteness from the outset and asks what structure is consistent with that assumption alongside the established requirements of relativity.

Causal set theory was proposed by Luca Bombelli, Joohan Lee, David Meyer, and Rafael Sorkin in 1987. Its foundation is a theorem proved by David Malament in 1977, building on earlier work by Hawking, King, and McCarthy. The theorem states that the causal structure of a spacetime — the partial ordering of events by which can causally influence which — almost entirely determines the spacetime geometry. Given only the causal relations between events, you can reconstruct the full metric structure up to an overall scaling factor. The only additional information needed is how much spacetime there is in each region – the volume.

Sorkin and collaborators drew a radical conclusion from this theorem. If causal structure plus volume is all that is needed to reconstruct geometry, and if volume can be recovered by counting discrete elements, then continuous spacetime is not fundamental. It is a large-scale approximation to an underlying discrete causal order. A causal set is a collection of elementary events, partially ordered by a causal precedence relation, with the volume of any spacetime region proportional to the number of elements it contains. From order and number, geometry emerges. This is summarised in the theorem’s founding slogan: order plus number equals geometry.

The causal set approach makes one property central that is absent from most other discrete spacetime proposals: Lorentz invariance. Special relativity requires that the laws of physics look the same to all observers moving at constant velocity. Most proposals for discrete spacetime, such as a regular lattice of spacetime points, immediately violate this requirement because a lattice picks out preferred directions. A lattice that is aligned horizontally in one reference frame appears rotated in another. Lorentz invariance is broken.

Causal set theory avoids this problem by requiring that the elementary events be distributed randomly within any region they approximate, following a Poisson distribution. A Poisson distribution has no preferred directions. The discreteness is real and fundamental, but because the elements are sprinkled randomly rather than arranged in a regular pattern, no reference frame sees a preferred structure. Lorentz invariance is preserved in the continuum approximation. This is a non-trivial result, and it is one of the causal set theory’s most important theoretical achievements.

A Prediction Made Before the Observation

Causal set theory has produced one concrete quantitative prediction that preceded the experimental result it anticipates by nearly a decade.

In 1990, Rafael Sorkin argued that the most natural consequence of a discrete spacetime, in which the cosmological constant and the spacetime volume are complementary quantities, is that the cosmological constant should fluctuate around a value of order one in Planck units divided by the square of the size of the observable universe. In the Planck units that physicists use for quantum gravity, this corresponds to a value of approximately 10⁻¹²² — an extraordinarily small number.

The cosmological constant is the term in Einstein’s field equations associated with the energy density of empty space. Before 1998, most physicists assumed it was exactly zero. In 1998, observations of distant Type Ia supernovae used as standard candles to measure cosmic distances revealed that the expansion of the universe is accelerating. The acceleration required a positive cosmological constant. The measured value was approximately 10⁻¹²² in Planck units.

Sorkin’s prediction, made eight years before the observation and at a time when the cosmological constant was widely believed to be zero, anticipated the correct order of magnitude. This is arguably the most striking predictive success of any quantum gravity programme, and it is rarely discussed in proportion to its significance. The prediction does not rest on detailed dynamical calculations — it follows from order-of-magnitude arguments about the relationship between volume and the cosmological constant in a discrete spacetime. Its significance is therefore contested. But the coincidence between a prediction made in 1990 and an observation made in 1998 is not easy to dismiss.

The Dimensional Reduction Convergence

One of the most unexpected results in contemporary quantum gravity research is the convergence of several independent theoretical approaches on a specific prediction about the effective dimension of spacetime at the Planck scale.

The spectral dimension of a space is a measure of its effective dimension that can be defined without reference to coordinates. It is determined by asking how a diffusing particle spreads through the space over time. In ordinary flat four-dimensional space, the spreading follows a specific law that depends on the dimension. The spectral dimension is four.

In loop quantum gravity, calculations of the spectral dimension at Planck-scale distances give a value close to two. The same result has been found in asymptotic safety, as noted in Article VI, where the renormalisation group flow near the Reuter fixed point reduces the effective dimension from four to two. It has also been found in causal dynamical triangulations — a separate approach to quantum gravity that constructs spacetime by assembling four-dimensional triangular building blocks and summing over all possible assemblies.

Three different theoretical frameworks, making different fundamental assumptions and using different mathematical methods, all find that the effective dimension of spacetime approaches two at the Planck scale. This convergence is either a significant and genuine prediction about the structure of quantum spacetime, or a coincidence arising from some shared but unstated mathematical assumption. Determining which it is remains an open problem.

If the dimensional reduction is real, it has a direct physical implication. In two-dimensional gravity, the theory has no propagating degrees of freedom. It is dramatically simpler than four-dimensional gravity. If spacetime effectively becomes two-dimensional at the Planck scale, the ultraviolet divergences that plague four-dimensional quantum gravity may be naturally suppressed without requiring any new fundamental ingredients. Dimensional reduction at the Planck scale might be the physical mechanism that resolves the quantum gravity problem, independent of which specific theoretical framework turns out to be correct.

What Experiments Can Say

Spacetime discreteness at the Planck scale is not directly accessible to any existing or foreseeable experiment. The Planck length is sixteen orders of magnitude below the reach of the Large Hadron Collider. No accelerator of conventional design could probe it within any plausible technological future.

There are, however, indirect experimental tests that constrain specific predictions of discrete spacetime models.

The most developed test uses gamma-ray bursts — extremely energetic explosions at cosmological distances that emit photons across a wide range of energies simultaneously. If spacetime is discrete, high-energy and low-energy photons may travel at slightly different speeds, because the granular structure of spacetime at the Planck scale acts like a medium whose refractive index depends on the photon’s energy. Photons that have travelled across billions of light-years would accumulate this difference into a measurable time delay.

The Fermi Large Area Telescope has observed gamma-ray bursts with sufficient precision to test this prediction. Using observations of the gamma-ray burst GRB090510, which occurred at a redshift of 0.9 — corresponding to a light travel time of approximately eight billion years — the Fermi collaboration placed limits on energy-dependent photon speeds at the Planck scale. The result was a null detection: no energy-dependent time delay was found at the level of linear energy dependence. The data constrain the energy scale at which such effects would appear to be at least as large as the Planck energy, ruling out the simplest models in which the speed of light varies linearly with photon energy at the Planck scale.

NASA’s Fermi Gamma-ray Space Telescope, illustrated here, scans the entire sky every three hours as it orbits Earth.
Credit: NASA’s Goddard Space Flight Center/Chris Smith (USRA/GESTAR)

More recent observations of the extremely energetic gamma-ray burst GRB221009A, detected in 2022 and observed by the LHAASO collaboration in China, extended these constraints to photons of unprecedented energy — up to 12 TeV. The constraints on Lorentz invariance violation are now among the most stringent in physics.

Images taken in visible light by Swift’s Ultraviolet/Optical Telescope show how the afterglow of GRB 221009A (circled) faded over the course of about 10 hours. The explosion appeared in the constellation Sagitta and occurred about 1.9 billion years ago. The image is about 4 arcminutes across.
Credit: NASA/Swift/B. Cenko

NASA’s Neil Gehrels Swift Observatory detected X-rays from the initial flash of GRB 221009A for weeks as dust in our galaxy scattered the light back to us, shown here in arbitrary colors.
Credit: NASA/Swift/A. Beardmore (University of Leicester)

These null results are important but they do not rule out spacetime discreteness. Causal set theory and loop quantum gravity both predict that discreteness is compatible with Lorentz invariance, precisely because the elementary events are randomly distributed rather than arranged in a regular pattern. In these frameworks, energy-dependent photon speeds are either absent or suppressed to a level well below the current experimental sensitivity. The experimental constraints are consistent with the theoretical predictions of the leading discrete spacetime frameworks. They constrain naive models of discreteness — those that predict systematic violations of Lorentz symmetry — while leaving the more sophisticated frameworks unconstrained.

The Deeper Difficulty: What Does Discreteness Mean?

There is a conceptual issue at the heart of the question of spacetime discreteness that is rarely stated plainly in popular discussions.

In quantum mechanics, saying that the energy levels of a hydrogen atom are discrete does not mean that space is discrete. The atom moves through continuous space. Its energy is quantised, but its position is not. Similarly, when loop quantum gravity says that area and volume have discrete spectra, it does not automatically follow that space itself is discrete in the sense of being composed of distinct, countable, separated pieces.

What it means for spacetime to be discrete — as opposed to for geometric observables to have discrete spectra — is a subtle question that different approaches answer differently. In loop quantum gravity, the discreteness is in the spectrum of geometric operators. The underlying mathematical space of the theory is still a continuum. In causal set theory, the discreteness is more fundamental: the elementary events are the atoms of reality, and the continuum is entirely emergent. There is no underlying manifold.

This distinction matters. In loop quantum gravity, the question is whether any measurement could detect the gaps between discrete area eigenvalues. In causal set theory, the question is whether any observation could detect the granular structure of the causal set from which spacetime emerges. Both questions are currently unanswerable by experiment, and answering one would not answer the other.

Where the Question Stands

Whether spacetime is fundamentally discrete or continuous is not a question that physics currently has the tools to answer definitively. No experiment has probed the Planck scale. No theory of quantum gravity has been confirmed. The question is open in the deepest sense: not merely unresolved, but currently unresolvable by available methods.

What the theoretical evidence suggests is that the smooth continuum of general relativity is unlikely to be the correct description of spacetime at the Planck scale. Three independent theoretical frameworks find that spacetime becomes effectively two-dimensional at that scale. The heuristic argument from quantum mechanics and black hole physics suggests that sub-Planckian distances are physically meaningless. Loop quantum gravity derives discrete spectra for geometric observables from first principles. Causal set theory predicted the cosmological constant’s order of magnitude before it was measured.

None of this constitutes proof. The convergence of independent lines of theoretical reasoning on a picture in which the smooth spacetime of general relativity is a large-scale approximation to something more fundamental is, however, a result that each of these frameworks must individually account for.

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