The Standard of Motion

The first law carries the geometry

Heller is a cosmologist and philosopher in Kraków. In a short paper, “Evolution of Space-Time Structures” (Concepts of Physics 3, 2006, 117–131), he reconstructs the space-times of Aristotle, Newton and Einstein in the language of modern differential geometry, and finds a pattern in the sequence. His starting point is that a dynamics must first say how to tell whether a force is acting. It does this by naming the motions that need no force:

This can be thought of as a “standard of motion”. A deviation from a free motion witnesses to the appearance of a force. To establish such a standard of motion is a task of the first law of dynamics.

Heller adds that in many theories the free motions are set by the geometry of space-time. Then the first law carries information about that geometry, and “suitably interpreted geometry becomes a part of physics, and as such it can be subject to empirical verification.”

Aristotle. Heller states Aristotle’s first law in Newton’s form: “A body, non acted upon by any force, remains in the absolute rest.” Absolute rest lets one speak of the same place at different times, since a body at rest stays in its place. That gives a projection from space-time onto absolute space, which sends each event to its place. Absolute simultaneity, which Aristotle also assumed, gives a projection onto absolute time. With both projections, space-time is a Cartesian product, A = T\times S: an event is a pair (when, where). Aristotle’s second law, in Heller’s stylization, is F = m\cdot v: force is proportional to the velocity it produces. A body with no force on it therefore has zero velocity relative to absolute space: it is at rest. Heller stresses that this is a reconstruction. Aristotle spoke of a “moving factor” and a “body”, not of force and mass. The operational concepts took a long evolution of science to arrive.

Newton. Newton’s first law says that a body with no force on it stays in uniform motion. That makes rest relative. Each inertial observer means something different by being at rest. The law, in Heller’s words, “introduces a full democracy among all inertial reference frames.” The projection onto absolute space is gone, replaced by many private projections, one per inertial observer. The projection onto absolute time survives, because Newtonian mechanics allows signals of unlimited speed, which can synchronize distant clocks. So Newtonian space-time N is no longer a product. Heller notes that Newton himself kept absolute space, and that he “misinterpreted his own first law of dynamics as implying the existence of absolute space.”

The second law becomes F = m\cdot a. The change from velocity to acceleration looks small and is decisive: “Now, it is acceleration that is absolute.” Whether two distant bodies have parallel velocities is then a fact, the same in every inertial frame. Newtonian space-time therefore has absolute parallelism at a distance, which Heller calls teleparallelism. Geometry expresses it through the frame bundle: the space of all local reference frames at all events, together with the group that takes one frame to another, here the group G of rotations and Galilean boosts. Teleparallelism says this bundle is a product, F(N) = N\times G. The product structure lost from space-time has reappeared one level up, in the space of frames.

Newton with gravity. All bodies fall with the same acceleration, so freely falling frames can be treated as locally unaccelerated. But free-fall paths converge: bodies dropped around the Earth meet at its center. Parallel transport then depends on the path. Teleparallelism is lost: no field of frames is parallel everywhere. The frame bundle acquires curvature. This is Newtonian gravity as a curvature, which Élie Cartan worked out in 1923–24 and Andrzej Trautman developed in the 1960s. What curves is a connection, not a metric, because Newtonian space-time has no space-time metric to curve. Absolute time survives.

Einstein. Special relativity removes both projections: there is neither absolute space nor absolute time. Minkowski space-time is flat, so it has teleparallelism, and its frame bundle under the Lorentz group is again a product. General relativity curves space-time, and teleparallelism is lost again. Heller then observes that the product reappears one level higher still. A connection on space-time, curved or not, supplies a global field of frames on the frame bundle itself, so the frame bundle of the frame bundle is a product. Heller notes this for general relativity. The same holds for Newton with gravity, whose free fall also defines a connection. The mathematics page opens with a short primer on bundles and connections, then works this construction out, along with the rest of the ladder, and says in what sense “product” is meant.

Heller reads the sequence, which his figures draw as a stack of boxes, as “a logic inherent in the evolutionary chain of physical space-time theories.” My own reading of the pattern is this. Each new first law makes relative something that was absolute. The product that reappears higher up marks what stays absolute after the step: acceleration for Newton, and the free-fall connection once gravity enters, in Newton–Cartan theory and in general relativity alike.

Aristotle’s rung is not a mistake to be outgrown. Carlo Rovelli has argued that Aristotle’s physics is “a correct and non-intuitive approximation of Newtonian physics in the suitable domain (motion in fluids)” (Rovelli 2015). A body falling through air or water reaches a steady speed set by its weight and the medium’s resistance, and Aristotle’s claims about such motion hold up. In Rovelli’s words, “What Aristotle does not have is only the square root.” His first law was a true report about the world he could observe, where friction stops everything. The sequel, The Demon Is a Learner, returns to that world, because learning machines live in it.

One bracket, three mechanics

A seminar Piotr Biler ran around 2003 opened with the qualitative theory of differential equations and then put classical, statistical and quantum mechanics into one Hamiltonian framework. The qualitative theory returns in the next section. Here is the framework, in the form I need.

A classical system lives in a phase space of positions x and momenta p. Observables are functions on it, and any two of them have a Poisson bracket, \{f, g\} = \sum_i \left(\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial x_i}\right). The bracket makes the observables a Lie algebra: it is antisymmetric, bilinear, and satisfies the Jacobi identity, \{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0. A single function H, the Hamiltonian, generates time evolution through the bracket.

In the usual cases H is the system’s total energy, written in terms of positions and momenta. Energy can be pictured as stored capacity for work, but the picture that fits this essay is another. Energy is what the laws conserve because they do not change with time, as momentum is what they conserve because they are the same at every place. Momentum has a reading of its own. Written as a constrained problem, with velocity a separate variable, the Lagrangian needs a multiplier to enforce “velocity is the rate of change of position.” That multiplier is the momentum: the price of the constraint, in the sense of How Does a Policy Improve?. The Hamiltonian is what remains of the Lagrangian once the price is paid. The pairing of each position with its price is where the bracket comes from. The mathematics page defines the Hamiltonian, derives Hamilton’s equations from the bracket, and proves the statements of this section.

The three mechanics differ in what H moves.

The quantum row replaces the Poisson bracket by the commutator divided by i\hbar. This replacement can be made precise as a deformation: on phase-space functions, the Moyal bracket equals the Poisson bracket plus corrections of order \hbar^2, and it encodes the commutator exactly. Any density that is a function of H alone is stationary, because H commutes with itself. The Gibbs state e^{-\beta H}/Z of statistical mechanics is the example that matters.

The framework names a freedom of its own. Hamiltonian flow preserves phase-space volume. This is Liouville’s theorem. A cloud of initial conditions can be stretched and folded but never compressed, so the entropy -\int\rho\ln\rho of an evolving density stays constant. Unitary quantum evolution likewise preserves the von Neumann entropy. So closed evolution loses no information. The first law said which motions need no force; Liouville’s theorem says that evolution which needs nothing from outside forgets nothing. Forgetting needs something outside the system, and what it costs is the subject of The Demon Is a Learner.

General relativity strains this framework. For a closed universe its Hamiltonian is a constraint, H\approx 0. The generator of time evolution then generates nothing observable: there is no external time for evolution to happen in. This is the problem of time of canonical quantum gravity (Isham 1993). For an isolated system in asymptotically flat space-time, the Hamiltonian is the total energy measured far away, and evolution goes on as usual.

The algebra of free motions

The bracket also carries the symmetries of a first law. A first law picks out the free motions. Its symmetry group is the set of transformations that carry free motions to free motions while keeping clocks and rulers: rotations, translations in space and time, and changes of velocity, called boosts. Each has a generator, and the generators form a Lie algebra: rotations J_i, spatial translations P_i, time translation H, boosts K_i. A generator is a motion taken in an instant. The Lie algebra records how two such motions fail to commute: a small turn about one axis, then about a second, then both undone, leaves a small turn about the third. The mathematics page starts from the simplest case, the group of time translations.

Heller’s ladder shows up in this algebra. Aristotle’s first law is preserved only by transformations that keep the rest frame. Its algebra has rotations and translations and no boosts at all. José Figueroa-O’Farrill and Stefan Prohazka call homogeneous space-times of this kind Aristotelian in their classification of spatially isotropic homogeneous space-times (2019). In their Aristotelian space-times, as in Heller’s, rest is absolute. Newton’s first law adds the boosts, so the step from Aristotle to Newton enlarges the symmetry. The steps after Newton deform it.

The bracket shows the deformation directly. For a free particle, the boost generator in special relativity is K_i = (H/c^2)\,x_i - t\,p_i, where H = \sqrt{p^2c^2 + m^2c^4} is its energy. Bracketing a boost with a translation gives \{K_i, p_j\} = \frac{H}{c^2}\,\delta_{ij}. Two boosts bracket to a rotation, \{K_i, K_j\} = -\varepsilon_{ijk}L_k/c^2, where L is the angular momentum. Now let c\to\infty. The boosts commute with each other and the Poincaré algebra goes over to the Galilei algebra. This is the contraction of İnönü and Wigner (1953). And since H/c^2 \to m, the first bracket becomes \{K_i, p_j\} = m\,\delta_{ij}. In Newton’s world, the bracket of a boost with a translation is the body’s mass. The Galilei group proper has these two commuting. Mass enters as a central extension, the one Valentine Bargmann found in the projective representations of the Galilei group (Bargmann 1954). A body’s inertia is thus written into the algebra of its free motions. Mass measures how much a change of velocity fails to commute with a change of place. The companion page derives these brackets.

The contraction runs the other way as well. Letting c\to 0 closes the light cones and gives the Carroll algebra, introduced by Jean-Marc Lévy-Leblond in 1965. A free Carroll particle cannot move at all (Bergshoeff, Gomis and Longhi 2014), so Aristotle’s first law returns at the far end. Henri Bacry and Lévy-Leblond classified the possible kinematical algebras and the contractions that connect them in “Possible kinematics” (1968).

Why should the ladder run upward, from Galilei to Lorentz? Ludvig Faddeev gave an answer in a 1989 talk, “A mathematician’s view of the development of physics”. An algebraic structure is stable if every small deformation of it is equivalent to it. Take the rotations and boosts alone. In Newton’s world the boosts commute. That algebra is not stable: the bracket of two boosts can be switched on as a small multiple of a rotation. With parameter -1/c^2 the result is the Lorentz algebra. It is semisimple: it has no set of mutually commuting generators that the rest of the algebra maps back into the set, as rotations map Newton’s commuting boosts among themselves. Semisimple algebras are stable, and in Faddeev’s words, “the Lorentz group is stable.” Classical mechanics is not stable either. Quantum mechanics deforms it with parameter \hbar. Faddeev concludes that “the two main revolutions in physics … are deformations from unstable structures into stable ones.”

Stability does not settle everything. The same deformation with the opposite sign of the parameter gives the rotation algebra of four-dimensional Euclidean space instead of the Lorentz algebra. Stability selects a semisimple algebra, and experiment selects the sign (Chryssomalakos and Okon 2004). And the ladder has not reached a stable top. The Poincaré algebra, Lorentz plus translations, still deforms to the de Sitter algebras when a cosmological constant is switched on. R. Vilela Mendes found that the algebra of relativistic quantum mechanics, Poincaré together with the canonical commutators, is unstable as a whole, and that it admits a stabilizing deformation with two parameters (Mendes 1994), which he later named a fundamental length and a curvature. Faddeev’s criterion gives Heller’s “logic inherent in the evolutionary chain” an engine. It also points to a further rung, with a fundamental length in the symmetry of free motion.

Mendes defines stable theories as those that “do not change in a qualitative manner under a small change of parameters.” The idea comes from the qualitative theory of differential equations, the subject Biler’s seminar opened with. There a system is structurally stable if every small perturbation of its equations leaves its phase portrait topologically the same, a notion due to Aleksandr Andronov and Lev Pontryagin (1937). Applied to dynamics rather than to symmetry algebras, the criterion points the other way. A frictionless pendulum swings on closed orbits around its rest point. The slightest friction turns those orbits into spirals, so conservative systems are structurally unstable among all systems. Gradient flows, in which every motion runs downhill, are the robust ones: a generic gradient flow is structurally stable (Stephen Smale 1961; Jacob Palis and Smale 1970). Within the class of Hamiltonian systems, the Kolmogorov–Arnold–Moser theorem shows that most of the regular motion of an integrable system survives small perturbations. Outside that class, dissipation destroys it: a flow that shrinks phase-space volume cannot keep tori that fill a region. At most a few survive, as attractors.

So stability depends on which perturbations one allows. Among symmetry algebras, Faddeev’s ladder climbs toward Lorentz. Among all dynamics, Aristotle’s first law is the generic one and Newton’s is fragile. Newton’s first law holds where a symmetry protects it: invariance under spatial translations, which by Noether’s theorem conserves momentum. Liouville’s theorem is protected by a structure of its own, the Hamiltonian form of the equations. Friction breaks both, and the laws go with them.

A computational physics must earn its first law

The Wolfram Physics Project models the universe as a hypergraph that a rule rewrites, one local event at a time (an exposition on this blog). Space is the large-scale structure of the hypergraph. Space-time is the causal graph of rewriting events: one event depends on another if it consumes something the other produced. A rule is causal invariant if every order of applying it produces the same causal graph. Jonathan Gorard proved that causal invariance acts as a discrete general covariance, from which he derived a discrete Lorentz covariance and, with further assumptions, the vacuum Einstein equations (Gorard 2020). In Heller’s terms, that is the geometry, derived under assumptions the paper lists, a well-behaved continuum limit among them. The first law is the other half, and it is open.

The project’s technical introduction gives motion a short entry: “motion: possible because of causal invariance; associated with change of causal foliations” (Wolfram 2020, §8.3). A foliation slices the causal graph into successive moments, and a moving observer slices it at a tilt. Causal invariance guarantees that every slicing describes the same causal graph. Gorard’s Lorentz argument re-slices one fixed causal graph in this way. I call this passive invariance: the description does not depend on the observer’s frame.

Newton’s first law needs more. It needs active invariance: for a body at rest there must be a moving twin, the same body at another speed. Speeds may come in steps when space does. What matters is that the steps are small and the body survives them: a small boost must change the body only a little, so that nowhere in the range of speeds does boosting mean becoming a different kind of thing. Passive invariance does not supply this, for the reason Life showed at the start. In a continuum theory, a boost acts on states, so a symmetry of the description carries every body to its moving twin. In a rewriting system, a moving body is a pattern that the rule rebuilds at each new place. Nothing guarantees that such a pattern exists at each speed. Life’s problem is not that its speeds are quantized but that they have gaps. The ships nearest to the glider in speed are different patterns, of tens to hundreds of thousands of cells, and nothing carries one into another. A small boost of a glider is not a slightly faster glider but no glider at all. A hypergraph has no fixed lattice, which leaves room for other carriers, but none comes for free.

Stephen Wolfram has said as much. In “On the Concept of Motion” (2022) he writes of “carriers of pure motion”, collective features of space persistent enough to “just move” without changing:

At the outset it’s not obvious that any such features should exist at all, and that pure motion should ever be possible. Unlike in the traditional “pure geometrical” view of space, in our Physics Project it’s something one has to explicitly derive from the underlying structure of the model—though it seems quite likely that it’s ultimately an inevitable and ubiquitous consequence of rather general “topological” features of hypergraph rewriting.

The candidate carriers are topological. If a rule preserves the planarity of a graph, a tangle of non-planarity inserted into a planar graph can never be removed, only moved or annihilated with another. Wolfram and Gorard both offer such tangles as models of particles. The project’s own readings of mass and time show what a derivation would have to connect. Energy is the flux of causal edges through moments of time, in excess of a background flux that maintains space. Rest mass is the part of that flux that reuses the same hypergraph elements rather than reaching out to new ones (Wolfram 2020). Time dilation becomes a budget: “if the object moves, then in effect it has to be ‘re-created at a different place in space’—and this process takes up a certain number of rewritings, leaving fewer for the intrinsic evolution of the object itself” (Wolfram 2024). This picture is active, but it presupposes the carrier it describes. The only exhibited particle-like structures live on one-dimensional or grid-like backgrounds. In 2022 Wolfram wrote that “we know less about the specifics of how particles work.” I found no work since 2024 that changes this.

So the test is a theorem: a natural class of rules under which persistent structures have moving twins across the range of speeds below c, with no gaps, so that neighboring speeds are carried by nearly the same body. The qualitative theory suggests what such a class needs. Friction is the generic outcome, so a discrete substrate left to itself should give Aristotelian motion. The rules must protect motion the way translation symmetry protects it in the continuum, through a discrete analogue of conserved momentum that rules out drag. “Natural” carries weight as well. In Life the answer is engineering: a recipe with the speed as a construction parameter. Random noise gives six speeds. A physics in which every speed of every particle had to be built would not be our physics.

Which speed is c? In his derivation, Gorard takes the light cones of the causal graph as the light cones of the emergent space-time: the constancy of c, he writes, “is enforced axiomatically by our definition of the edge lengths in causal graphs.” That is a convention of the derivation, not a claim that nothing can outrun it. Wolfram’s own account in “Faster than Light in Our Model of Physics” (2020) is looser. He calls it “at some level a tautology to say that the light cone in the spatial hypergraph advances at the speed of light,” since that is how c is defined. The definition is statistical: c is the speed of the “average border” of the causal cone. He writes that he increasingly suspects “that going faster than light is not a physical impossibility.” What he has in mind is a “space demon,” “a kind of analog of Maxwell’s demon not for molecules in a gas, but for atoms of space,” which with enough computation could move faster than the speed so defined. On this account the substrate allows faster influence than the typical carrier achieves. Life shows the same gap. Life’s own c, one cell per generation, bounds all influence, but no spaceship in empty space exceeds c/2. Patterns reach c only inside an active medium, a periodic background called an agar, along which signals can run at one cell per generation. The first such “lightspeed wire” was found in 2002 (LifeWiki). If the carriers of motion in a rewriting universe are capped below the substrate’s limit, observers built from them would measure, with their rods and clocks, an invariant speed equal to the carriers’ cap, in Life the c/2 of its fastest ships, not the rule’s own c.

That is how relativity emerges wherever physics has found it emerging. Sound in a flowing fluid obeys a wave equation in an effective metric whose light cones are set by the speed of sound (Unruh 1981). Electrons in graphene behave as massless Dirac particles with an effective “speed of light” of about 10^6 m/s, roughly c/300 (Novoselov et al. 2005). In each case Lorentz symmetry holds for slow excitations, with a speed below the substrate’s own, and fails at high energies. Wolfram expects a version of this in his model: “‘maximum boosts’ for sufficiently light particles” (Wolfram 2021).

An emergent c comes with a condition: every carrier of information must share the same cap, or any faster channel must stay unusable. Life fails the condition, in ways its inhabitants could exploit. Its caps depend on direction: c/2 along the axes, c/4 along the diagonals, which in Euclidean terms is about 0.35 cells per generation. They depend on the medium: a lightspeed wire carries signals twice as fast as any ship in empty space, so inhabitants could build a network that outruns every vacuum messenger once the wire is laid. And because Life updates every cell at once, its caps are fixed in the lattice’s frame. Velocities therefore add as Galileo said they do. A machine riding a ship eastward at c/4 can send nothing forward faster than c/4 relative to itself, while signals sent backward recede at 3c/4. From the difference it can compute its own velocity relative to the lattice. In Life, the Michelson–Morley experiment would succeed. Our world shows none of this. Its caps agree in every direction, and no medium carries a signal faster than the vacuum does. Light in glass is slower. Phase velocities can exceed c, but signal velocities never do, as Arnold Sommerfeld and Léon Brillouin showed. Wolfram’s space demon would be a faster channel of exactly the kind the condition forbids. His answer takes the second option: the faster channel stays unusable. As he puts it, “to scale up to anything substantial requires resources beyond what we—or even the universe—could ever muster.” If that holds, the condition is met for computationally bounded observers, and relativity becomes a law about what such observers can do. The sequel, The Demon Is a Learner, follows the price a demon pays for its computation.

Life’s media also point to a requirement on the vacuum itself. Wolfram’s vacuum is active: most causal edges go into “maintaining the structure of space,” the background against which energy is measured. In that respect his space resembles an agar more than Life’s empty grid. Our own vacuum, with its quantum fluctuations, is active too. What an agar cannot be is the vacuum of relativity, because a periodic medium has a rest frame, as a crystal does. A rewriting universe needs an active medium that heals around its excitations, carries them across the range of speeds without gaps, under one isotropic cap, and has no rest frame. The obstacle is what such a medium costs.

The obstacle. Exact Lorentz symmetry may be out of reach for a discrete substrate. Luca Bombelli, Joe Henson and Rafael Sorkin proved that no measurable rule can pick out a timelike direction from a random scattering of points in Minkowski space while respecting Lorentz symmetry (2009). As a corollary, no graph in which each point has finitely many neighbors can be attached to such a scattering in a Lorentz-invariant way. The causal graphs of rewriting rules have finitely many edges at each event.

Gorard addresses the theorem directly, and claims compatibility “provably” (Gorard 2020b, §3). His argument has two steps. The first is that a preferred direction would imply a preferred foliation, “which is incompatible with the hypothesis that the causal network is invariant under different choices of foliation.” That is passive invariance again. It shows that no slicing is needed to generate the graph, not that the finished graph has no structure that picks out a frame. The regular grid Gorard uses as his worked example does have such structure, since a boost does not map a grid onto itself. Tommaso Bolognesi and Alexander Lamb, testing algorithmically generated causal sets, observe that regular ones “would not pass the key test for Lorentz invariance: the absence of preferred reference frames” (2014). The second step concedes the corollary for the microscopic graph. Lorentz invariance is to hold for an effective network, coarse-grained into classes of events so large that “the out-degree of vertices in the effective causal network may be unbounded, even if the out-degree of vertices in the underlying (microscopic) [network] is bounded.” The classes are defined by a choice of foliation, so the coarse-graining starts from a frame. The paper does not show that the result is the same for every choice. What survives is a claim that Lorentz violation lives below any resolution an observer can reach. That is approximate Lorentz invariance, and approximate invariance can be tested. The experimental margin is thin. Photon arrival times from the gamma-ray burst GRB 221009A rule out a speed of light that varies linearly with energy, at any scale below ten times the Planck energy (LHAASO 2024).

Precedents. Discrete models of motion have kept relativity at a price. The precedents show three kinds of price. A substrate with a lattice has a rest frame and shows the generic outcome. A defect in a crystal, such as a dislocation, glides through the lattice against a periodic resistance, the Peierls–Nabarro barrier, which falls off exponentially with the defect’s width (Braun and Kivshar 1998). A wide defect moves almost freely; a narrow one is pinned. A moving defect also radiates vibrations into the lattice and slows down. On such a substrate the first law holds up to an exponentially small friction against the lattice’s rest frame: Aristotle at the Planck grain. A random substrate, a causal set, keeps exact Lorentz symmetry and has no rest frame. Fay Dowker, Henson and Sorkin modeled a massive particle on it and found that it swerves: its momentum diffuses in a Lorentz-invariant way, so the first law holds only on average (2004). Relic neutrinos bound the diffusion constant below 10^{-61}\,\text{GeV}^3 (Kaloper and Mattingly 2006), for neutrino masses above about 0.01 eV. Quantum cellular automata show the third price. Alessandro Bisio, Giacomo Mauro D’Ariano and Paolo Perinotti showed that a quantum walk’s symmetries include the Lorentz group as an exact limit at low energies, “while at the Planck scale the covariance is nonlinearly distorted,” in the manner of doubly special relativity (2016). Their nonlinear representation is a different construction from Mendes’s new Lie algebra, but it lands where the stability argument points: relativity modified at a fundamental scale.

A deterministic rewriting rule sits between these cases. It is discrete like a lattice. If its causal graph has a preferred slicing, its particles should feel Aristotelian friction. If the graph has no structural frame at all, its particles might swerve instead. Either outcome is a prediction with an experimental bound attached. Neither has been derived.

An observer-based reply. Wolfram ties pure motion to observers as well as to rules. In 2022 he wrote that “pure motion is in a sense an operational consequence not necessarily of an actual continuum world, but of a continuum perception of the world by an embedded observer like us.” One way to develop this, my reading rather than his argument, is as selection: only observers who persist while they move can do physics, so any observer finds itself where motion works. That would make the first law a postulate about observers rather than a property of a rule. It may be true, since persistence does need carriers of motion. But a postulate cannot be checked against the rule, and a theorem can. A later essay on observers takes up the selection argument.

Heller’s ladder ends in a structure of its own. His next rung replaces space-time by a groupoid built from the frame bundle, with a noncommutative algebra of functions on it, so that points and instants lose meaning below the Planck scale. Its dynamics comes from the Tomita–Takesaki theorem, the device Alain Connes and Rovelli used for their thermal time hypothesis. That dynamics depends on the state, though Heller and his coauthors expect the state dependence to disappear in realistic models (Heller, Pysiak and Sasin 2005). The Wolfram model and Heller’s groupoid agree that space-time emerges at low energies from a structure without points. Neither says yet which motions in that structure need no cause.

What is free

The first laws in this essay were discoveries, not conventions. Each could be tested, and Heller’s point is that testing the first law tests the geometry. Aristotle’s held in the fluids he could observe, and fails in the void. Newton’s held in the void, and its geometry, absolute time with relative rest, fails near the speed of light.

A computational physics inherits the question in its sharpest form. Its substrate gives nothing for free. Every motion is a rebuilding. Recovering our world means deriving which rebuildings look like no change at all. The geometry has been derived, under assumptions. The standard of motion has not. The Demon Is a Learner takes up the other side of the ledger: what everything that is not free costs, and why a learning machine pays it.