Physical histories and the temporal organization of a participant

A physical history contains change. Geometry, lawful evolution and a complete record of what occurs do different explanatory work. This chapter develops the Newtonian, relativistic and quantum distinctions needed to discuss agency without confusing a worldline with predetermination or a local present with universal simultaneity. Physical equations constrain the comparison; they do not decide every interpretation of the chooser.

3 Physical histories and the temporal organization of a participant

The complete-history picture has a serious physical basis, but several claims that are often bundled together must be separated. A four-dimensional representation assigns physical quantities to events. Determinism concerns uniqueness across admissible histories under specified data. Reversibility concerns recovering a state from its evolution. Time-reversal invariance concerns a symmetry of the laws under an appropriate reversal operation. Eternalism concerns the reality of events at different times. None is a definition of the others. Their separation improves the argument: a limitation on authorship should depend on the premise that actually carries it, while the positive account should survive wherever its physical organization is possible.

3.1 Newtonian histories already have a four-dimensional representation

A Newtonian system with configuration $q$, momentum $p$ and Hamiltonian $H(q,p,t)$ obeys

$$\dot q=\partial_pH,\qquad \dot p=-\partial_qH.$$

When the resulting vector field is continuous in time and locally Lipschitz in state, locally uniformly in time, initial data have locally unique solutions. Additional bounds or completeness assumptions are needed for a solution over the whole interval under discussion. Where evolution exists uniquely in both directions, the evolution maps $\phi_{t,s}$ satisfy

$$\phi_{t,s}^{-1}=\phi_{s,t},\qquad \phi_{t,u}\phi_{u,s}=\phi_{t,s}.$$

This is an invertibility statement. Time-reversal symmetry additionally requires an appropriate operation, commonly momentum reversal, to relate solutions of the same laws. A time-dependent protocol can have invertible evolution without being invariant under reversing its unchanged schedule.

Uniqueness is not guaranteed merely by writing Newton’s second law. For the elementary mathematical equation $\ddot x=\sqrt{x}$, $x\geq0$, data $x(0)=\dot x(0)=0$ admit the zero solution and, for each $T\geq0$,

$$x_T(t)= \begin{cases} 0,&t\leq T,\\ (t-T)^4/144,&t\gt T. \end{cases}$$

The function and its first two derivatives join at $T$, and for $t\gt T$ both $\ddot x_T$ and $\sqrt{x_T}$ equal $(t-T)^2/12$. The force fails the usual local Lipschitz condition at zero. This is a countermodel to an unrestricted mathematical inference, not a claim that a human choice is such a solution or that the force is a complete realizable apparatus.

A solution can be represented as a worldline or field history in $\mathbb R\times\mathbb R^3$ even when Newtonian time supplies a preferred simultaneity. Thus four-dimensional representation predates relativity’s challenge to a privileged present. Thyssen explicitly distinguishes representation from a proof of eternalism (Thyssen 2020, 9-10). The stronger relativistic contribution concerns which temporal structures are physically available, rather than the mere possibility of drawing an entire trajectory.

3.2 Relativity makes local order objective without making simultaneity universal

In an inertial chart on Minkowski spacetime,

$$ds^2=-c^2dt^2+dx^2+dy^2+dz^2.$$

A boost along $x$ gives

$$t'=\gamma(t-vx/c^2),\quad x'=\gamma(x-vt),\quad \gamma=(1-v^2/c^2)^{-1/2}.$$

For spatially separated events simultaneous in one chart, $\Delta t=0$ need not imply $\Delta t'=0$. In contrast, proper orthochronous Lorentz transformations preserve the future order of timelike and null related events. There is no contradiction between frame-dependent ordering of spacelike separated operations and an invariant fact that a message must reach the assessor before it can influence assessment.

For a timelike worldline $z$, a clock measures proper time

$$\tau[z]=\frac1c\int_z\sqrt{-g_{\mu\nu}\,dz^\mu dz^\nu}.$$

Different paths between meetings can accumulate different durations. For example, in units $c=1$, a clock resting for coordinate duration $10$ records $10$, whereas a two-leg path at speed magnitude $3/5$ records $10\sqrt{1-9/25}=8$. The instantaneous reversal idealizes a short acceleration interval; it is not a discontinuity of the clock’s proper-time order. The relevant point is the physical availability of local duration without a unique universe-wide present.

An embodied organization occupies an extended worldtube, not a coordinate point. Its sensory inputs, recurrent processing, motor outputs and archives have spatial and temporal extent. Its experienced present therefore need not correspond to a global hypersurface of simultaneity. A finite integration process can unite recent records and anticipation at a local stage, with earlier contributions arriving along causal paths. Relativity constrains those paths. It does not by itself say which integration constitutes experience, or why an episode has the phenomenal temporal character it does.

The same point applies to deliberation. A decision region can contain a sequence of comparisons, bodily changes and commitment events. A later action is not independent of that sequence merely because the whole sequence admits a spacetime description. Temporal ordering and change are relations within the history. To infer their unreality from the absence of a moving global now would confuse the absence of one proposed temporal structure with the absence of all temporal organization.

3.3 General relativity supplies conditional initial-value uniqueness

In general relativity the metric is dynamical:

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac{8\pi G_{\!N}}{c^4}T_{\mu\nu}.$$

One cannot generally prescribe arbitrary metric and matter values independently. Gauge freedom also prevents treating different coordinate expressions as different physical futures. For smooth vacuum initial data with $\Lambda=0$ on a connected three-manifold $\Sigma$, the spatial metric $h$ and extrinsic curvature $K$ satisfy

$$R(h)-|K|_h^2+(\operatorname{tr}_hK)^2=0,\qquad \nabla^iK_{ij}-\nabla_j\operatorname{tr}_hK=0.$$

The classical maximal-development result states that such data possess a maximal globally hyperbolic development, unique up to an isometry preserving the initial embedding. Sbierski supplies a direct reconstruction of the Choquet-Bruhat-Geroch result; the precise statements used here are his Theorems 2.6-2.8 (Sbierski 2016, PDF pp. 11-12, 22-24). This is uniqueness within the globally hyperbolic class. It does not assert geodesic completeness, exclude extensions beyond a Cauchy horizon, or establish a corresponding result for arbitrary matter theories. Matter extensions require appropriate local existence and uniqueness.

A Cauchy hypersurface intersects every inextendible timelike curve once. Its complete admissible data can therefore support a global prediction question in an appropriate model. A small part of that surface, such as a brain-sized region, is not complete Cauchy data. Its future domain of dependence consists of events through which every past-inextendible causal curve meets that part; the past domain is defined with future-inextendible curves. Confusing these local domains with the development of complete data changes the theorem being invoked.

Nor is global hyperbolicity by itself a dynamical uniqueness theorem. It supplies causal structure on which suitable equations can have a well-posed initial-value problem. Non-globally-hyperbolic models, singular boundaries and Cauchy horizons require further conditions; global extrapolation is not automatic. In the deterministic regime that is actually justified, the natural histories are equivalence classes under the relevant gauge identifications. There is no reason to count a diffeomorphism of one development as a new agential alternative.

The constraint equations also discourage an image of many agents independently writing arbitrary boundary values. Legal changes must remain jointly consistent. Section 6 makes that problem explicit. It does not follow that legal changes are impossible or that only an observer outside the universe can specify an intervention. An embedded intervention is another physical operation, whose admissible comparison must respect the equations, boundaries and resources.

3.4 Quantum theory limits the inference to a unique future

For an isolated quantum system with a specified time-independent self-adjoint Hamiltonian, the unitary evolution $U(t)=\exp(-iHt/\hbar)$ determines its quantum state from the earlier quantum state. Whether that state is a complete physical state, and whether a particular outcome is uniquely fixed, are additional questions. A deterministic quantum-state equation cannot simply be substituted for an argument that every individual outcome is uniquely determined. Conversely, stochastic outcome laws do not supply independent personal authorship merely by leaving more than one outcome possible.

Relativistic covariance does not settle this interpretation dispute in favor of determinism. Tumulka’s relativistic collapse model gives a Lorentz-invariant stochastic law for spacetime flashes without a preferred foliation. The 2006 construction concerns distinguishable noninteracting particles; its ontology and modified dynamics are substantive hypotheses (Tumulka 2006, sec. 1, preprint pp. 1-3). A later interacting construction assumes suitable interaction-local unitary evolution and still does not constitute a general theory of all actual matter (Tumulka 2020, sec. 1). These models matter here as counterexamples to a claimed logical incompatibility between relativistic structure and objective stochasticity, not as established descriptions of this person’s brain.

The limited scope of these surveyed models gives no basis for identifying the unknown Ground with a classical manifold. The spacetime models used here are nominated physical descriptions in specified regimes. The source-readout programme can ask how those descriptions arise or fail to close without claiming that a presently available mathematical source describes the Ground exhaustively.

3.5 The strongest physical support for a complete-history interpretation

A Minkowski model represents distant and temporally separated events through one spacetime structure, invariant causal relations, local clock durations and no preferred inertial simultaneity. Suitable classical equations relate admissible data to entire developments. These features make a tenseless complete-history interpretation powerful and economical. They explain why treating an individual’s decision as an extra addition from beyond physical reality is unnecessary within a successful realized account.

The inference to eternalism nevertheless requires an ontological bridge. A manifold used to represent events is not, just by being written down, a proof that all represented times are equally real. Putnam, Rietdijk and Thyssen examine arguments for that bridge, with importantly different conclusions. We next reconstruct their strongest steps, then ask which limitation on authorship they actually support. The positive account will require only that the participant and its records can be represented as related physical occurrences, together with whichever further constitution and experiential premises are explicitly invoked.