Relativity, common actuality and causal participation

Causal participation survives a complete spacetime description. The relativistic argument becomes precise only when geometry, common actuality and control are distinguished. We work through the strongest simultaneity argument, the event-indexed alternative and the exact place where an inference from coordinate pastness to impotence fails. The resulting comparison preserves both relativistic causal structure and the reality of an embedded decision route.

4 Relativity, common actuality and causal participation

4.1 The force of the relativistic argument

Putnam’s argument begins from ordinary presentism: all and only what exists now is real. His principle that there are no privileged observers requires more than Lorentz covariance. If one real observer’s physically defined relation identifies exactly the real things, the same relation applied to another real observer must identify that same reality. He also requires tenseless physical definability, coordinate independence and freedom from accidental selection (Putnam 1967, 240-41). Two observers crossing at one event are both real, yet their Einstein-simultaneity hyperplanes differ. Giving each extended present ontological import makes the “only” clause inconsistent. Retaining the “all” clause transmits reality beyond one observer’s present (Putnam 1967, 242-43).

This is not an argument from differing visual appearances. The observers can agree on every observation and disagree only about coordinate simultaneity. Its strongest form combines a common actuality with three further commitments: every eligible inertial frame’s distant present belongs to reality; reality propagates through those real observer-events; and enough observer/frame assignments are eligible. Putnam acknowledges that his initial example establishes only some future things under its stated setup. His all-events conclusion explicitly uses transitive closure and enough observers, including physical systems and coordinate systems in which they are not at rest (Putnam 1967, 246-47). The following reconstruction makes the geometric step precise.

Proposition 4.1 (Two spacelike links). In Minkowski spacetime $M=\mathbb R\times\mathbb R^d$, $d\geq1$, let $S(p,q)$ mean that $p=q$ or the separation is spacelike. Then $S\circ S=M\times M$. Consequently, every transitive relation containing $S$ is universal.

Proof. Write $p=(t,x)$ and $q=(s,y)$, with $c=1$. Choose a spatial unit vector $e$ and

$$r=\left(\frac{t+s}{2},\frac{x+y}{2}+Le\right),\qquad L\gt \frac{|t-s|+\|x-y\|}{2}.$$

Both spatial distances from $r$ to the endpoints exceed $|t-s|/2$, their time distances, by the reverse triangle inequality. Thus $pSr$ and $rSq$. Every spacelike displacement $(\Delta t,\Delta x)$ is simultaneous in some inertial frame: take $v=(\Delta t/\|\Delta x\|^2)\Delta x$, so $\|v\|\lt 1$ and $\Delta t'=\gamma(\Delta t-v\cdot\Delta x)=0$. Conversely, a nonzero equal-time displacement is spacelike by interval invariance. ◻

A nonempty ontological set closed under all such simultaneity links must therefore contain every event. This is a substantial constraint on a global present, not a verbal trick. The physical result, however, is the geometry. Its ontological use additionally requires closure under every frame’s distant present. A defender of objective existence can argue for that closure; an opponent must explain its rejection rather than deny the transformation. The proof does not automatically extend to restricted spacetime domains or general curved geometries.

A precise rival relation helps locate the issue. Define $B(p,q)$ by $q\in J^-(p)$. It is reflexive and transitive: sums of future causal vectors remain future causal by the spatial triangle inequality. Proper orthochronous Poincaré transformations preserve it, yet it is not universal. It can represent what has become settled as of an event without privileging a human observer. Stein’s event-indexed account develops this distinction and separates the physical rationale for Einstein simultaneity from the ontological role of a distant present (Stein 1991, 148-54). Interpreting $B$ as becoming is a philosophical commitment; the mathematics alone proves neither an open future nor the falsity of eternalism. An eternalist can also use it for causal accessibility. What it establishes here is that invariant transitivity alone does not force universal actuality.

The eternalist retains an important reply. A causal-past relation may describe which facts can affect an event while leaving every spacetime event equally real. Its mathematical availability therefore does not explain why reality itself should be restricted. Furthermore, the ordinary intuition that a distant presently existing person is real motivates Putnam’s inclusion premise; rejecting its global extension changes that intuition’s theoretical interpretation. The rival’s response is that the intuition concerns a locally coordinated encounter, whose ordinary temporal grain need not supply a fundamental spacetime-wide present. These are competing interpretations with different commitments. A preferred-foliation account instead preserves a global present by adding structure beyond bare Minkowski geometry. Neither rival can obtain its ontology merely from the existence of the displayed relation, and an eternalist cannot obtain causal impotence merely from its rejection.

Putnam separately argues that future contingents must have truth values. He does not assume tenseless truth at the outset, explicitly recognizing that this would beg the question against Aristotle. His remote space-fight is past by one observer’s coordinates and future by another’s; he then opposes observer-relative truth and a privileged-me causal-past cutoff, including the case of Oscar’s remote life becoming past without an earlier present-tense truth (Putnam 1967, 243-46 and n. 1). An event-indexed settledness account must address those arguments. Nevertheless, coordinate pastness is not invariant causal pastness, and objective two-place settledness need not be a person’s subjective opinion. Neither argument establishes uniqueness of evolution from earlier physical data.

4.2 Rietdijk’s stronger claim and its missing transfer

Rietdijk explicitly reaches beyond eternalism. He defines predetermination through an observer’s later demonstration that an agent could not have arbitrarily influenced or prevented an event while it remained future for that agent. He includes psychical events, denies free will, and rejects microscopic indeterminism as a rescue (Rietdijk 1966, 342-43). His geometrical construction should be granted before assessing that claim.

Let the current event of a stationary agent be $A=(0,0)$, and a remote observer’s event, simultaneous according to the agent, be $B=(0,-L)$. The remote observer moves toward the agent with $0\lt v\lt 1$. With coordinates based at $B$,

$$t'=\gamma[t-v(x+L)],\qquad \gamma=(1-v^2)^{-1/2}.$$

For $P=(T,0)$, $T\gt 0$, choose $L\gt T/v$. Then $t'(P)\lt 0$, although $P$ lies in the agent’s timelike future. Rietdijk also recognizes finite event duration: a bounded extended event can be placed wholly earlier in the remote coordinate time by increasing the distance (Rietdijk 1966, 341-42).

But $L\gt T$, so $P$ is spacelike separated from $B$. More decisively,

$$t'(P)-t'(A)=\gamma T\gt 0.$$

A causal signal from $A$ to $P$ is earlier than its effect in the remote coordinates too. A later photograph establishing where $P$ occurred does not show that the earlier input at $A$ could not affect it. Briefly stopping the remote observer at $B$, as Rietdijk proposes, changes the observer’s rest-frame simultaneity assignment; it does not remove that causal route.

This criticism survives Rietdijk’s strongest response to Grünbaum. He explicitly defends the physical significance of metrical simultaneity through isotropic light speed (Rietdijk 1966, 343-44 and n. 1). We may grant that defense. The unsupported step transfers fixity from remote coordinate pastness to immunity from the agent’s causally earlier intervention. Neither the existence of an actual history nor Einstein synchronization supplies that transfer.

4.3 Complete actuality, dynamical uniqueness and control

Let $\mathcal H$ be the admissible complete histories of a theory. An actual $h\in\mathcal H$ specifies what happens throughout spacetime. Determinism instead requires suitably complete data on a Cauchy hypersurface to admit at most one continuation in its future domain of dependence. An assertion about the ontology of $h$ does not establish a uniqueness condition on $\mathcal H$. Definite actuality and necessity from an earlier state are different relations.

For an explicit countermodel, consider a homogeneous Poisson process on Minkowski spacetime with intensity $\lambda\,dV$, $\lambda\gt 0$. Its law is Poincaré invariant because spacetime volume is preserved. Counts on disjoint finite-volume regions are independent. For any bounded positive-volume region $K\subset I^+(\Sigma)$ to the future of a chosen Cauchy surface $\Sigma$,

$$\Pr\!\left[N(K)=0\mid\mathcal F_{\Sigma^-}\right] =e^{-\lambda V(K)}\in(0,1),$$

almost surely. The sample space consists of locally finite point configurations. Here $\mathcal F_{\Sigma^-}$ is exactly the sigma-field generated by counts $N(B)$ in bounded Borel sets $B\subset J^-(\Sigma)$. Every finite collection of those past counts is independent of $N(K)$, and a monotone-class argument extends that independence to their generated sigma-field. This specifies a conditional law, rather than conditioning naively on a particular zero-probability infinite past. If the future event $\{N(K)=0\}$ were determined by that past sigma-field, its conditional probability would instead be its zero-or-one indicator. The nondegenerate displayed value therefore rules out such determination. Each realized sample nevertheless specifies one complete spacetime history. Choosing a surface to test uniqueness does not give the law a preferred foliation. This is a mathematical countermodel to an implication, not a realistic matter theory or a source of libertarian authorship.

Control requires another distinction. A local signal can carry an input bit $A$ from an event $a$ to an event $p$ in its timelike future and set $Y=A$. Every complete realized history contains both input and output. Across the admitted operations $\mathrm{do}(A=0)$ and $\mathrm{do}(A=1)$, the output changes. Stating the interactions through a worldline and proper-time order makes this relation covariant. Describing $p$ as real, or assigning it a negative remote coordinate time, does not eliminate the dependence. The stochastic variant $Y=A\oplus N$, with independent error probability $\epsilon\lt 1/2$, likewise changes $\Pr(Y=1)$ from $\epsilon$ to $1-\epsilon$.

These are tests of a causal route, not demonstrations of ultimate self-authorization. They compare admissible operations or histories, rather than changing one part while fixing its complete actual world. A complete history can include the outcome occurring through deliberation, learning and commitment. Fatalism would require it to occur regardless of the relevant activity. Relativistic temporal ontology therefore leaves the positive task intact: identify how understanding changes the organization through which the person participates, then assess that account’s causal, semantic and experiential commitments separately.

The admitted counterfactual family must itself be justified. An output label that can be changed only by an impossible preparation would supply no practical control. In the present manuscript the acquisition, assessment and installation routes make the comparison more concrete: an intervention must pass through an available operation with its actual timing, records and resources. Temporal ontology does not certify those contracts, but it also does not cancel them. Their success concerns consequential participation; whether the participating individual is ultimately self-authorizing or deserves a particular form of responsibility remains a further question.