Relativity, Local Presents, and Observers

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Local present

In general relativity there is no observer‑independent global “now.” Each timelike worldline carries its own proper timeTime measured by a clock moving with the observer; invariant along a timelike worldline. $\tau$, locally given by

$$ d\tau^{2} \,=\, -\frac{1}{c^{2}}\, g_{\mu\nu}\, dx^{\mu} dx^{\nu}. $$

At an event $p$ on a worldline with 4‑velocityUnit tangent to a timelike worldline, normalized so $u^{a}u_{a}=-c^{2}$; components $u^{a}=dx^{a}/d\tau$. $u^{a}$, a precise operational notion of the local present is the instantaneous rest‑spaceThe 3‑dimensional subspace orthogonal to $u^{a}$ at an event; the observer’s instantaneous “space.” through $p$:

$$ H_{p} \,=\, \{\, v^{a} \mid g_{ab}\, u^{a} v^{b} = 0 \,\}, $$

i.e., the 3‑dimensional subspace orthogonal (with respect to $g_{ab}$) to the observer’s motion. In a small neighborhood one can extend this to a spacelike slice using Einstein synchronizationClock‑sync by exchanging light signals between nearby comoving observers, assuming isotropic light speed. or Fermi normal coordinatesLocally inertial coordinates constructed along a worldline via Fermi–Walker transport; flatten the metric on the line.. Curvature (tidal gravity)Nonzero spacetime curvature (Riemann tensor); tidal effects that cannot be removed by any coordinate choice. prevents stitching such slices into a unique global present except in special cases (e.g., adopting a conventional “cosmic time” in FLRW cosmologies).

In the weak‑field, slow‑motion limit with Newtonian potential $\Phi$ and speed $v$ relative to a static frame, the combined gravitational–kinematic dilation reads

$$ d\tau \;\approx\; dt\, \sqrt{\,1 + \tfrac{2\Phi}{c^{2}} \; - \; \tfrac{v^{2}}{c^{2}}\,}\,. $$

For an entire congruenceA family of worldlines filling a region of spacetime (one worldline through each point). of observers $u^{a}(x)$, a global foliation whose slices are everywhere orthogonal to $u^{a}$ exists iff the vorticityThe local rotation of the congruence (antisymmetric part projected orthogonal to $u^{a}$). Zero vorticity implies hypersurface‑orthogonality. vanishes ($\omega_{ab}=0$; Frobenius conditionIntegrability criterion: orthogonal hypersurfaces exist iff $u_{[a}\nabla_{b}u_{c]}=0$ (equivalently $\omega_{ab}=0$).). Rotating congruences ($\omega_{ab}\neq 0$) admit no such global simultaneity surfaces.

Integrability (plain‑language breakdown)

• Goal: Build “sheets of now” (spacelike surfaces) that are perpendicular to every local time‑arrow $u^{a}$.
• No twist: If the time‑arrows don’t swirl, you can lay down one continuous sheet. Mathematically this is the Frobenius condition: \(u_{[a}\nabla_{b}u_{c]}=0\) ⇔ \(\omega_{ab}=0\).
• With twist: If there’s swirl (nonzero vorticity), any sheet you try to build will tear or double back; there is no single, global “now” surface orthogonal to all $u^{a}$.
• Picture: Think of combing hair on a sphere. If there’s a cowlick (twist), you can’t make all strands lie flat in one direction without a discontinuity.

• Time dilation: Kinematic and gravitational. Clocks deeper in a potential or at higher speed tick slower relative to distant or comoving clocks.
• Simultaneity is slicer‑dependent: Different observers (and foliations) define different “nows”; there is no canonical global simultaneity.
• Observation vs. presence: Telescopes probe your past light cone (you see the past); travelling places you in that system’s local present. Both are consistent with causality.

Observers (operational)

An observer is any physical interaction that entangles with a system (no consciousness required). Without interaction, evolution is unitary and supports interference; persistent, value-revealing interactions suppress interference (decoherence / quantum Zeno).

Quantum Observers and a No-Global-Observer Theorem

Theorem (No–global, ever–measuring, noncontextual observer). Assume: (A1) Standard quantum mechanics holds in the lab: unitary evolution for isolated systems, the Born rule, observed interference; experiments violate Bell/CHSH and the Kochen–Specker theorem rules out global noncontextual value assignments. (A2) There exists a single agent G that, for every system and all times, either (i) performs projective, value–revealing measurements without altering the statistics of isolated experiments, or (ii) assigns sharp, noncontextual values to all observables without disturbance. Then such a G cannot exist.

Proof sketch. Under (A2.i), continuous measurement would universally suppress interference in carefully isolated setups, contrary to observation. Under (A2.ii), global noncontextual value assignments contradict Bell/CHSH and Kochen–Specker. Hence no such G.

Corollary (No noncontextual, ever-measuring omniscience)

A physics-loaded reading of singular, ever-measuring, noncontextual, value-revealing omniscience is incompatible with quantum mechanics.

Remark (What remains compatible)

This rules out a specific class of models. Models that (a) do not continuously measure, (b) allow contextual knowledge (no global noncontextual assignments), (c) “know” the quantum state without adding extra collapse, or (d) intervene locally with ordinary physical footprints, remain compatible with quantum mechanics.

Mapping to the ⊠-Algebra

• Future = admissible set. The world yields an option set R (e.g., set-valued division; 0/0 = S denotes the admissible option set, not a numeric infinity).
• Present = selection. A selector (policy/intent) chooses one element of R to realize.
• Past = record. The realized outcome is written into the environment; records proliferate and are hard to erase as entropy increases.
• Relativity link. Different observers left-fold their worldlines with different local conditions; there is no single global evaluation order, only frame-dependent proper time.

Remark (Compatibility with Hellenic polytheism)

A plural, bounded, contextual pantheon fits naturally. Each deity acts as a policy/selector over an admissible option set R, intervening locally and sporadically. This avoids any global, ever-measuring, noncontextual omniscience while modeling oracular bias and “fate” as weights over R.