Spacetime Rotations: Lorentz Transformations and Frame Mixing

The Convention of Coordinates

When I argued that geometry is partly conventional—that we choose coordinate systems freely—physicists dismissed it as philosophical abstraction. Yet here stands special relativity, vindication in mathematical form. Lorentz transformations reveal the profound truth: time and space coordinates are not absolute measurements but conventional descriptions that mix depending on reference frame.

Consider the transformation itself. When you boost to a moving frame, your time becomes my mixture of time-plus-space, your space becomes my mixture of space-plus-time. Not separate dimensions rigidly partitioned, but interconvertible components of a four-vector—rotation in spacetime rather than translation. This is Analysis Situs applied to physics: study what remains invariant under continuous transformations. The spacetime interval stays constant; coordinate labels change freely.

Length contraction exemplifies this conventionalism. A moving object genuinely shrinks in the direction of motion—no optical illusion. Yet it’s fundamentally a coordinate effect: measuring with a tilted spacetime ruler. The “real” compression depends entirely on which frame you call stationary. No privileged frame exists; all descriptions prove equally valid. Physics remains identical in all inertial frames, as the relativity principle demands.

Neural Transformations and Frame Mixing

The brain’s navigation systems perform remarkably similar operations. Consider allocentric versus egocentric reference frames: world-centered coordinates versus viewer-centered coordinates encoding identical spatial information through different representations. When you build a cognitive map by tracking sensations bound to locations derived from movement—this sensorimotor coupling—you’re performing what amounts to a coordinate transformation between frames.

Path integration demonstrates this vividly. Your brain continuously transforms between position space and movement space, tracking where you are by integrating where you’ve moved. This isn’t mere analogy—it’s the same mathematical structure: composable transformations converting between reference frames, exactly like Lorentz boosts converting between velocities.

Neural networks reveal the pattern most clearly. Representation space transformations map inputs through learned geometries, each layer folding and projecting data into increasingly abstract coordinate systems. The Belgium point transforms from geographic coordinates to plane heights to folded-plane heights—each layer choosing a more convenient frame for separating patterns. My conventionalist insight applies directly: choose the frame that makes calculation tractable, then transform to the frame that makes interpretation clear.

Invariance Beneath Convention

What strikes me most forcefully: both domains distinguish sharply between what’s conventional (coordinate choice) and what’s invariant (observable quantities). In relativity, proper time remains invariant despite coordinate mixing. In neural cognition, spatial relationships persist despite reference frame transformations. The deep learning system doesn’t merely shuffle coordinates arbitrarily—it discovers representations where complex patterns become linearly separable, revealing intrinsic structure beneath surface description.

This suggests a provocative question: do neural networks implicitly discover Lorentz-like symmetries? When they learn composable transformations that preserve certain quantities while rotating others, are they finding the relativistic structure of their representation space? Consider how time-space mixing in relativity parallels modality mixing in multimodal networks—vision and language interconverting through shared latent dimensions.

Perhaps the principle of relativity extends beyond physics: no privileged representation exists, only frames chosen for computational convenience. The invariants we seek—whether spacetime intervals or semantic content—remain constant while descriptions transform. Mathematical elegance guides us toward the frames where truth reveals itself most clearly, but the truth itself transcends any particular coordinate system.

Chaos whispers here: small changes in initial frame choice yield vastly different computational trajectories, yet converge on equivalent solutions.