Where this sits — and what it is

This is the series' capstone: a step back from filaments to the geometry of spacetime itself. The empirical posts asked whether a position-plus-orientation lift finds and shapes the cosmic web (Part 1, Part 2). This one is theoretical, not a data study — its figures are interactive diagrams, not measurements — and it develops, with the actual machinery, the claim the whole program kept circling: the geometry we used for filaments is the geometry Einstein used for gravity. Both are what you get when you take the symmetries of flat space and make them local. The route: the Euclidean group done properly (§2), Cartan's recipe for curving a symmetry (§3), gravity as the gauged version (§4), what curvature does to falling matter (§5) — and then back down to Earth, or rather to the web: filament theory as the caustics of free fall, in the eigenframe of the very tensor that ran the whole series (§6).

The thread the series kept pulling

Every post here has been about frames and directions. The tidal eigenframe picks the filament axis — the direction gravity squeezes last (Appendix B2). The orientation lift carries a full direction at every point, \(\mathbb{R}^3 \times S^2 \cong \mathrm{SE}(3)/\mathrm{SO}(2)\). The Geometry of Seeing series did the 2D version, \(\mathrm{SE}(2)\), and traced shortest paths under a turning constraint.

Attach a preferred frame to every point of a space, and ask how it twists as you move from point to point. That single question — how does the local frame turn? — is the one Einstein’s gravity is built from. The cosmic web handed us frames; gravity is the theory of how frames turn. The rest of this post makes that link precise, and it ends somewhere concrete: the tensor whose eigenframe oriented every experiment in this series is the Newtonian face of the Riemann curvature of spacetime, and the filaments are the caustics of its free-fall flow.

The Euclidean group, all the way down

Start with the group this series has been living on. A rigid motion of ordinary 3-space is a rotation followed by a translation: it moves points as

\[\mathbf{x} \;\longmapsto\; R\,\mathbf{x} + \mathbf{t}, \qquad R \in \mathrm{SO}(3),\; \mathbf{t} \in \mathbb{R}^3 .\]

Composing two of them shows the structure immediately: do \((R_1, \mathbf{t}_1)\) first, then \((R_2, \mathbf{t}_2)\), and you get

\[(R_2, \mathbf{t}_2)\,(R_1, \mathbf{t}_1) \;=\; \bigl(R_2 R_1,\; R_2\,\mathbf{t}_1 + \mathbf{t}_2\bigr).\]

The rotations multiply among themselves, but they also act on the translations — the \(R_2\,\mathbf{t}_1\) term. That twist is what the symbol \(\mathrm{SE}(3) = \mathbb{R}^3 \rtimes \mathrm{SO}(3)\) records: a semidirect product,¹ not a plain product. Concretely, every rigid motion is one \(4\times 4\) matrix,

\[g \;=\; \begin{pmatrix} R & \mathbf{t} \\ 0 & 1 \end{pmatrix},\]

acting on points written as \((\mathbf{x}, 1)^{\top}\); matrix multiplication reproduces the composition law above. This is the group the cosmic-web lift and the visual-cortex model are built on (in two dimensions, \(\mathrm{SE}(2)\), rotations about one axis).

Klein’s dictionary: the space is the group

Felix Klein’s Erlangen program (1872) turns this around: don’t start with the space and find its symmetries — start with the symmetry group and reconstruct the space from it. Fix a point, say the origin. The subgroup that leaves it fixed is the rotations, \(H = \mathrm{SO}(3)\). Every other point of space is reached from the origin by some motion, and two motions land on the same point exactly when they differ by a rotation about it. So the points of space are the cosets:

\[\mathbb{R}^3 \;\cong\; \mathrm{SE}(3)\,/\,\mathrm{SO}(3).\]

A geometry, in Klein’s dictionary, is a pair \((G, H)\) — a group and the subgroup fixing a basepoint — and the space is the quotient \(G/H\), a homogeneous space.² Geometric notions are exactly the \(G\)-invariant ones: distance and angle survive because rigid motions preserve them; “left of” does not, because rotations scramble it. Euclidean geometry is \((\mathrm{SE}(3), \mathrm{SO}(3))\). Special relativity’s flat spacetime is the same construction one row down: Minkowski space is \(\mathrm{ISO}(3,1)/\mathrm{SO}(3,1)\) — the Poincaré group (translations, rotations, boosts) quotiented by the Lorentz group fixing an event.

The lift is a quotient of the same group

Now the observation that ties the whole series to this machinery. Quotient \(\mathrm{SE}(3)\) not by all rotations but only by the rotations about a chosen axis, \(\mathrm{SO}(2)\), and you keep two pieces of data: where you are and which way that axis points:

\[\mathrm{SE}(3)\,/\,\mathrm{SO}(2) \;\cong\; \mathbb{R}^3 \times S^2 .\]

That is precisely the state space of the orientation lift of Part 2 — position plus direction — and \(\mathrm{SE}(2)\) itself (no quotient at all) is the state space of the V1 model. There is a cleaner way to say it: \(\mathrm{SE}(3)\) is the bundle of oriented orthonormal frames of flat space — a point plus a full right-handed triad at it. The lift keeps the triad’s first leg and forgets the spin about it. So when the tidal analysis of Appendix B2 attaches an eigenframe to every voxel, it is literally handing you a section of the frame bundle — a copy of the group’s own geometry, spread over space. “The web hands us frames” is not a metaphor; it is the statement that the data lives on \(\mathrm{SE}(3)\).

The algebra: where flatness hides

Groups are unwieldy; their Lie algebras³ — the infinitesimal motions — carry the same information linearly. For \(\mathfrak{se}(3)\) the generators are three infinitesimal translations \(P_1, P_2, P_3\) and three infinitesimal rotations \(J_1, J_2, J_3\), with brackets

\[[J_i, J_j] = \epsilon_{ijk} J_k, \qquad [J_i, P_j] = \epsilon_{ijk} P_k, \qquad \boxed{\;[P_i, P_j] = 0.\;}\]

Read them in words. Rotations compose like rotations. Rotations turn translations into other translations (that is the semidirect twist again). And — the boxed line — translations commute: go east then north, or north then east, and you arrive at the same point. That innocuous-looking zero is the flatness of Euclidean space, written algebraically. Hold that thought: in §4 the entire difference between flat space and a gravitating universe with a cosmological constant will live in whether \([P, P] = 0\).⁴

Cartan’s move: curve the model

Klein’s picture is perfect and rigid — and useless for a lumpy universe, because a lumpy space has no global symmetries at all: no motion of the actual matter distribution maps it to itself. Élie Cartan’s generalisation (1923) keeps Klein’s dictionary but applies it infinitesimally: a Cartan geometry is a space that looks like the Klein model \(G/H\) at each point, together with a rule for comparing the model copies at neighbouring points. All the geometry is stored in that rule — the Cartan connection — and curvature is the precise measure of its failure to be the flat model globally.

The connection is one object with two parts, mirroring the algebra’s split \(\mathfrak{se}(3) = \mathbb{R}^3 \oplus \mathfrak{so}(3)\). It is a 1-form⁵ \(\omega\) with values in the algebra, and it decomposes as

\[\omega \;=\; \underbrace{e}_{\text{translation part}} \;\oplus\; \underbrace{\hat\omega}_{\text{rotation part}} .\]

• The translational piece \(e\) — the solder form, or frame/coframe — says which infinitesimal step in the model corresponds to each infinitesimal step in the actual space. It is what welds the abstract model to the manifold, and it carries the metric: lengths and angles are read through \(e\).
• The rotational piece \(\hat\omega\) — the spin connection — says how the local frame turns as you slide from a point to its neighbour. It is the mathematical answer to the series’ running question, “how does the frame rotate from voxel to voxel?”

One curvature measures everything. The field strength of a connection is

\[F \;=\; d\omega + \tfrac{1}{2}[\omega, \omega],\]

and because the algebra splits, \(F\) splits too — into two failures with two names:

\[\Theta \;=\; de + \hat\omega \wedge e \quad(\text{translation part: } \textbf{torsion}), \qquad R \;=\; d\hat\omega + \hat\omega \wedge \hat\omega \quad(\text{rotation part: } \textbf{curvature}).\]

Torsion is the failure of infinitesimal parallelograms to close: step along \(X\), then \(Y\), then back along \(X\) and \(Y\) — torsion is the gap. Curvature is the failure of frames to return: carry a frame around the same little loop and curvature is the rotation it comes back with. Riemannian geometry is the special case \(\Theta = 0\) with \(\hat\omega\) then fixed uniquely by \(e\) (the Levi-Civita choice); general relativity, the orientation lift, and the rolling-ball problem below are all Cartan geometries, differing only in the model \(G/H\) they are glued from.

Holonomy: curvature you can watch

The rotation-around-a-loop definition of curvature is not an abstraction — it is the one thing about curved space you can demonstrate on a ball. Parallel transport means: carry a vector so that it never rotates as far as the local frame can tell — no twisting relative to the surface. On a flat sheet the vector comes back unchanged around any loop. On a sphere it comes back rotated, and the rotation angle is exactly the curvature integrated over the enclosed area — for the unit sphere, the enclosed solid angle. That net rotation is the loop’s holonomy, and the figure below lets you watch it accumulate.

colatitude θ    transport progress 

Parallel transport measures curvature. A tangent vector (orange) is carried around a circle of constant latitude on the unit sphere, never rotating as far as the surface can tell. The ghost arrow marks its starting direction. After a full loop the vector returns rotated — by the holonomy angle $$\Delta\psi = 2\pi(1 - \cos\theta)$$, which is exactly the solid angle the loop encloses: curvature integrated over area. Drag the loop toward the pole and the loop encloses little area — small rotation; push it toward the equator and the enclosed cap approaches a hemisphere — deficit approaching $$2\pi$$, a full turn back to identity, as befits the equator being a geodesic. On a flat sheet the same procedure returns every vector unchanged: holonomy is the operational meaning of the curvature 2-form $$R = d\hat\omega + \hat\omega\wedge\hat\omega$$ of the text.

Rolling the model: Cartan meets the sub-Riemannian series

There is a second, wonderfully physical way to say what a Cartan geometry is: roll the Klein model along the actual space without slipping or twisting. The connection is the rolling map; a straight line in the model, rolled out, develops onto the curved space as its natural “straightest” curve. Cartan’s rolling-ball problem — one sphere rolling on another — is simultaneously the textbook example of this and a bona-fide sub-Riemannian system: the no-slip constraint restricts the allowed velocities to a small set of directions whose brackets restore full controllability, exactly the Chow–Rashevskii mechanism the Geometry of Seeing series used on \(\mathrm{SE}(2)\), with the same Pontryagin principle generating the optimal paths. Gravity, read as Cartan geometry, rolls a copy of flat Minkowski spacetime along the curved universe (Wise 2010). One machine, three incarnations: the visual cortex, the rolling ball, and gravity.

And the bracket has one more name here. In sub-Riemannian geometry \([X, Y]\) measures the failure of a frame of allowed moves to close — go out along \(X\), then \(Y\), then back, and you miss your starting point. In frame-based gravity that same failure-to-close of the frame field, its anholonomy, is the field strength (it is the torsion of the teleparallel formulation). A frame that closes like a graph-paper grid is flat and gravity-free; the amount by which the cosmic web’s tidal frame fails to close from voxel to voxel is, in this language, its connection made visible.

Gauging: how gravity appears

Now the third move. Special relativity is the Klein geometry \((\mathrm{ISO}(3,1), \mathrm{SO}(3,1))\): one global Poincaré symmetry for the whole of spacetime — the same ten motions (four translations, three rotations, three boosts) applied everywhere at once. Demand instead that the symmetry hold independently at every event — that each observer may choose their own frame, at their own point, with no god’s-eye alignment — and rigid symmetry becomes impossible to state without new structure: you need fields that connect the frame choices at neighbouring points before you can compare them. Those compensating fields are forced on you, and they are precisely Cartan’s two pieces:

• gauging the translations forces in the frame field \(e\) (the tetrad), which carries the metric \(g_{\mu\nu} = \eta_{ab}\, e^a_\mu e^b_\nu\);
• gauging the rotations/boosts forces in the spin connection \(\hat\omega\), which says how frames turn between events.

This is not an analogy for gravity; it is a construction of it (Utiyama 1956; Kibble 1961; Sciama 1962). The two field strengths are the torsion and curvature of §3, and Noether’s theorem pairs each gauged symmetry with the source that excites it:

Symmetry, made local	Conserved source	Geometric field	Its “field strength”
Translations	energy–momentum	the frame / tetrad \(e\)	torsion \(\Theta\)
Rotations & boosts	spin	the spin connection \(\hat\omega\)	curvature \(R\)

The dynamics comes from the most economical invariant you can build from these pieces, the Einstein–Cartan (Palatini) action

\[S \;=\; \frac{1}{16\pi G}\int \epsilon_{abcd}\; e^a \wedge e^b \wedge R^{cd},\]

read: sum over spacetime the curvature, measured in the units the frame provides. Varying the frame \(e\) gives Einstein’s equation \(G_{\mu\nu} = 8\pi G\, T_{\mu\nu}\) — curvature sourced by energy–momentum. Varying the connection \(\hat\omega\) gives a second, purely algebraic equation: torsion sourced by spin density. Where spin density vanishes — everywhere outside quantum matter in extreme states — the second equation says \(\Theta = 0\), the connection collapses to Levi-Civita, and the construction lands exactly on general relativity. Nothing in the gauge reading changes Einstein’s predictions; it explains where the two fields come from.⁶

What curvature does: the tidal equation

So far, machinery. Here is the observable — and the series’ punchline.

A single freely falling particle feels nothing: it follows a geodesic,⁷ the developed straight line of §3, and by the equivalence principle its local experience is indistinguishable from floating in empty flat space. Curvature only becomes measurable with two nearby free-fallers. Let \(\xi\) be the separation vector between neighbouring geodesics with 4-velocity \(u\). It obeys the geodesic deviation (Jacobi) equation

\[\frac{D^2 \xi^{\mu}}{d\tau^2} \;=\; -\,R^{\mu}{}_{\alpha\nu\beta}\, u^{\alpha}\, \xi^{\nu} u^{\beta} ,\]

read: the relative acceleration of free-falling neighbours is curvature contracted with their separation. The piece of the Riemann tensor doing the work, in the rest frame of the pair, is its electric part

\[E_{ij} \;=\; R_{i0j0}, \qquad \frac{d^2 \xi^{i}}{d\tau^2} \;=\; -\,E_{ij}\,\xi^{j} :\]

a symmetric \(3\times 3\) matrix that stretches and squeezes a ball of test particles into an ellipsoid along its eigenvectors. Its trace focuses volume — by Einstein’s equation, \(\mathrm{tr}\,E = 4\pi G(\rho + 3p)\), the seed of gravitational collapse — while its trace-free part deforms shape without changing volume: the tides.

Now take the Newtonian limit — weak fields, slow motion, potential \(\Phi\). The electric Riemann tensor degenerates to

\[E_{ij} \;\longrightarrow\; \frac{\partial^2 \Phi}{\partial x_i\, \partial x_j} \;=\; T_{ij},\]

the Hessian of the gravitational potential — which is, symbol for symbol, the tidal tensor \(T_{ij}\) of Appendix B2, the matrix whose ordered eigenvalues \(\lambda_1 \ge \lambda_2 \ge \lambda_3\) and eigenframe \(\{e_1, e_2, e_3\}\) oriented every experiment in this series. The compass that pointed our detectors down the filaments is the eigenframe of the electric part of spacetime curvature. The cosmic web’s tidal frame and Einstein’s gravitational field are not cousins after all — they are the same tensor, read at two levels of the same theory.

λ₁   λ₂   growth D 

The tidal ellipsoid — geodesic deviation as web morphology. A ring of freely falling test particles (grey circle: initial; blue: now) deforms under the tidal tensor: each principal axis scales by $$1 - D\lambda_i$$, where $$D$$ is the growth factor playing the role of time and $$\lambda_1 \ge \lambda_2$$ are the eigenvalues of $$T_{ij} = \partial_i \partial_j \Phi$$ in the plane shown (the third axis, $$e_3$$, points out of the screen). Positive eigenvalue = compression along that eigenvector (arrows). Signs give the morphology of Appendix B2: both positive → a filament cross-section (axis $$e_3$$; a node if $$\lambda_3$$ is positive too); mixed → one axis collapsing, one expanding — a wall seen edge-on; both negative → a void. Push $$D\lambda_1 \to 1$$ and the ellipse degenerates to a line: the first caustic — a Zel'dovich pancake — forms (§6). The readout tracks the density factor $$1/\prod(1-D\lambda_i)$$ in this plane, which diverges exactly at the caustic.

Filaments: free fall until the map folds

With the tidal dictionary in hand, the web’s own theory — the one running underneath Appendix B1’s transport models — reads as a chapter of the same geometry.

Zel’dovich: geodesics with a frozen tide

The Zel’dovich approximation (1970) moves matter from its initial (Lagrangian) position \(\mathbf{q}\) to its evolved position by a displacement that never updates:

\[\mathbf{x}(\mathbf{q}, t) \;=\; \mathbf{q} \;-\; D(t)\, \nabla_{q} \Phi^{(1)}(\mathbf{q}),\]

with \(D(t)\) the linear growth factor⁸ serving as the clock and \(\Phi^{(1)}\) the (suitably scaled) initial potential. In growth-factor time this is free flight: each parcel receives one initial push \(-\nabla\Phi^{(1)}\) from the primordial potential and then coasts on a straight ray — the ballistic transport of Appendix B1, now recognisable as the first-order statement of geodesic motion in the perturbed spacetime, with the tide frozen at its initial value. The whole map is one gradient flow; everything that happens next is differential geometry of that map.

The deformation tensor: the tide decides the shape

How does a small blob of matter deform under this map? Differentiate:

\[\frac{\partial x_i}{\partial q_j} \;=\; \delta_{ij} \;-\; D(t)\, T_{ij}(\mathbf{q}), \qquad T_{ij} = \frac{\partial^2 \Phi^{(1)}}{\partial q_i \,\partial q_j},\]

the identity minus growth times — the tidal tensor again. In its eigenframe the map is diagonal: an initial cube becomes a brick with edge factors \((1 - D\lambda_1)\), \((1 - D\lambda_2)\), \((1 - D\lambda_3)\), and mass conservation gives the density in closed form:

\[\frac{\rho}{\bar\rho} \;=\; \frac{1}{(1 - D\lambda_1)(1 - D\lambda_2)(1 - D\lambda_3)} .\]

This one formula is the skeleton of large-scale structure. As \(D\) grows, the axis with the largest eigenvalue collapses first: at \(D\lambda_1 = 1\) the brick flattens to zero thickness along \(e_1\) and the density diverges — a pancake (a wall of the web) with normal \(e_1\). Collapse along \(e_2\) follows: the pancake drains into a filament whose axis is \(e_3\), the direction squeezed last — exactly the eigenframe reading of Appendix B2 that our detectors exploited. Collapse of the third axis makes a node; regions with all eigenvalues negative expand into voids. The interactive tidal ellipsoid above plays this sequence in the \((e_1, e_2)\) plane.

Caustics: where free fall folds

The divergence at \(D\lambda_1 = 1\) is not a physical infinity — it is the signature that the Lagrangian map \(\mathbf{q} \mapsto \mathbf{x}\) has folded: three streams of matter now pass through the same point, and the boundary where \(\det(\partial \mathbf{x}/\partial \mathbf{q}) = 0\) is a caustic — the same mathematical object as the bright lines of focused light on the bottom of a coffee cup.⁹ Arnold, Shandarin and Zel’dovich (1982) classified the caustics of these gravitational maps with catastrophe theory: generic folds (\(A_2\)) making the walls, cusps (\(A_3\)) stiffening them into filament-like edges, swallowtails and umbilics (\(A_4\), \(D_4\)) decorating the nodes — a complete local taxonomy of the web’s skeleton, refined into the modern “caustic skeleton” of large-scale structure (Feldbrugge, van de Weygaert et al. 2018).

Readers of the companion series will recognise the machinery: the same singularity theory names the Maxwell strata and conjugate/caustic wavefronts of the \(\mathrm{SE}(2)\) shortest-path problem — where optimality forks on the group, matter multi-streams in the universe. In both series, the load-bearing objects are not the smooth solutions but the singularities of a map built from a variational principle: the exponential map there, the gravitational Lagrangian map here.

After the fold, ballistic flight and reality part ways — Zel’dovich streams sail through the caustic while real matter, bound by gravity, stays. The adhesion model of Appendix B1 patches this with vanishing viscosity (Burgers’ equation), gluing streams at the caustics and preserving the skeleton. And this series’ own measured correction slots in precisely here: the transverse-damping model found (E4–E5c, Part 2) that the leading error of pure free flight is cured by damping \(\beta \approx 60\%\) of the velocity components perpendicular to the local filament axis — that is, transversally in the tidal eigenframe — while leaving the along-axis flow untouched. In the language of this post: the next-order effect of the real, un-frozen tide is a partial arrest of exactly the motions that geodesic deviation squeezes, applied in the eigenframe of \(E_{ij}\). The empirical correction the program measured is a tidal-tensor term.

Where the geodesic language honestly stops

One refutation from the program belongs in this picture, because it sharpens it. The theory question T1 asked whether filament spines are themselves shortest paths — geodesics of some effective metric — and the answer was no (Part 1): filaments are pile-ups of the flow, not paths of it. The Cartan/gauge reading of this post is consistent with that verdict, and states it better: matter follows geodesics; filaments are the caustics of the geodesic flow — the folds of the free-fall map, organised by the tidal eigenframe. The web is not made of straightest lines; it is made of the places where families of straightest lines focus. That is also why the caustic machinery of the Seeing series (wavefronts, conjugate points) kept resurfacing here: both projects study focusing, not paths.

Where the Euclidean group sits — dial the speed of light

The figure below places the group of §2 in its family. The Euclidean group is the non-relativistic corner of the spacetime symmetries: the limit where the speed of light is infinite, so “now” is shared everywhere and space is just \(\mathrm{SE}(3)\). Slide \(c\) down and the light cone tilts shut, passing through Minkowski (finite \(c\)) on the way to the Carroll limit (\(c \to 0\)), where the cone closes completely — the geometry of a black-hole horizon. Each corner, gauged by the recipe of §4, gives a theory of gravity.

space (c → ∞) horizon (c → 0)

One family, dialled by the speed of light. The wedge is the set of events a signal can reach from the centre — the future "light cone". At left (c → ∞) it opens flat: you can reach anywhere now, time is absolute, and space is the Euclidean group $$\mathrm{SE}(3)$$ — the geometry this series used. In the middle (finite c) it is the 45° Minkowski cone of special relativity, symmetry group Poincaré. At right (c → 0) it closes onto the time axis — the Carroll limit, the intrinsic geometry of a black-hole horizon. Gauging each corner gives a theory of gravity: Newton–Cartan (strictly, one gauges the Bargmann central extension of the Galilei group), Einstein(–Cartan), and Carrollian respectively. Your Euclidean symmetries are the c → ∞ face of the same object that curves into gravity.

Does any of this change Einstein’s equations?

Standard general relativity: no. Set torsion to zero and the whole gauge/Cartan construction reproduces Einstein’s equations exactly. It is a re-reading, not a rewrite.

Einstein–Cartan: minimally, yes. Keep the rotational piece honest and the spin of matter sources torsion through a second equation. Because that equation is algebraic — torsion does not propagate on its own — it vanishes wherever spin density vanishes, so ordinary space, the Solar System, and all current tests are untouched. Corrections switch on only at colossal spin density — the early Universe, neutron-star cores — where they act as an effective repulsion that can replace the Big-Bang singularity with a bounce (Popławski’s scenario).

The honest status. Torsion has never been detected; laboratory and astrophysical searches (spin-polarised matter, Lorentz/torsion-coupling constraint tables) bound specific couplings without seeing anything. Versions in which torsion propagates are further constrained — among other things by the absence of extra gravitational-wave polarisations in current observations. The unification here is therefore conceptual and classical: it says gravity and the geometry of this series are the same kind of object, and it extends Einstein’s equations by a term that is, so far, a whisper — loud only where the cosmic web itself was born.

One blueprint, three geometries

The synthesis the series has been building, in one table:

	Visual cortex (Seeing series)	Cosmic web (this series)	Gravity
Base space	image plane \(\mathbb{R}^2\)	comoving space \(\mathbb{R}^3\)	spacetime \(M^4\)
Symmetry group	\(\mathrm{SE}(2)\)	\(\mathrm{SE}(3)\)	Poincaré \(\mathrm{ISO}(3,1)\)
Lifted / bundle space	\(\mathrm{SE}(2)\) itself	\(\mathbb{R}^3{\times}S^2 = \mathrm{SE}(3)/\mathrm{SO}(2)\)	frame bundle, connection \((e, \hat\omega)\)
Frame field from data	orientation columns of V1	tidal eigenframe \(\{e_1,e_2,e_3\}\)	tetrad \(e^a\)
“Straightest” curves	SR geodesics (cuspidal; smooth face: elastica)	Zel’dovich rays (free fall, frozen tide)	geodesics
Governing tensor	pendulum curvature (elastica face: \(2k\,\mathrm{cn}\))	tidal tensor \(T_{ij} = \partial_i\partial_j\Phi\)	Riemann; electric part \(E_{ij}\)
Where smoothness fails	Maxwell strata, conjugate caustics	pancake caustics, multi-stream folds	conjugate points, horizons
Singularity bookkeeping	Jacobi elliptic clock \(K(k^2)\) (ties at \(4K\), SR cut at \(2K\))	catastrophes \(A_2, A_3, A_4, D_4\)	focusing theorems

Same blueprint each time: a homogeneous model, a frame at every point, a variational flow along it, and the interesting physics concentrated on the singular set where the flow’s map folds.

What is solid, what is active, what is open

• Solid (textbook). Gravity is the gauge theory of the local spacetime symmetry group; Cartan geometry is the common language of Riemannian geometry, the orientation lift, and general relativity (Sharpe 1997; Wise 2010; Blagojević & Hehl 2013). Zel’dovich transport, its caustic classification, and adhesion are the standard analytic theory of the web (Zel’dovich 1970; Arnold–Shandarin–Zel’dovich 1982; Shandarin & Zel’dovich 1989).
• Active. The caustic skeleton — using the catastrophe taxonomy as a practical web-finder in the tidal eigenframe — is a live research line (Feldbrugge, van de Weygaert et al. 2018). Sub-Lorentzian geometry — the sub-Riemannian construction with a Lorentzian metric on the allowed directions — is young and moving (Grochowski and others). Carrollian horizons are a live topic (Donnay & Marteau 2019).
• Open, and close to home. The exact \(\mathrm{SE}(2)\) results of the Geometry of Seeing series — Maxwell strata, cut loci via elliptic functions — have sub-Lorentzian analogues on the relativistic cousins of \(\mathrm{SE}(2)\) that, as far as I can tell, nobody has worked out. And in the other direction: whether the measured transverse-damping coefficient \(\beta\) of this series can be derived — as the leading post-Zel’dovich tidal term in the eigenframe, rather than fit — is exactly the kind of question the dictionary of §5–6 makes well-posed.

Glossary

• Euclidean group \(\mathrm{SE}(3) = \mathbb{R}^3 \rtimes \mathrm{SO}(3)\) — rigid motions of space: rotations acting on translations in a semidirect product; equally, the bundle of oriented orthonormal frames of \(\mathbb{R}^3\).
• Semidirect product — a product of two groups in which one acts on the other; the reason \((R_2,\mathbf{t}_2)(R_1,\mathbf{t}_1) = (R_2R_1,\, R_2\mathbf{t}_1 + \mathbf{t}_2)\).
• Klein geometry — a geometry presented as a symmetry group \(G\) and a subgroup \(H\) fixing a basepoint; the space is the homogeneous quotient \(G/H\).
• Cartan geometry — a curved space modelled infinitesimally on a Klein geometry, with a connection comparing neighbouring model copies; contains Riemannian geometry, the orientation lift, and general relativity as instances.
• Solder form / frame / tetrad — the translational part \(e\) of the Cartan connection: an orthonormal set of directions at each point, carrier of the metric.
• Spin connection — the rotational part \(\hat\omega\): how the frame turns from point to point; its field strength is the curvature.
• Torsion — the translational field strength \(\Theta = de + \hat\omega \wedge e\); failure of infinitesimal parallelograms to close; sourced by spin in Einstein–Cartan.
• Holonomy — the net rotation a vector acquires under parallel transport around a closed loop; curvature integrated over the enclosed area.
• Anholonomy — failure of a frame to be a coordinate basis; the same object as the sub-Riemannian bracket in Chow–Rashevskii, and the field strength of teleparallel gravity.
• Geodesic deviation — the relative acceleration of neighbouring free-fallers, \(D^2\xi/d\tau^2 = -R(u,\xi)u\); the operational meaning of curvature.
• Electric part of Riemann / tidal tensor — \(E_{ij} = R_{i0j0}\), in the Newtonian limit the potential’s Hessian \(T_{ij} = \partial_i\partial_j\Phi\); its eigenframe is the tidal frame of Appendix B2.
• Zel’dovich approximation — free flight in growth-factor time with the initial tide as the only push: \(\mathbf{x} = \mathbf{q} - D\,\nabla_q\Phi^{(1)}\).
• Deformation tensor — \(\partial\mathbf{x}/\partial\mathbf{q} = \mathbb{1} - D\,T\); its eigenvalue factors \((1-D\lambda_i)\) set the collapse sequence and the density \(1/\prod(1-D\lambda_i)\).
• Caustic / catastrophe — the fold set of a Lagrangian map, where its Jacobian vanishes and streams cross; classified generically by catastrophe theory (\(A_2\) folds, \(A_3\) cusps, …); the walls and filaments of the web.
• Einstein–Cartan theory — general relativity plus torsion, from gauging the full Poincaré group; reduces to GR wherever spin density is zero.
• Sub-Lorentzian geometry — a Lorentzian metric on a bracket-generating distribution; the relativistic sibling of the sub-Riemannian geometry used throughout this series.
• Carroll limit — the \(c \to 0\) contraction of Minkowski geometry; the light cone closes; the intrinsic geometry of null surfaces and black-hole horizons.

References

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2. É. Cartan (1923). "Sur les variétés à connexion affine et la théorie de la relativité généralisée." Ann. Sci. ENS 40, 325–412.
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6. Ya. B. Zel'dovich (1970). "Gravitational instability: an approximate theory for large density perturbations." Astron. Astrophys. 5, 84–89.
7. F. W. Hehl, P. von der Heyde, G. D. Kerlick & J. M. Nester (1976). "General relativity with spin and torsion: foundations and prospects." Rev. Mod. Phys. 48, 393–416.
8. V. I. Arnold, S. F. Shandarin & Ya. B. Zel'dovich (1982). "The large scale structure of the universe I: general properties; one- and two-dimensional models." Geophys. Astrophys. Fluid Dyn. 20, 111–130.
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1. A product \(N \rtimes K\) of two groups where the second acts on the first: elements are pairs, but composition twists the \(N\)-component by the \(K\)-action, \((n_2,k_2)(n_1,k_1) = (n_2 \cdot k_2(n_1),\, k_2k_1)\). For \(\mathrm{SE}(3)\): rotations act on translation vectors by rotating them. ↩

2. A space on which a group acts transitively — any point can be moved to any other. Equivalently \(G/H\) for \(H\) the stabiliser of a basepoint; all points look alike because the group says so. ↩

3. The tangent space of a Lie group at the identity: the infinitesimal versions of the group’s motions, with the commutator bracket \([X,Y] = XY - YX\) recording how little motions fail to commute. Appendix A1 of the Seeing series builds it from scratch for \(\mathrm{SE}(2)\). ↩

4. Replace the flat model by de Sitter space (the maximally symmetric solution with cosmological constant \(\Lambda\)) and the “translations” no longer commute: \([P_i, P_j] \propto \Lambda\, J_{ij}\). The cosmological constant is, in Cartan language, the curvature you build into the model itself (MacDowell–Mansouri; Wise 2010). ↩

5. A 1-form eats a direction and returns a number (here, an algebra element) — the right gadget for “what happens if I step this way”. A 2-form eats a little parallelogram; that is why curvature and torsion, which measure loop defects, are 2-forms: \(d\) and \(\wedge\) assemble loop-answers from step-answers. ↩

6. Two loose ends the construction also explains: the metric is derived (from \(e\)), not fundamental; and coupling fermions to gravity — which textbook GR does awkwardly — is automatic, since spinors talk to the tetrad and spin connection directly. ↩

7. The curve a free particle follows: straightest possible, extremal proper time. In Cartan’s rolling picture, the development of a straight line of the flat model. ↩

8. The factor \(D(t)\) by which small density fluctuations grow in linear theory; using it as the time variable absorbs the cosmic expansion so that Zel’dovich motion is uniform and straight. See Appendix B1. ↩

9. The optical analogy is exact, not decorative: light rays are geodesics of an effective metric, the bright curves are where the ray map folds, and the classification of stable fold shapes (fold, cusp, swallowtail…) is the same Arnold catastrophe list that organises the web’s walls and filament edges. ↩