Causal Geodesy: Geometry & Inference

• Causal geodesy is the integration of geometric, analytic, and statistical methods to encode, reconstruct, and infer causal structures and geodesic behavior across various geometries.
• It extends classical Lorentzian models with Randers/Finsler, discrete, and noncommutative frameworks, enabling causal inference for outcomes in complex spaces like networks and probability measures.
• Methodologies include variational principles, null distance metrics, and geodesic causal inference to quantify intervention effects and analyze data expressed as random objects.

Causal geodesy encompasses a suite of geometric, analytic, information-theoretic, and statistical frameworks that encode, reconstruct, and infer causal structure, geodesic behavior, and intervention effects in settings where the geometry is Lorentzian, Finslerian, noncommutative, topological, or even purely metric or metric-measure-theoretic. Across these domains, causal geodesy provides powerful tools for relating causal structure to geometry and for generalizing causal inference to data whose outcomes are "random objects" in complex spaces, such as networks, probability measures, or compositional data.

1. Causal Structure and its Geometric Encoding

Classical causal geodesy arises from the encoding of the causal (i.e., conformal) structure of a spacetime in its geometry. For Lorentzian manifolds, the causal structure is fully determined by the pattern of null cones and can be reconstructed (e.g., via the Hawking-King-McCarthy-Malament theorem) from the causal order relation $(M, \prec)$ and local volume data $\epsilon$, up to a conformal factor (Eichhorn et al., 2018).

Parameterized Randers/Finsler structures generalize this by demonstrating that for stably causal spacetimes, the causal structure is isomorphic to a time-parameterized family of spatial Randers norms derived from the ADM decomposition (Skakala et al., 2010). The crucial object is

$$R^+(t, x, dx) = \sqrt{h_{ij}(t, x)\,dx^i\,dx^j} + V_i(t, x)\,dx^i,$$

where $(h_{ij}, V_i)$ encode the nonstationary aspects of the geometry. Null geodesics, causal curves, and their boundaries (the null cones) can be reconstructed from random spatial slices equipped with this Finslerian data, and light propagation becomes a Fermat-type variational principle.

Further generalizations dispense with the quadratic structure of Lorentzian geometry, introducing regular conical (e.g., Legendrian/contact) subbundles defined by $G(x, v) = 0$, where $G$ is a smooth, homogeneous function, not necessarily quadratic (Holland et al., 2011). Null geodesics are then defined as Legendrian flows on the projectivized tangent bundle, and the full machinery of contact geometry is deployed to analyze tidal forces, the generalized Raychaudhuri equation, and the extension of Weyl curvature.

2. Causality in Nonstandard and Discrete Geometries

Causal geodesy is not restricted to smooth Lorentzian geometries. In causal set theory, a spacetime is replaced by a partially ordered set (poset) with a count measure as the volume element. Here, spatial distance is reconstructed from the causal order and the volume, using suspended-causal-beam constructions and path minimization with mesoscale cutoffs, and can accurately recover induced geometry under sufficient scale separation (Eichhorn et al., 2018). In the context of almost commutative and noncommutative geometries, the causal structure can be "split" between spacetime and finite-dimensional internal degrees of freedom, and causal ordering restricts both the trajectory in spacetime and evolution in the internal space, favoring unitary transformations that preserve causality (Franco et al., 2013).

Notably, on the conformal boundary (ambient infinity), only horismos (null-cone) order is topologically compatible with the Zeeman topology (Antoniadis et al., 2016), yielding that only null (massless) geodesics are meaningful on the boundary, with ordinary (timelike) temporal structure erased by conformal invariance.

3. Metric and Topological Approaches to Causality

A major advance is the formalization of causal spaces equipped with a "null distance" and a regular cosmological time function $\tau$ (Sakovich et al., 2022). In this construction,

$$\hat{d}_\tau(p, q) := \inf\left\{ \sum_{i} |\tau(x_i) - \tau(x_{i-1})| : \text{piecewise causal curves from } p \to q \right\},$$

where $\hat{d}_\tau(p, q) = \tau(q) - \tau(p)$ if and only if $q$ is in the causal future of $p$. If a bijection preserves both the null distance and cosmological time function, it must be a Lorentzian isometry, yielding a canonical metric space structure that fully encodes the original causal geometry.

Order-theoretic and cone-theoretic approaches clarify that much of a spacetime's geometry and topology can be recovered from the causal order alone, through K-causality, domain-theoretic intervals, and cone-preserving maps (Saraykar et al., 2014). In these frameworks, maximal causal curves (geodesics) are reconstructed from the transitive closure of the order relation.

4. Causal Inference in Geodesic Metric Spaces ("Geodesic Causal Inference")

Causal geodesy in the context of statistical inference extends causal methods to outcomes valued in general geodesic metric spaces (Kurisu et al., 28 Jun 2024). Let $(\mathcal{M}, d)$ be a uniquely geodesic space (e.g., networks, covariance matrices, probability measures, compositional data). The paper introduces:

• Geodesic Calculus: Generalizes classical vector arithmetic. Given $\alpha,\beta\in\mathcal{M}$, their connecting geodesic $\gamma_{\alpha,\beta}$ is the fundamental object, and operations such as concatenation, inversion, and scalar multiplication ($\rho\odot\gamma_{\alpha,\beta}$) are defined using two-sided extensions of geodesics.
• Geodesic Average Treatment Effect (GATE): For potential outcomes $Y_i(0), Y_i(1)\in\mathcal{M}$, the individual effect is the geodesic $\gamma_{Y_i(0),Y_i(1)}$. The population GATE is $\gamma_{E_+[Y(0)], E_+[Y(1)]}$, where $E_+[Y(t)]$ is the Fréchet (intrinsic) mean of $Y(t)$.
• Estimation and Uncertainty Quantification: Estimators are constructed via doubly robust procedures involving (i) propensity score modeling $e(x)$ and (ii) Fréchet regression for metric-valued outcomes $m_t(x) = E_+[Y(t)|X=x]$. Doubly robust estimators minimize

$$Q_{n,t}(\nu;\mu_t,\phi) = \frac{1}{n} \sum_{i=1}^n d^2\left(\nu, \gamma_{\mu_t(X_i),Y_i}(\dotsc)\right)$$

for $\nu\in\mathcal{M}$, yielding $\Theta_t^{DR}$. Consistency and quantifiable convergence rates are proven under standard overlap and regularity assumptions. Adaptive subsampling (HulC) methods construct confidence regions for the geodesic distance between Fréchet means.

• Applications: The methodology is demonstrated on US energy source compositional data, New York taxi trip network Laplacians (capturing COVID-19 effects), and brain functional connectivity networks in Alzheimer's disease.

This approach allows for rigorous inference on causal contrasts (treatment effects) where traditional vector space subtraction does not exist, generalizing causal statistical methodology to "random object" outcomes such as shapes, graphs, or probability measures, with nontrivial geometry.

5. Analytic and Variational Principle Approaches

In variational and analytical approaches to causal structures—especially in theories beyond standard Lorentzian geometry, such as the causal variational principle—global solutions to field equations are constructed by gluing local weak solutions and introducing causal Green's operators (Finster et al., 2022). The cone structure ("causal cones") is encoded in the propagation properties of these solutions, and exact sequences involving causal fundamental solutions formalize the analogs of null geodesics.

In Lyra geometry, the modification of Einstein's field equations by the inclusion of a displacement vector (arising from a gauge function) changes the causal and geodesic structure, allowing, for instance, entirely causal solutions of the Gödel-type metrics even in the absence of matter sources—novel in comparison to standard Riemannian theory (Jesus et al., 2018).

6. Information Geometry and Causal Emergence

Causal geometry, in the sense of information geometry, considers the congruence between effect and intervention manifolds equipped with Fisher information metrics (Chvykov et al., 2020). The effective informativeness of a causal model—the "effective information" (EI)—depends on the geometric alignment between intervention-induced and effect-induced variability. Coarse-grained (macroscopic) models may, due to better alignment with feasible interventions, have higher EI than microscopic models, an effect termed causal emergence.

Causal inference frameworks for geoscientific and remote sensing data leverage regression (additive noise) models, kernel sensitivity maps, and independence criteria (e.g., HSIC), along with causal graphs, to robustly infer causal direction and quantify treatment effects even in complex, observational earth systems data (Pérez-Suay et al., 2020, Pérez-Suay et al., 2020, Massmann et al., 2021). These methods have enabled the detection of directionality in compositional, functional, and network data directly linked to geodetic and environmental outcomes.

7. Summary Table: Major Formalisms in Causal Geodesy

Framework	Causal Structure Input	Geodesic/Distance Concept	Application Domains
Lorentzian/Randers geometry	Metric/signature, ADM, etc	Null geodesics, Fermat, Randers	Relativity, optical geometry
Contact/Legendrian geometry	Conical bundle $G(x,v)=0$	Legendrian dynamics, null spray	Generalized gravity, Finslerian settings
Causal sets	Order $(M, \prec)$, vol.	Discrete spatial distance	Discrete/quantum gravity
Null distance encoding	Cosmological time $\tau$	Null length $\hat{d}_\tau$	Metric representation of causality
Geodesic causal inference	Covariates + metric space	Geodesic between Fréchet means	Causal inference for random objects
Information geometry	Model/interventions	Geometric effective information	Statistical modeling, causal emergence
Causal Green’s operators	Variational field eqns	Causal cones via solution support	Causal variational principles

• Causal structure of spacetime via parameterized Randers geometry (Skakala et al., 2010)
• Generalized causal geometries, null geodesics, Legendrian dynamics (Holland et al., 2011)
• Null and horismos order on the ambient boundary (Antoniadis et al., 2016)
• Encoding causality with null distance and cosmological time (Sakovich et al., 2022)
• Geodesic causal inference for random object outcomes (Kurisu et al., 28 Jun 2024)
• Causal information geometry and causal emergence (Chvykov et al., 2020)
• Induced spatial geometry from discrete or causal set order (Eichhorn et al., 2018)
• Gluing solutions and causal cones in causal variational principles (Finster et al., 2022)

Causal geodesy is thus the unification of geometric and information-theoretic approaches to understanding and quantifying causal structure, propagation, and inference, encompassing settings from differential to discrete geometry, from theoretical physics to statistical data science.