Centralized Reference Frames

• Centralized reference frames are systems of spatial and temporal coordinates maintained by a central authority to ensure unique and stable positioning for celestial and terrestrial applications.
• They employ rigorous mathematical techniques such as block-graph Laplacian construction, singular value decomposition, and generalized least-squares inversion to reconcile diverse observational data.
• Their implementation underpins high-precision astrometry, geodesy, and navigation by addressing systematic, statistical, and relativistic effects for long-term stability.

A centralized reference frame is a system of spatial and temporal coordinates, together with well-defined transformation laws, constructed and maintained by a central authority or observing network to provide a stable, unique, and globally consistent basis for locating objects in space and time. Such frames are foundational in astrometry, geodesy, relativistic positioning, and celestial mechanics, and serve as the backbone of international standards such as the International Celestial Reference Frame (ICRF) and International Terrestrial Reference Frame (ITRF). Centralization entails not only the averaging and synthesis of independent observables, but also rigorous handling of statistical, systematic, and relativistic effects to ensure long-term stability and epistemic continuity.

1. Mathematical Foundation and Consistency

Centralized reference frames rest on a formal relationship between coordinate systems and the underlying physics or measurement network. In Euclidean and affine geometries, a reference frame can be represented by an invertible linear transformation, and the collection of transformations linking local frames must satisfy transitive consistency: for any nodes $i, j, k$, the transformation $T_{ik}$ must equal $T_{jk}T_{ij}$, and, in noisy practical scenarios, the central task is finding a set $\{T_i\}$ minimizing aggregate inconsistency relative to measured links $\widehat T_{ij}$ (Thunberg et al., 2015). Rigorous solutions rely on centralized least-squares methodologies, such as constructing a block-graph Laplacian or Hessian, and extracting global frame generators via singular value decomposition or eigendecomposition:

$$\min_{\{T_i\}} \sum_{(i,j)\in E} \|\widehat T_{ij} - T_j T_i^{-1}\|_F^2$$

This centralized synchronization approach is essential for establishing a unique, optimally consistent global reference frame from a network of relative measurements, with provable optimality and quadratic convergence under Gauss-Newton refinement.

Centralization is further formalized in the context of generalized least-squares inversion in VLBI-based reference frames, where normal equation systems (NES) from many observing sessions are stacked to yield a unified adjustment for both terrestrial (TRF) and celestial (CRF) frames. The full covariance is propagated through the process, allowing all variance-covariance information to be retained at the global level (Krásná et al., 2022, Karbon et al., 2019).

2. Realization in Celestial and Terrestrial Contexts

Global reference frames are constructed and maintained via central processing of data from geodetic networks (GNSS, VLBI) and astrometric observatories. The ICRF (radio) and Gaia-CRF (optical) represent the canonical celestial reference frames, anchored either to emission centroids of extragalactic radio sources or to quasars observed in the optical band (Eubanks, 2020, Krásná et al., 2022). Centralized realization is achieved via unified least-squares fitting across all VLBI sessions, simultaneously adjusting station positions, radio source locations, Earth orientation parameters (EOP), and nuisance parameters, with datum definitions (typically no-net-rotation and no-net-translation conditions) applied to selected fiducial sources and stations.

As an example, the VIE2022b solution (Krásná et al., 2022) centrally adjusts $N\approx 120$ stations and $>5,000$ radio sources, using a seven-parameter Helmert transform to tie the solution to ITRF2020/ICRF3. This tightly couples the terrestrial and celestial frames, ensuring mutual consistency and mm/μas-level alignment:

$$X_\text{VIE} = (1+s)\cdot R(\varepsilon)\cdot X_\text{ITRF} + T$$

where $R(\varepsilon)$ encodes small rotations, $s$ is a scale parameter, and $T$ is translation.

3. Centralized Astrometric Reference Frames: Advanced Examples

Recent advances have enabled the construction of centralized reference frames in difficult regions (e.g., the Galactic center), merging heterogeneous data. The "HST-Gaia" frame (Jr. et al., 24 Jun 2025) centrally integrates near-IR HST data (14 epochs) with Gaia-DR3, using a two-stage approach: (1) reference stars are cross-matched and a full polynomial transformation to Gaia-CRF is determined (with propagation of errors via bootstrap resampling), and (2) proper motions are fit via Gaussian Processes that explicitly model systematic errors. Centralization enables these epochs and measurement systems to be merged into a single, ICRS-tied reference frame with an accuracy floor of $\sim$0.03 mas/yr (proper motion) and $\sim$0.04 mas (position).

Similarly, multifrequency VLBI catalogs are combined by stacking their normal equations and imposing no-net-rotation boundary conditions, ensuring that the resulting frame has no global deformations and preserves inter-catalog covariances (Karbon et al., 2019). This multifrequency approach mitigates source structure systematics by leveraging independent networks and frequency bands.

Network/Instrument	Epoch Spanned	Core Methodology	Accuracy Achieved
VIE2022b/IVS	1979–2022.5	Global least squares	$\sim$143 μas (α*), 250 μas (δ)
HST-Gaia frame	2010–2023.5	Poly/GP+ICRS tie	$\sim$0.03 mas/yr, 0.045 mas
Multifrequency CRF	1980s–2015	NES stacking	$\sim$0.1 mas global errors

4. Relativistic and Physical Anchoring

In relativistic contexts, the physical definition of a reference frame requires specifying not just coordinate charts, but observer congruences and tetrad transport laws. In exact general relativity, centralized frames are constructed from congruences of worldlines (observers), equipped with spatial triads that are Fermi–Walker transported to guarantee nonrotation relative to distant inertial frames (Costa et al., 8 Oct 2025). The mathematical requirement for nonrotating central frames is that the observer congruence be shear-free, with vorticity and acceleration vanishing at spatial infinity:

$$\lim_{r\to\infty} \omega_{\alpha\beta} = 0, \qquad \lim_{r\to\infty} a^\alpha = 0$$

This definition ensures the spatial axes are anchored to “fixed stars.” In practice, the IAU BCRS and GCRS (post-Newtonian) are specific instantiations designed to approximate these principles in the solar system, serving as central systems for relativity-corrected astrometry (Costa et al., 8 Oct 2025).

A key controversy addressed is the use of zero angular momentum observers (ZAMOs) which, while locally nonrotating and irrotational, admit nonzero shear and so do not provide the spatial rigidity relative to distant inertial directions required of centralized, astronomically meaningful frames.

5. Challenges, Limitations, and Advanced Anchors

Centralized reference frames encounter intrinsic limitations when their fiducials (e.g., AGN jet centroids) are unstable. The ICRF and Gaia-CRF2 are limited by source structure variability (jet bending, “core shift”) and frequency-dependent systematics, with current systematic floors of 30–100 μas and stability compromised at the 5–10 μas/yr level (Eubanks, 2020). This has motivated anchoring frames directly to physically immutable features such as SMBH shadows, imaged by the EHT as emission rings of 10–40 μas diameter. The use of photon ring centers as CRF fiducials, enabled by a dedicated space-VLBI array (with sub-μas astrometric capability), is projected to yield a 0.3 μas accuracy floor and <0.01 μas/yr drift—improvement by two orders of magnitude over jet-based frames (Eubanks, 2020).

Current limitations also arise in multifrequency combinations: core shift is frequency-dependent and can reach ~100 μas unless directly modeled; multifrequency NES stacking can reduce deformations but only if cross-band systematics are adequately controlled (Karbon et al., 2019).

Further, in the GNSS context, the centralized, earth-fixed approach (ITRF, GCRF) is inherently limited by uncorrected dynamical effects (Earth rotation, plate tectonics, tides). Proposed alternatives such as the Autonomous Basis of Coordinates (ABC) concept use only inter-satellite emission coordinates and relativistic modeling to achieve mm-level, locally inertial, fully autonomous realization, but these are inherently non-centralized (Kostić et al., 2014).

6. Centralization: Operational Considerations and Reference Frame Updating

Centralized reference frames require substantial infrastructure, both in terms of observing networks (VLBI arrays, GNSS ground stations, optical astrometry satellites), rigorous calibration (clock stability, atmospheric modeling, station motion), and repeated, system-wide reprocessing and validation. The inclusion of new data (e.g., additional VGOS stations, deeper sky coverage, multi-epoch datasets) invariably necessitates recomputation of the global adjustment so as to absorb and redistribute estimation improvements throughout the entire frame (Krásná et al., 2022, Karbon et al., 2019, Jr. et al., 24 Jun 2025).

The handling of systematic error terms—be it radio-source structure evolution, tropospheric gradients, or galactic aberration—demands continuous methodological innovation (e.g., Gaussian Process modeling of proper motions (Jr. et al., 24 Jun 2025), vector spherical harmonic decomposition (Karbon et al., 2019)) to diagnose and mitigate frame distortions at the μas level.

7. Implications for Science and Technology

Centralized reference frames underpin fundamental and applied science, enabling high-precision astrometry, relativistic geodesy, orbital dynamics, navigation, and cosmology. The transition to physically anchored, sub-μas-precision frames (e.g., SMBH-shadow-based) will enable detection of phenomena such as galactic aberration ($\sim6$ μas/yr) or cosmological proper motions ($\sim0.1$–$1$ μas/yr), previously lost in the noise floor (Eubanks, 2020). The alignment of multifrequency and multi-instrument frames is vital for cross-wavelength astrophysics and the validation of reference frames in extreme environments (Galactic Center, radio/optical discrepancies) (Jr. et al., 24 Jun 2025).

However, extreme caution must be exercised in the interpretation and use of frames that are not globally nonrotating in the asymptotic sense, as inadequately anchored or misused constructs (e.g., using ZAMOs as inertial) yield physically spurious artifacts and undermine the long-term stability required for extranatural metrology and future mission planning (Costa et al., 8 Oct 2025).