Dynamical Reference Frames (DRFs)

• Dynamical reference frames are physical systems whose dynamic degrees of freedom, such as clocks and rods, define observables relationally.
• They span applications in general relativity, quantum mechanics, and celestial mechanics, enabling background independence and self-calibrating coordinates.
• Approaches include utilizing dynamical frame fields in GR, unitary transformations in quantum systems, and satellite constellations in relativistic positioning.

Dynamical Reference Frames (DRFs) are reference systems whose defining rods, clocks, coordinate labels, or embedding variables are themselves physical degrees of freedom rather than fixed background structures. Across the literature, the term spans several closely related but non-identical uses: in background-independent and generally covariant theories, a DRF is a material subsystem or field configuration used to localize observables relationally; in quantum-reference-frame formalisms, it is a quantum system whose external and internal degrees of freedom define spatial origin and clock variables; in relativistic positioning, it is a self-calibrating constellation defining coordinates through signal exchange; and in celestial mechanics, it is a frame realized by dynamical laws such as planetary ephemerides or spin-orbit resonances (Smolin, 2020, Bamonti, 2023, Kostić et al., 2014, Liu et al., 2022, Stark et al., 2017). The unifying idea is that the frame is treated as part of the physical system, so the distinction between observer, clock, and observed subsystem becomes contingent rather than fundamental.

1. Definitions and conceptual scope

In a background-independent theory, nothing—including clocks, rods, or observers—may be treated as fixed and non-dynamical. On that basis, a DRF is any subsystem whose internal degrees of freedom, when suitably decoupled or “frozen” to some approximation, serve to define readings of time, space, momentum, or other quantities. In the canonical language described by Smolin, variables ordinarily treated as external parameters, such as the time parameter $s$, the symplectic structure, and the Hilbert-space norm, are promoted to dynamical operators or functions (Smolin, 2020).

In General Relativity, Bamonti distinguishes Idealised Reference Frames, Dynamical Reference Frames, and Real Reference Frames. In that taxonomy, a DRF is a material system $\Phi$ whose own equations of motion are retained, while its stress-energy contribution is neglected in Einstein’s equations. The geometry acts on the frame, but the frame does not gravitate back on the geometry (Bamonti, 2023). A closely related classification appears in a later analysis of symmetry principles and observability: DRFs satisfy the frame-field equations of motion coupled to $g_{ab}$, but have $T_{ab}[\phi]=0$ in the metric field equation; Real Reference Frames retain both equations of motion and stress-energy, whereas Idealised Reference Frames neglect both (Bamonti et al., 2024).

In relativistic quantum theory, a DRF is associated to a quantum system whose position and momentum Hilbert space fixes the reference frame and whose internal Hilbert space provides a clock. The frame is therefore both a spatial origin and a temporal standard, and covariance under frame changes is implemented by unitary transformations between quantum reference frames (Giacomini, 2021). In the earlier nonrelativistic quantum-reference-frame formalism, a dynamical quantum reference frame is a quantum system $A$ with its own Hilbert space and Hamiltonian, whose center-of-mass degrees of freedom define the frame relative to which other systems are described (Giacomini et al., 2017).

A persistent theme in the GR literature is that a reference frame is not the same thing as a coordinate chart. In one formal definition, a reference frame is given by a main observer, a foliation by temporal layers $J_t$, and a congruence of local observers that agree on simultaneity within each layer. The adapted coordinates are then constructed from that physical structure rather than imposed independently (Drivotin, 2014). This distinction is sharpened by the claim that partial observables are relational even when gauge-variant, while complete observables require dynamical coupling to become gauge-invariant (Bamonti et al., 2024).

Context	DRF content	Characteristic feature
Background-independent mechanics	Dynamical clock/frame subsystem	External structures promoted to dynamical variables
General Relativity	Material frame fields with retained EOM	Relational localization of observables
Quantum reference frames	Quantum system + internal clock	Frame changes are unitary
Relativistic positioning	Satellite constellation	Coordinates from emission times
Celestial mechanics/astrometry	Ephemeris or resonant rotation model	Axes fixed by dynamics

2. Classical and relativistic structures

A formal classical and relativistic definition of reference frames starts from a spacetime manifold $M$, a main observer with worldline $L_0$, a congruence $C$ of local observers, and a foliation $D=\cup_t J_t$ by simultaneity layers. In the nonrelativistic case, the same foliation applies to all observers; on each layer $\Phi$0, a Riemannian 3-metric $\Phi$1 defines spatial distances. In the relativistic case, simultaneity is frame-dependent, but one can still introduce adapted coordinates in which $\Phi$2 is the time reading of the main observer and $\Phi$3 are constant along local-observer worldlines (Drivotin, 2014).

The key metric condition in such adapted relativistic coordinates is

$\Phi$4

This condition expresses the separation of temporal and spatial coordinates within the chosen frame. The same source states that the existence of a time function whose level sets are the simultaneity leaves is equivalent to an integrability condition involving the 1-form

$\Phi$5

with $\Phi$6 as the criterion for finding adapted coordinates in which the mixed components vanish identically (Drivotin, 2014).

Examples make the distinction between coordinates and frames explicit. In Minkowski space, standard inertial Cartesian coordinates trivially satisfy $\Phi$7. In a uniformly accelerated frame, one again obtains adapted coordinates with $\Phi$8, while $\Phi$9 becomes position-dependent. By contrast, standard rotating cylindrical coordinates yield $g_{ab}$0; this is presented as a case in which $g_{ab}$1 is not a pure spatial coordinate for the frame unless $g_{ab}$2 (Drivotin, 2014). A common misconception is therefore that any convenient coordinate transformation defines a relativistic reference frame in the same physical sense. The cited construction rejects that identification.

In the generally covariant setting, DRFs are used to build relational, gauge-invariant observables. If four scalar frame fields $g_{ab}$3 define a local diffeomorphism from spacetime to “physical coordinates,” then tensor fields can be re-expressed relationally; for example,

$g_{ab}$4

Because the frame fields are themselves dynamical variables, such quantities are invariant under active diffeomorphisms (Bamonti, 2023). The same logic underlies the more general “universal dressing space” formalism, in which a dynamical frame is an arbitrary subset of dressings $g_{ab}$5, and a relational observable is obtained by pulling back a covariant field along a frame map $g_{ab}$6: $g_{ab}$7 This framework is presented as fully non-perturbative and as encompassing matter frames, boundary-anchored geodesic frames, and minimal-surface constructions (Goeller et al., 2022).

The gravitational literature also distinguishes relational invariance from gauge invariance. A relational quantity can be invariant under the diagonal action of diffeomorphisms on geometry and frame fields while still failing to be deterministic if the frame fields are dynamically uncoupled. This distinction is central to later discussions of symmetry principles and the Hole Argument (Bamonti et al., 2024, Bamonti et al., 2024).

3. Quantum and spacetime quantum reference frames

In the quantum-reference-frame literature, the basic operation is a unitary change of perspective from one physical system to another. For systems $g_{ab}$8, the change from frame $g_{ab}$9 to frame $T_{ab}[\phi]=0$0 is implemented by a unitary $T_{ab}[\phi]=0$1 acting on the joint Hilbert space. In the simplest one-dimensional equal-time case,

$T_{ab}[\phi]=0$2

where $T_{ab}[\phi]=0$3 swaps $T_{ab}[\phi]=0$4 and $T_{ab}[\phi]=0$5 and flips sign appropriately. States, observables, and Hamiltonians transform by conjugation, with the transformed Hamiltonian given by

$T_{ab}[\phi]=0$6

This formalism is used to argue that entanglement and superposition are frame-dependent features, and that covariance of physical laws extends to “superpositions of coordinate transformations” (Giacomini et al., 2017).

A major extension treats spacetime quantum reference frames in a weak gravitational field. There, each particle carries external degrees of freedom in $T_{ab}[\phi]=0$7 and internal clock degrees of freedom in $T_{ab}[\phi]=0$8, with $T_{ab}[\phi]=0$9. The total physical state is constrained by particle dispersion relations, a global Page–Wootters-type energy constraint, and global spatial translation invariance (Giacomini, 2021). Taking the perspective of one particle as QRF involves a unitary $A$0 that maps to relational coordinates and a projection onto the condition that the chosen particle sits at the origin of the new frame. The resulting relational history state has the Page–Wootters form

$A$1

The global state remains “frozen,” while dynamics re-emerges in terms of the proper time of the frame system (Giacomini, 2021).

A central technical result is that proper times in different QRFs are related by operators rather than c-number transformations. The proper-time relation is written

$A$2

with

$A$3

In the special-relativistic limit, momentum superposition produces a superposition of time dilations; in the Newtonian gravitational limit, position superposition produces a superposition of gravitational redshifts (Giacomini, 2021). This suggests that “time coordinate transformations” between quantum frames are themselves quantum objects rather than classical functions.

Related work studies indefinite metrics generated by mass configurations in superposition. There, a QRF transformation

$A$4

is used to move to a frame in which a superposed mass distribution becomes definite, provided the branches satisfy a distance-preserving condition. Known classical dynamics can then be applied in that branch-definite frame, and transforming back yields superpositions of time dilations and probe dynamics associated with the superposed gravitational source (Hamette et al., 2021).

The dynamical aspect of QRFs also changes reduced dynamics. A subsystem that evolves unitarily in one frame can decohere in another. In Tuziemski’s analysis, the reduced state of a subsystem in the new frame acquires a decoherence factor

$A$5

and the reduced dynamics becomes completely positive and trace-preserving rather than unitary (Tuziemski, 2020). The same work argues that Quantum Darwinism and Spectrum Broadcast Structures can therefore be frame-dependent.

The transformation laws between quantum reference frames also possess a nontrivial group structure. Linear canonical transformations on the phase space of the two systems close a Lie algebra $A$6, generated by seven quadratic operators, and exponentiate to a seven-parameter Lie group $A$7. In the classical-frame limit $A$8, this structure reduces to the centrally extended Galilei algebra (Ballesteros et al., 2020).

4. Triality, matrix models, and emergent temporal frames

One background-independent proposal places DRFs inside a larger symmetry that exchanges the reference system with the degrees of freedom measured relative to it. In Smolin’s formulation, if $A$9 denote the usual $J_t$0 phase-space variables, a DRF adds a clock degree of freedom $J_t$1, so that the total phase space is $J_t$2-dimensional (Smolin, 2020). Instead of the ordinary symplectic two-form on a $J_t$3-dimensional manifold, the enlarged manifold $J_t$4 carries a closed, totally antisymmetric three-form

$J_t$5

with $J_t$6. This three-form supplies a cubic invariant that generalizes the quadratic symplectic invariant and supports a triality mixing clock, position, and momentum variables (Smolin, 2020).

The simplest realization is formulated in terms of three objects $J_t$7, where $J_t$8 encodes the frame degree of freedom, $J_t$9 the would-be positions, and $M$0 the would-be momenta. The permutation group $M$1, or its continuous analogue $M$2, acts on this triple, and any one of the three can play the role of clock while the remaining two form a duality pair (Smolin, 2020). This is presented as an extension of Born duality to a triality that also includes the temporal reference frame.

The corresponding cubic matrix model is based on a single traceless Hermitian matrix $M$3 with action

$M$4

After the decomposition

$M$5

the action becomes

$M$6

which is manifestly invariant under discrete triality, global $M$7, and a large gauge symmetry $M$8. The equations of motion are

$M$9

or, with quadratic mass-like terms and Lagrange multipliers,

$L_0$0

A particular solution in which

$L_0$1

selects a preferred clock operator $L_0$2 and breaks the triality to the ordinary duality between the remaining directions. This is interpreted as spontaneous symmetry breaking that freezes one DRF degree of freedom into the background and recovers the canonical two-form on the residual $L_0$3-dimensional sector (Smolin, 2020).

The same framework is claimed to accommodate both classical and quantum mechanics. In the commuting or diagonal-matrix limit, the cubic action reproduces classical mechanics of oscillators or free particles on a circle. In the large-matrix limit, introducing an operator $L_0$4 with $L_0$5 yields Heisenberg equations

$L_0$6

The proposal therefore treats quantization as a limit of purely classical matrix dynamics (Smolin, 2020). A plausible implication is that, in this framework, the emergence of time, canonical structure, and quantization are all tied to the same symmetry-breaking mechanism.

5. Operational realizations and empirical uses

A concrete relativistic realization of a dynamical frame is the Autonomous Basis of Coordinates for GNSS. Four satellites, labeled $L_0$7, broadcast their proper times $L_0$8, and a user receiving all four signals at one event obtains emission coordinates

$L_0$9

These coordinates are related to the reception event $C$0 by four null conditions

$C$1

Solving them yields the coordinate transformation from emission coordinates to standard spacetime coordinates (Kostić et al., 2014).

The satellite worldlines are modeled in a perturbed Earth spacetime with

$C$2

and Hamiltonian

$C$3

Inter-satellite links enforce consistency through relations of the form

$C$4

and deviations are minimized through an action $C$5 over orbital elements (Kostić et al., 2014). The reported outcome is that, after minimization, the orbital elements are known to $C$6 relative and the system defines a primary DRF without Earth-station updates (Kostić et al., 2014). This is an explicitly dynamical, relativistically self-calibrating frame.

In celestial astrometry, a dynamical reference frame is one in which the axes are defined so that Solar-System bodies exhibit no apparent acceleration due to frame rotation. In practice, such a frame is materialized by a numerical planetary ephemeris such as the JPL DE series, INPOP, or EPM (Liu et al., 2022). Pulsars provide a way to compare this dynamical frame with kinematic frames such as Gaia-CRF3 and VLBI-CRF. The comparison is based on epoch propagation and a small-rotation model

$C$7

$C$8

Weighted least squares then estimates the frame orientation offsets (Liu et al., 2022). The cited analysis reports orientation offsets of $C$9 mas in the DE200 frame relative to Gaia and VLBI, while more modern ephemerides are sub-mas to a few mas depending on subset and systematics (Liu et al., 2022).

Planetary cartography supplies another operational notion. Mercury’s dynamical reference frame is defined by the 3:2 spin-orbit resonance and Cassini-state alignment. The prime meridian is chosen to point at the Sun exactly at every second perihelion passage, while the $D=\cup_t J_t$0-axis follows the Cassini-state spin axis (Stark et al., 2017). The inertial-to-body transformation is written through Euler angles $D=\cup_t J_t$1 and rotation matrix

$D=\cup_t J_t$2

This dynamical frame is then related to MESSENGER cartographic, principal-axes, and ellipsoid frames through explicit rotations and translations (Stark et al., 2017).

These examples show that, outside foundational discussions, DRFs can be realized as satellite constellations, ephemerides, or rotational states. The shared feature is that the frame is generated by physical evolution rather than imposed as an external coordinate scaffold.

6. Symmetry, determinism, and current debates

A recurrent issue is whether DRFs restore determinism or merely relocate gauge freedom. In Bamonti’s account, using a DRF in the Hamiltonian formulation amounts to supplying four gauge-fixing conditions, thereby turning first-class constraints into second-class ones and yielding deterministic relational observables such as $D=\cup_t J_t$3 (Bamonti, 2023). In the Hole-Argument analysis, coupled frame fields $D=\cup_t J_t$4 are introduced through the action

$D=\cup_t J_t$5

which yields Einstein’s equations with source $D=\cup_t J_t$6 and Klein–Gordon equations $D=\cup_t J_t$7. The relational metric

$D=\cup_t J_t$8

is then invariant under diffeomorphisms and evolves uniquely, so the standard Hole Argument is said to be neutralized once one works with these coupled-frame observables (Bamonti et al., 2024).

By contrast, work on Earman’s SP1 principle argues that uncoupled or partially coupled frame fields enlarge the dynamical symmetry group beyond ordinary spacetime diffeomorphisms. If geometry and frame fields can be reshuffled independently, then one obtains a dynamical symmetry not representable by a single spacetime diffeomorphism, and SP1 fails (Bamonti et al., 2024). The claim is summarized there as

$D=\cup_t J_t$9

for dynamically uncoupled sectors. This debate turns on how much of the frame dynamics is retained: equations of motion only, or equations plus backreaction.

A related misconception is that “relational” and “gauge-invariant” are synonymous. One cited analysis denies that equivalence. Partial observables can be relational and gauge-variant, and only complete observables obtained through dynamical coupling and constraint flow become gauge-invariant in the Dirac sense (Bamonti et al., 2024). The distinction matters because a frame may label events physically without yet producing deterministic observables.

Recent work also extends DRFs to covariant phase space with fluctuating boundaries and soft cutoffs. There, a DRF is a field-dependent identification of points in physical spacetime $\Phi$00 with an auxiliary relational manifold $\Phi$01, expressed via an embedding

$\Phi$02

Soft cutoffs are introduced by replacing hard Heaviside boundaries with smearing functions $\Phi$03, but this initially breaks diffeomorphism covariance. Covariance is recovered only by restricting the DRFs, their associated and linear moving coordinate frame functions, and the boundary conditions so that $\Phi$04 (Liu et al., 12 Mar 2026). In the covariant phase-space formalism, this leads to integrable Noether charges for fluctuating boundaries and, in asymptotically AdS gravity, to agreement with holographic renormalization after suitable matching conditions (Liu et al., 12 Mar 2026).

Taken together, these debates show that the phrase “dynamical reference frame” does not denote a single universally fixed formalism. It denotes a family of constructions centered on the same physical demand: the frame must belong to the dynamics. The principal disagreements concern how much coupling is required, how relational observables should be defined, and whether determinism and covariance are recovered by neglecting, retaining, or fully coupling the frame’s own dynamics (Bamonti, 2023, Bamonti et al., 2024, Bamonti et al., 2024, Liu et al., 12 Mar 2026).