A reference frame can be modeled mathematically as a coordinate chart on a manifold.
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refutedsupported
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2 sources for · 1 against
While some sources identify reference frames with coordinate charts on a manifold, other recent work introduces distinctions and classifications that differentiate physical reference frames from purely mathematical coordinate systems.
This study develops a canonical framework that relates gauge-fixing procedures with the dynamics of reference frames in general relativity. The central idea is that gauge-fixing conditions can be reinterpreted as the dynamical equations obeyed by the physical systems used for spacetime localisation. Within the Hamiltonian formalism, this link provides a principled criterion for determining when relationally expressed quantities qualify as Dirac observables. The analysis distinguishes three regimes of reference frames: idealised frames, whose own equations of motion and stress-energy are neglected; dynamical frames, which evolve according to specified equations but do not backreact on the metric; and fully coupled frames, which contribute to the gravitational dynamics. In the idealised regime the relational metric remains gauge-variant under the canonical flow, reflecting the persistence of diffeomorphism freedom. By contrast, once the reference fields satisfy suitable dynamical equations, they supply four functionally independent conditions that combine with the Hamiltonian and momentum constraints to form a second-class set. The resulting Dirac bracket defines a non-degenerate reduced symplectic structure in which relationally local quantities, such as the metric expressed in the reference-frame variables, strongly commute with all constraints and thereby become genuine Dirac observables. This construction also clarifies the correspondence between gauge-fixed quantities and complete observables: the frame dynamics internalises the gauge choice, making gauge-fixed quantities coincide with complete observables whenever the gauge conditions arise from the physical evolution of the reference fields. Familiar coordinate gauges, including harmonic and transverse-traceless gauges, thereby admit a relational interpretation: they arise from, and are constrained by, the dynamics of the physical frames used for localisation. This work thereby clarifies the interplay between diffeomorphism symmetry and operational localisation and offers a unified canonical characterisation of classical reference frames.
Evidence against · 1
What is a reference frame in General Relativity?
2023 · cited by 4
This work introduces a novel three-fold classification of reference frames in General Relativity, distinguishing between Idealised Reference Frames (IRFs), Dynamical Reference Frames (DRFs), and Real Reference Frames (RRFs). By defining a reference frame as a set of degrees of freedom instantiated by a physical system, the work contrasts this notion with that of coordinate systems-purely mathematical idealisations lacking physical instantiation. This classification addresses two longstanding challenges in GR: (P1) the difficulty of defining local and gauge-invariant observables, and (P2) how to interpret diffeomorphism gauge freedom in physical terms rather than as merely a mathematical redundancy. Overall, this work clarifies the conceptual foundations in classical General Relativity, enhancing our understanding of gauge-symmetries, observers and laying the groundwork for future investigations in both classical and quantum gravitational contexts.
{\displaystyle c} is equal to 1. A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event p {\displaystyle
In physics, spacetime, or the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects, such as how different observers perceive where and when events occur.
Until the turn of the 20th century, the assumption had b
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold
(
M
,
g
)
{\displaystyle (M,g)}
. This means the smooth Lorentz metric
g
{\displaystyle g}
has signature
(
3
,
1
)
{\displaystyle (3,1)}
. The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates
(
x
,
y
,
z
,
t
)
{\displaystyle (x,y,z,t)}
are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light
c
{\displaystyle c}
is equal to 1.
A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event
p
{\displaystyle p}
. Another reference frame may be identified by a second coordinate chart about
p
{\displaystyle p}
. Two observers (one in each reference frame) may describe the same event
p
{\displaystyle p}
but obtain different descriptions.
Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing
p
{\displaystyle p}
(representing an observer) and another containing
q
{\displaystyle q}
(representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate…
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