Gravity as temporal structure and shape dynamics

01What is being traded

The shape-dynamics construction of Gomes, Gryb and Koslowski relates a formulation using spatial conformal symmetry to general relativity in a suitable constant-mean-curvature setting. The correspondence depends on the gauge and domain. It is a precise relation between mathematical descriptions, not a claim that all physical length has vanished or that every spacetime admits the same global treatment.

DTT shares a relational motivation: the observed geometry should be described through accessible comparisons rather than an external absolute stage. Its extra ambition is to explain how those comparisons are produced by a nonspatial source. The six nested inner levels organize that production, while outer completion gives the observer a geometric record. Shape dynamics does not supply this source mechanism, but it offers a useful example of how apparently different temporal descriptions can encode related gravitational content. ^[1]

02Worked conformal example

Let a three-dimensional spatial metric transform as qab→e^(4φ)qab. For tangent vectors u and v at the same point, their inner product and each squared norm acquire the same factor e^(4φ). The normalized angle cosθ=q(u,v)/√[q(u,u)q(v,v)] is therefore unchanged. The local volume element instead scales as √q d³x→e^(6φ)√q d³x.

Thus conformal geometry preserves local angles while changing local scale. A constant φ rescales all lengths uniformly; a varying φ changes distances more subtly because geodesic lengths integrate the local scale along a path. Volume-preserving conformal transformations impose an additional global restriction. This calculation explains why shape and size can be treated differently without confusing a conformal transformation with a physical evolution in time.

03A source-based meaning of relational geometry

SMM starts from one nonspatial generative source. In the DTT reading, six nested inner functional levels prepare the distinctions that become available in the seventh, outer completion role. These are levels of one generative organization; counting them as six additional time coordinates would change the proposal. Its constructive ambition is to explain why an observed spatial relation is a completed relation among generated records.

A quaternionic carrier offers a concrete partial result: the purely imaginary quaternions form a real three-dimensional space. With a suitable positive quadratic readout they support rotationally organized directions. That is a useful algebraic selection mechanism. It does not by itself select a continuum of positions, prove Euclidean distance, or establish Lorentzian spacetime. Those further questions are where comparison with a quantum-gravity model becomes productive.

04The role of temporal variables

A change of time parameter can make the same trajectory look faster or slower without changing relational predictions. DTT's inner production must therefore be distinguished from a lapse choice or a reparametrization. If the inner variable can be eliminated entirely without changing any preparation or observable record, it may be a useful description but not an additional physical degree of freedom.

A source model can become more restrictive by generating a material clock whose relation to geometry cannot be freely rescaled. This is where the comparison with shape dynamics is productive: different formulations can teach us which temporal features are gauge and which are observable. The six nested levels may remain meaningful as a constructive architecture even when their internal labels are not measurable. The physical claim concerns the relations among completed records, not the absolute numerical value of an inner counter.

If DTT predicts a preferred completion slicing, it must show whether that preference is gauge, hidden but physically inert, or observable. These options have different implications for Lorentz symmetry and clock tests. Naming the slicing outer time does not decide the issue. ^{[1] [2]}

05The local inertial benchmark

Write z=ct+jx and apply z′=e^(−jη)z, with e^(−jη)=coshη−j sinhη. The real components become ct′=ct coshη−x sinhη and x′=x coshη−ct sinhη. Multiplication gives (ct′)²−(x′)²=(ct)²−x², so the interval is preserved algebraically.

If tanhη=v/c=0.6, then coshη=1.25 and sinhη=0.75. A clock moving with x=vt has t′=t/1.25. This reproduces the standard time-dilation factor after identifying the coordinates with physical inertial measurements. Without that identification, it is only a norm-preserving transformation. The example isolates exactly what a temporal-source derivation has supplied and what its matter-and-clock dynamics must still establish. ^{[2] [3]}

06Where a new source result would lie

A useful DTT theorem would generate conformal or metric data and a clock relation from one update, then show how the resulting observable sector maps to a known gravitational formulation. That would be a genuine reconstruction result. Merely expressing an existing theory in temporal vocabulary would establish an interpretation rather than a new physical law.

The relational affinity is nevertheless valuable. It suggests that the source need not generate absolute coordinates before it can generate physics. Ratios, angles, transport and clock correlations can be the first outputs. A controlled metric reconstruction can follow, with the residual scale and time freedoms made explicit.

THE SAME QUESTION, DIFFERENT STARTING POINTS

The comparison at a glance

Continue on SMONAD

• Relational time: order, clocks, and shared duration →

  Develops the source construction or operational distinction used in this comparison.

• When geometric symmetries become physical boosts →

  Develops the source construction or operational distinction used in this comparison.

• One clock direction plus three readout directions →

  Develops the source construction or operational distinction used in this comparison.