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We present the anchor core (v1.0) of the Temporal Theory of Gravitation (TTG), a minimal, falsifiable framework in which a scalar field τ(x) represents physical time and universally couples to matter via a conformal factor A(τ) = 1+βτ. The action consists of the Einstein-Hilbert term plus a canonical kinetic term and potential V(τ). In the Einstein frame this implies a universal non‑conservative coupling ∇_μ T^{μ}{}_ν = −β T ∇_ν τ, yielding a testable fifth‑force phenomenology and energy exchange between matter and the temporal field. We derive the field equations, the Newtonian limit with a Yukawa correction Φ(r) = −GM/r [1+ε e^{−m_τ r}] (ε≃β²), the post‑Newtonian predictions (γ_PPN = β_PPN = 1), and the gravitational‑wave speed c_gw = c. A concise list of falsifiable signatures is provided (MICROSCOPE, spacecraft tracking, LIGO/Virgo/KAGRA). We keep v1.0 strictly minimal: no screening, no nonlinear A(τ) (e.g., cosh), and no TTU extensions. Variants and cosmological specializations lie outside the scope of this anchor document and do not affect the equations and predictions presented here | ||
We present the anchor core (v1.0) of the Temporal Theory of Gravitation (TTG), a minimal, falsifiable framework in which a scalar field (x) represents physical time and universally couples to matter via a conformal factor A() = 1+. The action consists of the EinsteinHilbert term plus a canonical kinetic term and potential V(). In the Einstein frame this implies a universal nonconservative coupling _ T^{}{}_ = T _ , yielding a testable fifthforce phenomenology and energy exchange between matter and the temporal field. We derive the field equations, the Newtonian limit with a Yukawa correction (r) = GM/r [1+ e^{m_ r}] ('), the postNewtonian predictions (_PPN = _PPN = 1), and the gravitationalwave speed c_gw = c. A concise list of falsifiable signatures is provided (MICROSCOPE, spacecraft tracking, LIGO/Virgo/KAGRA). We keep v1.0 strictly minimal: no screening, no nonlinear A() (e.g., cosh), and no TTU extensions. Variants and cosmological specializations lie outside the scope of this anchor document and do not affect the equations and predictions presented here.
temporal field; scalartensor gravity; fifth force; postNewtonian parameters; gravitational waves; Yukawa potential; MICROSCOPE; spacecraft tracking; falsifiability; energy nonconservation
1. Postulates
2. Action and Field Equations
3. Newtonian Limit and Yukawa Correction
4. PostNewtonian and GravitationalWave Predictions
5. FifthForce Phenomenology
6. Falsifiable Signatures (v1.0)
7. Potential V() and Phenomenology
8. Conclusion
9.References
Appendix A. Notation
P1. There exists a fundamental scalar field (x) that encodes physical time (not coordinate time).
P2. Matter is minimally coupled to the Jordan metric g_{} = A()^2 g_{} with a strictly linear coupling A() = 1 + in v1.0.
P3. The dynamics of g_{} and follow from the action below; all predictions quoted in this document are derived from it without additional mechanisms (no screening in v1.0).
We use c = = 1 unless shown explicitly; c is kept only where it aids phenomenology (e.g., Eq. (9))
We work in the Einstein frame with metric g_{} and assume a canonical normalization of ; any kinetic prefactor can be absorbed by a field redefinition, hence we set = 1 throughout v1.0.
(1) S = dx -(-g) [ (1/(16G)) R + (1/2) ()' V() ] + S_m[; A()^2 g_{}], with A() = 1+.
Variation yields the Einstein equations with the stress and the scalar equation:
(2) G_{} = 8G ( T^{(m)}_{} + _ _ (1/2) g_{} ()' g_{} V() ).
(3) V() = + T, where T = T^{(m)} = g^{}T^{(m)}_{}.
The nonconservative matter coupling in the Einstein frame follows as:
(4) _ T^{(m) }{}_ = T _ .
Consider weak, static fields: g_{00} (1+2/c'), g_{ij} (12/c')_{ij}, = + with ||1. Linearizing Eqs. (2)(3) around a homogeneous background and taking nonrelativistic matter (T- c') yields:
(5) ' = 4G + O(' '), and (' m_') = + c', with m_' V().
For a point mass M, the scalar profile is (r) (e^{m_ r})/r and the total potential becomes:
(6) (r) = G M / r " [ 1 + e^{ m_ r} ], with ' ( = O(1) depends on conventions).
Equation (6) is the standard Yukawa correction: sets the amplitude and m_^{-1} the range of the fifth force. Solarsystem and laboratory constraints bound (, ).
(7) _PPN = 1, _PPN = 1 (v1.0 core with linear A()).
(8) c_gw = c (identical tensor sector propagation speed).
Deviationse.g., dipole radiation in binariesarise only via the universal coupling through Eq. (4) and depend on the compositiondependent sensitivities of compact objects; these are constrained by GW data.
In quasistatic backgrounds with a gentle spatial gradient , a test mass experiences an anomalous acceleration:
(9) a_anom = c' .
This is directly constrained by MICROSCOPE (Weak Equivalence Principle), spacecraft tracking (Pioneerlike anomalies), and planetary ephemerides. Equation (6) connects these constraints to (, ).
Observational channel | Signature | TTG parameters | Primary datasets |
Equivalence principle (laboratory/orbit) | Universal compositiondependent acceleration | (via Eq. 4), , | MICROSCOPE; torsion balances |
Spacecraft tracking | Small constant a_anom or Yukawa deviations | , , | Voyager, Cassini, New Horizons |
Gravitational waves | Dipole phase deficit at low PN order | and sensitivities of bodies | LIGO/Virgo/KAGRA; LISA (future) |
Galaxy rotation / clusters | Allowed (, ) envelopes from dynamics | = ' (=O(1)), = m_^{-1} | SPARC; cluster mass profiles |
TTG v1.0 remains agnostic about V(), but two benchmark choices are useful:
Massive scalar: V() = (1/2) m_' ' finite range = m_^{-1} for the Yukawa term in Eq. (6).
Exponential/quintessencelike: V() = V e^{_Q } slowroll cosmology (beyond v1.0).
In this anchor document we do not commit to a specific V(); Eq. (6) shows how a mass term alone already maps to data.
We have fixed TTG v1.0 as a clean, minimal, and falsifiable anchor: linear matter coupling A() = 1+, canonical scalar dynamics, GRlike PPN and GW propagation, and a standard Yukawa correction in the Newtonian limit. The nonconservative coupling (Eq. 4) organizes the phenomenology and experimental tests. Extended couplings (e.g., screening) and cosmological specializations exist but lie outside the scope of this document and do not affect the equations or predictions presented here.
9.References
Symbol | Meaning | Units |
(x) | temporal scalar field | dimensionless (v1.0 normalization) |
A() | matter coupling (Jordan metric factor) | ; v1.0: A() = 1+ |
dimensionless universal coupling | ||
V() | scalar potential | J/m |
m_'V() | mass of fluctuations around | 1/m' (c = = 1) |
Yukawa amplitude (') | ||
Yukawa range | m | |
Newtonian potential | m'/s' | |
_PPN, _PPN | postNewtonian parameters | |
c_gw | GW speed | m/s |
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