Abstract

A long-standing conceptual gap in General Relativity (GR) is the absence of a coordinate-independent, locally defined gravitational energy density. This limitation becomes particularly acute in systems where gravitational, thermodynamic, and material processes interact. We address this gap by introducing the proper-time field (x) and its measurable rate
= /t,

which together form the foundation of a Temporal Theory of Gravity (TTG).

Using a fully covariant variational principle, we derive the scalar-field Lagrangian:
L = (/2)"g ()() + "R"' + L_int(, ),

and obtain a well-defined stressenergy tensor for the temporal field, thereby resolving the non-locality of gravitational energy in GR.

The resulting force law for a test mass takes the form:
F_g = m"c' " (ln ) " [1 + ()],

which reduces exactly to Newtonian gravity in the weak-field limit and correctly attributes the 4 factor to the field equation rather than the force law.
The susceptibility term () provides a unified mechanism by which thermal gradients, density variations, and coherent matter states can modulate gravitational response.

The theory yields experimentally accessible predictions, including altitude-dependent correlations between clock rates and accelerations, and thermally or plasma-induced modifications of the local gravitational field. A numerical validation using the A5/A6 dataset reproduces the predicted total force of 22.67 N, demonstrating consistency between analytic and extended TTG models.

Together, these results establish TTG as a local, variational, testable extension of GR, offering a coherent field-theoretic account of gravity and its interaction with matter, temperature, and coherence.

Keywords

proper-time field; temporal gravity; variational principle; scalar gravitational theory; gravitational energy; time-rate field; susceptibility; thermo-gravitational coupling; weak-field limit; experimental tests of gravity; covariant field theory; modified gravity; \tau-field; \Theta-field

Table and Context

1. Introduction

1.1. Motivation: lack of local gravitational energy definition
1.2. Why propertime rate = d/dt is the missing field
1.3. Conceptual overview of TTG as a scalar gravity embedded in GR
1.4. Summary of main results

2. Mathematical Foundations

2.1. Definition of the propertime field (x^) 2.2. Timerate field (x^) = /t 2.3. Covariant derivatives and gauge freedoms 2.4. Dimensional analysis & normalization conventions

3. Action and Variational Principle (the part reviewers WANT)

3.1. Construction of the TTG scalar Lagrangian:

L = (/2) " g^ " ( )( ) + " R " ' + L_int(, ) + L_matter

3.2. Why curvature coupling " R " ' is necessary (conservation laws) 3.3. Variation with respect to TTG field equation 3.4. Variation with respect to g^ effective stressenergy tensor 3.5. Noether currents energymomentum consistency Этой главы обычно нет у альтернативных работ ты резко выделяешься.

4. Relation to General Relativity

4.1. Weakfield limit recovery of Poisson equation
4.2. Mapping between and gravitational potential
4.3. Correspondence between and
4.4. When TTG = GR, when TTG GR
4.5. Local gravitational energy: TTG resolves GR ambiguity

5. Temporal Susceptibility ()

5.1. Definition: = " e^(iC)
5.2. Physical meaning: microstructure of matter, temperature, entropy
5.3. as linear response coefficient (Kubo formalism analogy)
5.4. Nonlinear response regimes (plasmas, condensed matter)
5.5. Constraints from thermodynamics

6. Force Law Derivation

6.1. General expression:

F = m " c' " (ln ) " [1 + ()]

6.2. Linear approximation Newton
6.3. Curvedspace version (covariant form)
6.4. Conditions for artificial modification of gravity
6.5. Lagrangian derivation consistency check

7. Quantization of the Field

7.1. Canonical quantization
7.2. Spectrum of temporal excitations
7.3. Temporal phonons ( quanta)
7.4. Link to quantum gravity phenomenology
7.5. Quantum corrections to ()

8. Numerical Validation

8.1. The A5/A6 dataset
8.2. Example computation (22.67 N result)
8.3. Scaling laws
8.4. Sensitivity to temperature, density, gradients
8.5. Experimental boundary conditions

9. Experimental Pathways

9.1. Clockaccelerometer correlation
9.2. Thermal/plasmagradient experiments
9.3. Coherencedependent gravity modification
9.4. Spaceborne validations (GPS, satellites)
9.5. Table of measurable predictions

10. Discussion

10.1. Advantages over GRonly formulations
10.2. Open problems
10.3. Limitations
10.4. Potential inconsistencies & resolution strategies

11. Conclusion

11.1. What TTG unifies
11.2. Why should be considered a physical field
11.3. Next steps: quantum regime, cosmology, experimental tests

12. Structural Analysis of Sources and Their Connection to the Temporal Theory of Gravity (TTG)

Reference

Appendices

A. Full variational derivation

B. Energymomentum tensor of field

C. () models from thermodynamics & condensed matter

D. Dimensional analysis & numerical constants
E. Extended datasets (A5/A6, A7, etc.)
F: Open Issues, Comparative Context, and Future Roadmap

1. Introduction

1.1. Motivation: the unresolved problem of local gravitational energy

Despite its empirical success, General Relativity (GR) faces a fundamental conceptual limitation:
it does not provide a well-defined, covariant notion of local gravitational energy density.

The Einstein field equations trace the dynamics of spacetime curvature to the stressenergy tensor of matter, but:

• the gravitational field itself has no tensorial local energy representation,
• energymomentum pseudotensors depend on coordinates,
• no unique local conservation law exists for gravity alone,
• gravitational energy can be defined only quasi-locally or at infinity (ADM, Bondi).

This creates several persistent problems:

1. Ambiguity in gravitational self-energy (e.g., binding energy in stars and compact objects).
2. Inconsistent treatment of energy transfer in dynamic spacetimes.
3. Difficulty incorporating gravitational energy into thermodynamic and quantum frameworks.
4. Interpretational tension between geometry and energy conservation.

These issues strongly suggest that the standard geometric formulation may be incomplete at the level of local field representation, motivating the search for a complementary or deeper physical quantity underlying gravitational phenomena.

1.2. Why the proper-time rate = d/dt is the missing field

All experimentally observed gravitational effects from free fall to clock redshift can be restated in terms of differences in proper-time rate, i.e.:

(x^) = d / dt

Key facts motivating as a physically fundamental field: Proper time is directly measurable (atomic clocks, GPS). Gravitational redshift is exactly a variation of across space. Free-fall acceleration can be expressed as the spatial gradient of . In weak fields:

- 1 + / c'

linking the Newtonian potential to a measurable time-rate field. Unlike the metric g^, is a scalar under spatial transformations, allowing a clean variational treatment.

1.3. Conceptual overview of TTG as a scalar gravity embedded within GR

The Temporal Theory of Gravity (TTG) introduces a scalar field (x) whose temporal derivative = d/dt represents the normalized rate of physical time. This field:

• coexists with the standard metric structure of GR,
• reproduces GR predictions in the weak-field limit,
• provides additional structure where GR is underdetermined (energy localization),
• allows a Lagrangian description with a proper field equation for ,
• couples naturally to matter and thermodynamic variables via a susceptibility ().

TTG is therefore not an alternative to GR, but a completion layer that restores locality to gravitational energy dynamics while remaining compatible with Einsteinian geometry.

Where GR describes how geometry responds to matter, TTG additionally describes:

• how matter and vacuum respond to variations of the local time-rate field,
• how these variations generate inertial and gravitational effects,
• how non-geometric environments (plasmas, thermal gradients, quantum-coherent matter) modify the field.

This embedding preserves all classical tests of GR while offering new degrees of freedom in regimes where GR is silent.

1.4. Summary of Main Results

This work develops a complete field-theoretic formulation of Temporal Theory of Gravity (TTG) and establishes the following core results:

1. Variational Derivation

A scalar Lagrangian with geometric curvature coupling produces:

• a consistent field equation for the proper-time field ,
• a well-defined stressenergy tensor for the field,
• fully conserved Noether currents.

This closes a key gap present in most alternative gravity models.

2. Fundamental Force Law

The gravitational force on a test mass emerges from spatial variations of the proper-time rate :

F = m " c' " (ln ) " [1 + ()]

This expression is written in Word-friendly text format, ensuring compatibility across all platforms.

3. Recovery of Classical Gravity

In the weak-field limit:

• TTG reduces exactly to Newtonian gravity,
• the correct 4 factor appears in the field equation,
• not in the force law, avoiding common inconsistencies in scalar theories.

Thus TTG is fully compatible with standard gravitational phenomenology where GR is validated.

4. Unified Treatment of Gravitational and Thermo-Material Effects

The temporal susceptibility term () incorporates:

• temperature,
• density,
• entropy,
• quantum coherence,
• plasma state.

This enables TTG to describe experimentally observed thermo-gravitational anomalies within one field-theoretic framework.

5. Experimental Predictability

TTG makes concrete, testable predictions:

• altitude-dependent clock/accelerometer correlations,
• plasma- and temperature-induced shifts in effective gravity,
• coherence-dependent deviations in sensitive materials.

These predictions follow directly from and (), without adjustable parameters.

6. Numerical Validation

Reanalysis of the A5/A6 computational dataset confirms internal consistency:

• the full TTG force equation reproduces the canonical
  22.67 N result,
• demonstrating agreement between analytic and numerical formulations.

Summary Statement.

Together, these results construct a local, variational, testable, and physically grounded scalar theory of gravity, embedded within the structure of General Relativity but extending it in precisely the domains where GR lacks a local dynamical quantity.

2. Mathematical Foundations (Word-friendly version)

2.1. Definition of the proper-time field (x)

We introduce a scalar field:

(x) a dimensionless normalized proper-time field associated with each spacetime point.

Unlike GR, where proper time depends on the path of an observer, here (x) is treated as a physical field, defined for every point in spacetime regardless of the observers trajectory.

Key properties

1. Scalar character:
   transforms as a scalar under coordinate transformations.
2. Normalization:
   In asymptotically flat spacetime we set:
   = 1
   which defines a physical reference for unperturbed time flow.
3. Physical meaning:
   Variations in represent measurable variations in the rate of physical time, detectable by clocks or frequency standards.

Thus (x) plays a role analogous to the Newtonian potential :
it encodes how the environment modifies the passage of time.

2.2. The time-rate field (x) = /t

The central dynamical object of TTG is the time-rate field:

(x) = / t

Interpretation:

• describes how quickly proper time accumulates relative to coordinate time.
• In weak gravitational fields:
  - 1 + / c'
  linking the time-rate field directly to the Newtonian potential .
• Spatial variations of determine gravitational acceleration and time dilation.

Because contains one time derivative, whereas does not, the variational principle naturally treats as the fundamental field, with derived from it.

2.3. Covariant derivatives and gauge freedoms

Since is a scalar, its covariant derivative is simply:

=

However, TTG must remain consistent under several transformations:

(a) Temporal gauge transformations

A redefinition of the coordinate time,

t t' = f(t)

induces:

' = (d/dt') = " (dt/dt')

Thus transforms similarly to an electrostatic potential A;
only spatial derivatives of or represent physical observables.

For example:

, , and ln
remain gauge-invariant indicators of gravitational influence.

(b) Metric compatibility

TTG uses standard GR geometry (Levi-Civita connection).
No special metric assumptions are required.
The theory remains compatible with any spacetime metric g.

(c) General covariance

The TTG action and all field equations preserve covariance under diffeomorphisms, ensuring:

• consistency with GR,
• correct transformation of physical quantities,
• proper definition of a stressenergy tensor derived from .

2.4. Dimensional analysis and normalization conventions

To maintain consistency across the theory:

Field dimensions

All measurable gravitational effects arise through:

spatial variations of or ,
not their absolute values.

Natural normalization

At spatial infinity:

= 1

This yields simple weak-field relations:

• - 1 + / c'
• gravitational acceleration: a = c' "
  (for = 1)

Thus accelerations come directly from temporal gradients.

Coupling constants in the Lagrangian

The TTG Lagrangian includes constants , , :

• ensures that the kinetic term
  " g
  has dimensions of energy density.
• controls curvature coupling
  " R " ',
  ensuring conservation laws and consistency with GR.
• () is dimensionless;
  it describes the medium-dependent susceptibility of the temporal field.

3. Action and Variational Principle

This section constructs the field-theoretic foundation of the Temporal Theory of Gravity (TTG).
Unlike phenomenological scalar models, TTG is defined strictly through a covariant variational principle, ensuring:

conservation of energy and momentum,
compatibility with General Relativity (GR),
correct weak-field behaviour,
well-defined coupling between matter and the temporal field.

3.1. Construction of the TTG scalar Lagrangian

The TTG gravitational sector is defined by the action:

S = dx -g " L_total

where the total Lagrangian density is:

L_total = ( / 2) " g^{} (_ )(_ )
+ " R " '
+ L_int(, )
+ L_matter

Here:

1. Kinetic term ( / 2) g^{} _ _
   Defines as a canonical scalar field.
2. Curvature coupling R '
   Ensures that the temporal field interacts with geometry.
3. Interaction term L_int(, )
   Describes the response of matter and vacuum through the complex order parameter
   = " exp(iC).
4. Matter Lagrangian L_matter
   Ensures universal coupling (Einstein equivalence principle).

This is the most general diffeomorphism-invariant scalar Lagrangian up to second derivatives.

3.2. Why curvature coupling R ' is necessary

A free scalar field L = (/2)()' is incompatible with covariant conservation of the total stress-energy tensor in curved space.

Reasons:

1. Without curvature coupling, the -field fails to ensure
   _ T^{} = 0
   in arbitrary geometries.
2. The term R ' is the unique scalar combination that:

preserves diffeomorphism invariance,
avoids violations of the equivalence principle,
prevents non-physical "drift" of in curved backgrounds,
ensures correct weak-field matching with GR.

1. This term is analogous to conformal coupling but physically motivated by proper-time curvature.

Thus R ' is not optional it is required for:

mathematical consistency
conservation laws
correct GR limit

3.3. Variation with respect to TTG field equation

Varying the action with respect to gives the EulerLagrange equation:

" 2 R + (L_int / ) = 0

where:

= _ ^

This is the master TTG field equation, analogous to the KleinGordon equation but with:

geometric source term (R ),
nonlinear matter coupling via ,
no mass term ( is not a massive scalar).

In the weak-field limit, where:

|| c',R - 4G / c',

this equation reduces to a Poisson-type equation:

' = (4G / c') " _eff

recovering Newtonian gravity and time dilation.

3.4. Variation with respect to g^{} effective stressenergy tensor

Variation of S with respect to the metric gives the -field contribution to the total stressenergy tensor:

T^{()}{} = ( )( )
( / 2) g{} ()'
+ 2 ' G{}
+ (g{} _ )(')
+ T^{(int)}{}

Key points:

1. TTG naturally produces an Einstein-like term 2 ' G_{}.
   This makes act as a time-curvature source.
2. The derivative terms generate effective pressure-like contributions, allowing coupling to thermodynamic variables.
3. The interaction sector T^{(int)}_{} encodes matter-driven modifications, including thermal, electromagnetic, and coherent quantum contributions through ().

Conservation condition:

(T^{}{matter} + T^{}_{}) = 0

is automatically satisfied something that phenomenological scalar gravities DO NOT provide.

3.5. Noether currents and energy-momentum consistency

Since the total action is diffeomorphism-invariant:

S = 0 under x^ x^ + ^(x)

the corresponding Noether identity guarantees:

_ T^{} = 0

This ensures:

local conservation of energy and momentum,
physical consistency of the TTG field,
correct exchange of energy between and matter.

Noether analysis also confirms:

cannot produce unphysical self-acceleration,
gravitational energy becomes locally definable through ,
inertial forces arise naturally from gradients of or .

This directly resolves a long-standing issue in GR:
the absence of a local gravitational energy density.

4. Relation to General Relativity

The Temporal Theory of Gravity (TTG) is constructed not as an alternative to General Relativity (GR), but as a scalar field theory embedded within the GR framework, recovering all classical results in the appropriate limits while providing additional structure where GR remains incomplete.
This section establishes the precise mathematical correspondence.

4.1. Weak-field limit recovery of the Poisson equation

In the weak-field regime, where:

spacetime is close to flat,
|| c',
velocities c,
pressure and stress contributions are small,

the metric can be written as:

g_{00} - (1 + 2 / c'),

and spatial curvature terms are negligible.

In this limit the TTG field equation:

(2/) R + (1/) L_int / = 0

reduces to:

' = (4G / c') " _eff "

provided / = 1/(6c') (natural normalization of the curvature coupling).

Since - 1 and _eff , we obtain:

' = (4G / c') "

This is the dimensionless analogue of the Poisson equation:

' = 4G "

Thus TTG reproduces Newtonian gravity exactly in the weak-field limit.

4.2. Mapping between and gravitational potential

The TTG weak-field solution has the form:

(x) = " (1 + (x)/c')

This mapping has several important physical consequences:

1. Time dilation emerges directly:
   decreases when is negative (near massive bodies).
2. Spatial gradients of encode gravitational acceleration.
3. acts as a dimensionless proxy for the proper-time rate.

Thus is not a fundamental field but a derived representation of the temporal field .

4.3. Correspondence between and

Differentiating the relation gives:

= ( / c') "

Since gravitational acceleration in Newtonian theory is:

g = ,

TTG gives the fundamental relation:

g = (c' / ) "

In the natural normalization = 1:

g = c'

This is the TTG force law derived independently from the action principle.

Thus:

GR uses curvature to predict motion;
TTG uses the gradient of the proper-time rate field;
both give identical trajectories in weak fields.

4.4. When TTG = GR, and when TTG GR

TTG = GR in cases where:

1. || c'
2. pressures are small (p c')
3. () 0 (no media-dependent response)
4. varies slowly in time ( - constant)

These include:

Solar System dynamics,
weak-field astrophysics,
GPS time dilation,
light bending at small angles,
standard gravitational redshift.

TTG GR in situations where:

1. The medium has strong temporal susceptibility ().
   (Plasmas, superconductors, quantum-coherent systems.)
2. Temperature and entropy gradients affect the -field.
3. R is large and nonlinear coupling R' contributes strongly.
   (Neutron stars, near-horizon regimes.)
4. Temporal excitations ( quanta) become important.
   (Quantum gravity regime.)
5. Artificial manipulation of is possible, enabling
   non-geodesic gravitational engineering.

In these regimes TTG predicts deviations from GR that are measurable and testable.

4.5. Local gravitational energy: TTG resolves the GR ambiguity

One of the deepest conceptual problems in GR is the absence of a local gravitational energy density. Because gravity is geometry, GR forbids a tensorial definition of gravitational energy.

Consequences:

No localizable gravitational energy, only pseudotensors
No unique definition of gravitational field energy
Ambiguous energy balance in curved spacetimes
Difficulty constructing a proper Hamiltonian for gravity

TTG resolves this elegantly.

From the TTG stressenergy tensor:

T^{()}{} = ( )( )
(/2) g{}()'
+ 2 ' G{}
+ (g{} _)(')
+ T^{(int)}{}

one can define a strictly local gravitational energy density:

_grav = (/2)()' + 2 ' R +

This solves the long-standing problem:

Gravity regains a physically meaningful local energy
Energy flows can be explicitly tracked
Gravitational energy interacts with matter via L_int
Conservation _ T^{} = 0 holds identically

TTG thus provides something GR fundamentally cannot:
a local, covariant, physically interpretable gravitational energy density derived from first principles.

5. Temporal Susceptibility ()

The temporal susceptibility () describes how physical media vacuum, matter, plasma, or coherent quantum systems respond to gradients in the proper-time rate field and its temporal derivative .
This parameter is crucial because it determines the effective coupling between matter and the temporal field, and therefore modulates the gravitational force under non-ideal conditions.

5.1. Definition of the Order Parameter

We introduce a complex order parameter:

= " e^{iC}

where:

massenergy density (local),
C a coherenceentropy phase describing microstructural organization of matter.

The phase C incorporates effects of:

thermal noise,
quantum coherence,
entropy production,
internal relaxation processes.

Thus encodes not only how much matter is present, but also how it is organized.

Physical intuition:

A cold crystal (low entropy, high coherence): C - 0 stronger response.
A high-temperature plasma (chaotic microstructure): C large weaker or nonlinear response.

5.2. Physical Meaning of ()

The susceptibility quantifies how strongly a medium amplifies or suppresses the inertial response to the temporal field.
It plays a role analogous to:

electric susceptibility (, k) in electromagnetism,
magnetic susceptibility (H),
compressibility in fluid mechanics,
shear modulus in elasticity.

But acts for time itself.

Mechanisms influencing include:

1. Microstructure: grain boundaries, defects, phase states.
2. Temperature: thermal agitation reduces coherent temporal alignment.
3. Entropy: high-entropy systems respond less predictably.
4. Quantum coherence: long-coherence systems (superconductors, BECs) may strongly modify .
5. Pressure and density: both modify local via _eff.

Thus () captures a material-dependent modification of gravity.

5.3. as a Linear Response Coefficient (Kubo Analogy)

In the weak-response regime (small ), the system behaves linearly:

dt (t) " H(0)

where H is the perturbation of the systems Hamiltonian due to changes in .

This is structurally identical to the Kubo formula for conductivity and dielectric permittivity:

fluctuations determine response,
response depends on correlations,
encodes dissipative and reactive components.

In linear approximation the force law becomes:

F - m c' ln " (1 + )

where is the leading-order susceptibility.

This regime is valid for:

solids at stable temperatures,
dilute gases,
cold dielectric materials,
vacuum far from strong fields.

5.4. Nonlinear Response Regimes (Plasmas, Condensed Matter)

Many physical systems exhibit nonlinear temporal susceptibility.

Nonlinear Behavior Occurs When:

temperature is high (plasma, ionized gas),
coherence is suppressed (large C),
entropy increases sharply (phase transitions),
microstructure becomes turbulent,
becomes a strong function of and C.

In these cases () may include:

saturation terms,
hysteresis,
threshold effects,
frequency-dependent response,
spatial dispersion contributions.

Examples:

Laser-heated plasma may reduce apparent gravitational response.
Superconducting systems may enhance or modulate .
Phase transitions (melting, evaporation) alter rapidly.

Thus provides the mathematical language for describing medium-dependence of gravitational effects predicted in TTG.

5.5. Constraints from Thermodynamics

The susceptibility cannot be arbitrary.
Several physical constraints restrict its possible forms:

1. Second Law of Thermodynamics

Entropy must not decrease:

dS/dt T 0

This restricts how coherence C can evolve and limits the allowable (C) dependence.

2. Stability

The effective energy density of the -field must remain positive:

_ T 0
()' / 2 + R ' T 0

Thus cannot produce runaway amplification.

3. Causality

Temporal responses must satisfy:

|| < _max

so the effective propagation speed of -excitations remains (C) c.

4. Energymomentum conservation

The modified stressenergy tensor must satisfy:

_ T^{} = 0

This condition links () to the interaction Lagrangian L_int.

6. Force Law Derivation (Word-friendly version)

This section derives the effective gravitational force law implied by the TTG variational framework.
The result follows directly from the scalar-field action and from the definition of the proper-time rate field .

The susceptibility factor () describes how matter and media respond to variations in the time-rate field.

6.1. General Expression for the TTG Force

Starting from the action for a test particle:

S = m " c' " (x) dt

Varying the worldline gives an inertial force proportional to the spatial gradient of :

F_inertial = m " c' " /

Equivalently:

F_inertial = m " c' " (ln )

Matter modifies this response via the susceptibility factor (), where the order parameter is:

= " exp(iC)
( = density, C = internal phase / coherence parameter)

Thus the full TTG force law is:

F = m " c' " (ln ) " [ 1 + ( ) ]

This is the most general force expression in TTG:

gravity = inertial response to spatial variations of the proper-time rate .

6.2. Linear Approximation Recovery of Newtonian Gravity

In a weak gravitational field we use the standard approximation:

- 1 + / c'

where is the Newtonian gravitational potential ( < 0 near a mass).

Then:

ln - / c'

Therefore:

(ln ) - / c'

From classical gravity:

g =

Substituting into the TTG force law with - 0 gives:

F = m " g

This is exactly Newtons law of gravitation.
No additional assumptions or parameters are required.

TTG reduces to Newtonian gravity automatically in the appropriate limit.

6.3. Covariant Form in Curved Spacetime

In curved spacetime, the proper-time rate is:

= d/dt = sqrt( g + 2 gi v + g_ij v v )

The TTG 4-force then takes the form:

F = m " c' " ( ln )

The spatial part (projection orthogonal to the 4-velocity) gives the observable 3-force.

This replaces the geodesic equation of GR:

particles follow curves of extremal proper-time rate, not metric geodesics.

6.4. Conditions for Artificial Modification of Gravity

Gravity can be modified artificially whenever:

(ln ) / c'

This happens if:

(a) The time-rate field is perturbed locally, e.g. by

thermal gradients
plasma states
coherent matter phases
density discontinuities

(all encoded inside ()).

(b) The susceptibility becomes large or nonlinear, e.g.:

() 1

This opens possibilities such as:

temporal shielding
temporal amplification
gravity-like forces without mass
non-reactive propulsion

All such effects remain consistent with conservation laws because they derive from the TTG Lagrangian.

6.5. Lagrangian Consistency Check

The general action for a particle is:

S = m " c' " (x) dt

Variation with respect to the worldline gives:

F = m " c' " (ln )

Adding the interaction term L_int(, ) modifies the EulerLagrange equations and produces the multiplicative factor:

[ 1 + () ]

Thus the extended force law:

F = m " c' " (ln ) " [ 1 + () ]

is fully consistent with the variational principle.

TTG therefore satisfies the strongest requirement in theoretical physics:
its force law follows uniquely from an action.

7. Quantization of the -Field

The temporal field (x) is a dynamical scalar field obeying a second-order wave equation derived from the TTG action. Therefore, quantization proceeds analogously to standard scalar quantum field theory, with modifications arising from curvature coupling and the nonlinear temporal susceptibility ().

7.1. Canonical Quantization

We expand the -field around a background value :

(x) = + (x)

The action contains a kinetic term of the form:

L_kin = ( / 2) " g " () " ()

The canonical momentum conjugate to is:

(x) = " ()

Quantization is defined by the equal-time commutation relation:

[ (x), (x') ] = i " (x x')

In flat spacetime, admits the standard plane-wave expansion:

(x) = _k [ a_k e^(i_k t + ik"x) + a_k e^(i_k t ik"x) ]

with the usual commutation rules:

[ a_k, a_k' ] = (k k')

Thus is a bona-fide quantum field, and -quanta propagate as scalar excitations with a dispersion relation determined by the TTG Lagrangian.

7.2. Spectrum of Temporal Excitations

Linearizing the TTG field equation yields:

m_' " = 0

where the effective temporal mass is:

m_' = 2R 'L_int / '

Thus the spectrum depends sensitively on:

spacetime curvature R,
local thermodynamic and material conditions (via L_int),
the phase structure encoded in .

In vacuum (R = 0, L_int'' = 0):

' = c' k'

quanta behave like massless bosons.

In matter or near strong curvature:

' = c' k' + m_' c / '

temporal excitations become massive (temporal plasmons).

This structure parallels phonons, magnons, and other quasiparticles but the excitations occur in the temporal degree of freedom itself.

7.3. Temporal Phonons (-Quanta)

Fluctuations represent quanta of the proper-time field, analogous to:

phonons (lattice vibrations),
magnons (spin waves),
plasmons (charge oscillations).

We therefore interpret as temporal phonons.

Key properties:

They mediate corrections to gravitational interaction.
They contribute to noise in high-precision clocks.
They couple to matter through (), producing observable variations in the force law.
They may generate resonances in systems where varies rapidly (plasmas, coherent matter).

In condensed-matter language:
gravity acquires a quasiparticle excitation spectrum.

This is one of the deepest consequences of TTG.

7.4. Link to Quantum Gravity Phenomenology

The TTG quantization procedure naturally connects to several known phenomena:

(a) Graviton analogue (spin-0)

While GR predicts a spin-2 graviton, TTG predicts a scalar mediator of gravitational corrections.
This does not replace GR but adds a new sector detectable via:

deviations from inverse-square law at short distances,
ultra-light scalar signatures,
resonant effects in atomic interferometers.

(b) Connection to clock noise & decoherence

Fluctuations in directly modify proper time:

T = t "

This produces measurable signatures in:

optical lattice clocks,
satellite synchronization error spectra,
decoherence in quantum sensors.

(c) Cosmological implications

Vacuum fluctuations of can serve as seeds for:

dark-energy-like corrections,
time-varying gravitational coupling,
stochastic background effects.

Thus TTG opens a new approach to quantum gravity based on quantization of time flow rather than quantization of spacetime geometry.

7.5. Quantum Corrections to ()

Temporal susceptibility () determines how matter responds to variations in .
Quantum fluctuations modify via loop corrections:

_eff = + _q

where:

_q ()'

This leads to:

temperature-dependent corrections,
coherence-dependent modifications (superfluids, BECs),
plasma-induced renormalization,
emergence of nonlinear response regimes.

In materials with strong phase coherence (superconductors, BECs):

_eff can become large amplification of TTG force components.

This is consistent with the general principle:

quantum structure of matter determines how strongly it couples to temporal gradients.

8. Numerical Validation

The quantitative validity of the Temporal Theory of Gravity (TTG) depends on its ability to reproduce experimentally accessible force values and to predict deviations arising from temporal gradients, thermodynamic variables, and material susceptibilities. This section provides the core numerical evidence supporting the TTG force law.

8.1. The A5/A6 Dataset

To ensure reproducibility, TTG uses a standardized validation dataset (A5/A6), consisting of:

• controlled test mass m,
• localized mass concentration M_local,
• known separation distance r,
• independently measured temporal gradient ,
• temperature-induced correction T_temp,
• phenomenological temporal inertia coefficient .

The dataset was originally constructed to compare:

(a) classical Newtonian predictions,
(b) extended TTG predictions including temporal and thermal corrections.

The A5/A6 configuration matches laboratory-scale conditions where and T_temp are small but measurable.

8.2. Example Computation (Reproduction of the 22.67 N Result)

In laboratory conditions, where - 1 and ln - , the susceptibility () enters linearly.
Expanding
(1 + ) - 1 + ()
we recover the phenomenological form
F_ = m ( + T_temp).
Thus used in the A5/A6 dataset is the effective low-frequency, low-gradient limit of ().

The extended TTG force law is:

F_total = (G " M_local " m) / r' + m " " ( + T_temp )

Using the A5/A6 parameter set:

• Test mass: m = 10 kg
• Local mass concentration: M_local = 1.0 10 kg
• Distance: r = 500 m
• Temporal gradient: = 1.0 10
• Thermal correction: T_temp = 2.0 10
• Coefficient = 1.0 10

Step 1. Classical Newtonian component

F_GR = (G " M_local " m) / r'
F_GR - 2.67 N

Step 2. Temporal-gradient component

( + T_temp ) = 1.010 + 2.010
- 2.0110

F_ = m " " (2.0110)
F_ - 20.0 N

Step 3. Total force

F_total = 2.67 N + 20.0 N
F_total - 22.67 N

Interpretation

The value T_temp = 210 corresponds not to vacuum, but to a high thermal-gradient configuration (as used in the A5/A6 dataset), where temperature contributes directly to the local temporal rate.
It is not intended as a universal value but as a demonstration of the sensitivity of the extended TTG force law to thermodynamic condition

• The result exactly reproduces the earlier A5/A6 numerical output.
• The temporal/thermal contribution dominates over Newtonian gravity by nearly an order of magnitude, despite being extremely small in absolute -variation.
• This demonstrates the high sensitivity of the TTG framework to and T_temp in laboratory conditions.

8.3. Scaling Laws

The structure of the TTG force law yields clear scaling behavior:

(a) Newtonian component

F_GR M_local / r'

This serves as the baseline for comparison.

(b) Temporal correction

F_ = m " " ( + T_temp )
Therefore:

• Linear scaling with test mass (m)
• Linear scaling with , which encodes coupling strength
• Linear dependence on , enabling detection of extremely small gradients
• Linear dependence on T_temp, enabling thermo-gravimetric measurements

Since g / c' in gravitational systems, TTG predicts:

F_ 1 / c'

explaining why gravitational effects are small in vacuum
but can be amplified in matter where () increases effectively.

8.4. Sensitivity to Temperature, Density, and -Gradients

The extended TTG force is sensitive to three measurable environmental parameters:

(1) Temperature (via T_temp)

Thermal gradients modify local proper-time rate:

T_temp /T

This effect aligns with:

• plasma experiments,
• thermal convection chambers,
• superconducting coherence effects.

(2) Density and material phase (via and )

The temporal susceptibility () reflects how strongly matter couples to :

= " e^{iC}

where encodes density and C encodes coherence/entropy.

High-coherence materials (superfluids, superconductors) modify dramatically.

(3) Direct -gradients ()

Even extremely small gradients (1010) produce measurable forces when amplified by .

This opens a path to detecting nonclassical forces with sensitive balances or interferometers.

8.5. Experimental Boundary Conditions

For rigorous validation, TTG predictions require control over the following experimental variables:

1. Temperature stability
   Experiments must stabilize T_temp to <10 for sub-newton precision.
2. Environmental density and coherence
   () depends strongly on material phase experiments should log humidity, plasma state, or superconducting fraction.
3. Distance r and geometry
   Newtonian baseline must be computed with high precision.
4. Clock-based monitoring of
   Optical clocks provide a direct measurement of -variations:

f/f - " h

1. Shielding from conventional forces
   TTG corrections are small experiments must exclude:
   • electromagnetic interactions,
   • thermal convection,
   • mechanical vibrations.
2. Replication under multiple regimes
   TTG predicts different behavior under:
   • vacuum vs. air,
   • cryogenic vs. room temperature,
   • coherent (superfluid) vs incoherent (normal matter) states.

When these conditions are satisfied, TTG yields quantitatively falsifiable predictions a crucial requirement for any physical theory.

9. Experimental Pathways

A central strength of the Temporal Theory of Gravity (TTG) is that it produces clear, quantitative, and falsifiable experimental predictions.
Unlike purely geometric reformulations of gravity, TTG introduces a physical scalar field (x) whose spatial derivatives generate inertial effects, enabling direct measurement of through laboratory and astrophysical techniques.

This section outlines the key experimental avenues capable of validating or falsifying the TTG framework.

9.1. ClockAccelerometer Correlation Experiments

TTG predicts a strict proportionality:

a = c' "

and

f / f - / - / c'

Therefore, changes in clock rate must correlate with measured accelerations in vertically separated positions or environments with controlled -gradients.

Core measurable prediction

f / f = (g " h) / c'
a = g

Thus:

(f / f) / a = h / c'

This relation is a unique TTG signature:
General Relativity predicts gravitational redshift but does not require simultaneous inertial acceleration measurements.

Experimental designs

• Dual atomic clocks + superconducting gravimeter.
• Cold-atom clocks separated by variable height in vacuum towers.
• Transportable optical clocks in elevator-like vertical motion.

Expected sensitivity:
Height difference of 1 meter f/f - 1.1 10, already measurable by modern optical clocks.

This is currently the cleanest test for TTG.

9.2. Thermal and Plasma Gradient Experiments

The extended TTG force law:

F_ = m " " ( + T_temp )

predicts that the effective gravitational force on a test mass changes in the presence of:

• strong temperature gradients,
• heated plasmas,
• cryogenic transitions,
• rapid thermal cycling.

Mechanism

Temperature modifies the local proper-time rate:

T_temp /T

This effect is small but amplifiable through and (), especially in:

• laser-heated plasma columns,
• high-Q thermal cavities,
• superconducting environments (coherence term C enhancement).

Suggested experiments

• Controlled heating of a mass near precision torsion balances.
• Plasma discharge chambers where density and T are tunable.
• Cryogenic transitions of materials placed near a gravimeter.

Expected effect magnitude:
1010 fractional force variations (detectable).

9.3. Coherence-Dependent Gravity Modification

The susceptibility:

() with = " e^{iC}

predicts that coherence of matterexpressed by phase Cmodifies gravitational coupling.

Environments of interest

• Superconductors
• Superfluids
• BoseEinstein condensates
• Photonic and magnonic coherent states
• Crystalline phases with low entropy

Measurable predictions

• Weak anomalies in weight when a nearby material transitions into a coherent phase.
• -dependent modulation of local gradients.
• Enhanced sensitivity in suspended resonators and optical cavities.

Why this is important

TTG is the first model to predict coherence-dependent gravitational response through ().
Even a null result strongly constrains the theory.

9.4. Spaceborne Validations (GPS, GNSS, Satellite Orbits)

Space systems provide continuous real-time measurement of:

• gravitational time dilation (),
• inertial effects (orbital accelerations),
• altitude-dependent clock drift.

TTG predictions accessible in orbit

1. GPS clock drift
   f/f - /c'
   TTG provides a direct -field reinterpretation.
2. Clockaccelerometer dual measurements on satellites
   Correlated deviations would signal anomalies.
3. Solar plasma environment
   () and T_temp may produce detectable deviations during solar storms.
4. Low-Earth vs. high-Earth orbits
   Differences in scaling can test TTG corrections to geometric gravity.

Advantages of spaceborne tests

• Low noise
• Large altitude differences (h up to 20,000 km)
• Natural environmental -gradient variations

9.5. Table of Measurable TTG Predictions

10. Discussion

10.1. Advantages over GR-only Formulations

Although General Relativity (GR) is a remarkably successful geometric theory, it exhibits several well-known conceptual limitations that the Temporal Theory of Gravity (TTG) naturally addresses.

(1) Local gravitational energy becomes well-defined

GR does not allow a tensorial, coordinate-independent definition of local gravitational energy.

TTG introduces a scalar field (x), whose spatial gradients generate inertial effects through:

g = c' "
(weak-field relation)

This immediately yields a local energy density:

_ = (c / 8G) " ||'

This quantity:

• is coordinate-independent,
• directly measurable through clocks and accelerometers,
• and can be mapped experimentally via altitude/time-rate variations.

(2) Unified description of gravitational, thermal, and material responses

TTGs force law includes the susceptibility factor ():

F = m " c' " (ln ) " [ 1 + () ]

This couples gravity to:

• temperature gradients,
• material density,
• coherence phases,
• plasma and condensed-matter effects.

GR treats gravity purely geometrically and contains no such mechanism.

(3) More direct connection to measurable quantities

GR works through the metric g, whereas TTG connects directly to measurable observables:

• clock rates,
• accelerations,
• -gradients,
• temperature-induced corrections.

Thus TTG is better suited for laboratory-scale tests.

(4) Compatibility with GR in validated regimes

When the potential satisfies || c':

• Poisson equation is recovered,
• Newtonian gravity appears automatically,
• gravitational time dilation matches GR predictions.

TTG is conservative where GR is tested, and innovative where GR is silent.

10.2. Open Problems

TTG opens several promising directions but also highlights challenges that remain unresolved.

(1) Full nonlinear behavior of in strong fields

The nonlinear TTG Master Equation, including the R' curvature term, must be explored in:

• neutron star interiors,
• near-black-hole environments,
• early universe cosmology.

We lack exact solutions for in these regimes.

(2) Dynamics of susceptibility ()

While () provides a powerful unification of gravitational and material behavior, open questions include:

• What is the microscopic derivation of from first principles?
• Does depend on entropy flow, coherence, or quantum phases more fundamentally?
• How does behave in non-equilibrium systems?

(3) Quantization of and relation to quantum gravity

Section 7 establishes a canonical quantization scheme, but several problems remain:

• Is the -field renormalizable?
• What is the physical meaning of -quanta in curved spacetime?
• How does TTG interface with existing quantum-gravity approaches (LQG, asymptotic safety, causal sets)?

(4) Cosmological implications

TTG may provide natural explanations for:

• cosmic acceleration,
• dark energy analogues (via -evolution),
• early universe inflation-like regimes.

But these require dedicated cosmological models beyond the weak field.

10.3. Limitations

Academic honesty requires clearly defining the current boundaries of TTG.

(1) The theory is scalar; GR is tensorial

TTG introduces a scalar gravitational field , whereas GR gravity emerges from spacetime curvature.
Although TTG reproduces the weak-field limit correctly, it may diverge from GR predictions in:

• gravitational wave structure,
• strong-field tests,
• binary pulsar timing.

(2) No complete cosmological model yet

TTG provides mechanisms but not a full CDM replacement.
Key components still missing:

• full Friedmann-like equations in -variables,
• perturbation theory for structure formation.

(3) Susceptibility () is phenomenological

Currently, is modeled analogously to Kubo response theory, but:

• microscopic foundations,
• state-dependence,
• scale dependence

remain to be derived rigorously.

(4) Limited experimental precision in -gradient measurements

Although TTG predicts correlations between clocks and accelerometers,
state-of-the-art experimental capabilities (1010) are only approaching the required sensitivity.

10.4. Potential Inconsistencies & Resolution Strategies

Issue 1 Logarithmic force term in strong gradients

The TTG force law:

F = m " c' " (ln )

may appear to diverge when 0.

Resolution:
The full nonlinear field equation includes the " R " ' term, which stabilizes and prevents singular behavior.

Issue 2 Energy conservation under () modulation

If depends on a coherence phase C, could this break thermodynamic constraints?

Resolution:
The stressenergy tensor derived from the TTG action automatically satisfies:

T = 0
or
_ T^{} = 0

Thus can modulate forces without violating conservation laws.

Issue 3 Compatibility with gravitational waves

Scalar theories typically predict monopole radiation not seen by LIGO.

Resolution:
TTG does not replace GR tensorial modes; it coexists with them.
-excitations may:

• couple only to matter,
• be suppressed by ,
• or be extremely weak.

A full wave analysis is future work.

Issue 4 Frame dependence of

Because is a scalar but = /t depends on slicing, does the force law remain invariant?

Resolution:
The covariant TTG 4-force is:

F = m " c' " h " (ln )

with h the projection orthogonal to u.
This ensures coordinate-invariant physical predictions.

11. Conclusion

11.1. What TTG Unifies

The Temporal Theory of Gravity (TTG) establishes a unified field-theoretic framework in which gravitation, thermodynamic responses of matter, and variations of the local proper-time rate emerge as manifestations of a single scalar field (x) and its time-rate = /t.
Within this formulation:

• Newtonian gravity appears as the linear response of ln to weak perturbations.
• General Relativity is recovered as the geometric limit where is slowly varying and curvature dominates.
• Thermal, density, and coherence-dependent contributions enter through the temporal susceptibility (), providing a natural bridge between gravitational and material phenomena.
• The extended force law reproduces classical results while predicting measurable deviations in non-equilibrium media, plasmas, and precision timekeeping systems.

Thus, TTG unifies gravitational, thermodynamic, and material-response behaviors in a single variationally grounded frameworka feature absent in both GR and scalar-tensor alternatives.

11.2. Why Should Be Considered a Physical Field

Three independent lines of reasoning justify treating as a bona fide physical field:

1. Observational: All precision experimentsGPS, clocks on satellites, optical-lattice clocksdirectly measure variations of proper-time rate. These variations behave like a field, not a coordinate artifact.
2. Theoretical: GR lacks a local gravitational energy density due to its geometric formulation. Introducing with curvature coupling resolves this by producing a well-defined stress-energy tensor for the time-rate field.
3. Variational Consistency: When is included in the action, its dynamics follow from the EulerLagrange equation, and gravitational force arises naturally as
   F = mc' ln ,
   rather than being inserted by hand.

Taken together, these arguments show that (x) carries independent physical content and should be treated on the same footing as other fundamental fields in physics.

11.3. Next Steps: Quantum Regime, Cosmology, Experimental Tests

The TTG framework opens several concrete research directions:

Quantum Regime

• Quantization of predicts temporal phonons and vacuum fluctuations that modify ().
• Possible links to quantum gravity phenomena: decoherence rates, Planck-scale time dispersion.

Cosmology

• A slowly varying cosmological -field produces expansion dynamics without requiring dark energy.
• Temporal susceptibility may provide a new mechanism for structure formation and early-universe thermal asymmetries.

Experimental Tests

Several tests fall squarely within current technological capability:

• Clockaccelerometer correlation: co-located clocks and accelerometers must show proportional responses to ln .
• Thermal/plasma-induced modifications of gravity: controlled laboratory tests for () 0.
• Coherence-dependent anomalies: superconductors, lasers, and BoseEinstein condensates provide natural platforms.
• Spaceborne timing: satellites and deep-space probes can constrain temporal-field gradients with exquisite precision.

Final Statement

TTG provides a coherent, variationally consistent, experimentally testable reformulation of gravity in which the proper-time rate replaces the metric as the primary dynamical variable.
By embedding gravitational, thermodynamic, and material responses into a single scalar field , the theory offers a clear pathway toward a unified description of classical and quantum phenomenasomething long missing from contemporary physics.

12. Structural Analysis of Sources and Their Connection to the Temporal Theory of Gravity (TTG)

The Temporal Theory of Gravity (TTG) does not arise in a vacuum. It is constructed as a deliberate synthesis of ideas from five distinct domains of contemporary physics, each addressing a specific facet of the gravitational puzzle. The following analysis maps the intellectual architecture of TTG, demonstrating how classical foundations and modern advances are woven into a coherent theoretical framework. This reflective overview precedes the bibliography to clarify the logical role of each reference and to position TTG within the ongoing scientific discourse.

I. The EnergyMomentum Problem in General Relativity The Core Motivation

Key TTG concepts derived from this domain: Introduction of the propertime field  as a localizable carrier of gravitational energy; construction of a welldefined stressenergy tensor T().
Relevant TTG sections: Section 1.1 (Motivation), Section 4.5 (Local Gravitational Energy), Appendix B (StressEnergy Tensor of the Field).
Core sources: Szabados (2009) [1]; Wald (1984) [2]; Misner, Thorne & Wheeler (1973) [3].
Purpose: To establish the fundamental inadequacy that TTG aims to resolve. The nonlocalizability of gravitational energymomentum in standard GR is a wellknown conceptual limitation, thoroughly discussed in foundational texts [2, 3] and modern reviews [1]. TTG addresses this directly by promoting propertime rate to a dynamical scalar field , whose associated tensor (B.3) provides a covariant and local description of gravitational energy, thereby filling a gap explicitly identified in the GR literature.

II. ScalarTensor and Alternative Theories of Gravity The Formal Context

Key TTG concepts derived from this domain: Variational principle with curvature coupling; Lagrangian structure L_; parameterized postNewtonian (PPN) consistency.
Relevant TTG sections: Chapter 3 (Action and Variational Principle), Chapter 4 (Relation to General Relativity).
Core sources: Fujii & Maeda (2003) [4]; Will (2014) [5]; Clifton et al. (2012) [6]; Poisson & Will (2014) [7].
Purpose: To embed TTG within the rigorous tradition of metric theories of gravity. TTG can be viewed as a specific scalartensor theory where the scalar field is identified with normalized proper time [4]. Its Lagrangian construction follows established norms, and its weakfield behaviour is explicitly designed to meet the stringent constraints of the PPN formalism [5, 7], ensuring compatibility with all solarsystem tests. Reviews of modified gravity [6] provide the broader landscape into which TTG is integrated.

III. Time as a Physical and Thermodynamic Quantity The Conceptual Foundation

Key TTG concepts derived from this domain: Interpretation of (x) and = d/dt as fundamental, nongeometric fields; linking temporal flow to irreversibility and gravitational interaction.

The propertime field is not only a dynamical scalar but also a natural candidate for the physical foundation of the arrow of time. Its gradients govern gravitational acceleration, while its rate = d/dt directly encodes irreversible processes. Through the entropylinked susceptibility _S (Appendix C), couples to thermodynamic irreversibility, ensuring that the temporal field aligns the gravitational arrow with both the thermodynamic and cosmological arrows of time. In this way TTG provides a unified physical basis for irreversibility across scales.

Relevant TTG sections: Section 1.2 (Why is the Missing Field), Section 2 (Mathematical Foundations), discussion of the arrow of time.

Core sources: Rovelli (2004) [8]; Prigogine (1997) [9]; Carroll (2010) [10]; Maudlin (2012) [11].

Purpose: To provide a deep conceptual rationale for treating time as a dynamical agent. Contemporary physics increasingly questions the passive, geometric role of time [8, 11]. Works on irreversibility [9] and the thermodynamic arrow [10] underscore times active, driverlike character. TTG absorbs this perspective, postulating that spatial variations in the rate of time () are the direct cause of inertial and gravitational phenomena, thereby offering a physical mechanism for what is merely a coordinate parameter in GR.

IV. Precision Gravity and Metrology The Experimental Bridge

Key TTG concepts derived from this domain: Prediction of a lockstep correlation f/f g; quantitative estimates for laboratory and spacebased tests.
Relevant TTG sections: Chapter 9 (Experimental Pathways), Appendix E (Sensitivity Analysis).
Core sources: Pound & Rebka (1960) [12]; Chou et al. (2010) [13]; Adelberger, Heckel & Nelson (2003) [14]; Tino et al. (2020) [15]; Peters, Chung & Chu (1999) [16].
Purpose: To demonstrate the falsifiable nature of TTG and its accessibility to modern experiment. The classic gravitational redshift measurement [12] is reinterpreted in TTG as a direct measurement of . Stateoftheart atomic clocks [13] and atom interferometers [16] have reached the sensitivity (10 in fractional frequency, 10' m/s' in acceleration) required to test the unique clockaccelerometer correlation predicted by TTG (Section 9.1). Proposals for spaceborne tests [15] and reviews of fifthforce searches [14] define the experimental frontier where TTG makes concrete predictions.

V. Condensed Matter Physics and Linear Response Theory The MicroMacro Coupling Mechanism

Key TTG concepts derived from this domain: Definition of the complex order parameter  = "e^(iC) and the temporal susceptibility (); phenomenological models linking  to material state.
Relevant TTG sections: Chapter 5 (Temporal Susceptibility ()), Appendix C (Models for ()).
Core sources: Tinkham (2004) [17]; Pethick & Smith (2008) [18]; Kubo, Toda & Hashitsume (1991) [19]; Landau & Lifshitz, Vol. 9 (1980) [20].
Purpose: To ground the materialdependent modulation of gravity in established manybody physics. The formalism of macroscopic quantum order parameters [17, 18, 20] provides the language to describe how coherence (e.g., in superconductors or BECs) can influence the field. The Kubo formula [19] elevates () from a phenomenological factor to a genuine response function, theoretically derivable from microscopic correlations. This connects TTG to the vast toolbox of condensedmatter theory and allows specific predictions for anomalous gravitational effects in quantumcoherent materials.

Conclusion: An Interdisciplinary Synthesis

The accompanying bibliography is therefore a curated map of TTG's intellectual origins. Each reference is not merely a citation but a structural pillar supporting a specific part of the edifice: from identifying the foundational problem (I) and establishing the formal rules (II), through providing the conceptual premise (III) and the means of verification (IV), to detailing the mechanism of interaction with complex matter (V). This interdisciplinary approach ensures that TTG is theoretically sound, conceptually novel, empirically testable, and deeply connected to active research fields beyond gravitation alone.

References (Numbered according to appearance in the analysis)

[1] Szabados, L. B. (2009). Quasi-local energy-momentum and angular momentum in GR: A review article. Living Reviews in Relativity, *12*(1), 4.

[2] Wald, R. M. (1984). General Relativity. University of Chicago Press.

[3] Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman.

[4] Fujii, Y., & Maeda, K.-I. (2003). The Scalar-Tensor Theory of Gravitation. Cambridge University Press.

[5] Will, C. M. (2014). The confrontation between general relativity and experiment. Living Reviews in Relativity, *17*(1), 4.

[6] Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. (2012). Modified gravity and cosmology. Physics Reports, *513*(1-3), 1189.

[7] Poisson, E., & Will, C. M. (2014). Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press.

[8] Rovelli, C. (2004). Quantum Gravity. Cambridge University Press.

[9] Prigogine, I. (1997). The End of Certainty: Time, Chaos, and the New Laws of Nature. Free Press.

[10] Carroll, S. M. (2010). From Eternity to Here: The Quest for the Ultimate Theory of Time. Dutton.

[11] Maudlin, T. (2012). Philosophy of Physics: Space and Time. Princeton University Press.

[12] Pound, R. V., & Rebka, G. A. (1960). Apparent weight of photons. Physical Review Letters, *4*(7), 337341.

[13] Chou, C. W., Hume, D. B., Rosenband, T., & Wineland, D. J. (2010). Optical clocks and relativity. Science, *329*(5999), 16301633.

[14] Adelberger, E. G., Heckel, B. R., & Nelson, A. E. (2003). Tests of the gravitational inverse-square law. Annual Review of Nuclear and Particle Science, *53*, 77121.

[15] Tino, G. M., et al. (2020). SAGE: A proposal for a space atomic gravity explorer. The European Physical Journal D, *74*(8), 164.

[16] Peters, A., Chung, K. Y., & Chu, S. (1999). Measurement of gravitational acceleration by dropping atoms. Nature, *400*(6747), 849852.

[17] Tinkham, M. (2004). Introduction to Superconductivity (2nd ed.). Dover Publications.

[18] Pethick, C. J., & Smith, H. (2008). BoseEinstein Condensation in Dilute Gases (2nd ed.). Cambridge University Press.

[19] Kubo, R., Toda, M., & Hashitsume, N. (1991). Statistical Physics II: Nonequilibrium Statistical Mechanics (2nd ed.). Springer-Verlag.

[20] Landau, L. D., & Lifshitz, E. M. (1980). Statistical Physics, Part 2 (Vol. 9). Pergamon Press.

[21] Lemeshko, A. (2025). Temporal Theory of the Universe. Zenodo (TTG Series).
$$Online resource, link: `https://zenodo.org/communities/ttg-series/`$$

Appendix A: Full Variational Derivation of the TTG Field Equations

This appendix provides a stepbystep derivation of the Temporal Theory of Gravity (TTG) field equations from the covariant action principle. The goal is to show explicitly how the field equation and its stressenergy tensor arise, and to verify energymomentum conservation.

A.1. The TTG Action

We consider the total action for gravity, the field, and matter:

S = dx -g " L_total

where the Lagrangian density is:

L_total = (/2) g^ ( )( ) + R ' + L_int(, ) + L_matter(A.1)

Here:

• (x) is the dimensionless propertime field,
• , are dimensionless coupling constants (in natural units),
• R is the Ricci scalar,
• L_int describes the interaction of with the order parameter = e^(iC),
• L_matter is the Lagrangian of ordinary matter minimally coupled to g^.

A.2. Variation with Respect to

Varying S with respect to gives:

S = dx -g [ g^ ( ) () + 2 R + (L_int/) ](A.2)

Integrating the first term by parts yields:

S = dx -g [ (g^ ) + 2 R + L_int/ ] + boundary terms(A.3)

Since g^ = ^ and ^ = , we obtain the EulerLagrange equation:

2 R + L_int/ = 0(A.4)

In the weakfield limit this reduces to:

' = (4G / c') _eff

A.3. Variation with Respect to the Metric g^

The metric variation is more involved because both the kinetic term and the curvature coupling depend on g^.

A.3.1. Kinetic term

(-g (/2) g^ ) = -g [ ( )( ) (1/2) g^ ()' ] g^(A.7)

A.3.2. Curvaturecoupling term

(-g R ') = -g ' [ R^ (1/2) g^ R + g^ ] g^(A.8)

This produces the Einsteinlike contribution 2 ' G^ plus divergence terms ensuring conservation.

A.3.3. Interaction term

If L_int does not depend explicitly on the metric:

(-g L_int) = (1/2) -g g^ L_int g^(A.9)

A.3.4. Full field StressEnergy Tensor

Combining the above, we obtain:

T^() = [ ( )( ) (1/2) g^ ()' ] + 2 ' G^ + ( g^ ) ' + T^(int)(A.10)

where G^ = R^ (1/2) g^ R is the Einstein tensor.

A.4. Conservation Law

From diffeomorphism invariance:

_ T^() = 0(A.11)

For the full system: _ [ T^() + T^(m) ] = 0

A.5. WeakField Limit and Coupling Constants

In the weakfield, stationary limit:

• g - (1 + 2/c')
• g_ij - _ij
• - (1 + /c')
• R - 4G / c'

The equation becomes:

' 2 (4G / c') = 0

Choosing = 1 and the coupling ratio:

2 / = 1 / (6 c')(A.12)

yields:

' = (4G / c')

and with = 1 + /c' we recover:

' = 4G

A.6. Summary

• The TTG field equation (A.4) follows from a covariant action.
• The stressenergy tensor (A.10) is consistent with diffeomorphism invariance.
• Energymomentum is locally conserved (A.11).
• The Newtonian limit is recovered with the correct factor 4G when the coupling constants satisfy (A.12).

Thus TTG satisfies the requirements of a classical field theory coupled to gravity.

Appendix B: StressEnergy Tensor of the Field

This appendix provides a self-contained derivation and physical interpretation of the stressenergy tensor for the temporal field (x). It shows that carries localized gravitational energy, momentum, and stresses resolving the long-standing problem of defining gravitational energy in General Relativity (GR).

B.1. Definition from the Variational Principle

The stressenergy tensor of the field is defined as:

T_{}() = (2 / -g) " S_ / g^{}(B.1)

with the -sector Lagrangian:

L_ = (/2) g^{} (_ )(_ ) + R ' + L_{int}(, )(B.2)

The full variation (derived in Appendix A) gives:

T_{}() = [(_ )(_ ) (1/2) g_{} ()']
+ 2 ' G_{}
+ ( g_{} _ _ ) '
+ T_{}^{int}(B.3)

where G_{} = R_{} (1/2) g_{} R.

B.2. Physical Interpretation of Each Term

B.2.1. Canonical Scalar Field Part

The canonical kinetic contribution of the field is:

T_can^ = [ (^ )(^ ) (1/2) g^ ()' ]

Energy density:

_can = T_can^00 = (/2) [ ()' + ()' ]

Momentum density:

p_i^can = " " _i

Weakfield, static limit

For weak gravitational fields we take:

- 0, - 1 + / c'

Then:

- (1/c') ()' - (1/c) ()'

Hence:

_can - ( / 2c) ()'

Choosing the Newtoncompatible normalization:

= c / (8G)

yields:

_can - (1 / 16G) ()'

Together with the geometric term contribution _geom (Sec.B.2.2), the total field energy density in the weakfield limit becomes:

_ = _can + _geom = (1 / 8G) ()'

which is exactly the classical Newtonian gravitational field energy density.

Note on the normalization of

The choice = c / (8G) is not dictated by the variational principle; the only condition required for the correct Newtonian limit is the ratio:

2 / = 1 / (6c')

The absolute scale of (and ) remains free, as in any scalartensor or conformal theory. The value adopted here is a convenient normalization that makes the field energy density coincide with the familiar Newtonian expression:

= (1 / 8G) ()'

This facilitates direct comparison with standard gravitational energetics, but any consistent with the ratio 2/ = 1/(6c') is equally valid in principle.

B.2.2. Geometric (Einstein-like) term

T_{}^{geom} = 2 ' G_{}

In weak fields:

• G_{00} - ' / c'
• - 1

Hence:

_{geom} - 2 (' / c') - (8G / c')

This couples directly to matter density.

B.2.3. Divergence (improvement) term

T_{}^{div} = ( g_{} _ _ ) '

This term:

• is a total divergence,
• does not contribute to integrated energy,
• enforces exact local conservation _ T^{}()=0.

B.2.4. Interaction term

T_{}^{int} = (2/-g) " (-g L_{int}) / g^{}

For the model L_{int} = () :

T_{}^{int} = (1/2) g_{} ()

This describes material-dependent modifications (thermal, coherent, plasma-induced).

B.3. Weak Field, Static Limit

The dominant terms yield:

_ = (/2)()' + 2 ' G_{00}(B.4)

Using:

• - 1 + /c'
• = (1/c')
• G_{00} = ' / c'
• condition from Appendix A: 2 / = 1 / (6c')

we obtain:

_ - ( / 2c)()' + (8G / c')

Thus total -energy naturally splits into gradient energy + matter coupling.

B.4. Local Energy Conservation

Diffeomorphism invariance gives:

_ [ T^{}() + T^{}(m) ] = 0(B.5)

In weak fields:

_t _ + "S_ = g " j_(B.6)

where S_ is the -field energy flux.

B.5. Comparison with GR Pseudotensors

Unlike GRs energy pseudotensors, T_{}():

1. is a true tensor,
2. is locally defined,
3. obeys covariant conservation,
4. has clear physical meaning.

Thus TTG supplies the missing local gravitational energy density that GR lacks.

B.6. Example: Static Spherical Solution

Gravitational potential: (r) = GM / r

Propertime field: (r) = 1 + (r)/c' = 1 GM / (c' r)

Radial derivative: d/dr = (1/c') " d/dr = GM / (c' r')

Canonical field energy density: (r) - (/2) " (d/dr)' = ( G' M') / (2 c r)

Normalization: Choosing = c / (8G), we obtain

(r) = (GM)' / (8G c r)(B.7)

Interpretation

Equation (B.7) has a double significance:

1. Newtonian gravitational field energy density. It exactly reproduces the classical Newtonian expression: Newton = (1 / 8G) " ()' = (GM)' / (8G r) since = GM / r'. This shows that TTG is fully consistent with the traditional fieldenergy formulation of Newtonian gravity.
2. PostNewtonian GR energy density. The same expression appears in the postNewtonian limit of GR when using the LandauLifshitz pseudotensor. The key advantage of TTG is that here this energy density arises from a true covariant stressenergy tensor, not from a coordinatedependent pseudotensor.

Conclusion: TTG not only reproduces the Newtonian and postNewtonian energy densities but provides a tensorial, locally defined origin for gravitational field energy.

B.7. Summary

The stressenergy tensor of the field:

• provides a local and tensorial definition of gravitational energy,
• ensures exact conservation when combined with matter,
• yields the correct Newtonian limit,
• unifies gradient, geometric, and material (-dependent) effects.

TTG thus resolves the ambiguity of gravitational energy in GR while remaining experimentally consistent in all tested regimes.

Appendix C: Models for the Temporal Susceptibility () from Thermodynamics and Condensed Matter

This appendix provides explicit phenomenological and semimicroscopic models for the temporal susceptibility (), where = "e^(iC) is the complex order parameter introduced in the main text. The goal is to illustrate how thermodynamic variables (temperature, entropy, pressure) and condensed matter properties (coherence, phase transitions, quantum order) can modulate the response of matter to gradients of the propertime rate field .

C.1. General Structure of the Susceptibility

The susceptibility () appears in the TTG force law as a multiplicative factor:

F = m c' (ln ) " [1 + ()](C.1)

It is a dimensionless quantity that depends on the local state of matter through the order parameter:

= e^(iC)(C.2)

where:

• effective massenergy density (including possible pressure contributions),
• C coherenceentropy phase, encoding the degree of microscopic order (C - 0 for maximal coherence, C /2 for maximal disorder).

In the linear response regime, is assumed to be small (|| 1) and can be expanded as:

() = + () + (C) + (, C) + (C.3)

C.2. Thermodynamic Models

C.2.1. Temperaturedependent model

If the dominant environmental variable is temperature T, we can model as a function of T relative to a characteristic scale T*:

T(T) = [1 tanh((T T) / T)] + *(C.4)

where T* is a characteristic temperature scale (e.g., Debye temperature, critical temperature of a phase transition).

Interpretation:

• At low temperatures (T T*), - _ + (enhanced response).
• At high temperatures (T T*), - (baseline response).
• T controls the sharpness of the transition.

C.2.2. Entropydependent model

_S(S) = _max exp(S / S)(C.5)

where S is a reference entropy scale. High entropy suppresses ; low entropy enhances it.

C.3. Condensed Matter and Coherence Models

C.3.1. Superconducting / Superfluid coherence

Order parameter: _coh = n_c e^(i), where n_c is the condensate density and is the macroscopic quantum phase.

Model: _coh(n_c) = (n_c / n) [1 (T / T_c)^](C.6)

Parameters:

• ~ 1010 (weak coupling),
• n reference density,
• T_c critical temperature,
• exponent ( = 4/3 for BCS superconductors, = 1 for Bose condensates).

C.3.2. Plasma and ionized media

_plasma(n_e, T_e) = p / [1 + (k_D )'](C.7)

where:

• _p plasma amplitude,
• k_D = -(n_e e' / ( k_B T_e)) Debye wave number,
• _ temporal screening length.

The value of _p obtained from this scaling relation should be regarded as a representative estimate for given plasma parameters (e.g., laboratory tokamak conditions), not as a fixed constant.

C.3.3. Glassy and disordered systems

Effective susceptibility:

_glass = ^{2} (C) P(C) dC(C.8)

with P(C) a wrapped Gaussian distribution around C with width . As increases (disorder), decreases.

C.4. Kubotype Linear Response Formulation

For small perturbations:

(, k) = (1 / i) ^ dt e^(it) [(t), (0)](C.9)

Static limit ( 0, k 0):

- _T ()'(C.10)

with _T = 1 / (k_B T).

Here is the operator that couples the system to the gradient of the temporal field . In many cases, it can be identified with the Hamiltonian density or the number density operator.

C.5. Constraints from Symmetry and Conservation Laws

Any admissible () must respect:

1. Causality (analyticity in upper half plane).
2. Stability (positive definite energy).
3. Second Law (nonnegative entropy production).
4. Galilean/relativistic invariance.

Generic quadratic model near equilibrium:

(, C) = a + a( ) + a C' + (C.11)

C.6. Estimated Orders of Magnitude

C.7. Summary

The susceptibility () provides a flexible phenomenological bridge between the field and material properties. The models presented here show how:

• Thermodynamic variables (T, S, P) modulate gravitational response.
• Quantum coherence (superconductivity, superfluidity) can enhance by several orders of magnitude.
• Collective states (plasmas, glasses) introduce characteristic scaling.
• Linear response theory connects TTG to standard fluctuationdissipation relations.

Appendix D: Dimensional Analysis and Numerical Constants

This appendix provides a systematic dimensional analysis of all quantities in the Temporal Theory of Gravity (TTG) and lists the fundamental and derived numerical constants used throughout the paper.

D.1. Fundamental Dimensions and Notation

We employ the standard SI system with three base dimensions: Length [L], Mass [M], and Time [T]. The speed of light c and the gravitational constant G are treated as fundamental conversion constants.

D.2. Dimensional Consistency of the TTG Equations

D.2.1. Field Equation (weak field) ' = (4G / c') _eff

• Left side: ['] = [L'] (since is dimensionless).
• Right side: [G/c'] = [L M], [] = [M L], so [G/c'] = [L'].
• Consistent.

D.2.2. Force Law F = m c' (ln ) " [1 + ()]

• [m c'] = [M L' T'] (energy).
• [(ln )] = [L] (since ln is dimensionless).
• Product: [M L' T'] " [L] = [M L T'], which is force.
• Consistent.

D.2.3. Lagrangian Density L_ = (/2) g^ ( )( ) + R ' + L_int

• [g^ ()'] = [L'] (since is dimensionless).
• The Lagrangian must have energy density dimension: [M L T'].

For the first term: [] " [L'] = [M L T'] [] = [M L T'] (i.e., has dimensions of force or energy per length).

For the second term: [R] = [L'], thus: [] " [L'] = [M L T'] [] = [M L T']

So and share the same dimensional class, ensuring consistency.

D.3. Natural Normalization and Constants

In the weakfield limit, to reproduce the Poisson equation it is sufficient that:

2 / = 1 / (6 c').

The absolute values of and can be chosen arbitrarily provided this ratio is maintained, since is dimensionless and field rescaling absorbs the normalization. This ensures dimensional consistency and the correct Newtonian limit.

• Asymptotic field: = 1 (dimensionless).
• Fundamental constants: CODATA 2022 values.
• Derived constants: used consistently in numerical examples.

D.4. Conversion Factors for Laboratory Measurements

For experiments with small gradients:

• Temporal gradient to acceleration: a = c' = a / c'

For Earths gravity g = 9.81 m/s': _Earth - 9.81 / (3 10)' - 1.09 10 m

• Clock frequency shift: f / f = = / c' = g h / c'

For h = 1 m: f / f - 1.09 10

D.5. Orders of Magnitude for ()

From Appendix C, is dimensionless. Expected ranges:

• Vacuum / classical matter: || < 10'
• Coherent matter (superconductors): || ~ 10 10
• Plasma / thermal gradients: || ~ 10 10

Appendix E: Extended Datasets (A5/A6, A7, B1)

This appendix describes the numerical datasets used to validate the Temporal Theory of Gravity (TTG) and provides additional configurations beyond the A5/A6 example presented in the main text.

E.1. Dataset Naming Convention

Datasets are labeled as follows:

• Letter: Series identifier (A = laboratory scale, B = astrophysical, C = extreme conditions).
• Number: Configuration within series.
• Subscript: Variant (e.g., thermal, plasma, coherence).

Example: A6 denotes laboratory configuration #6 with thermal gradient emphasis.

E.2. Core Dataset A5/A6 (Reference Configuration)

Results:

• Newtonian component: F_N = 2.67 N
• Temporal component: F_ = 20.0 N
• Total force: F_total = 22.67 N

E.3. Extended Laboratory Dataset A7 (Cryogenic/Coherence)

Predicted anomaly: F = m c' (ln ) " _eff - 4.5 10 N

Order of magnitude justification:

• m c' - 0.1 9 10 = 9 10 J
• (ln ) - = 10 m
• Product: 9 10 10 = 0.9 J/m = 0.9 N
• Including _eff = 5 10: 0.9 5 10 - 4.5 10 N

Detectability: A force of ~4.5 10 N is at the edge of sensitivity of modern torsion balances (10 N) and cryogenic resonators (10 N). Reliable detection would require enhanced gradients or materials with larger _eff.

E.4. Plasma Dataset B1 (High Temperature Plasma)

Scaling relation: _p 1 / [1 + (D / )'](C.7)

The value _p = 2.3 10 is obtained from the scaling relation (C.7) for the given plasma parameters and serves as a representative estimate for laboratory tokamak conditions.

For tokamak conditions (D ): _p ~ 10.

E.5. Astrophysical Dataset C1 (Neutron Star Crust)

Note: In this regime, the full nonlinear field equation must be solved; the linear Poisson approximation breaks down.

E.6. Sensitivity Analysis Dataset D1

E.7. Data Format and Availability

All datasets are available in machinereadable format (JSON/CSV). Example:

json

{

"dataset_id": "A6",

"description": "Laboratory reference configuration",

"parameters": {

"m": {"value": 10, "units": "kg", "uncertainty": 0.001},

"M_local": {"value": 1.0e10, "units": "kg", "uncertainty": 1.0e7},

"r": {"value": 500, "units": "m", "uncertainty": 0.01}

},

"results": {

"F_newtonian": {"value": 2.67, "units": "N"},

"F_temporal": {"value": 20.0, "units": "N"},

"F_total": {"value": 22.67, "units": "N"}

}

}

E.8. Summary

These extended datasets demonstrate:

1. Reproducibility of the core A5/A6 result across different configurations.
2. Scalability from laboratory to astrophysical regimes.
3. Testability through specific experimental proposals.
4. Consistency with known physics in limiting cases.

The datasets provide a foundation for:

• Numerical validation of TTG codes,
• Design of laboratory experiments,
• Comparison with alternative theories,
• Extrapolation to unexplored parameter regimes.

Appendix F: Open Issues, Comparative Context, and Future Roadmap

This appendix addresses key conceptual, phenomenological, and theoretical questions that naturally arise in the formulation and interpretation of the Temporal Theory of Gravity (TTG). Its purpose is to clarify the current status of the theory, highlight welldefined directions for future development, and situate TTG within the broader landscape of gravitational physics. This distinguishes TTG from adhoc modifications and frames its falsifiable predictions.

F.1. Physical Interpretation of the ProperTime Field

• Weakfield regime: - 1 + / c' = d/dt - 1 + / c' where is the Newtonian potential. This directly links to measurable clockrate variations.
• General static spherically symmetric metric: ds' = A(r) dt' + B(r) dr' + r' d' Identification: (r) = A(r), (r) = ((r)/c) dr (in a suitable gauge). Here acts as a potential for the timerate field .
• Strongfield and horizon behavior: Near horizons (A(r) 0), 0 but may remain finite, suggesting could be horizonpenetrating. This offers a novel description of black hole interiors without a singularity in .
• Observable consequence: is the direct observable in clock comparisons, while g is derived. This reverses the usual geometric perspective.

F.2. Microscopic Foundation of the Susceptibility ()

• Order parameter: = " e^(iC), where is density and C encodes coherence/entropy.
• Linear response (Kubo formalism): - (1 / k_B T) " Var(h), linking directly to energydensity fluctuations.
• Examples: Ideal Fermi gas: - (3' / _F) " k_B T Bose gas near Tc: - [(5/2)/(3)] " (Tc/T)^(3/2) Superconductor (BCS): _SC(T) - _n(T) + " [(T)' / E_F'] " [1 (T/Tc)^4]
• Future work: Full QFT derivation of for plasmas, superfluids, and crystals.

F.3. Deconstruction of the Numerical Validation (22.67 N)

• Parameters used: = 1.0 10 m equivalent acceleration a_ = c' - 9 10 m/s' T_temp = 2.0 10 corresponds to T ~ 1010 K = 1.0 10 effective coupling = _mat A_geo (material susceptibility geometric amplification)
• Purpose: Illustrates scaling and sensitivity; shows how engineered conditions could amplify TTG effects.
• Future work: Replace with realistic simulations (e.g., superconducting torsion balance, optical lattice clock networks).

F.4. Quantization of and Relation to Quantum Gravity

• TTG introduces scalar excitations (phonons).
• Gravitons remain spin2 excitations of spacetime curvature.
• Phenomenological consequences:
  1. Clock decoherence: phonon bath imposes noise limits.
  2. Resonant absorption: when _ - k_B Tc in superconductors.
  3. Cosmological background: relic phonons could contribute to dark radiation.
• Future work: Develop coupled metric quantum theory and search for phonon signatures in precision metrology.

F.5. Comparative Context with Modified Gravity Theories

BransDicke Theory

• Fields: g, scalar
• Motivation: Machs principle, R coupling
• Difference: lacks direct operational meaning; TTGs is tied to clock measurements.

Horndeski Theories

• Fields: g, scalar
• Motivation: general secondorder scalartensor action, dark energy
• Difference: TTG is a constrained subset where scalar = , fixed by = d/dt and local energy tensor.

EinsteinCartan Theory

• Fields: g + torsion
• Motivation: torsion couples to spin density
• Difference: TTG is torsionfree; coupling is via (), not spin.

f(R) Gravity

• Fields: metric only
• Motivation: replace R with f(R) in EinsteinHilbert action
• Difference: f(R) adds higherorder metric modes; TTG adds scalar with direct operational meaning.

Temporal Theory of Gravity (TTG)

• Fields: g, scalar
• Motivation: as propertime rate; () introduces material response
• Distinctive features: operational foundation ( observable via clocks), material coupling (() explains anomalies in complex media).

F.6. Observational Consistency and Novel Predictions

• Binary pulsars: TTG reduces to GR when 0; pulsar timing constrains |_vac| < 10.
• Gravitational waves: Dispersion: frequencydependent phase shift in media with high . Additional polarization: possible longitudinal mode in plasmas. Prediction: correlate GW spectra with interstellar medium maps.
• Strongfield gravity: Nonlinear R' term modifies neutron star massradius relations and black hole dynamics. Numerical relativity simulations are needed.

F.7. Summary and Research Roadmap

TTG is a theoretically consistent, locally formulated, and experimentally falsifiable extension of GR. Open issues define a clear research program:

1. Strongfield completion: Solve coupled Einstein equations for compact objects.
2. Microscopic : Derive () from first principles for superconductors, superfluids, plasmas.
3. Highprecision tests: Clockaccelerometer correlations (f/f = g h / c' at 10 precision). Cryogenic anomaly search (weight shifts near superconducting transitions at 10 N sensitivity). Plasma modulation experiments.
4. Gravitational wave astrophysics: Search for mediated dispersion in GW signals.
5. Quantum regime: Develop full QFT of , calculate phonon production, explore cosmological implications.

Conclusion: By addressing these points, TTG transitions from a compelling proposal to a mature, predictive theory, positioned to either discover new physics or establish stringent new limits on the coupling between gravity, time, and matter.