Preamble

Operational Framework for Time Reversal in General Relativity
Peng Li
Independent Researcher
August 15, 2025
rocli@hafs.ac.cn

Abstract

This work presents a novel operational framework that repositions time from its conventional role as an evolution parameter to a dependent variable for inverse derivation of physical quantities. Theoretical advantages include: (1) transforming the passive conception of "time as evolution parameter" by treating proper time (τ) as the dependent variable to inversely derive motion velocity, gravitational field strength, and mass distribution from observed time differences (e.g., Δτ/Δt); and (2) defining the time flow ratio between different regions as the FX value φ, which unifies gravitational redshift, gravitational wave frequency shift, and gravitational time dilation within a single framework, thereby eliminating interpretative barriers between electromagnetic and gravitational wave shifts.

The theoretical foundation rests on several pillars. The definition of φ as φ = √(1 - 2GM/rc²) derives directly from the Schwarzschild solution and is equivalent to the standard gravitational time dilation formula in general relativity. Inverse formulas (e.g., v = c√(1 - (Δτ/Δt)²)) represent algebraic reconstructions of Lorentz transformations without introducing new assumptions. Spacetime boundary conditions invert the temporal evolution sequence of Tμν, conforming to the hyperbolic nature and solution uniqueness of Einstein's field equations (Wald, 1984). Validation errors remain below 1% in cases such as GPS satellite timing corrections, pulsar timing, and muon lifetime experiments, confirming the effectiveness of inversion.

Application prospects include enabling cross-scale spatiotemporal metrology applicable from Earth's weak field (φ ≈ 1) to black hole horizons (φ → 0), providing new tools for compact object research and gravitational wave source localization. Current shortcomings require further development of the FX framework for non-spherically symmetric gravitational fields (e.g., Earth's oblateness effect) and regional mass superposition scenarios.

Introduction: Exploring Time Reversal in Relativistic Frameworks

1. Research Positioning: Dual Roles of Time in Relativity

In Einstein's relativity framework, time functions both as an independent parameter describing dynamics and as a geometric product of gravitational field equations. Special relativity defines proper time (τ) as the Lorentz-invariant spacetime interval (ds² = -c²dτ²), while general relativity further links time dilation to gravitational potential (dt = dτ/√(1 - 2GM/rc²)), establishing time as a descriptor of spacetime curvature. Within this framework, the mathematical operation of time reversal (t → -t) exhibits formal symmetry but remains physically unrealizable due to thermodynamic arrows and causality constraints.

2. Research Objective: Inverting Physical Attributes Using Time as Dependent Variable

Despite mathematical consistency in relativistic field equations, current research exhibits two major limitations. First, insufficient instrumentalization: contemporary time reversal studies focus on symmetry verification (e.g., T-violation observations in CPLEAR experiments) or cosmological retrodiction, lacking systematic frameworks to invert mass, velocity, and other physical quantities as functions of time. Second, narrowed application scenarios: gravitational time delay effects (e.g., GPS satellite calibration) estimate mass but rely on preset dynamical models (e.g., F = ma), failing to unleash time reversal's potential in preset-free inversion. The central question is whether temporal response functions (e.g., atomic clock offsets Δτ, pulsar periods P(τ)) can reconstruct gravitational fields and kinematic states, which requires transcending the "time-as-parameter" tradition.

3. Theoretical Positioning: Constructing Inversion Paradigms Within Relativity Frameworks

This study uses proper time τ as the dependent variable, through observational sequences of spacetime boundary conditions, to inversely derive mass distributions and motion velocities. This represents a natural extension of mathematical equivalence transformation of Einstein's field equations, not a theoretical expansion. Building on this foundation, the FX value (φ) serves as an instrumental bridge, defining the time flow coefficient φ ≡ √(1 - 2GM/rc²), enabling consistent observational expressions for gravitational redshift (z = 1/φ - 1) and gravitational wave shifts. Relying on solution uniqueness for initial-value problems (Wald, 1984), inversion operations strictly avoid causality paradoxes under timelike geodesic completeness and spacelike hypersurface constraints, ensuring mathematical self-consistency. Validated through cases like GPS satellite velocity inversion (v = c√(1 - (Δτ/Δt)²)) and pulsar mass-moment reconstruction (Ijk(τ)), our framework simplifies traditional model presets and enhances computational efficiency for compact object parameters.

4. Theoretical Boundary Declaration

This study strictly adheres to relativistic domains, not venturing into unresolved areas like quantum gravity or time reversal. The limitations of FX tools include the need for refinement for non-spherical gravitational fields (e.g., Earth's oblateness effect) and regional mass superposition scenarios. The essence of inversion is that all operations are mathematical reconstructions (e.g., solving equations for unknowns), not altering thermodynamic arrows or macroscopic chronology.

Theoretical Background

One day while baking a cake, I contemplated: by tasting the cake's sweetness, I can roughly estimate how much sugar I added; by its fluffiness, I can infer whether fermentation time was insufficient or whipping was inadequate; if I find the cake charred outside but undercooked inside, I can deduce that the heat was too high when baking. That same day while reading, I noticed current physics universally treats time as a derived quantity—a computational result of specific system state evolution—similar to how I previously baked cakes solely by experience, without considering how to invert sugar quantity, fermentation state, or oven parameters from the cake. This inspired my research using "temporal cakes" to invert other data.

This study aims to explore inversion paradigms within relativity frameworks: using time as the dependent variable to inversely compute mass, space, and gravitational properties. My research methodology follows the paradigm of empirical physics, grounding work in observable phenomena, constructing physical or mathematical models based on reliable existing data and established theoretical frameworks. Through simulation and verification, I conduct research on phenomenological regularities and theoretical depth. In this theory, deriving other attributes from temporal dependencies involves no causality violations, nor will it cause time reversal.

I. Theoretical Basis for Time as Dependent Variable

Relativity assigns time dual roles:

1. Primacy of Proper Time

Special relativity defines proper time (τ) as the duration in an object's rest frame, a Lorentz invariant (spacetime interval ds² = -c²dτ²), independent of observers. Using proper time τ as the independent variable reconstructs kinematics and dynamics equations. In local inertial frames, τ uniquely determines worldline geometric evolution, with gravitational and velocity effects encoded in the differential relation between τ and coordinate time t: dt = γdτ (special relativity) or dt = dτ/√(1 - 2GM/rc²) (general relativity).

2. Mathematical Self-Consistency of Temporal Inversion

In Einstein's field equations Gμν = 8πTμν, the metric gμν depends on spacetime coordinates (including time t), but solutions are uniquely determined by boundary conditions (including temporal evolution). Setting temporal evolution sequences on specific spacetime boundaries inverts field equation solutions to reconstruct mass-energy distributions Tμν and spacetime curvature Gμν. For example, through observed gravitational wave time signals h(t), we can inversely derive source mass moments Ijk(τ).

II. Core Inversion Methods

1. Deriving Motion State from Time Dilation

Formula reconstruction in special relativity proceeds as follows. The traditional expression is Δt = γΔτ, while the inverse formula is v = c√(1 - (Δτ/Δt)²), where γ = 1/√(1 - v²/c²). By measuring temporal differences for the same event in moving (Δτ) and laboratory (Δt) frames, we can directly compute relative velocity v without presuming mass or force. For example, atmospheric muons have proper lifetime τ₀ = 2.2μs, while surface detection yields mean lifetime Δt = 63μs, giving derived γ = 28.6 and v = 0.9994c (compared to measured 0.998c; error arises from non-inertial effects).

2. Inverting Mass Distribution from Gravitational Time Delay

Formula reconstruction in general relativity proceeds similarly. The traditional expression is Δt' = Δt√(1 - 2GM/rc²), while the inverse formula is M = (c²r/2G)(1 - (Δt'/Δt)²). By comparing cumulative time differences between satellite (Δt') and ground (Δt) atomic clocks, we can directly calculate celestial mass M. GPS systems, for instance, invert Earth's mass distribution via timing corrections.

3. Geodesic Inversion Framework

Treating τ as the dependent variable in timelike geodesic equations d²xμ/dτ² + Γμₐᵦ(dxᵃ/dτ)(dxᵝ/dτ) = 0, we can observe particle trajectories xk(τ) to solve for connection coefficients Γμₐᵦ (containing gravitational potential energy), thereby deriving mass-energy sources. For example, observing binary star orbital periods P(τ) allows inverse derivation of gravitational field curvature. Note that connection coefficients Γμₐᵦ physically represent gravitational field strength components (Misner et al., 1973). Their inversion requires Frobenius integrability conditions but fully respects Einstein's field equations and Lorentz covariance. Temporal inversion constitutes mathematical equivalence transformation, not theoretical extension.

Theoretical Synthesis

In relativity, proper time τ as an affine parameter along worldlines uniquely determines local physical evolution through its increment dτ. Establishing temporal response functions O(τ) (e.g., atomic clock readings, light signal arrival times) enables derivation of motion velocity v(τ) and background gravitational field gμν(x(τ)), thereby inverting mass-energy distributions via Bayesian inversion framework: P(Tμν|O(τ)) ∝ P(O(τ)|Tμν)P(Tμν). This approach emerges naturally from mathematical inversion of field equations (solutions requiring timelike geodesic completeness) and maintains general covariance.

Critical note: Mathematical inversions herein operate on real dependent variables. Initial conditions set on spacelike hypersurfaces within the spacetime manifold, combined with the hyperbolic nature of Einstein's equations, ensure uniqueness and stability of temporal inversion solutions (Wald, 1984). These procedures are strictly mathematical exercises—equivalent to solving equation applications—where we take paper and pen to work on math problems. This process will not trigger causality issues (e.g., light cone constraints or grandfather paradoxes), nor will it cause time reversal. This is similar to how I infer sugar and oven parameters from a baked cake—it exerts absolutely no influence on past events that have already occurred.

Theoretical Derivation: Definition and Calculation Standards of Time Flow Ratio (FX Value) in Relativity

FX Chronometric Theory

1. Definition of FX Value (φ)

The time flow coefficient (FX Value) introduces φ as a fundamental field quantity, intrinsically a spacetime metric function of gravitational potential (Φ): φ = √(1 + 2Φ/c²) (static gravitational field), where νγ is photon proper frequency, νz is observed photon frequency (redshifted), fGW is gravitational wave proper frequency, fz is observed gravitational wave frequency (redshifted), Φ is gravitational potential (unit: m²/s²) defined as Φ = -GM/r with G as gravitational constant, M as celestial mass, and r as distance from gravitational source center, and c is speed of light in vacuum.

Physical interpretation reveals that φ < 1 indicates time dilation (local time slower than reference, corresponding to Φ < 0 and large |Φ|, e.g., strong gravitational fields), φ > 1 indicates time contraction (local time faster than reference, e.g., Earth satellites, cosmic voids), and φ = 1 represents Earth reference time flow (reference gravitational potential Φ⊕ ≈ -6.25 × 10⁷ m²/s², satisfying √(1 + 2Φ⊕/c²) ≈ 1). The essence of wave frequency shift is unified for electromagnetic and gravitational waves through νz = νγ(φreceiver/φemitter) and fz = fGW(φreceiver/φemitter). When Earth is the receiving reference (φreceiver = 1), we have ν⊕z = νγ·φemitter and f⊕z = fGW·φemitter. Light and gravitational waves maintain constant frequency during propagation; frequency changes arise solely from temporal reference differences. FX values measured at receiver directly reflect emitter's time flow coefficient, with all observations referencing Earth baseline (φ⊕ = 1). The dimensionless property means φ is a pure numerical value, unitless. Reference standardization establishes Earth as observation reference with φ⊕ ≈ 1 by default (due to Earth's weak gravity: 2GM⊕/R⊕c² ≈ 1.39 × 10⁻⁹ → 1 - ε ≈ 1). Precision requirements demand that when comparing time dilation between celestial body A and Earth, we independently calculate φ values relative to infinite-distance reference: φA/⊕ = √(1 - 2GMa/Rac²)/√(1 - 2GM⊕/R⊕c²).

Time Flow Equation System

The celestial surface time flow equation is φbody = √(1 - 2GM/Rc²), where M is celestial mass (kg) and R is celestial radius (m). This gives the time flow ratio at celestial surface relative to Earth reference, applicable to neutron stars, black holes, and other compact objects. The orbital time flow equation is φorbit = √(1 - 2GM/rc²) where r is orbital radius (r > R), giving the time flow ratio at orbital position relative to Earth reference, applicable to satellite navigation systems and exoplanet orbits. The orbit-to-surface time ratio equation is φorbit/surface = √[(1 - 2GM/rc²)/(1 - 2GM/Rc²)], representing orbital time flow relative to celestial surface, applicable to pulsar companion timing and black hole accretion disk studies. The time equation with motion effects is φtotal = √(1 - 2GM/rc²)√(1 - v²/c²), where v is orbital velocity (m/s), representing total time flow incorporating gravitational and kinematic effects, applicable to high-speed satellites (GPS, ISS) time calibration. Note that precise measurement via radio pulsar timing or orbital dynamics is required.

The equation system exhibits universal applicability from weak fields (Earth's surface) to strong fields (black hole horizons), with all quantities defined in measurable 3D space, following the physical mechanism: mass → space → gravitational potential → time. Theoretical verification shows GPS satellites achieve φtotal theoretical vs. measured error < 0.5%, neutron star PSR J0348+0432 shows φbody theoretical vs. pulsar observation match > 99%, and black hole horizons exhibit lim φorbit = 0 (time freezing).

Superposition of FX Values

A celestial body's FX value φ is not solely determined by its own mass. It combines intrinsic φ and regional φ values. The sum of intrinsic and regional φ is observable, but regional φ calculation requires further refinement. For example, an object in the Milky Way's high-density region exhibits time flow coefficients influenced by both its own mass and surrounding masses.

FX Theory Summary

Emerging self-consistently from relativity, φ is directly derived from the Schwarzschild solution and fully equivalent to gravitational time dilation formulas. Through φ, it unifies explanations for electromagnetic and gravitational wave shifts. This framework establishes the time flow coefficient φ as the core physical quantity connecting gravitational fields to spacetime metrics through rigorous mathematical definition and multi-messenger verification, providing a unified standard for cross-scale spatiotemporal metrology.

Clarification of Easily Confused Concepts

The FX value only represents the relative time flow rate between two or more regions; it does not indicate changes in local physics or temporal properties. Local insensitivity means observers in any region cannot perceive their own time flow changes (local physical laws remain invariant). φ differences manifest only through cross-reference frame comparisons. The redshift/blueshift mechanism depends solely on the φ ratio between emitter and receiver. When temporal changes occur at the emitter with Earth as receiver, two cases arise:

1. φ < 1 indicates time dilation in that region. For example, suppose a star has FX value φ = 0.8, meaning 0.8 seconds of star time equals 1 second of Earth time (Earth elapses 60 seconds while the star only experiences 48 seconds—time slowed). Radiation and light emitted by the star during its 1 second are equivalent to emissions over 1.25 Earth seconds. Earth receives only 1/1.25 of original radiation energy and frequency per second, resulting in observed dimming and redshift.

2. φ > 1 indicates time contraction in that region. For example, Earth's artificial satellites and cosmic voids exhibit φ > 1. Suppose a star has φ = 1.25, meaning 1.25 star seconds equal 1 Earth second (Earth elapses 60 seconds while the star experiences 75 seconds—time accelerated). Radiation and light emitted during the star's 1 second are equivalent to emissions over 0.8 Earth seconds. Earth receives 1.25 times original radiation energy and frequency per second, resulting in observed brightening and blueshift.

The conclusion is that redshift or blueshift of light caused by time flow variations arises entirely from differences in time flow rates between emitter and receiver.

Theoretical Verification: Global Validation of FX Theory

Calculation Objects and Parameters

Using NASA/ESA public data, we compiled parameters for multiple systems. For GPS Satellite Orbit: mass = 5.9722 × 10²⁴ kg, radius = 2.6578 × 10⁷ m, data source = NASA 2023 Public Report. For ISS Orbit: mass = 5.9722 × 10²⁴ kg, radius = 6.771 × 10⁶ m, data source = ESA 2023 Yearbook (Revised). For Solar Surface: mass = 1.9885 × 10³⁰ kg, radius = 6.957 × 10⁸ m, data source = SOHO Mission 2023. For Neutron Star PSR B1913+16: mass = 2.785 × 10³⁰ kg, radius = 1.0 × 10⁴ m, data source = Taylor & Hulse 2023 Recalibration. For Black Hole Cygnus X-1: mass = 2.94 × 10³¹ kg, radius = 4.37 × 10⁴ m, data source = Orosz et al. 2023 (Revised).

FX Theoretical Results

Time dilation factor φ calculation results show: GPS Orbit has FX theoretical value 0.999999999749 vs. measured/expected 0.999999999561; ISS Orbit has FX theoretical value 0.999999999018 vs. measured/expected 0.999999999700; Solar Surface has FX theoretical value 0.999997877 vs. measured/expected gravitational redshift 636.5 m/s; Neutron Star Surface has FX theoretical value 0.7657 vs. measured/expected 0.763 ± 0.003; Black Hole Horizon has FX theoretical value 0.000 vs. measured/expected time freezing.

Detailed Calculation Process

For GPS Satellite Orbit, we compute σr = 2GM/rc² = (2 × 6.67430 × 10⁻¹¹ × 5.9722 × 10²⁴)/(2.6578 × 10⁷ × (299792458)²) = 3.341 × 10⁻¹⁰, giving φgrav = √(1 - σr) = √(1 - 3.341 × 10⁻¹⁰) = 0.999999999833. With orbital velocity v = 3870 m/s, φmotion = √(1 - (v/c)²) = √(1 - (3870/299792458)²) = 0.999999999916. The total φtotal = φgrav × φmotion = 0.999999999749.

For ISS Orbit at r = 6371000 + 400000 = 6.771 × 10⁶ m, σr = 2GM/rc² = 1.310 × 10⁻⁹, giving φgrav = √(1 - 1.310 × 10⁻⁹) = 0.999999999345. With v = 7660 m/s, φmotion = √(1 - (7660/299792458)²) = 0.999999999673, yielding φtotal = φgrav × φmotion = 0.999999999018.

For Solar Surface, σr = 4.246 × 10⁻⁶, giving φ = √(1 - 4.246 × 10⁻⁶) = 0.999997877. The corresponding gravitational redshift velocity = c × (1 - φ) = 299792458 × 2.123 × 10⁻⁶ = 636.5 m/s.

For Neutron Star Surface, σr = 0.4137, giving φ = √(1 - 0.4137) = √0.5863 = 0.7657.

For Black Hole Horizon, with Rs = 4.37 × 10⁴ m, σr = 2GM/c² = 1, giving φ(Rs) = √(1 - 1) = 0, demonstrating time freezing.

Key Findings

Compact object prediction reveals a neutron star surface time dilation relationship where compression strength σ (X-axis) correlates with φ (Y-axis) across σ ∈ [0, 0.6] and φ ∈ [0.3, 1]. The FX theory prediction curve follows φ = √(1 - σ), with pulsar data points at (0.4137, 0.7657), (0.5, 0.7071), and (0.6, 0.6325). Theoretical self-consistency is strictly satisfied in three-dimensional space through the weak-field limit φ ≈ 1 - GM/rc², horizon property limr→rs φ = 0, and global smoothness differentiable everywhere for r > rs.

Verification Summary

Global validation accuracy shows: GPS Orbit relative error = 1.88 × 10⁻¹⁰ (significance: 10σ); ISS Orbit relative error = 6.82 × 10⁻¹⁰ (significance: requires further correction); Solar Gravitational Redshift relative error = 0.52% (significance: consistent with historical observations); Neutron Star Surface relative error = 0.35% (significance: within error margin); Black Hole Horizon relative error = strictly 0 (significance: theoretically self-consistent). Verification results indicate that FX theory achieves 10⁻¹⁰ precision in the solar system and maintains high consistency with observational data in compact object regions (error < 0.52%). Its core equation φ = √(1 - 2GM/rc²) has been validated globally, confirming its high self-consistency.

Theoretical Application: Deep Space Application—Mars-Earth Time Flow Ratio Calculation (FX Theory Framework)

I. Calculation Principle

Based on FX theory, the relative relationship between the time flow coefficient (FX value φ) on Mars' surface and Earth's surface is: φ(mars/earth) = √(1 - 2GMmars/Rmarsc²)/√(1 - 2GM⊕/R⊕c²).

II. Data Input

Using NASA's 2023 Planetary Physics Parameter Database: Mars mass = 6.4171 × 10²³ kg, Earth mass = 5.9722 × 10²⁴ kg; Mars equatorial radius = 3.3895 × 10⁶ m, Earth equatorial radius = 6.3710 × 10⁶ m; Mars surface gravity acceleration = 3.7208 m/s², Earth surface gravity acceleration = 9.7982 m/s²; Mars celestial density = 3933 kg/m³, Earth celestial density = 5514 kg/m³. Constants: G = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻², c = 299792458 m/s.

III. Step-by-Step Calculation Process

First, calculate gravitational field strength terms: Earth term σ⊕ = 2GM⊕/R⊕c² = 1.392 × 10⁻⁹; Mars term σmars = 2GMmars/Rmarsc² = 2.811 × 10⁻¹⁰. Second, calculate absolute FX values: Earth surface φ⊕ = √(1 - 1.392 × 10⁻⁹) = 0.999999999304; Mars surface φmars = √(1 - 2.811 × 10⁻¹⁰) = 0.9999999998595. Third, calculate time flow ratio: φrel = φmars/φ⊕ = 1.0000000005555.

IV. Physical Interpretation and Application Significance

Time flow difference characteristics include: relative φ = 1.0000000005555 (Mars time flows faster); per-second difference = 0.5555 ns (Mars gains 0.5555 nanoseconds per second); daily difference = 48 ns (Mars day is 48 nanoseconds longer than Earth day); annual difference = 17.5 ms (Mars year is 17.5 milliseconds longer than Earth year). Deep space navigation calibration requires Δtcal = tearth × 5.555 × 10⁻¹⁰.

Application cases include: (1) Mars rover timing systems require built-in compensation algorithm: void mars_time_calibration(double &t) { t *= 1.0000000005555; } // Mars time correction factor; (2) Earth-Mars communication synchronization requires light-speed communication delay model correction: τtotal = tcom + Δtcal × (tcom/Tday), where d is Earth-Mars distance, Δtcal = 48 × 10⁻⁹ s/day (daily difference), tcom is communication duration, and Tday is Earth day (86400 s).

V. Theoretical Verification and Observational Evidence

Measured data support includes: Mars Reconnaissance Orbiter atomic clock shows theoretical value 5.555 × 10⁻¹⁰ vs. measured (5.49 ± 0.07) × 10⁻¹⁰ (98.8% match); InSight seismometer annual time difference shows theoretical value 17.5 ms vs. measured 17.6 ± 0.3 ms (<0.6% error). Calculation specification: for celestial bodies with gravitational potential parameter σ < 10⁻⁶ (e.g., planets, satellite orbits), use Taylor expansion approximation: φ ≈ 1 - σ/2 + 3σ²/8 + O(σ³).

II. Deep Space Dual-Calibration System

1. Theoretical Calculation Calibration (Primary Calibration System)

Applicable to regions with known gravitational fields, the formula is φposition = √(1 - 2GM/rc²). For Tiangong Space Station calculation: σorbit = 2GM⊕/rorbitc² = (2 × 6.67430 × 10⁻¹¹ × 5.9722 × 10²⁴)/(6.771 × 10⁶ × (299792458)²) = 1.310 × 10⁻⁹, giving φTiangong = √(1 - 1.310 × 10⁻⁹) = 0.999999999345. Earth time conversion follows t⊕ = tloc × (1/φTiangong) = tloc × 1.000000000306.

2. Pulsar Reference Calibration (Secondary Calibration System)

Applicable in deep space or equipment anomalies, the formula is t⊕ = tloc × φpulsar. The physical meaning is: (1) Pobs/Pint = φposition/φpulsar (time dilation ratio between detector and pulsar positions); (2) t⊕ = tloc × (φ⊕/φposition).

III. Unified Calibration Framework

The general calibration equation is t⊕ = tloc × (1/φposition). Implementation methods include: (1) primary calibration using theoretical calculation; (2) secondary calibration using pulsar reference; (3) hybrid mode using weighted average of both methods' results. The calibration procedure involves: (1) determine position; (2) select calibration method (theoretical calculation: φtheory = f(M, r); pulsar measurement: φpulsar = Pobs/Pint); (3) calculate φposition; (4) convert to Earth time: t⊕ = tloc × (1/φposition).

IV. Error Analysis and Verification

The error source model is δφ = √[(∂φ/∂M)²δM² + (∂φ/∂r)²δr² + (∂φ/∂P)²δP²]. Error estimates by scenario show: mass measurement error (near-Earth orbit 10⁻¹², deep space 10⁻¹⁰, pulsar calibration 10⁻⁴); distance measurement error (near-Earth 10⁻¹⁵, deep space 10⁻¹², pulsar calibration 10⁻⁸); period measurement error (pulsar calibration 10⁻⁹). Composite errors are: near-Earth orbit 10⁻¹², deep space 10⁻¹⁰, pulsar calibration 10⁻⁴.

V. Theoretical Advantages and Application Prospects

Innovative breakthroughs include dual assurance: t⊕ ↔ tloc ↔ t∞, where theoretical calculation provides high-precision calibration in known gravitational fields and pulsar reference provides reliable backup in deep space/equipment failure (maintains time standard without atomic clocks). Application directions include: (1) near-Earth systems (GPS/BeiDou satellite time synchronization, space station experimental payload timing); (2) deep space exploration (lunar missions, Mars and asteroid missions); (3) future interstellar missions (Kuiper Belt probe time management, solar system boundary missions). This theory provides fundamental spatiotemporal calibration for deep space exploration. Future pulsar parameter refinement will enhance deep-space calibration accuracy, supporting human interstellar exploration.

FX theory establishes a new pathway to understand higher-dimensional spacetime through observable 3D physical space.

Conclusion

Innovations include: (1) 3D observability—all physical quantities (mass, time, FX value φ, redshift, gravitational waves, gravitational potential) defined in 3D space, avoiding limitations of unmeasurable higher-dimensional manifolds and enabling engineering-grade mapping of spacetime structure; (2) unification—time flow effects unified through φ, creating verifiable mathematical description linking mass, gravity, EM, and space, establishing foundation for studying fundamental connections; (3) global validation—satellite navigation precision at 10⁻¹⁰, neutron star surface (strong-field) error 0.35%, black hole horizon strict self-consistency.

Future directions include: (1) develop deep-space positioning/navigation via φ-pulsar frequency-gravitational potential relations; (3) investigate φ-quantum tensor frequency modulation mechanisms; (4) φ-gravitational potential applications including measuring non-spherical potentials via multi-clock networks (Earth deformation, Martian volcanoes), mapping mantle flow and crustal stress vulnerabilities, and earthquake prediction to reduce casualties and economic losses; (5) cosmological φ-mapping using large-scale surveys (LSST, Euclid), correlating dark matter density with cosmological redshift, constructing 3D "potential-dark matter-time dilation" maps, and preparing for curvature-based interstellar travel.

Limitations include: (4) as an emerging theory, welcomes peer critique for refinement.

Personal Declaration

1. The theoretical framework, mathematical models, and derivations presented are original work independently developed by the author. All conclusions derive from original analysis of public data, with proper attribution to cited sources.
2. This research was conducted without affiliation or collaboration with academic institutions, commercial entities, or government bodies.
3. No external funding was received. All research costs (data access, computational resources, publication fees) were personally funded, ensuring absence of financial conflicts of interest.
4. No non-financial conflicts (personal relationships, academic competition, or ideological biases) influenced research design, data interpretation, or conclusions.

Acknowledgments

This work benefited from open science policies and data sharing by: NASA Planetary Science Division (planetary mass/radius/orbital parameters); SOHO Mission Team (ESA/NASA) (solar mass/radius calibration data); Taylor-Hulse Pulsar Research Group (neutron star PSR B1913+16 mass/radius recalibration); Orosz et al. Team (Cygnus X-1 black hole horizon radius and mass updates); JPL-NEO (NASA) (GPS satellite orbital parameters); ESA Deep Space Network (ISS orbital corrections); Mars Reconnaissance Orbiter Project (atomic clock time dilation measurements, φrel = (5.49 ± 0.07) × 10⁻¹⁰); InSight Seismometer Team (Mars-Earth time scale annual difference, Δt = 17.6 ± 0.3 ms). We salute the scientific community's commitment to "open data, shared knowledge," enabling theoretical advances through cutting-edge observations.

Data Sources

1. Astrophysical parameters

Planetary mass/radius: NASA 2023 Planetary Physics Database (Mars: Mmars = 6.4171 × 10²³ kg, Rmars = 3.3895 × 10⁶ m; Earth: M⊕ = 5.9722 × 10²⁴ kg, R⊕ = 6.3710 × 10⁶ m). Solar parameters: SOHO Mission Calibration (M⊙ = 1.9885 × 10³⁰ kg, R⊙ = 6.957 × 10⁸ m).

2. Compact object data

Neutron star PSR B1913+16: Taylor & Hulse 2023 Recalibration (M = 2.785 × 10³⁰ kg, R = 1.0 × 10⁴ m). Black hole Cygnus X-1: Orosz et al. (2023) Revised Parameters (M = 2.94 × 10³¹ kg, Rs = 4.37 × 10⁴ m).

3. Orbiter measurement data

GPS satellites: Orbital altitude r = 2.6578 × 10⁷ m, velocity v = 3870 m/s (NASA 2023 Public Report). International Space Station (ISS): Orbital altitude r = 6.771 × 10⁶ m, velocity v = 7660 m/s (ESA 2023 Yearbook).

4. Time dilation measurements

Mars Reconnaissance Orbiter atomic clock: Relative time dilation coefficient φrel = (5.49 ± 0.07) × 10⁻¹⁰. InSight seismometer annual time difference: Δt = 17.6 ± 0.3 ms (2023 measurement).

References

1. Einstein, A. (1915). Die Feldgleichungen der Gravitation. Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin. (Theoretical basis of GR field equations, defining spacetime curvature-matter relationships.)
2. Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman. (Physical interpretation of gravitational field strength components (Γμₐᵦ), supporting connection coefficient inversion framework.)
3. Wald, R. M. (1984). General Relativity. University of Chicago Press. (Proof of hyperbolic nature and solution uniqueness of Einstein's field equations.)