A geoid is the surface where the total potential of the Earth’s gravitational and rotational effects is constant. In the classical definition, this is:
Φ(𝐫) = Φg (𝐫) + Φc (𝐫) = Φ0
It is expressed as follows: Here, Φg is the gravitational potential, and Φc is the centrifugal potential due to rotation.
Within the framework of the entropic impedance law I developed, the geoid is defined not only by classical potential but also by phase coherence and the energy-geometry relationship. In this case, the geoid can be written as an entropic equipotential surface as follows:
Definition of Entropic Geoid
1. Combined entropic potential:
Ψ(𝐫) = 𝛼 Φ(𝐫) + 𝛾 arg 𝑍E (𝜔0 , 𝐫)
- Φ(𝐫): classical gravity + centrifugal potential
- arg 𝑍E: phase angle of entropic impedance (LC–RC balance)
- 𝛼, 𝛾: scale factors (determine energy and phase contribution)
Geoid surface:
Ψ(𝐫) = Ψ0
2. Using the alternative superposition metric:
Ψ(𝐫) = 𝛼 Φ(𝐫) + 𝛿 (∣ 𝑍LC − 𝑍RC ∣ / ∣ 𝑍LC + 𝑍RC ∣)
Here, the difference/sum ratio of the LC and RC components measures the superposition regime. The geoid is the surface where this value is constant.
Interpretive Commentary
- Concave regions (LC-e dominant): Dense cratons, mountainous areas → the geoid rises.
- Convex regions (RC-īt dominant): Oceanic trenches, low-density regions → the geoid descends.
- Superposition regions (phase coherence): Continental margins, areas of balanced spreading and concentration → the geoid surface becomes more stable and flattened.
Conclusion
While the geoid is classically a gravitational equipotential, the law of entropic impedance, along with phase coherence and energy-geometry balance, comes into play. Thus, the geoid becomes an entropic equipotential surface defined not only by mass distribution but also by the combined effect of LC–RC–i regimes.
Blue area → LC-e (concaveness, focusing, and storage effect) Red area → RC-īt (convexity, diffusion, and damping) Gray neutral area → Superposition (phase matching, LC+RC combination)
The combined entropic impedance equation is shown below:
Ψ(𝐫) = 𝛼Φ(𝐫) + 𝛾arg 𝑍E (𝜔0 , 𝐫)
This diagram shows how geoid fluctuations are shaped by phase coherence and energy-geometry balance, in addition to the classical gravitational equipotential surface.
Comparison of Classical and Entropic Definitions of Geoid
| Feature | Classical Geoid Definition | Entropic Impedance Definition |
|---|---|---|
| Definition | Surface where gravitational + centrifugal potential is constant | Entropic surface where the energy–geometry–phase triad is constant |
| Potential expression | Φ(r) = Φg + Φc = Φ0 | Ψ(r) = αΦ(r) + γ arg ZE(ω0, r) |
| Physical basis | Mass distribution and rotational effect | Impedance, curvature, and phase coherence (LC–RC–i regimes) |
| Geometric contribution | Surface shape and gravity anomalies | Concavity/convexity (K < 0 / K > 0) and superposition (K = 0) |
| Energy component | None (passive potential) | Active energy transport modes: LC (focusing), RC (diffusion), i (phase) |
| Phase component | None | arg ZE: phase angle, jitter, resonance matching |
| Application domain | Geodesy, mapping, sea level measurement | Geodesy + chemistry, biology, information systems, optics |
| Time / frequency sensitivity | Static surface (time-independent) | Frequency-selective: impedance phase measured at ω0 |
| Validation method | Satellite measurements, gravimetry | Spectroscopy, conductivity, emission, phase measurements |
Key Differences
- The classical geoid relies solely on potential energy; it does not consider the internal dynamics of the system (phase, impedance, resonance).
- The entropic geoid incorporates both geometric and dynamic properties: the mode of energy transport, phase stability, and curvature are considered together.
- The classical model is static; the entropic model is frequency and phase sensitive, making it more suitable for time-varying systems.
Conceptual Overlap
- In both models, the surface defines the locations where a potential is constant.
- The Φ0 value of the classical model is expanded by Ψ0 in the entropic model.
- The entropic model can be considered a special case of the classical geoid: when ε = 0, the entropic geoid → the classical geoid.
Transition from entropic geoid to height and ellipsoid.
Geodesy height calculations are performed using three quantities: ellipsoidal height ℎ (GNSS), geoid undulation −, and orthometric height −. The basic relationship is:
ℎ = 𝐻 + 𝑁
In the entropic framework, we expand this trio with my Ψ potential (energy–geometry–phase).
Entropic potential and equipotential surface.
- Classical total potential: Φ(𝐫) = Φg (𝐫) + Φc (𝐫).
- Entropic potential (expanded): Ψ(𝐫) = 𝛼 Φ(𝐫) + 𝛾 arg 𝑍E (𝜔0 , 𝐫)
- Entropic geoid: surface Ψ(𝐫) = Ψ0.
Here, 𝛼 is the energy scale, 𝛾 is the phase contribution; 𝜔0 is the selected geophysical band (e.g., tidal/normal mode).
Entropic geopotential number and orthometric height
In classical geopotential, the geopotential number is C = 𝑊0 − 𝑊(𝑃) and the orthometric height is H = C/g¯. Its entropic equivalent is:
- Entropic geopotential number:
𝐶E (𝑃) = Ψ0 − Ψ(𝑃)
- Entropic orthometric height:
𝐻E (𝑃) = (𝐶E (𝑃) / 𝑔¯E (𝑃)) = (Ψ0 − Ψ(𝑃)) / 𝑔¯E (𝑃)
𝑔¯E (𝑃) : Measured/modeled entropic mean gravity (corrected for phase-rheology effects).
This definition makes the classical H sensitive to phase matching and LC–RC regimes when 𝛾 ≠ 0; it is reduced to its classical form when 𝛾 = 0.
Relationship between entropic geoid undulation and ellipsoidal height
- Entropic geoid undulation:
𝑁E (𝑃) = ℎ(𝑃) − 𝐻E (𝑃)
- Directly from the potential:
𝑁E (𝑃) = ℎ(𝑃) − (Ψ0 − Ψ(𝑃)) / 𝑔¯E (𝑃)
This is calculated using ℎ from GNSS, Ψ(𝑃) from the entropic potential map, Ψ0 from calibration, and 𝑔¯E.
Ellipsoid transition: reference surface and transformations
1. Reference ellipsoid selection: (e.g., GRS80/WGS84) semi-major axis 𝑎, flattening 𝑓.
2. Ellipsoid projection of entropic potential:
- Calculate Φ(𝐫ell) on the ellipsoid.
- Use the LC/RC/i regime map (rheology, Q, conductivity, heat flux) for arg 𝑍E (𝜔0 , 𝐫ell).
- Generate Ψ(𝐫ell) = 𝛼 Φ + 𝛾 arg 𝑍E.
3. Calibration of the entropic geoid constant Ψ0:
- Adjust 𝛾 and Ψ0 using least squares at the common checkpoints with the classical geoid:
ℎi − (Ψ0 − Ψi) / 𝑔¯E,i ≈ 𝑁iobs
4. Height conversions:
- GNSS → orthometric (entropic): 𝐻E = ℎ − 𝑁E .
- Orthometric → ellipsoidal: ℎ = 𝐻E + 𝑁E .
- Classical–entropic comparison: Δ𝑁 = 𝑁E − 𝑁, Δ𝐻 = 𝐻E − 𝐻 .
Calculation flow (practical steps)
1. Data:
- GNSS h(P), gravity g¯(P), classical geoid N(P).
- Geophysics: seismic Q, conductivity (MT/EM), heat flux, density models.
2. Parameter fields:
- 𝑎(𝐫), 𝛽(𝐫)(LC/RC weights), 𝐿(𝐫), 𝐶(𝐫), 𝑅(𝐫).
3. Entropic phase:
Θ(𝐫) = arg 𝑍E (𝜔0 , 𝐫) = arctan , (ℑ𝑍E / ℜ𝑍E)
4. Potential:
Ψ(𝐫) = 𝛼 Φ(𝐫) + 𝛾 Θ(𝐫)
5. Height:
𝐻E (𝑃) = (Ψ0 − Ψ(𝑃)) / 𝑔¯E (𝑃) , 𝑁E (𝑃) = ℎ(𝑃) − 𝐻E (𝑃)
6. Calibration and verification:
- 𝛾, regional fit for Ψ0.
- Quality control with residues ΔN, ΔH.
Physical comments and expected effects
- LC-e dominant (concave, high Q): Θ stable → Ψ increases → HE increases, NE decreases (surface elevations behave more “stiffly”).
- RC-īt dominant (convex, low Q): Θ diffuse → Ψ decreases → HE decreases, NE increases (spreading-damping zones pull the geoid down).
- Superposition (phase coherence): Θ gradients decrease → NE field is smoother, transitions are stable.
Brief summary:
- Entropic orthometric height:
𝐻E = (Ψ0 − (𝛼 Φ + 𝛾 arg 𝑍E)) / 𝑔¯E
- Entropic geoid undulation:
𝑁E = ℎ − 𝐻E
- Transition to ellipsoid: Calculate Ψ on the ellipsoid, calibrate Ψ0 and 𝛾; then use the transformation ℎ = 𝐻E + 𝑁E .
Gravity Explanation According to the Law of Entropic Impedance
Within the framework of the entropic impedance law, gravity should be interpreted not merely as gravitational attraction in the classical sense, but as a component of the energy-geometry-phase triad.
1. Classical Definition
- Gravity: The potential field created by the distribution of mass.
- Geoid: The surface where this potential is constant.
- Mathematical expression:
Φ(𝐫) = Φg (𝐫) + Φc (𝐫)
2. Entropic Impedance Frame
In the entropic impedance law, gravity is defined as the combination of reactive and resistive components with geometric curvature in impedance space:
- LC-e (concaveness): Gravity focusing effect → increased mass density, resonance storage.
- RC-īt (convexity): Gravity spreading effect → mass distribution widens, damping predominates.
- Superposition (phase alignment): Gravity-balanced phase carrier → mass-phase alignment provides maximum capacity.
Mathematical expression:
Ψ(𝐫) = 𝛼Φ(𝐫) + 𝛾arg 𝑍E (𝜔0 , 𝐫)
Here, Φ(𝐫) is the classical gravity potential, and arg 𝑍E is the phase angle of the impedance.
3. Entropic Interpretation of Gravity
- Energy component: Gravity is not just the gravitational pull; it is the total energy of the LC (storage) and RC (emission) regimes.
- Geometry component: Curvature (K<0 concave, K>0 convex) determines the focusing or emission character of gravity.
- Phase component: Gravity measurements directly affect capacitance through phase jitter and resonance matching.
4. Experimental Connections
- Spectroscopy: Line width → LC quality factor → focusing component of gravity.
- Electrical conductivity: RC damping parameters → propagation component of gravity.
- Phase measurements: Phase angle of gravity fluctuations → superposition regime.
5. Conclusion
According to the law of entropic impedance, gravity is:
- In the classical model, gravity is defined solely by gravity and rotational effects.
- In the entropic model, however, gravity is seen as the combined potential of the energy-geometry-phase triplet in impedance space.
Gravity = Energy (LC+RC) + Geometry (K) + Phase (i)