The Unified Relativistic Potential: A Re-examination of the GPS Time Dilation Problem

 J. Rogers, SE Ohio, 02 Aug 2025, 1515

Abstract: The daily time correction for the Global Positioning System (GPS) is the canonical real-world example of relativistic effects, conventionally understood as a sum of two distinct phenomena: gravitational time dilation from General Relativity (GR) and velocity time dilation from Special Relativity (SR). This paper re-examines the standard derivation and demonstrates, through direct algebraic substitution, that these two components are not independent. They can be combined into a single, unified expression dependent only on the central mass and the radii of the observer and the satellite. This unified formula reveals that both effects are consequences of a single underlying potential, challenging the conventional view of their separateness and suggesting a deeper unity in the structure of relativistic physics.

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1. Introduction: The Standard Model of GPS Time Dilation

The remarkable precision of the Global Positioning System is contingent upon corrections for relativistic effects. Without these adjustments, the system would accumulate errors rendering it useless within minutes. The standard textbook explanation, validated by decades of successful operation, attributes the net daily time gain of approximately 38 microseconds to two separate physical principles:

1. General Relativity (GR): Clocks in a stronger gravitational potential (closer to a massive body) run slower than clocks in a weaker potential. A clock on Earth's surface therefore loses time relative to a satellite in orbit.

2. Special Relativity (SR): A clock moving at high velocity runs slower than a stationary clock. A GPS satellite's orbital velocity therefore causes it to lose time relative to an observer on Earth.

These two effects are calculated independently and summed to find the net effect. The GR effect results in a time gain for the satellite clock, while the SR effect results in a time loss. This paper does not dispute the validity of these calculations but aims to investigate the fundamental relationship between them. Our central question is: Are these truly two independent phenomena, or are they two facets of a single, unified effect?

2. The Conventional Derivation: A Sum of Two Effects

Let us begin by formally stating the standard derivation. We consider a satellite in a stable circular orbit around the Earth.

2.1 Gravitational Time Dilation (GR)

The formula for gravitational time dilation compares the gravitational potential, Φ = -GM/r, at two locations. For a time period T, the accumulated time difference Δt_GR is given by the difference in potential divided by c²:

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>G</span>}\mathrm{<span>R</span>}}<span>=</span>\mathrm{<span>T</span>}<span>(</span>\frac{{\mathrm{<span>\Phi </span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}<span>-</span>{\mathrm{<span>\Phi </span>}}_{\mathrm{<span>e</span>}\mathrm{<span>a</span>}\mathrm{<span>r</span>}\mathrm{<span>t</span>}\mathrm{<span>h</span>}}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>)</span><span>=</span>\mathrm{<span>T</span>}<span>(</span>\frac{<span>-</span>\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>-</span>\frac{<span>-</span>\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}{\mathrm{<span>r</span>}}_{\mathrm{<span>e</span>}\mathrm{<span>a</span>}\mathrm{<span>r</span>}\mathrm{<span>t</span>}\mathrm{<span>h</span>}}}<span>)</span>$$

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>G</span>}\mathrm{<span>R</span>}}<span>=</span>\mathrm{<span>T</span>}\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>(</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>e</span>}\mathrm{<span>a</span>}\mathrm{<span>r</span>}\mathrm{<span>t</span>}\mathrm{<span>h</span>}}}<span>-</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>)</span><span>(</span>\mathrm{<span>1</span>}<span>)</span>$$

Here, M is the mass of the Earth, r_earth is the Earth's radius, and r_sat is the satellite's orbital radius. Since r_sat > r_earth, this term is positive, indicating the satellite's clock runs faster.

2.2 Velocity Time Dilation (SR)

For an object moving at velocity v, the time dilation factor is given by γ = 1/sqrt(1 - v²/c²). For velocities much less than c, a common and highly accurate approximation is used:

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>S</span>}\mathrm{<span>R</span>}}<span>\approx </span><span>-</span>\mathrm{<span>T</span>}<span>(</span>\frac{{\mathrm{<span>v</span>}}^{\mathrm{<span>2</span>}}}{\mathrm{<span>2</span>}{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>)</span><span>(</span>\mathrm{<span>2</span>}<span>)</span>$$

The negative sign indicates the moving clock runs slower. At this stage, the calculation appears entirely independent of the GR effect, relying only on the satellite's velocity v.

3. The Unseen Constraint: The Physics of a Stable Orbit

The conventional approach treats the velocity v in equation (2) as an independent variable. However, for a GPS satellite, v is not arbitrary. A satellite must maintain a specific velocity to remain in a stable circular orbit. This velocity is determined by the balance between the gravitational force and the required centripetal force:

$$\frac{{\mathrm{<span>m</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}{\mathrm{<span>v</span>}}^{\mathrm{<span>2</span>}}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>=</span>\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}{\mathrm{<span>m</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}^{\mathrm{<span>2</span>}}}$$

Solving for the orbital velocity squared, v², we find a crucial relationship:

$${\mathrm{<span>v</span>}}^{\mathrm{<span>2</span>}}<span>=</span>\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>(</span>\mathrm{<span>3</span>}<span>)</span>$$

This equation reveals that the kinetic parameter v² is not independent of the gravitational parameters G, M, and r_sat. The satellite's velocity is dictated by the very gravitational potential it resides in. This suggests a deeper connection between the SR and GR calculations.

4. Derivation of the Unified Formula

We can now perform a direct algebraic substitution to unify the two effects. By substituting the expression for v² from equation (3) into the SR time dilation formula (2), we can re-express the velocity effect in terms of gravitational parameters:

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>S</span>}\mathrm{<span>R</span>}}<span>\approx </span><span>-</span>\mathrm{<span>T</span>}<span>(</span>\frac{\mathrm{<span>1</span>}}{\mathrm{<span>2</span>}{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>\cdot </span>\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>)</span><span>=</span><span>-</span>\mathrm{<span>T</span>}\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{\mathrm{<span>2</span>}{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>(</span>\mathrm{<span>4</span>}<span>)</span>$$

Equation (4) is remarkable. It shows that the velocity time dilation for a stable orbit is not fundamentally about velocity itself, but can be expressed entirely in terms of the gravitational potential at that orbit.

The total time dilation, Δt_total, is the sum of the GR and SR effects. Let us now sum our re-expressed terms from equations (1) and (4):

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>t</span>}\mathrm{<span>o</span>}\mathrm{<span>t</span>}\mathrm{<span>a</span>}\mathrm{<span>l</span>}}<span>=</span>\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>G</span>}\mathrm{<span>R</span>}}<span>+</span>\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>S</span>}\mathrm{<span>R</span>}}$$

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>t</span>}\mathrm{<span>o</span>}\mathrm{<span>t</span>}\mathrm{<span>a</span>}\mathrm{<span>l</span>}}<span>=</span><span>[</span>\mathrm{<span>T</span>}\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>(</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>e</span>}\mathrm{<span>a</span>}\mathrm{<span>r</span>}\mathrm{<span>t</span>}\mathrm{<span>h</span>}}}<span>-</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>)</span><span>]</span><span>+</span><span>[</span><span>-</span>\mathrm{<span>T</span>}\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{\mathrm{<span>2</span>}{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>]</span>$$

We can factor out the common term T * (GM/c²):

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>t</span>}\mathrm{<span>o</span>}\mathrm{<span>t</span>}\mathrm{<span>a</span>}\mathrm{<span>l</span>}}<span>=</span>\mathrm{<span>T</span>}\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>[</span><span>(</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>e</span>}\mathrm{<span>a</span>}\mathrm{<span>r</span>}\mathrm{<span>t</span>}\mathrm{<span>h</span>}}}<span>-</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>)</span><span>-</span>\frac{\mathrm{<span>1</span>}}{\mathrm{<span>2</span>}{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>]</span>$$

Combining the terms involving r_sat:

$$\mathrm{<span>\Delta </span>}{\mathrm{<span>t</span>}}_{\mathrm{<span>t</span>}\mathrm{<span>o</span>}\mathrm{<span>t</span>}\mathrm{<span>a</span>}\mathrm{<span>l</span>}}<span>=</span>\mathrm{<span>T</span>}\frac{\mathrm{<span>G</span>}\mathrm{<span>M</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>(</span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>r</span>}}_{\mathrm{<span>e</span>}\mathrm{<span>a</span>}\mathrm{<span>r</span>}\mathrm{<span>t</span>}\mathrm{<span>h</span>}}}<span>-</span>\frac{\mathrm{<span>3</span>}}{\mathrm{<span>2</span>}{\mathrm{<span>r</span>}}_{\mathrm{<span>s</span>}\mathrm{<span>a</span>}\mathrm{<span>t</span>}}}<span>)</span><span>(</span>\mathrm{<span>5</span>}<span>)</span>$$

5. Discussion and Significance

We have arrived at Equation (5), the Unified Relativistic Potential Formula. This was achieved not through new physics, but by rigorously following the logical consequences of the standard model's own equations. The significance of this unified form is threefold:

1. Conceptual Unity: The formula demonstrates that for a stable orbital system, gravitational and velocity time dilation are not two separate phenomena. They are two components of a single, unified potential. The distinction between them is an artifact of treating velocity as an independent parameter, when in fact it is determined by the gravitational field itself.

2. The Centrality of  The equation's structure is governed by the terms GM/c², 1/r_earth, and 1/r_sat. This reveals that the underlying physical quantity driving all relativistic time effects is the potential, which can be thought of as a scaled representation of the fundamental ratio M/r. The entire complex interplay of SR and GR reduces to evaluating this single potential at different radii.

3. Reframing Physical Law: This result encourages a re-evaluation of how we perceive physical laws. What we often treat as separate domains (e.g., kinematics in SR, gravity in GR) may, in fact, be different observational consequences of a single, deeper principle. The goal should be to find the unified expression, which often reveals a simpler and more profound underlying structure.

6. Unified Time Dilation Between Any Two Circular Orbits

Building upon the unified expression previously derived for a ground clock and an orbital clock, we now generalize the formula to compare the rate of time between any two stable circular orbits at radii $r_1$ and $r_2$ around a central mass $M$.

6.1. Generalized Formula Derivation

Each clock in a circular orbit experiences both gravitational and velocity time dilation, both determined by its radius. Using the earlier approach:

• Gravitational time dilation for radius $r$:

  $$\mathrm{\Delta }{t}_{\text{GR}}=T\cdot \frac{GM}{c^2}\cdot (\frac{1}{r_{\text{ref}}}-\frac{1}{r})$$

• Velocity (kinetic) time dilation for a circular orbit (where $v^2=GM\mathrm{/}r$):

  $$\mathrm{\Delta }{t}_{\text{SR}}\approx -T\cdot \frac{v^2}{2c^2}=-T\cdot \frac{GM}{2c^{2}r}$$

Sum these to get the total time rate difference between orbits:

$$\mathrm{\Delta }{t}_{\text{total, orbit}}=T\cdot \frac{GM}{c^2}(\frac{1}{r_{\text{ref}}}-\frac{3}{2r})$$

To compare two orbits at $r_1$ and $r_2$, subtract the total time dilation for each:

$$\mathrm{\Delta }{t}_{12}=\mathrm{\Delta }{t}_{\text{total, orbit 2}}-\mathrm{\Delta }{t}_{\text{total, orbit 1}}$$$$\mathrm{\Delta }{t}_{12}=T\cdot \frac{GM}{c^2}(\frac{3}{2r_2}-\frac{3}{2r_1})$$$$\boxed{{\displaystyle \mathrm{\Delta }{t}_{12}=T\cdot \frac{GM}{c^2}\cdot \frac{3}{2}(\frac{1}{r_2}-\frac{1}{r_1})}}$$

6.2. Physical Interpretation

• This formula unifies both gravitational and kinetic time dilation into a single, symmetric expression between any two orbits.

• The result depends only on the mass of the central body, the radii of the two orbits, and universal constants.

• The “3/2” coefficient is a direct consequence of orbital mechanics, signifying the sum of 1 part gravitational potential and 0.5 part kinetic (velocity) contribution for any circular orbit.

6.3. Special Cases

• Earth’s surface to GPS orbit (one clock at rest):
  Set $r_1=r_{\text{earth}}$, $r_2=r_{\text{GPS}}$; the formula exactly reproduces the unified GPS correction found earlier.

• Both clocks in orbit:
  The formula applies directly with $r_1$ and $r_2$ as the orbital radii—no reference to the Earth’s surface is necessary.

6.4. Example Calculation

Comparing time rates between two satellite orbits:

• $r_1=20,200,000$m (typical GPS)

• $r_2=42,164,000$m (typical geostationary)

$$\mathrm{\Delta }{t}_{12}=T\cdot \frac{GM}{c^2}\cdot \frac{3}{2}(\frac{1}{42,164,000}-\frac{1}{20,200,000})$$

This gives the exact time difference accumulated per time period $T$ between clocks in these two orbits.

In summary: For any two objects in stable circular orbits, the unified relativistic time difference is determined by simply computing the difference in the inverses of their radii, scaled by a universal “3/2” factor. This generalizes the principle of the previously unified formula, further demonstrating the deep connection between orbital mechanics and relativistic time dilation, and eliminating the need for separate gravitational and velocity effect calculations in comparative orbital timing.

7. Conclusion

While the conventional method of calculating GPS time dilation by summing two separate effects is numerically correct, it obscures a deeper physical truth. By acknowledging the constraint that orbital velocity is dependent on gravitational potential, we have shown that the two effects can be seamlessly unified into a single expression. This unified formula, Δt = T * (GM/c²) * (1/r_earth - 3/(2*r_sat)), demonstrates that the total time dilation experienced by an orbiting body is a singular phenomenon, governed entirely by the mass of the central body and the radii involved. This perspective challenges us to look beyond the historical divisions in physics and seek the simpler, unified principles that govern our universe.

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Appendix A: Numerical Verification for stationary observer on Earth

The results: 

=== GPS Time Dilation: Standard vs. Unified Method ===

--- Input Parameters ---

Earth Mass (M):       5.9720e+24 kg

Earth Radius (r_earth): 6.3710e+06 m

Sat. Radius (r_sat):  2.6600e+07 m

Time Period (T):      86400 s

--- Method 1: The Standard Model (Separate Effects) ---

a) GR Effect (Time Gain):         +45.7386 microseconds/day

b) SR Effect (Time Loss):         -7.2025 microseconds/day

----------------------------------------------------------

Total Dilation (Standard Method): +38.5361 microseconds/day

--- Method 2: The Unified Substrate Law ---

This method uses a single formula derived from the Inertial Stress (m/r).

FORMULA: Δt = T * (GM/c²) * (1/r_earth - 3/(2*r_sat))

Projection Factor (GM/c²):        4.4349e-03

Radius Factor (1/r_e - 3/2r_s):   1.0057e-07

----------------------------------------------------------

Total Dilation (Unified Method):  +38.5361 microseconds/day

=======================================================

                    VERDICT

=======================================================

Standard Method Result:   0.000038536068 s/day

Unified Method Result:    0.000038536068 s/day

Absolute Difference:      6.78e-21 s/day

CONCLUSION: The results are identical. The two methods are mathematically equivalent.

The standard model's 'separate effects' are just two parts of a single, unified substrate phenomenon.

The program:

import scipy.constants as const

from math import sqrt

# --- 1. Define Constants & Data ---

# We use standard SI constants because we are calculating an SI result.

print("=== GPS Time Dilation: Standard vs. Unified Method ===")

G = const.G

c = const.c

M_earth = 5.972e24  # Mass of Earth in kg

r_earth_surface = 6.371e6  # Radius of Earth in meters

# GPS satellites are in a semi-major axis orbit of 26,600 km from Earth's center

r_sat_orbit = 2.66e7 # Orbital radius from center of Earth in meters

T_day = 86400  # Seconds in one day

print("\n--- Input Parameters ---")

print(f"Earth Mass (M):       {M_earth:.4e} kg")

print(f"Earth Radius (r_earth): {r_earth_surface:.4e} m")

print(f"Sat. Radius (r_sat):  {r_sat_orbit:.4e} m")

print(f"Time Period (T):      {T_day} s\n")

# --- 2. The Old Way: Separate Relativity Calculations ---

print("--- Method 1: The Standard Model (Separate Effects) ---")

# a) General Relativity (Gravitational Time Dilation)

# Clock ticks FASTER in weaker gravity.

gr_factor_earth = (G * M_earth) / (c**2 * r_earth_surface)

gr_factor_sat = (G * M_earth) / (c**2 * r_sat_orbit)

delta_t_gr = T_day * (gr_factor_earth - gr_factor_sat)

print(f"a) GR Effect (Time Gain):         +{delta_t_gr * 1e6:.4f} microseconds/day")

# b) Special Relativity (Velocity Time Dilation)

# Clock ticks SLOWER when moving fast.

# First, find the orbital velocity v^2 = GM/r

v_sq = (G * M_earth) / r_sat_orbit

# Then, apply the SR formula: delta_t = -T * (v^2 / 2c^2)

delta_t_sr = -T_day * (v_sq / (2 * c**2))

print(f"b) SR Effect (Time Loss):         {delta_t_sr * 1e6:.4f} microseconds/day")

# c) Total Time Dilation (Standard Method)

total_dilation_standard = delta_t_gr + delta_t_sr

print("----------------------------------------------------------")

print(f"Total Dilation (Standard Method): +{total_dilation_standard * 1e6:.4f} microseconds/day\n")

# --- 3. The New Way: The Unified Substrate Formula ---

print("--- Method 2: The Unified Substrate Law ---")

print("This method uses a single formula derived from the Inertial Stress (m/r).\n")

print("FORMULA: Δt = T * (GM/c²) * (1/r_earth - 3/(2*r_sat))")

# The entire calculation in one unified step

# The term GM/c^2 is the projection of the substrate's m/r logic into SI

projection_factor = (G * M_earth) / (c**2)

radius_factor = (1 / r_earth_surface) - (3 / (2 * r_sat_orbit))

total_dilation_unified = T_day * projection_factor * radius_factor

print(f"Projection Factor (GM/c²):        {projection_factor:.4e}")

print(f"Radius Factor (1/r_e - 3/2r_s):   {radius_factor:.4e}")

print("----------------------------------------------------------")

print(f"Total Dilation (Unified Method):  +{total_dilation_unified * 1e6:.4f} microseconds/day\n")

# --- 4. Final Verification ---

print("="*55)

print("                    VERDICT")

print("="*55)

print(f"Standard Method Result:   {total_dilation_standard:.12f} s/day")

print(f"Unified Method Result:    {total_dilation_unified:.12f} s/day")

difference = abs(total_dilation_standard - total_dilation_unified)

print(f"Absolute Difference:      {difference:.2e} s/day")

if difference < 1e-12:

    print("\nCONCLUSION: The results are identical. The two methods are mathematically equivalent.")

    print("The standard model's 'separate effects' are just two parts of a single, unified substrate phenomenon.")

else:

    print("\nCONCLUSION: The results do not match. Check the derivation.")

Appendix B: Numerical Verification for time difference  between two orbits

The results: 

=== Time Dilation Between Two Orbits: Standard vs. Unified Method ===

--- Input Parameters ---

Earth Mass (M):           5.9720e+24 kg

Orbit 1 Radius (r1_gps):  2.6600e+07 m

Orbit 2 Radius (r2_geo):  4.2164e+07 m

Time Period (T):          86400 s

--- Method 1: The Standard Model (Relative to Infinity) ---

Total Dilation for Orbit 1 (GPS):  -21.6076 microseconds/day

Total Dilation for Orbit 2 (GEO):  -13.6316 microseconds/day

----------------------------------------------------------------

Net Time Difference (GEO relative to GPS): +7.9760 microseconds/day

--- Method 2: The Generalized Unified Formula ---

This method uses a single, direct formula between the two orbits.

FORMULA: Δt₁₂ = T * (3GM/2c²) * (1/r₁ - 1/r₂)

Unified Factor (T*3GM/2c²):       5.7476e+02

Radius Difference (1/r₁ - 1/r₂):    1.3877e-08

----------------------------------------------------------------

Net Time Difference (Unified Method):  +7.9760 microseconds/day

================================================================

                           VERDICT

================================================================

Standard Method Result:   0.000007976032 s/day

Unified Method Result:    0.000007976032 s/day

Absolute Difference:      1.69e-21 s/day

CONCLUSION: The results are identical. The generalization is correct.

The unified formula correctly predicts the time difference between any two

stable circular orbits in a single, elegant step.

The program: 

       

import scipy.constants as const

from math import sqrt

# --- 1. Define Constants & Data for Two Orbits ---

print("=== Time Dilation Between Two Orbits: Standard vs. Unified Method ===")

G = const.G

c = const.c

M_earth = 5.972e24      # Mass of Earth in kg

T_day = 86400          # Seconds in one day

# Orbit 1: Typical GPS Satellite

r1_gps = 2.66e7  # Orbital radius from center of Earth in meters (approx. 20,200 km altitude)

# Orbit 2: Geostationary (GEO) Satellite

r2_geo = 4.2164e7 # Orbital radius for a geostationary orbit in meters

print("\n--- Input Parameters ---")

print(f"Earth Mass (M):           {M_earth:.4e} kg")

print(f"Orbit 1 Radius (r1_gps):  {r1_gps:.4e} m")

print(f"Orbit 2 Radius (r2_geo):  {r2_geo:.4e} m")

print(f"Time Period (T):          {T_day} s\n")

# --- 2. The Old Way: Relative to a Distant, Stationary Observer (at Infinity) ---

print("--- Method 1: The Standard Model (Relative to Infinity) ---")

# -- Satellite 1 (GPS) --

# a) GR Effect for GPS (relative to infinity)

delta_t_gr_gps = -T_day * (G * M_earth) / (c**2 * r1_gps)

# b) SR Effect for GPS

v_sq_gps = (G * M_earth) / r1_gps

delta_t_sr_gps = -T_day * (v_sq_gps / (2 * c**2))

# c) Total Dilation for GPS

total_dilation_gps = delta_t_gr_gps + delta_t_sr_gps

print(f"Total Dilation for Orbit 1 (GPS):  {total_dilation_gps * 1e6:.4f} microseconds/day")

# -- Satellite 2 (GEO) --

# a) GR Effect for GEO (relative to infinity)

delta_t_gr_geo = -T_day * (G * M_earth) / (c**2 * r2_geo)

# b) SR Effect for GEO

v_sq_geo = (G * M_earth) / r2_geo

delta_t_sr_geo = -T_day * (v_sq_geo / (2 * c**2))

# c) Total Dilation for GEO

total_dilation_geo = delta_t_gr_geo + delta_t_sr_geo

print(f"Total Dilation for Orbit 2 (GEO):  {total_dilation_geo * 1e6:.4f} microseconds/day")

# d) Final Difference (Standard Method)

# Difference is (Time at GEO) - (Time at GPS)

# Since both are time losses (negative), a less negative value means a faster clock.

final_difference_standard = total_dilation_geo - total_dilation_gps

print("----------------------------------------------------------------")

print(f"Net Time Difference (GEO relative to GPS): +{final_difference_standard * 1e6:.4f} microseconds/day\n")

# --- 3. The New Way: The Generalized Unified Formula ---

print("--- Method 2: The Generalized Unified Formula ---")

print("This method uses a single, direct formula between the two orbits.\n")

print("FORMULA: Δt₁₂ = T * (3GM/2c²) * (1/r₁ - 1/r₂)")

# The entire calculation in one unified step

unified_factor = T_day * (3 * G * M_earth) / (2 * c**2)

radius_difference = (1 / r1_gps) - (1 / r2_geo)

final_difference_unified = unified_factor * radius_difference

print(f"Unified Factor (T*3GM/2c²):       {unified_factor:.4e}")

print(f"Radius Difference (1/r₁ - 1/r₂):    {radius_difference:.4e}")

print("----------------------------------------------------------------")

print(f"Net Time Difference (Unified Method):  +{final_difference_unified * 1e6:.4f} microseconds/day\n")

# --- 4. Final Verification ---

print("="*64)

print("                           VERDICT")

print("="*64)

print(f"Standard Method Result:   {final_difference_standard:.12f} s/day")

print(f"Unified Method Result:    {final_difference_unified:.12f} s/day")

difference = abs(final_difference_standard - final_difference_unified)

print(f"Absolute Difference:      {difference:.2e} s/day")

if difference < 1e-12:

    print("\nCONCLUSION: The results are identical. The generalization is correct.")

    print("The unified formula correctly predicts the time difference between any two")

    print("stable circular orbits in a single, elegant step.")

else:

    print("\nCONCLUSION: The results do not match. Check the derivation.")