Unity Through Transformation: A Lesson in Geometric Relativity

 J. Rogers, SE Ohio

Abstract

This lesson plan demonstrates that Special Relativity (SR) and General Relativity (GR) are not separate theories but different manifestations of a single geometric principle. By systematically transforming equations between frameworks, we prove that gravitational and kinematic time dilation are identical phenomena. The appearance of "separate effects" arises from treating constrained variables as independent. We show that constants like

        $G$
      

and

        $c$
      

are merely unit conversion factors, and that all relativistic physics reduces to dimensionless ratios of the form

        $m\mathrm{/}r$      

.

---

Lesson 1: The Fundamental Identity

Learning Objective

Students will algebraically transform the Schwarzschild metric into the Lorentz factor, proving mathematical identity between GR and SR.

The Schwarzschild Metric (General Relativity)

The time dilation near a mass

        $M$
      

at radius

        $r$
      

is:

        $$d\tau =dt\sqrt{1-\frac{2GM}{r{c}^{\mathrm{2}}}}$$      

This looks complicated. But let's identify what's hidden inside.

The Escape Velocity

The classical escape velocity at radius

        $r$
      

is:

        $$v_{esc} = \sqrt{\frac{2GM}{r}}$$      

Square both sides and normalize by

        $c^2$
      

:

        $$\frac{v_{esc}^2}{c^2} = \frac{2GM}{rc^2}$$      

The Transformation

Step 1: Define the dimensionless escape ratio:

        $$\beta_{esc} = \frac{v_{esc}}{c}$$      

Step 2: Substitute into Schwarzschild metric:

        $$d\tau = dt\sqrt{1 - \beta_{esc}^2}$$      

Step 3: Recognize the Lorentz factor:

        $$\gamma = \frac{1}{\sqrt{1-\beta^2}}$$      

Result:

        $$d\tau = \frac{dt}{\gamma_{esc}}$$      

What We've Proven

The Schwarzschild metric of General Relativity IS the Lorentz transformation of Special Relativity, where the velocity is the escape velocity at that radius.

Gravitational time dilation is not "like" kinematic time dilation—it IS kinematic time dilation with

        $v = v_{esc}$
      

.

Student Exercise 1

Calculate the time dilation at:

• Earth's surface (

          $r = 6.371 \times 10^6$
        

  m)

• GPS satellite orbit (

          $r = 2.66 \times 10^7$
        

  m)

• Sun's surface (

          $r = 6.96 \times 10^8$
        

  m)

Using both methods. Verify they give identical results.

---

Lesson 2: The GPS Paradox - Two Effects or One?

Learning Objective

Students will discover that "gravitational" and "velocity" time dilation in orbits are not independent effects but components of a single unified phenomenon.

The Traditional Approach

Textbooks calculate GPS time dilation by summing two "separate" effects:

1. Gravitational time dilation (GR):

        $$\Delta t_{GR} = T\frac{GM}{c^2}\left(\frac{1}{r_{earth}} - \frac{1}{r_{sat}}\right) = +45.74 \text{ μs/day}$$      

2. Velocity time dilation (SR):

        $$\Delta t_{SR} = -T\frac{v^2}{2c^2} = -7.20 \text{ μs/day}$$      

Total:

        $+38.54$
      

μs/day

This suggests two separate physical phenomena. But this is an illusion.

The Hidden Constraint

For a GPS satellite in circular orbit, velocity is not a free parameter. Orbital mechanics requires:

        $$\frac{mv^2}{r} = \frac{GMm}{r^2}$$      

Solving for velocity:

        $$v^2 = \frac{GM}{r_{sat}}$$      

The velocity is determined by the gravitational potential!

The Transformation

Step 1: Substitute

        $v^2 = GM/r_{sat}$
      

into the SR formula:

        $$\Delta t_{SR} = -T\frac{v^2}{2c^2} = -T\frac{GM}{2c^2 r_{sat}}$$      

The "velocity effect" is now expressed entirely as a gravitational potential term.

Step 2: Add the GR and transformed SR effects:

        $$\mathrm{\Delta }{t}_{total}=T\frac{GM}{c^2}(\frac{1}{r_{earth}}-\frac{1}{r_{sat}})-T\frac{GM}{2c^2{r}_{sat}}$$

Step 3: Factor out common terms:

        $$\Delta t_{total} = T\frac{GM}{c^2}\left(\frac{1}{r_{earth}} - \frac{1}{r_{sat}} - \frac{1}{2r_{sat}}\right)$$      

Step 4: Simplify:

        $$\Delta t_{total} = T\frac{GM}{c^2}\left(\frac{1}{r_{earth}} - \frac{3}{2r_{sat}}\right)$$      

The Unified Formula

        $$\boxed{\Delta t = T\frac{GM}{c^2}\left(\frac{1}{r_1} - \frac{3}{2r_2}\right)}$$      

This single equation replaces both "GR" and "SR" calculations.

What We've Proven

The apparent separation into "gravitational" and "velocity" effects is an artifact of treating

        $v$
      

as independent when it's constrained by

        $v^2 = GM/r$
      

.

There is only one effect: Time dilation in a gravitational potential, where orbital motion contributes a factor of 3/2 instead of 1.

Student Exercise 2

1. Calculate GPS time dilation using traditional method (two separate effects)

2. Calculate using unified formula

3. Verify they match to within numerical precision

4. Compare two satellites at different orbital radii using unified formula

Data:

•  $G = 6.674 \times 10^{-11}$m³/kg·s²

•  $M_{earth} = 5.972 \times 10^{24}$kg

•  $c = 2.998 \times 10^8$m/s
  

•  $r_{earth} = 6.371 \times 10^6$m
  

•  $r_{GPS} = 2.66 \times 10^7$m
  

---

Lesson 3: The Mystery of the 3/2 Factor

Learning Objective

Students will understand that the 3/2 coefficient emerges from the low-velocity expansion of the Lorentz factor, connecting the Virial Theorem to special relativity.

The Lorentz Factor Expansion

The relativistic energy is:

        $$E = \gamma mc^2 = \frac{mc^2}{\sqrt{1-v^2/c^2}}$$      

Taylor expansion for small

        $v/c$
      

:

        $$\gamma \approx 1 + \frac{1}{2}\frac{v^2}{c^2} + \frac{3}{8}\frac{v^4}{c^4} + ...$$      

Kinetic energy:

        $$KE = E - mc^2 = \gamma mc^2 - mc^2 \approx \frac{1}{2}mv^2$$      

The famous 1/2 factor in kinetic energy comes from the first-order Taylor expansion of

        $\gamma$
      

!

Circular Orbits and the Virial Theorem

For a stable circular orbit:

Centripetal force equals gravitational force:

        $$\frac{mv^2}{r} = \frac{GMm}{r^2}$$      

Orbital velocity:

        $$v^2 = \frac{GM}{r}$$      

Kinetic energy:

        $$KE = \frac{1}{2}mv^2 = \frac{GMm}{2r}$$      

Potential energy:

        $$PE = -\frac{GMm}{r}$$      

The Virial Theorem:

        $$2KE = |PE|$$      

or equivalently:

        $$KE=\frac{1}{2}\mathrm{\mid }PE\mathrm{\mid }$$      

The Geometric Origin

Why exactly 1/2? Because motion is orthogonal to the potential gradient.

In a circular orbit:

• Velocity vector: Tangent to the circle (azimuthal direction)

• Gravitational gradient: Radial (pointing inward)

• Angle between them: 90°

The kinetic component is the tangential projection of the total energy, which in the low-velocity limit gives the 1/2 factor from γ's expansion.

Connecting to Time Dilation

Gravitational time dilation at radius

        $r$
      

:

        $$\Delta\tau_{grav} \approx -\frac{GM}{c^2 r}$$      

Kinetic time dilation for orbital velocity

        $v^2 = GM/r$
      

:

        $$\Delta\tau_{kinetic} \approx -\frac{v^2}{2c^2} = -\frac{GM}{2c^2 r}$$

Ratio:

        $$\frac{\Delta\tau_{kinetic}}{\Delta\tau_{grav}} = \frac{1}{2}$$      

Total time dilation for orbiting clock compared to distant clock:

        $$\Delta\tau_{total} = -\frac{GM}{c^2 r} - \frac{GM}{2c^2 r} = -\frac{3GM}{2c^2 r}$$      

This is the origin of the 3/2 factor:

• 1 part from gravitational potential

• 1/2 part from orbital kinetic energy (first-order

          $\gamma$
        

  expansion)

What We've Proven

The Virial Theorem (

        $2KE = |PE|$
      

) is the weak-field limit of the Lorentz factor.

The 1/2 in

        $KE = \frac{1}{2}mv^2$
      

and the 1/2 in the orbital time dilation are the same 1/2—both arising from the Taylor expansion of γ.

Student Exercise 3

1. Derive the Taylor expansion of

           $\gamma = 1/\sqrt{1-\beta^2}$
         

   to third order

2. Show that

           $\gamma \approx 1 + \frac{1}{2}\beta^2 + \frac{3}{8}\beta^4$
         

3. For Earth's orbital velocity around Sun (

           $v \approx 30$
         

   km/s), calculate:

   • The exact

             $\gamma$
           

     factor

   • The first-order approximation

   • The second-order approximation

   • Compare errors

---

Lesson 4: Gravity as Time Gradient

Learning Objective

Students will derive Newton's law of gravitation from the gradient of the time dilation field, showing that acceleration is the slope of time.

The Time Field

Define the dimensionless gravitational potential (in natural units where

        $G=c=1$
      

):

        $$\Phi = \frac{m}{r}$$      

In SI units:

        $$\Phi = \frac{GM}{c^2 r}$$      

Time dilation at radius

        $r$
      

:

        $$\frac{d\tau}{dt} = \sqrt{1 - 2\Phi} \approx 1 - \Phi$$      

(using weak-field approximation for

        $\Phi \ll 1$
      

)

The Gradient

Derivative with respect to radius:

        $$\frac{d\Phi}{dr} = \frac{d}{dr}\left(\frac{GM}{c^2 r}\right) = -\frac{GM}{c^2 r^2}$$      

The negative sign indicates that time runs slower (higher

        $\Phi$
      

) as you go inward.

Free-Fall Acceleration

An object following the natural geodesic (free-fall) experiences acceleration equal to the gradient of the time field:

        $$a = -c^2 \nabla \Phi = -c^2 \frac{d\Phi}{dr} = -c^2 \left(-\frac{GM}{c^2 r^2}\right)$$      

        $$\boxed{a = \frac{GM}{r^2}}$$      

This is Newton's law of gravitation!

Physical Interpretation

What is "falling"?

• At radius

          $r$
        

  : time rate is

          $\tau(r) = 1 - GM/(c^2 r)$
        

• At radius

          $r - dr$
        

  : time rate is

          $\tau(r-dr) = 1 - GM/(c^2(r-dr))$
        

  (slower)

• The mismatch in time rates creates acceleration toward slower time

Free-fall = following the time gradient

Standing still = resisting the time gradient (requires force)

Mach's Principle Realized

"Motion is the object's time experience moving in a straight line in a universal time gradient. Force is what is felt when you are prevented from following your natural time path."

Free-fall (weightless):

• Following natural geodesic in

          $\Phi = \sum m_i/r_i$
        

  field

• Zero proper acceleration

• No force felt

Standing on ground (weight):

• Ground prevents natural motion down time gradient

• Non-zero proper acceleration

• Force =

          $ma = m \cdot \nabla \Phi$
        

What We've Proven

Gravitational acceleration is not a mysterious force—it's simply the gradient (slope) of the time dilation field.

        $$\text{Acceleration} = -c^2 \nabla\Phi = -c^2 \nabla\left(\frac{GM}{c^2 r}\right) = \frac{GM}{r^2}$$      

Student Exercise 4

1. Calculate

           $\nabla \Phi$
         

   for a point mass at the origin

2. Calculate

           $\nabla \Phi$
         

   for two masses (Earth + Moon) at arbitrary location

3. Find the point where

           $\nabla \Phi = 0$
         

   (L1 Lagrange point)

4. Verify that this matches the classical calculation

---

Lesson 5: Force as the Product of Time Fields

Learning Objective

Students will transform Newton's law of gravitation into a product of dimensionless time dilation factors, revealing that force is the interaction of time fields.

Planck Units Review

The original 1899 Planck units are defined by setting

        $G = c = \hbar = 1$
      

:

        $$l_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.616 \times 10^{-35} \text{ m}$$    

        $$m_P = \sqrt{\frac{\hbar c}{G}} \approx 2.176 \times 10^{-8} \text{ kg}$$     

        $$F_P=\frac{c^4}{G}$$

$$t_P=\; sqrt\; (\; m\_P\; l\_P^3/\; G)$$     

The Planck force is the conversion factor between natural units and SI units.

Newton's Law in Natural Units

In natural units (

        $G = c = 1$
      

), the gravitational force between two masses separated by distance r is:

        $$F_{nat} = \frac{m_1 m_2}{r^2}$$     

This is a dimensionless number—a pure geometric ratio.

Converting to SI Units

To get force in Newtons, multiply by the Planck force:

        $$F_{SI} = F_{nat} \times F_P = \frac{m_1 m_2}{r^2} \times \frac{c^4}{G}$$

But

        $m_1$
      

and

        $m_2$
      

in natural units need to be converted from SI:

        $$m_{nat} = \frac{m_{SI}}{m_P}$$    

And

        $r_{nat}$
      

needs conversion:

        $$r_{nat} = \frac{r_{SI}}{l_P}$$      

The Complete Transformation

Step 1: Write force in terms of dimensionless potentials:

        $$\Phi_1 = \frac{m_1}{r} \cdot \frac{l_P}{m_P} = \frac{Gm_1}{c^2 r}$$      

        $$\Phi_2 = \frac{m_2}{r} \cdot \frac{l_P}{m_P} = \frac{Gm_2}{c^2 r}$$      

Step 2: Force as product of potentials:

        $$F_{SI} = \Phi_1 \times \Phi_2 \times F_P$$      

        $$F_{SI} = \frac{Gm_1}{c^2 r} \times \frac{Gm_2}{c^2 r} \times \frac{c^4}{G}$$      

Step 3: Simplify:

        $$F_{SI} = \frac{G^2 m_1 m_2}{c^4 r^2} \times \frac{c^4}{G} = \frac{Gm_1 m_2}{r^2}$$      

✓ Newton's law recovered!

What We've Proven

Gravitational force is the product of two time dilation fields, scaled by the Planck force unit conversion.

        $$F = \left(\frac{m_1}{r}\right) \times \left(\frac{m_2}{r}\right) \times F_P$$      

Each factor

        $m/r$ in natural units
      

is the dimensionless time dilation field created by that mass.

Force = (time field₁) × (time field₂) × (unit conversion)

The Meaning of G

Newton's gravitational constant is not a fundamental property of gravity—it's a unit conversion factor:

   ${\mathrm{G\; =\; l}}_P^3\mathrm{/(}{m}_P^{\mathrm{}}\times$$t_P^2)$

It converts between:

• SI units (kg, meters, seconds)

• Natural dimensionless ratios (

          $m/r$
        

  )

In natural units,

        $G$
      

doesn't appear. There is no constant—only ratios.

Student Exercise 5

1. Calculate the dimensionless potential

           $\Phi = Gm/(c^2 r)$
         

   for:

   • Earth at its surface

   • Sun at its surface

   • Neutron star at its surface (

             $m \sim 2 M_{\odot}$
           

     ,

             $r \sim 10$ km)
     

2. Calculate the force between Earth and Moon using:

   • Traditional Newton's law:

             $F = Gm_1m_2/r^2$
           

   • Product of potentials:

             $F = \Phi_1 \Phi_2 F_P$
           

   • Verify they match

3. Show explicitly that

           $l_P^3\mathrm{/(}{m}_P^{\mathrm{}}\times$$t_P^2)=G$
         

---

Lesson 6: The Event Horizon as β = 1

Learning Objective

Students will understand that the Schwarzschild radius (event horizon) occurs exactly where escape velocity equals the speed of light.

The Schwarzschild Radius

The event horizon of a black hole is at:

        $$r_s = \frac{2GM}{c^2}$$     

This is traditionally derived from the Schwarzschild metric by finding where

        $g_{00}=0$      

.

Escape Velocity at the Horizon

At radius

        $r_s$
      

, the escape velocity is:

        $$v_{esc} = \sqrt{\frac{2GM}{r_s}}$$

Substitute

        $r_s = 2GM/c^2$
      

:

        $$v_{esc} = \sqrt{\frac{2GM}{2GM/c^2}} = \sqrt{c^2} = c$$      

At the event horizon, escape velocity equals the speed of light.

Time Dilation at the Horizon

Using our unified framework:

        $$d\tau = dt\sqrt{1 - \beta_{esc}^2}$$

At the horizon where

        $\beta_{esc} = 1$
      

:

        $$d\tau = dt\sqrt{1 - 1} = 0$$      

Time stops at the event horizon.

Physical Meaning

The event horizon is not a mysterious surface—it's simply the radius where:

• Escape velocity = speed of light

•         $\beta_{esc} = 1$
        

•         $\gamma_{esc} = \infty$
        

• Time dilation becomes infinite

Light emitted exactly at

        $r_s$
      

can't escape (gravitationally redshifted to zero frequency).

What We've Proven

The Schwarzschild radius is the natural consequence of the escape velocity formula:

        $$v_{esc} = c \implies \sqrt{\frac{2GM}{r}} = c \implies r = \frac{2GM}{c^2}$$      

The event horizon occurs exactly where the time dilation formula

        $d\tau = dt\sqrt{1-\beta_{esc}^2}$
      

predicts time should stop.

This is not a coincidence—GR and SR are the same equation.

Student Exercise 6

Calculate the Schwarzschild radius for:

1. Earth (

           $M = 5.97 \times 10^{24}$
         

   kg)

2. Sun (

           $M = 1.99 \times 10^{30}$
         

   kg)

3. Sagittarius A* (

           $M \sim 4 \times 10^6 M_{\odot}$
         

   )

4. A stellar-mass black hole (

           $M = 10 M_{\odot}$
         

   )

For each, calculate

        $\beta_{esc}$
      

at:

• The Schwarzschild radius

• 2× the Schwarzschild radius

• 10× the Schwarzschild radius

---

Lesson 7: Multi-Body Superposition

Learning Objective

Students will understand that gravitational fields from multiple masses simply add as

        $\Phi_{total} = \sum m_i/r_i$
      

, and that every object exists in nested orbits around all masses simultaneously.

The Superposition Principle

For

        $N$
      

masses at positions

        $\vec{r}_i$
      

with masses

        $m_i$
      

, the total gravitational potential at location

        $\vec{r}$
      

is:

        $$\Phi(\vec{r}) = \sum_{i=1}^{N} \frac{Gm_i}{c^2|\vec{r} - \vec{r}_i|}$$      

In natural units:

        $$\Phi = \sum_i \frac{m_i}{r_i}$$      

This is exact in the weak-field limit and a good approximation for most astronomical scenarios.

Time Dilation from All Masses

The total time dilation at any point is:

        $$\frac{d\tau}{dt} = \sqrt{1 - 2\Phi_{total}} \approx 1 - \Phi_{total} = 1 - \sum_i \frac{Gm_i}{c^2 r_i}$$      

Every mass in the universe contributes to your local time rate.

Example: Standing on Earth

At Earth's surface, you're simultaneously influenced by:

Earth:

        $$\Phi_{Earth} = \frac{GM_{Earth}}{c^2 R_{Earth}} \approx 6.95 \times 10^{-10}$$      

Sun:

        $$\Phi_{Sun} = \frac{GM_{Sun}}{c^2 (1 \text{ AU})} \approx 9.87 \times 10^{-9}$$      

Moon:

        $$\Phi_{Moon} = \frac{GM_{Moon}}{c^2 (384,400 \text{ km})} \approx 3.11 \times 10^{-11}$$      

Milky Way (approximate):

        $$\Phi_{Galaxy} \sim 10^{-6}$$      

(dominant at large scales!)

Total:

        $$\Phi_{total} = \Phi_{Earth} + \Phi_{Sun} + \Phi_{Moon} + \Phi_{Galaxy} + ...$$

Nested Orbits

You are never not in orbit. At any moment, you're in nested orbits around:

• Earth (24 hour period if at equator, rotating with planet)

• Sun (365.25 day period)

• Galactic center (~230 million year period)

• Local group center

• Virgo cluster

• Laniakea supercluster

Your time rate is determined by the sum of all these

        $m_i/r_i$ in natural units
      

contributions.

What We've Proven

There is no such thing as "empty space." Every location has a time rate determined by:

        $$\tau = 1 - \sum_{all masses} \frac{Gm_i}{c^2 r_i}$$      

Gravity is not a local interaction between two bodies—it's your position in the global time gradient field created by all mass in the universe.

This is Mach's principle realized: Your local physics is determined by the distribution of all matter.

Student Exercise 7

1. Calculate the relative contributions to

           $\Phi$
         

   at Earth's surface from:

   • Earth

   • Sun

   • Moon

   • Jupiter (approximate as r ∼ 5 AU)

2. At what distance from Earth does the Sun's contribution equal Earth's?

3. Estimate the galactic contribution assuming:

   •         $M_{galaxy}\sim {10}^{12}{}_{}$      

   • Distance to center:

             $r \sim 8$kpc

---

Lesson 8: Summary and Philosophical Implications

What We've Accomplished

Through systematic transformation of equations, we've proven:

1. GR = SR with escape velocity:

           $d\tau = dt\sqrt{1-\beta_{esc}^2}$
         

2. GPS "two effects" are one:

           $\Delta t = T(GM/c^2)(1/r_1 - 3/2r_2)$
         

3. The 1/2 factor is gamma: Virial theorem = first Taylor term of Lorentz factor

4. Acceleration = time gradient:

           $a = -c^2\nabla\Phi = GM/r^2$
         

5. Force = product of time fields:

           $F = (m_1/r)(m_2/r) F_P$
         

6. Event horizon = β=1: Schwarzschild radius is where

           $v_{esc} = c$
         

7. Superposition:

           $\Phi_{total} = \sum m_i/r_i$
         

   determines local time rate

The Core Insight

There is only one phenomenon: position and motion in the universal time gradient field.

What we've historically called "separate effects" are artifacts of:

• Using arbitrary measurement units (kg, meters, seconds instead of dimensionless ratios)

• Treating constrained variables as independent (

          $v$
        

  is determined by

          $\Phi$
        

  in orbits)

• Separating "gravitational" from "kinetic" when both are aspects of time geometry

Constants Are Unit Conversions

        $$G = \frac{l_P^2 c^4}{m_P^2}$$

converts SI units to natural ratios

        $$c$$     

converts time-delay (space) to time-rate (time)

Neither encodes fundamental physics—only our choice of measurement units.

Mach's Principle

Force is what you feel when prevented from following your natural path through the time gradient.

• Free-fall: Following

          $\nabla\Phi$
        

  → no force felt

• Standing still: Resisting

          $\nabla\Phi$
        

  → force felt

• Acceleration: Changing path through

          $\Phi$
        

  → force felt

Unity Through Transformation

The test of a unified theory is not philosophical argument—it's whether equations transform into each other algebraically.

We've shown:

• Schwarzschild → Lorentz (exact)

• Two GPS effects → One unified formula (exact)

• Newton's force → Time field product (exact)

• Acceleration → Potential gradient (exact)

These are not analogies. They are mathematical identities.

What Remains

This framework successfully unifies weak-field gravity (the observable universe outside black hole horizons). Open questions:

1. Strong-field regime: Does

           $\sum m_i/r_i$
         

   work near event horizons, or do non-linear corrections emerge?

2. Cosmology: How does homogeneous mass distribution affect

           $\Phi$
         

   ? Does expansion emerge naturally?

3. Gravitational waves: Can time-dependent

           $\Phi(t-r/c)$
         

   with retardation reproduce GR's predictions?

4. Quantum regime: How does this framework interface with quantum mechanics at the Planck scale?

These are questions for further research. But for 99.9999% of spacetime (weak-field regime), the unity is proven.

---

Final Assessment

Problem Set

Problem 1: Transform the following equation from SR to GR form:

        $$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$      

Problem 2: A satellite orbits at

        $r = 42,164$ km (geostationary orbit). 

Calculate its time dilation relative to Earth's surface using:

a) Traditional two-effect method
b) Unified formula
c) Show they match

Problem 3: Derive the Schwarzschild radius by setting

        $\beta_{esc} = 1$
      

in the escape velocity formula.

Problem 4: Calculate the gradient

        $\nabla\Phi$
      

for the Earth-Moon system at the L1 Lagrange point. Verify it equals zero.

Problem 5: Express Newton's law

        $F = Gm_1m_2/r^2$
      

in terms of dimensionless potentials

        $\Phi_1$
      

and

        $\Phi_2$
      

, showing explicitly how

        $G$
      

cancels.

Problem 6: Two satellites orbit at different altitudes. Using only the unified formula, predict their relative clock rates.

Problem 7: Show that the Taylor expansion of

        $\gamma$
      

gives

        $E = mc^2 + \frac{1}{2}mv^2 + \frac{3}{8}mv^4/c^2 + ...$
      

Problem 8: Calculate the time dilation at Earth's surface including contributions from Earth, Sun, Moon, and Galaxy. Which dominates?

---

Conclusion

Physics is not a collection of separate theories connected by coincidence. It is a unified geometric structure expressed in different coordinates.

By treating space as time-delay and velocity as a dimensionless ratio, we've shown that SR and GR are the same equation:

        $d\tau = dt\sqrt{1-\beta^2}$
      

, where

        $\beta$
      

can be either actual velocity or escape velocity.

Constants like

        $G$
      

and

        $c$
      

are not fundamental—they're conversion factors between arbitrary human units and natural dimensionless ratios.

The universe computes in ratios. We've just been adding unit labels and then treating them as physics.

This lesson plan teaches unity through transformation—showing that equations which appear different are algebraically identical. That is the signature of true understanding.

---

References for Further Study

1. The GPS satellite data for empirical verification

2. Taylor & Wheeler, "Exploring Black Holes" (for Schwarzschild geometry)

3. Misner, Thorne & Wheeler, "Gravitation" (for the full GR mathematical machinery)

4. Natural units and dimensional analysis in physics

5. Mach's principle and the origin of inertia

6. Weak-field limit of General Relativity

The student who understands transformation understands unity.