A Note on the Recurrence of the 3/2 Coefficient in Relativistic Orbit and Thermodynamic Equipartition

J. Rogers, SE Ohio

Abstract In a recent derivation of the time dilation experienced by Global Positioning System (GPS) satellites, a coefficient of

$3/2$

emerged naturally when unifying gravitational and velocity effects. This coefficient is structurally identical to the

$3/2$

coefficient found in the Equipartition Theorem of classical thermodynamics, which describes the average energy of a monatomic gas. This paper explores whether this numerical similarity is a matter of algebraic coincidence or if it indicates a deeper structural relationship between the degrees of freedom available to a particle in a gas and the geometric constraints acting on a satellite in orbit. We examine the derivation of both coefficients in terms of spatial dimensionality (

$D$

) to assess the nature of this parallel.

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1. Introduction

It is not uncommon in physics for the same numerical constant to appear in disparate fields, often hinting at an underlying unity in the mathematical structure of physical laws. A striking example of this phenomenon is the appearance of the ratio

$3/2$

in two seemingly unrelated contexts:

1. Relativistic Time Dilation (Circular Orbits): The total time dilation of a clock in a circular orbit, when expressed as a function of orbital radius, is governed by a factor of
   $3/2$
   .
2. Thermodynamic Equipartition: The average internal energy of a monatomic ideal gas is equal to
   $3/2$
   times the Boltzmann constant times the temperature (
   $E=\frac{3}{2}{k}_{B}T$
   ).

This note investigates the derivation of these two values. Rather than asserting a definitive causal link, we explore the possibility that both coefficients are generated by the same geometric input: the three dimensions of physical space.

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2. The Relativistic

$3/2$

In the context of GPS time dilation, the classical approach treats gravitational time dilation and velocity-based (kinematic) time dilation as separate additive terms. However, for a circular orbit, the velocity

$v$

is not an independent variable; it is a function of the gravitational potential (specifically, the radius

$r$

):

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

When we express the total time dilation

$\Delta t_{total}$

relative to a reference radius, we sum the General Relativistic (GR) effect and the Special Relativistic (SR) effect:

$\Delta t_{GR}\propto \frac{GM}{c^{2}r}(\text{Coefficient: }1)$

$\Delta t_{SR}\approx -\frac{v^2}{2c^2}\propto -\frac{GM}{2c^{2}r}(\text{Coefficient: }-1/2)$

Combining these into a single unified expression for the relative time rate between two orbits yields:

$\Delta t_{total}\propto \left(1-\frac{1}{2}\right)\frac{GM}{c^{2}r}$

Here, the term "1" represents the gravitational potential component, and the term "

$1/2$

" represents the kinetic component derived from

$v^2$

. The factor

$3/2$

emerges as the sum of these two contributions.

Observation: This factor of

$1/2$

in the kinetic term is a direct consequence of the quadratic relationship between velocity and energy (

$E_k\propto v^2$

).

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3. The Thermodynamic

$3/2$

In classical statistical mechanics, the Equipartition Theorem states that each degree of freedom contributes

$\frac{1}{2}{k}_{B}T$

to the average energy of a system.

For a monatomic gas (a point particle):

• The particle is free to translate in three independent spatial directions:
  $x$
  ,
  $y$
  , and
  $z$
  .
• Therefore, the number of degrees of freedom
  $f$
  is 3.

Summing the energy contribution from each degree of freedom:

$⟨E⟩={\sum }_{i=1}^3\frac{1}{2}{k}_{B}T=\frac{3}{2}{k}_{B}T$

Observation: Here, the coefficient is explicitly the number of spatial dimensions (

$D=3$

) divided by 2.

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4. Exploring the Structural Similarity

At first glance, the two derivations appear distinct. One deals with the curvature of spacetime and the velocity of a massive satellite; the other deals with the statistical distribution of energy among microscopic particles.

However, when we generalize the mathematics, a pattern emerges involving the number of spatial dimensions,

$D$

.

4.1. Dimensional Analysis of the Coefficients

If we generalize the Relativistic derivation for a hypothetical universe of

$D$

spatial dimensions where gravity follows an inverse-power law:

• The potential energy
  $U$
  generally scales with position
  $r$
  .
• The Virial Theorem dictates a fixed relationship between Kinetic Energy (
  $K$
  ) and Potential Energy (
  $U$
  ) for bound orbits.
• In a 3D universe (
  $D=3$
  ), the Virial Theorem yields
  $K=-\frac{1}{2}U$
  .
• The total energy scaling factor becomes
  $|K|+|U|=\frac{1}{2}U+U=\frac{3}{2}U$
  .

If we generalize the Thermodynamic derivation for a universe of

$D$

spatial dimensions:

• A point particle has
  $D$
  translational degrees of freedom.
• The energy is
  $⟨E⟩=\frac{D}{2}{k}_{B}T$
  .

In both cases, the coefficient takes the form

$D/2$

. In our specific universe, where

$D=3$

, the coefficient is

$3/2$

.

4.2. The Role of the Quadratic Term

Both phenomena rely fundamentally on quadratic relationships, which introduces the denominator of 2:

• Kinematics: Kinetic energy is defined as
  $\frac{1}{2}m{v}^2$
  . The relativistic correction factor in the low-velocity limit (Taylor expansion) is dominated by this quadratic term.
• Thermodynamics: The Equipartition Theorem applies to quadratic degrees of freedom (terms like
  $\frac{1}{2}m{v}_x^2$
  ).

It appears that the "2" in the denominator of the

$3/2$

coefficient in both fields may originate from the fundamental quadratic nature of kinetic energy, while the "3" corresponds to the three dimensions available for that motion.

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5. Interpretation and Discussion

The recurrence of the

$3/2$

coefficient invites speculation, though not definitive conclusions, regarding the nature of physical law.

Hypothesis A: Coincidence It is possible that the similarity is numerically coincidental. The orbital

$3/2$

is derived from the metric of spacetime and the dynamics of a specific force (gravity), whereas the thermodynamic

$3/2$

is derived from statistical probability distributions independent of the specific forces acting on the particles (barring their collision).

Hypothesis B: Geometric Constraint It is possible that both coefficients are distinct expressions of the same underlying geometric constraint: the dimensionality of space.

• The gas particle has 3 "ways" to move (degrees of freedom).
• The orbiting clock is affected by a gravity that "splits" its time dilation into a potential component and a kinetic component, where the ratio of these components is fixed by the 3-dimensional geometry of the field (Virial Theorem).

In this view, the

$3/2$

in the GPS formula is not merely a convenient sum of 1 and 1/2; it is the signature of 3-dimensional space manifesting in the time-rate of a bound object.

Hypothesis C: The "Freedom" of the Orbit One might characterize the orbiting clock as a particle with "3 degrees of freedom" constrained by a central potential. The

$3/2$

factor in the time dilation may then be interpreted as the system expressing its available dimensions of motion. The satellite is "free" to move in 3D space, and this freedom is encoded in the rate at which it experiences time.

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6. Conclusion

This paper has observed that the coefficient

3/2—found in the unified GPS time dilation formula (1 + 1/2

) and the thermodynamic equipartition energy (

$3/2k_{B}T$

)—can both be expressed as the ratio of spatial dimensions to a quadratic factor (

$D/2$

).

While the derivation from orbital mechanics is distinct from the derivation from statistical mechanics, the convergence on the value

$3/2$

suggests that both systems are sensitive to the same geometric parameter: the three dimensions of space. Further inquiry might be directed toward whether other physical coefficients, often treated as fundamental constants, can be reduced to simple integer ratios representing the dimensional architecture of the universe.