The Geometric Ideal Gas: Covariance of Volume and Frequency in Relativistic Frames

 J. Rogers, SE Ohio 

Abstract

Standard thermodynamics relies on arbitrary constants (Boltzmann’s constant,

        $k_B$
      

) and treats Temperature as a statistical abstraction. By redefining Temperature as Frequency (

        $T=f$
      

) and Volume as Spatial Delay (

        $V=L^3$
      

), we demonstrate that the Ideal Gas Law (

        $PV=Nf$
      

) is a geometrically invariant statement. We prove that for a relativistic observer, the Lorentz contraction of Volume and the Time Dilation of Frequency occur in "lockstep." This simultaneous scaling ensures that the physical relationship between Energy Density (Pressure) and Vibration (Temperature) remains consistent across all frames of reference, unifying Thermodynamics with Special Relativity.

1. The Natural Gas Law

We begin by stripping the Ideal Gas Law of historical constants to reveal its geometric nature.
Standard form:

        $PV = N k_B T$
      

In Natural Units, energy and frequency are equivalent (

        $E \equiv f$
      

). Therefore, Temperature is simply the frequency of the constituent particles. We set

        $k_B = 1$
      

(dimensionless).

The equation becomes:

        $$P V = N f$$

Where:

•         $P$
        

  : Pressure (Energy Density).

•         $V$
        

  : Volume (3-Dimensional Space).

•         $N$
        

  : Number of oscillators (particles).

•         $f$
        

  : Average Frequency (Temperature).

This states that the Total Energy Capacity of the container (

        $PV$      

) is equal to the Total Frequency of the matter inside it (

        $Nf$
      

).

2. The Observer's Frame (The Transformation)

Consider a container of gas at rest in Frame

        $S$
      

.
An observer moves past this container at velocity

        $v$
      

(Frame

        $S'$
      

).
We define the Lorentz Factor:

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

.

We must determine how the dimensions of Space (Volume) and Time (Frequency) transform for the moving observer.

A. Transformation of Volume (Length)

Length along the direction of motion contracts.

        $$L' = \frac{L}{\gamma}$$

Since the transverse dimensions (

        $y, z$
      

) are unaffected, the Volume transforms linearly with the length contraction:

        $$V' = \frac{V_0}{\gamma}$$
      

Geometric Consequence: The observer measures a physically smaller container.

B. Transformation of Temperature (Time)

Since

        $T = f$
      

, Temperature is a measure of time-rate (ticks per second).
According to Special Relativity, a moving clock ticks slower (Time Dilation).
Therefore, the observed frequency of the particles' thermal vibration decreases:

        $$f' = \frac{f_0}{\gamma}$$

Geometric Consequence: The observer measures a "colder" gas (lower frequency).

3. The Lockstep Proof

We now test the Ideal Gas Law in the moving frame (

        $S'$
      

).
Does the law

        $P'V' = N f'$
      

still hold?

Substitute the transformed variables into the equation:

        $$P' V' = N f'$$

        $$P' \left( \frac{V_0}{\gamma} \right) = N \left( \frac{f_0}{\gamma} \right)$$      

Notice that the Lorentz factor

        $\gamma$
      

appears in the denominator on both sides of the equation.

        $$\frac{1}{\gamma} (P' V_0) = \frac{1}{\gamma} (N f_0)$$     

Because the Space Axis (Volume) contracts in exact proportion to the Time Axis (Frequency) dilating, the scaling factors cancel out perfectly.

4. Interpretation: The Invariant Ratio

This derivation proves that the Ideal Gas Law is physically robust in Relativity because it describes a Ratio.

        $$P' = \frac{N (f_0 / \gamma)}{(V_0 / \gamma)} = \frac{N f_0}{V_0} = P_0$$      

The "Pressure" (Energy Density) remains invariant in this specific simplified formulation because the "box" shrinks exactly as fast as the "energy" slows down.

• Standard View: Confusion about whether Temperature transforms as

          $T/\gamma$
        

  or

          $T\gamma$
        

  .

• Geometric View (

          $T=f$
        

  ): Clarity. The Space contracts (

          $1/\gamma$
        

  ) and the Time dilates (

          $1/\gamma$
        

  ). They are effectively coupled.

5. Conclusion

The Ideal Gas Law is not merely a statistical approximation; it is a fundamental geometric identity relating the Spatial Extent of a system to its Temporal Frequency.

When an observer moves at relativistic speeds:

1. They see the box shrink (Length Contraction).

2. They see the particles vibrate slower (Time Dilation).

Because these two distortions happen in geometric lockstep, the fundamental relationship describing the gas remains unbroken. The universe distorts the stage (Volume) and the actors (Frequency) simultaneously, preserving the play.