The Decoupling of Geometry and Signal: Resolving the Relativistic Thermodynamic Paradox

 J. Rogers, SE Ohio

Abstract

For over a century, relativistic thermodynamics has been plagued by conflicting definitions of temperature transformation (Planck vs. Ott vs. Landsberg). These conflicts arise from a failure to distinguish between the Internal Geometric State of a system and the Observed Radiative Signal from that system. By defining Temperature as Frequency (

        $T=f$
      

) and Volume as Spatial Extent (

        $V=L^3$
      

), we demonstrate that the Ideal Gas Law (

        $PV=Nf$
      

) is invariant under Lorentz transformations. We show that while the observed temperature of an approaching gas appears hotter due to Doppler shift, the internal thermodynamic state remains colder due to time dilation. Crucially, we prove that Volume is a purely geometric quantity that undergoes Length Contraction but does not experience "Doppler shifting." This decoupling ensures that Pressure is a Lorentz Scalar (

        $P'=P_0$
      

), resolving the historical paradox.

1. The Geometric Ideal Gas

We begin with the fundamental identity of the Ideal Gas in Natural Units (

        $k_B=1, h=1$
      

):

        $$PV = Nf$$

Where:

•         $P$
        

  = Pressure (Energy Density)

•         $V$
        

  = Volume (Spatial Extent)

•         $f$
        

  = Average Particle Frequency (Thermodynamic Temperature)

•         $N$
        

  = Number of Particles

For the laws of physics to remain consistent across frames, an observer moving relative to the gas must calculate the same intrinsic pressure

        $P_0$
      

, provided the gas is isolated.

2. The Transverse Case: Pure Geometry

Consider an observer moving at velocity

        $v$
      

relative to the gas box, observing the box from a transverse angle (

        $90^\circ$
      

to the motion vector). In this frame, there is no longitudinal Doppler component.

2.1 Volume Transformation

The length of the box along the motion vector contracts by the Lorentz factor

        $\gamma$
      

:

        $$L' = \frac{L_0}{\gamma}$$

Since transverse dimensions are unchanged, the Volume contracts linearly:

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

2.2 Temperature (Frequency) Transformation

In the transverse frame, the observer sees the internal clocks of the gas particles running slower due to pure Time Dilation. There is no wave compression.

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

Observation: The gas appears Colder and Smaller.

2.3 The Invariant Pressure

Substituting these transformed values into the Gas Law:

        $$P'_{trans} = \frac{N f'_{trans}}{V'_{trans}} = \frac{N (f_0 / \gamma)}{(V_0 / \gamma)}$$      

The

        $\gamma$
      

factors cancel perfectly:

        $$P'_{trans} = \frac{N f_0}{V_0} = P_0$$      

This confirms that in the purely geometric limit, the "Lockstep" of Space contraction and Time dilation preserves the thermodynamic state.

3. The Longitudinal Case: The Doppler Trap

Now consider the observer moving directly towards the gas box (Head-On Approach).

3.1 The Transformation of Observed Frequency

The observer measures the radiation coming from the gas. Due to the approach, the wavefronts are compressed. The observed frequency follows the relativistic Doppler formula:

        $$f_{obs} = f_0 \sqrt{\frac{1+v}{1-v}}$$      

Using the algebraic identity

        $\sqrt{\frac{1+v}{1-v}} = (1+v)\gamma$
      

, we can express this equivalently as:

        $$f_{obs} = f_0 (1+v)\gamma$$      

Observation: To a spectrometer, the gas appears Hotter (Blue-shifted).

3.2 The "Non-Doppler" Nature of Volume

Crucially, Volume is not a wave. It does not propagate; it extends. Therefore, Volume does not experience "Doppler shifting." It only experiences geometric Length Contraction.
Regardless of whether you approach or recede, the space occupied by the box depends only on the magnitude of velocity

        $|\vec{v}|$
      

:

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

3.3 The Paradox

If one were to naively calculate pressure using the observed Doppler temperature (

        $f_{obs}$
      

) and the geometric volume (

        $V'$
      

):

        $$P_{naive} = \frac{N f_{obs}}{V'} = \frac{N (1+v)\gamma f_0}{(V_0/\gamma)} = P_0 \gamma^2 (1+v)$$

This result is catastrophic. It implies the pressure of a gas depends on the direction from which you look at it. This violates the principle that Pressure is a scalar state variable.

4. The Resolution: Internal vs. Radiative State

The error lies in using the Signal Frequency (

        $f_{obs}$
      

) to calculate the State Variable (

        $P$
      

).

4.1 The Internal Temperature (

        $T_{int}$
      

)

Thermodynamic temperature is defined by the eigenfrequencies of the system in its momentary rest frame. When viewed from a moving frame, these eigenfrequencies are simply slowed by Time Dilation. We denote this Internal Frequency as

        $f_{int}$
      

:

        $$f_{int} = \frac{f_0}{\gamma}$$      

This is the Thermodynamic Temperature. It represents the geometric time-rate of the matter.

4.2 The Radiative Temperature (

        $T_{rad}$
      

)

The photons leaving the box are Doppler shifted (

        $f_{obs}$
      

). This defines the Radiation Pressure (

        $P_{rad}$
      

) that the box would exert on the observer if they collided. It does not define the hydrostatic pressure inside the box.

4.3 Restoring Invariance

To calculate the Thermodynamic Pressure in the longitudinal frame, we must use the Internal Frequency:

        $$P'_{long} = \frac{N f_{int}}{V'} = \frac{N (f_0/\gamma)}{(V_0/\gamma)} = P_0$$      

5. Synthesis: The Rotation of Perspective

We can map the transition from Transverse to Longitudinal as a rotation of perspective, while the physical object remains invariant.

1. Geometric State (The Object):

   • The Box is always contracted:

             $V \to V/\gamma$
           

     .

   • The Atoms are always slowed:

             $f \to f/\gamma$
           

     .

   • Result:

             $P = P_0$
           

     (Always).

2. Radiative Signal (The View):

   • Transverse: Doppler Factor

             $k=1$
           

     . Signal is purely time-dilated (

             $f_{obs} = f/\gamma$
           

     ).

   • Longitudinal: Doppler Factor

             $k \approx 2\gamma$
           

     . Signal is compressed (

             $f_{obs} \approx 2f$
           

     ).

The confusion of the last century arose because physicists applied the Signal Transformation to the Geometric Container.

6. Covariant Interpretation

This geometric resolution is fully consistent with modern relativistic fluid dynamics. The confusion disappears entirely when we express the gas through the Stress-Energy Tensor:

        $$T^{\mu\nu} = (\rho + P) u^\mu u^\nu + P g^{\mu\nu}$$      

Here:

•         $u^\mu$
        

  is the 4-velocity of the gas.

•         $\rho$
        

  is the energy density in the rest frame.

•         $P$
        

  is the isotropic pressure (scalar).

•         $g^{\mu\nu}$
        

  is the metric tensor.

We can mathematically isolate the scalar pressure

        $P$
      

using the projection tensor

        $h_{\mu\nu} = g_{\mu\nu} + u_\mu u_\nu$
      

:

        $$P=\frac{1}{3}{h}_{\mu \nu }{T}^{\mu \nu }$$

The scalar pressure

        $P$
      

is invariant because it is derived from the full tensor contraction. It depends only on the fluid's internal state (

        $u^\mu$
      

), not on the observer's relative motion vector. The observed Doppler-shifted radiation (

        $f_{obs}$
      

) does not enter the

        $T^{\mu\nu}$
      

of the gas; it enters the stress-energy tensor of the photons received by the observer.

Thus, the invariance of pressure emerges automatically from covariance, and our geometric derivation (

        $P = Nf/V$
      

) serves as the intuitive physical mechanism underlying the formal tensor mathematics.

7. Conclusion

Physics is the study of Invariants. By identifying that Volume transforms Geometrically while Radiation transforms via Signal Propagation, we resolve the conflict.

• If you look at the gas (Light): It looks Blue/Hot.

• If you measure the gas (Volume): It looks Small.

• If you ask the gas (Pressure): It is Unchanged.

The "Colder" gas (Time Dilation) inside the "Smaller" box (Length Contraction) creates a stable, invariant physical reality (

        $P_0$
      

), completely decoupled from the "Hotter" appearance created by the Doppler shift of light.