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\title{The McGucken Invariance}
\author{}
\date{August 25, 2025}

\begin{document}

\maketitle

\begin{abstract}
In this paper, we introduce the McGucken Invariance, a novel relativistic invariant derived within the framework of special relativity. This invariant combines the time of arrival ((\tau)), observed frequency ((f’)), and observed brightness ((I’)) of light signals from stationary sources, as perceived by an observer in a moving inertial frame. Drawing from Einstein’s classic thought experiment on the relativity of simultaneity,\footnote{\label{fn:einstein1916} Einstein, A. \textit{Relativity: The Special and General Theory} (1916).} we extend the analysis to incorporate the relativistic Doppler effect and the transformation of radiation intensity. The invariant takes the form (\frac{\tau (f’)^2}{(I’)^{1/4}} = L \gamma \frac{f^2}{I^{1/4}}), where (L) is the half-distance between the signal sources, (\gamma) is the Lorentz factor, and (f) and (I) are the emitted frequency and brightness, respectively. We demonstrate its constancy for signals from two sources (A and B) at various relative velocities ((v = 0.2c), (0.5c), and (0.9c)). While the components of this invariant are rooted in established relativistic principles,\footnote{\label{fn:jackson} Jackson, J. D. \textit{Classical Electrodynamics} (3rd ed., 1998).}\footnote{\label{fn:rybicki} Rybicki, G. B., \& Lightman, A. P. \textit{Radiative Processes in Astrophysics} (1979).} the specific combination represents a new synthesis, potentially applicable to relativistic optics and astrophysical observations.
\end{abstract}

\section{Introduction}

Special relativity fundamentally alters our understanding of space, time, and the propagation of light. A pivotal illustration is Einstein’s thought experiment involving lightning strikes on a railway embankment observed from a moving train, which demonstrates the relativity of simultaneity.\footnote{\label{fn:einstein1916} Einstein, A. \textit{Relativity: The Special and General Theory} (1916).}\footnote{\label{fn:vicphysics} Vicphysics. “Einstein’s Train and other ‘Gedanken’ experiments.” \url{https://www.vicphysics.org/documents/teachers/unit3/EinsteinsTrainGedanken.pdf} (accessed August 25, 2025).} In this setup, simultaneous events in the stationary frame appear asynchronous in the moving frame due to the finite and invariant speed of light.

Beyond simultaneity, relative motion induces the relativistic Doppler effect, shifting the frequency and wavelength of light,\footnote{\label{fn:wikipedia_doppler} Wikipedia contributors. Relativistic Doppler effect.'' \url{https://en.wikipedia.org/wiki/Relativistic_Doppler_effect} (accessed August 25, 2025).}\footnote{\label{fn:openstax} OpenStax.5.8: Doppler Effect for Light.” \url{https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/05%3A__Relativity/5.08%3A_Doppler_Effect_for_Light} (accessed August 25, 2025).} and alters the observed intensity (or brightness) through Lorentz transformations of the electromagnetic field.\footnote{\label{fn:jackson} Jackson, J. D. \textit{Classical Electrodynamics} (3rd ed., 1998).}\footnote{\label{fn:mckinley} McKinley, J. M. Relativistic transformations of light power.'' \url{https://muj.optol.cz/~richterek/data/media/ref_str/mckinley1979a.pdf} (accessed August 25, 2025).} The intensity transformation arises from the Lorentz invariance of the photon phase-space density, leading to \(I' \propto (f'/f)^4\) for broadband radiation.\footnote{\label{fn:synge} Synge, J. L.Lorentz-invariance of the distribution in phase space.” \url{https://www.sciencedirect.com/science/article/pii/0031891469900056} (accessed August 25, 2025).}\footnote{\label{fn:pasricha} Pasricha, A. “Lorentz-invariant photon number density.” Physical Review A, 78(1), 012111 (2008).}

Here, we derive the McGucken Invariance, which unifies the time of arrival, frequency shift, and intensity transformation into a single invariant quantity. This invariant holds for pairs of signals from symmetric sources in the stationary frame, as observed from the midpoint of the moving frame, and scales with the Lorentz factor (\gamma = 1 / \sqrt{1 – \beta^2}), where (\beta = v/c).

\section{Theoretical Framework}

Consider points A and B on the embankment, separated by (2L), with lightning strikes occurring simultaneously at (t = 0) when the train’s midpoint M’ coincides with the embankment’s midpoint M. The train moves at velocity (v = \beta c) to the right. In the embankment frame (where (c = 1) for simplicity), A is at (x = -L) and B at (x = L).

The time of arrival (\tau) at M’ in the embankment frame is:

• For light from B (approaching): (\tau_B = L / (1 + \beta)),
• For light from A (receding): (\tau_A = L / (1 – \beta)).

The relativistic Doppler shift yields:

• (f’_B = f \sqrt{\frac{1 + \beta}{1 – \beta}}),
• (f’_A = f \sqrt{\frac{1 – \beta}{1 + \beta}}),

where (f) is the emitted frequency.\footnote{\label{fn:wikipedia_doppler} Wikipedia contributors. Relativistic Doppler effect.'' \url{https://en.wikipedia.org/wiki/Relativistic_Doppler_effect} (accessed August 25, 2025).}\footnote{\label{fn:openstax} OpenStax.5.8: Doppler Effect for Light.” \url{https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/05%3A__Relativity/5.08%3A_Doppler_Effect_for_Light} (accessed August 25, 2025).}

The observed brightness transforms as:

• (I’_B = I \left( \sqrt{\frac{1 + \beta}{1 – \beta}} \right)^4),
• (I’_A = I \left( \sqrt{\frac{1 – \beta}{1 + \beta}} \right)^4),

due to the invariance of (I_\nu / \nu^3).\footnote{\label{fn:jackson} Jackson, J. D. \textit{Classical Electrodynamics} (3rd ed., 1998).}\footnote{\label{fn:rybicki} Rybicki, G. B., \& Lightman, A. P. \textit{Radiative Processes in Astrophysics} (1979).}

Substituting these into (\frac{\tau (f’)^2}{(I’)^{1/4}}) yields (L \gamma \frac{f^2}{I^{1/4}}) for both signals, establishing the invariance.

\section{Calculations}

We compute the invariant for (L = 1) (in units where (c = 1)) at (\beta = 0.2, 0.5, 0.9).

\subsection{For (\beta = 0.2)}
(\gamma \approx 1.0206),
(\tau_B \approx 0.8333), (\tau_A = 1.25),
(f’_B \approx 1.2247 f), (f’_A \approx 0.8165 f),
(I’_B \approx 2.25 I), (I’_A \approx 0.4444 I).

Invariant for B: (\frac{0.8333 \cdot (1.2247 f)^2}{(2.25 I)^{1/4}} \approx 1.0206 \frac{f^2}{I^{1/4}}),
for A: (\frac{1.25 \cdot (0.8165 f)^2}{(0.4444 I)^{1/4}} \approx 1.0206 \frac{f^2}{I^{1/4}}).

\subsection{For (\beta = 0.5)}
(\gamma \approx 1.1547),
(\tau_B \approx 0.6667), (\tau_A = 2),
(f’_B \approx 1.7321 f), (f’_A \approx 0.5774 f),
(I’_B = 9 I), (I’_A \approx 0.1111 I).

Invariant for B: (\frac{0.6667 \cdot (1.7321 f)^2}{(9 I)^{1/4}} \approx 1.1547 \frac{f^2}{I^{1/4}}),
for A: (\frac{2 \cdot (0.5774 f)^2}{(0.1111 I)^{1/4}} \approx 1.1547 \frac{f^2}{I^{1/4}}).

\subsection{For (\beta = 0.9)}
(\gamma \approx 2.2942),
(\tau_B \approx 0.5263), (\tau_A = 10),
(f’_B \approx 4.3589 f), (f’_A \approx 0.2294 f),
(I’_B = 361 I), (I’_A \approx 0.00277 I).

Invariant for B: (\frac{0.5263 \cdot (4.3589 f)^2}{(361 I)^{1/4}} \approx 2.2942 \frac{f^2}{I^{1/4}}),
for A: (\frac{10 \cdot (0.2294 f)^2}{(0.00277 I)^{1/4}} \approx 2.2942 \frac{f^2}{I^{1/4}}).

In each case, the value is identical for signals from A and B.

\section{Discussion}

The McGucken Invariance encapsulates relativistic distortions in a frame-invariant manner for symmetric events, scaling with (\gamma) across velocities. This reflects the increasing asymmetry in perception at higher (\beta). Potential applications include analyzing Doppler-boosted emissions in relativistic jets or high-energy particle experiments.\footnote{\label{fn:thorne} Thorne, K. S. Relativistic radiative transfer - Moment formalisms.'' Monthly Notices of the Royal Astronomical Society, 194, 439 (1981).} While novel, it aligns with foundational relativistic radiative transfer.\footnote{\label{fn:blackman} Blackman, E. G.Basics of Radiation Transfer.” \url{https://www.pas.rochester.edu/~blackman/ast461/lectures.pdf} (accessed August 25, 2025).}\footnote{\label{fn:physics_se} Physics Stack Exchange. “Relativistic Doppler Effect: Intensity change.” \url{https://physics.stackexchange.com/questions/704674/relativistic-doppler-effect-intensity-change} (accessed August 25, 2025).}

\section{Conclusion}

We have introduced and verified the McGucken Invariance, extending Einstein’s framework to unify timing, spectral, and intensity effects. Future investigations may explore its generalization to curved spacetimes or quantum contexts.

\begin{thebibliography}{9}

\bibitem{einstein1916}
Einstein, A. \textit{Relativity: The Special and General Theory} (1916).

\bibitem{vicphysics}
Vicphysics. “Einstein’s Train and other ‘Gedanken’ experiments.” \url{https://www.vicphysics.org/documents/teachers/unit3/EinsteinsTrainGedanken.pdf} (accessed August 25, 2025).

\bibitem{wikipedia_doppler}
Wikipedia contributors. “Relativistic Doppler effect.” \url{https://en.wikipedia.org/wiki/Relativistic_Doppler_effect} (accessed August 25, 2025).

\bibitem{openstax}
OpenStax. “5.8: Doppler Effect for Light.” \url{https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/05%3A__Relativity/5.08%3A_Doppler_Effect_for_Light} (accessed August 25, 2025).

\bibitem{jackson}
Jackson, J. D. \textit{Classical Electrodynamics} (3rd ed., 1998).

\bibitem{mckinley}
McKinley, J. M. “Relativistic transformations of light power.” \url{https://muj.optol.cz/~richterek/data/media/ref_str/mckinley1979a.pdf} (accessed August 25, 2025).

\bibitem{synge}
Synge, J. L. “Lorentz-invariance of the distribution in phase space.” \url{https://www.sciencedirect.com/science/article/pii/0031891469900056} (accessed August 25, 2025).

\bibitem{pasricha}
Pasricha, A. “Lorentz-invariant photon number density.” Physical Review A, 78(1), 012111 (2008).

\bibitem{rybicki}
Rybicki, G. B., \& Lightman, A. P. \textit{Radiative Processes in Astrophysics} (1979).

\bibitem{thorne}
Thorne, K. S. “Relativistic radiative transfer – Moment formalisms.” Monthly Notices of the Royal Astronomical Society, 194, 439 (1981).

\bibitem{blackman}
Blackman, E. G. “Basics of Radiation Transfer.” \url{https://www.pas.rochester.edu/~blackman/ast461/lectures.pdf} (accessed August 25, 2025).

\bibitem{physics_se}
Physics Stack Exchange. “Relativistic Doppler Effect: Intensity change.” \url{https://physics.stackexchange.com/questions/704674/relativistic-doppler-effect-intensity-change} (accessed August 25, 2025).

\end{thebibliography}

\end{document}