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Refractive index for the mechanical refraction of a relativistic particle

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Abstract

We have analytically determined the refractive index for the mechanical refraction of a relativistic particle for its all possible speeds. We have critically analysed the importance of Descartes’ metaphysical theory and extended it in this regard. We have considered the conservation of the tangential component of the relativistic momentum and the relativistic energy of the particle in the process of the mechanical refraction within the optical-mechanical analogy. Our result for the mechanical refractive index exactly matches with the forms of both the Fermat’s result on Snell’s law of optical refraction at the ultra-relativistic limit and the Descartes’ metaphysical result on the pseudo-Snell law of optical refraction at the non-relativistic limit.

Graphic abstract

Mechanical refraction from medium-1 to medium-2 for \(U2>U1\)

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Data Availibility Statement

No new data were created or analysed in this study. This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.]

Notes

  1. In the limiting case of the geometrical optics the wavelength (\(\lambda \)) of a monochromatic wave in an optical medium is so small compared to the lowest dimension (d) of an aperture in the medium that it satisfies \(\lambda /d\rightarrow 0\).

  2. Here tangential means, tangential to the interface of the two media.

  3. Here, the time is truly not least. Maupertuis’ least abbreviated action principle (\(\delta \int _{q_1}^{q_2}p{\textrm{d}}q=0\) [24, 25]) can, however, be applied to a photon of (relativistic) energy \(E=\hbar \omega \) in an optical medium of refracting index n. The magnitude of the (relativistic) momentum of the photon is given by \(p=E/v_p\) where \(v_p\) is the phase speed of the photon in the medium. Since the energy is conserved in the process of refraction, we can recast Maupertuis’ least abbreviated action principle for the photon as \(\delta \int _{q_1}^{q_2}\frac{{\textrm{d}}q}{v_p}=0\). The infinitesimal displacement of the photon (\({\textrm{d}}q\)) over the infinitesimal time dt is not same as \(v_pdt\) because the phase speed does not represent the actual speed of a photon (particle) in a medium. The difference of these two, of course, was not known during Fermat’s time. However, we can define \(d\text {``}t\text {''}=\frac{{\textrm{d}}q}{v_p}\) as an infinitesimal abbreviated time. Thus we have \(\delta \int _{q_1}^{q_2}d\text {``}t\text {''}=0\). This is Fermat’s least “time” principle. By multiplying c both the sides, it can be further recast as \(\delta \int _{q_1}^{q_2}d\text {``}q\text {''}=0\) where \({\textrm{d}}\text {``}q\text {''}=\frac{c}{v_p}{\textrm{d}}q=n{\textrm{d}}q\) is an infinitesimal optical path-length. Thus the least “time” principle can also be called as the least optical path-length principle.

  4. With spin 0, we are considering the particle to be a scalar particle.

  5. Here \(E_{nr}=\frac{\hbar ^2[\textbf{k}\cdot {\hat{x}}]^2}{2m_0}\) and \(V(x)=-\frac{k^2}{2m_0}[n^2(x)-1]\).

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Acknowledgements

S. Biswas acknowledges partial financial support of the SERB, DST, Govt. of India under the EMEQ Scheme [No. EEQ/2023/000788]. B. K. Behera acknowledges the financial support through the Non-NET Fellowship Scheme of the University of Hyderabad. We thank the anonymous reviewers for their thorough review. We highly appreciate their comments which significantly contributed to improving the quality of the article.

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  1. Surendra Kumar Gour

    Present address: Department of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

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  1. School of Physics, University of Hyderabad, C.R. Rao Road, Gachibowli, Hyderabad, 500046, India

    Bikram Keshari Behera & Shyamal Biswas

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BKB and SKG together solved the problem and wrote the article. SB framed the problem, solved the problem, and wrote the article.

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Behera, B.K., Gour, S.K. & Biswas, S. Refractive index for the mechanical refraction of a relativistic particle. Eur. Phys. J. D 78, 60 (2024). https://doi.org/10.1140/epjd/s10053-024-00849-z

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