# Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics

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## Abstract

**:**

## 1. Introduction

**v**the velocity of a moving point M in an inertial reference system (IRS) ${K}_{0}$, by

**u**the velocity of the IRS ${K}_{0}$ with respect to the IRS $K$, and by

**w**the velocity of the moving point M as it is seen by an observer in $K$; u is the modulus of velocity

**u**, which determines the strength of the boost. Here, the velocities are scaled at c (i.e., c is taken 1 by choosing conveniently the length or time unit [31]). The vectors (Equation (1)) with the composition law $\oplus $ in Equation (2), have a “group-like” structure in the sense that this composition law ensures the closure condition (Equation (1)), but it is neither commutative nor associative.

## 2. Poincaré Vectors, Poincaré Sphere and P-Spheres

#### 2.1. Poincaré Vectors

- -
- ${\mathbf{s}}_{i},\text{\hspace{0.17em}\hspace{0.17em}}{\mathbf{s}}_{d},{\mathbf{s}}_{o}$ are the Poincaré vectors of the incident light, dichroic device, and output light, respectively,
- -
- ${\mathbf{n}}_{i},\text{\hspace{0.17em}\hspace{0.17em}}{\mathbf{n}}_{d},\text{\hspace{0.17em}\hspace{0.17em}}{\mathbf{n}}_{o}$—the corresponding unit vectors,
- -
- ${p}_{i},\text{\hspace{0.17em}\hspace{0.17em}}{p}_{o}$—the degrees of polarization of the incident and emergent light, ${p}_{d}$ the degree of dichroism [39] of the dichroic device (the strength of the boost), and:$${\gamma}_{d}=1/\sqrt{1-{p}_{d}^{2}}$$

#### 2.2. Poincaré Sphere

#### 2.3. P-Spheres

## 3. Mapping of the P-Spheres by Lorentz Boosts: P-Ellipsoids

**v**,

**u**, and

**w**precised in Equation (2).

**v**with a same, given, modulus v, it will be mapped by a pure boost of vector

**u**to an oblate ellipsoid. For demonstrating this assertion, we shall refer first to a diametrical section of the Poincaré sphere, determined by the Poincaré vector

**u**of the boost and some Poincaré vector

**v**and let us denominate by

**n**and

**m**the unit vectors parallel and perpendicular to

**u**, respectively, and by $\varphi $ the angle between

**u**and

**v**(Figure 1). The corresponding Poincaré vector

**w**(outcoming from the boost

**u**) is given by Equation (7). Its projection on

**u**is:

**u**:

**w**for a given

**u**and a given modulus of

**v**, i.e., the geometrical locus of the top of the resultant Poincaré vectors

**w**corresponding to all of the Poincaré vectors

**v**of modulus v and situated in the plane (

**u**,

**v**), or, equivalently, in the plane (

**n**,

**m**).The cartesian coordinates of this geometrical locus are:

**u**). Making the change of variables:

- -
- the center displaced from the origin of the coordinate system $Oxy$ by:$$\mathsf{\Delta}x=\frac{u(1-{\mathrm{v}}^{2})}{1-{u}^{2}{\mathrm{v}}^{2}}$$
**u**, - -
- minor semiaxis:$${a}_{x}=\frac{\mathrm{v}}{{\gamma}_{u}^{2}(1-{u}^{2}{\mathrm{v}}^{2})}=\mathrm{v}\frac{1-{u}^{2}}{1-{u}^{2}{\mathrm{v}}^{2}}$$
- -
- major semiaxis:$${a}_{y}=\frac{\mathrm{v}}{{\gamma}_{u}^{}{(1-{u}^{2}{\mathrm{v}}^{2})}^{1/2}}=\mathrm{v}{\left(\frac{1-{u}^{2}}{1-{u}^{2}{\mathrm{v}}^{2}}\right)}^{1/2},$$
- -
- eccentricity:$$e={\left(1-\frac{{a}_{x}^{2}}{{a}_{y}^{2}}\right)}^{1/2}={\left[1-\frac{1}{{\gamma}_{u}^{2}(1-{u}^{2}{\mathrm{v}}^{2})}\right]}^{1/2}=u{\left(\frac{1-{\mathrm{v}}^{2}}{1-{u}^{2}{\mathrm{v}}^{2}}\right)}^{1/2}.$$

**u,**or, equivalently, by the Poincaré vector’s composition law, Equation (2), we have to consider all the possible corresponding planes (

**v**,

**u**) intersecting along the direction

**u**, i.e., to rotate in Figure 1 the circular section (

**n**,

**m**) around the

**n**axis. The corresponding Lorentz modified P- surface will be an ellipsoid of revolution around

**u**, i.e., with the axis of symmetry along

**u**. Thus, the sphere ${\mathsf{\Sigma}}_{2}^{\mathrm{v}}$ of all Poincaré vectors of a given, fixed, modulus v, is mapped into an ellipsoid:

**u**.

**v**with a same, given, modulus, v, corresponding to the observer ${K}_{0}$, will be mapped by a pure boost of velocity

**u**to an oblate ellipsoid, i.e., it will be seen by the observer $K$ as an oblate ellipsoid.

## 4. Behavior of the Ellipsoid with the Parameters u and v

**n**. The corresponding approach in PT is to take a P-sphere ${\mathsf{\Sigma}}_{2}^{{p}_{i}}$ of SOP-s of the same degree of polarization ${p}_{i}$ and to visualize how it is deformed, mapped, by orthogonal dichroic devices of various degrees of dichroism ${p}_{d}$ (boost of various strengths ${p}_{d}$).

**w**resulting by the Poincaré vectors’ composition law for all

**v**with the boost strength

**u**are symmetrically gathered together around the direction of the boost

**u**. In SR, this is a holistic expression of the “head-light effect” [41] or “forward collimating effect” [42], emphasized in high energy elementary particle reactions [42]. Such a global view of the forward collimating effect is not known in SR.

**w,**are still oriented towards this rear surface (opposite to

**u**). Increasing u (v = 0.40, u = 0.40, Figure 3b), the last rear point of the ellipsoid touches the center of the sphere; for this point w = 0. For higher boost strength, all of the emergent Poincaré vectors corresponding to the given, initial, P-sphere ${\mathsf{\Sigma}}_{2}^{\mathrm{v}}$ are oriented forward with respect to

**u**, i.e., u is high enough to make this conversion. From Figure 3b, Equations (16) and (17) this happens for:

**w**is pushed farther and farther (Figure 3d). Referring to SR (but having in mind the problem of the specific of the Lorentz transformation in its whole generality discussed here), in the Galilean case, the sphere ${\mathsf{\Sigma}}_{2}^{\mathrm{v}}$ can be pushed at infinity in the velocity space without any deformation. Here, in the relativistic case, it can be pushed only up to the relativistic velocity enclosure, which is up to the wall of the Poincaré sphere. Therefore, its behavior when u increases is quite another one: the sphere is deformed to an ellipsoid and this velocity ellipsoid becomes smaller and smaller and flatter and flatter.

## 5. Nonlinearity and Indefinitness of the Ellipsoid Characteristics as Functions of u and v

- -
- $\mathsf{\Delta}x$ increases very slowly up to the critical value of u, and after this value $\mathsf{\Delta}x$ starts, suddenly, to grow very abruptly with u (Figure 8a); and,
- -
- similarly, ${a}_{x}$ decreases from the value v very slowly with u up to the critical value of u, and after this value ${a}_{x}$ becomes suddenly to decrease abruptly to zero (Figure 8b).

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Barakat, R. Theory of the coherency matrix for light of arbitrary spectral bandwidth. J. Opt. Soc. Am.
**1963**, 53, 317–323. [Google Scholar] [CrossRef] - Gil, J. Polarimetric characterization of light and media. Eur. Phys. J. Appl. Phys.
**2007**, 40, 1–47. [Google Scholar] [CrossRef] - Gil, J.J.; Ossikovski, R. Polarized Light and the Mueller Matrix Approach; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Savenkov, S.V.; Sydoruk, O.; Muttiah, R.S. Conditions for polarization elements to be dichroic and birefringent. J. Opt. Soc. Am. A
**2005**, 22, 1447–1452. [Google Scholar] [CrossRef] - Angelsky, O.V.; Hanson, S.G.; Zenkova, C.Y.; Gorsky, M.P.; Gorodyns’ka, N.V. On polarization metrology of the degree of coherence of optical waves. Opt. Express.
**2009**, 17, 15623–15634. [Google Scholar] [CrossRef] [PubMed] - Angelsky, O.V.; Polyanskii, P.V.; Maksimyak, P.P.; Mokhun, I.I.; Zenkova, C.Y.; Bogatyryova, H.V.; Felde, C.V.; Bachinskiy, V.T.; Boichuk, T.M.; Ushenko, A.G. Optical measurements: Polarization and coherence of light fields. In Modern Metrology Concerns—Monography; Coccco, L., Ed.; In Tech: Rijeka, Croatia, 2012. [Google Scholar]
- Barakat, R. Bilinear constraints between elements of the 4 × 4 Mueller-Jones transfer matrix of polarization theory. Opt. Commun.
**1981**, 38, 159–161. [Google Scholar] [CrossRef] - Takenaka, H. A unified formalism for polarization optics by using group theory. Nouv. Rev. Opt.
**1973**, 4, 37–41. [Google Scholar] [CrossRef] - Kitano, M.; Yabuzaki, T. Observation of Lorentz-group Berry phases in polarization optics. Phys. Lett. A
**1989**, 142, 321–325. [Google Scholar] [CrossRef] - Pellat-Finet, P. What is common to both polarization optics and relativistic kinematics? Optik
**1992**, 90, 101–106. [Google Scholar] - Opatrnỳ, T.; Peřina, J. Non-image-forming polarization optical devices and Lorentz transformations—An analogy. Phys. Lett. A
**1993**, 181, 199–202. [Google Scholar] [CrossRef] - Han, D.; Kim, Y.S. Polarization optics and bilinear representation of the Lorentz group. Phys. Lett. A
**1996**, 219, 26–32. [Google Scholar] [CrossRef] - Han, D.; Kim, Y.S.; Noz, M.E. Stokes parameters as a Minkowskian four-vector. Phys. Rev. E
**1997**, 56, 6065–6076. [Google Scholar] [CrossRef] - Kim, Y.S. Lorentz group in polarization optics. J. Opt. B
**2000**, 2, R1–R5. [Google Scholar] [CrossRef] - Morales, J.A.; Navarro, E. Minkowskian description of polarized light and polarizers. Phys. Rev. E
**2003**, 67, 026605. [Google Scholar] [CrossRef] [PubMed] - Tudor, T. Interaction of light with the polarization devices: A vectorial Pauli algebraic approach. J. Phys. A Math. Theor.
**2008**, 41, 1–12. [Google Scholar] [CrossRef] - Lages, J.; Giust, R.; Vigoureux, J.M. Composition law for polarizers. Phys. Rev. A
**2008**, 78, 033810. [Google Scholar] [CrossRef] - Vigoureux, J.M. Use of Einstein’s addition law in studies of reflection by stratified planar structures. J. Opt. Soc. Am. A
**1992**, 9, 1313–1319. [Google Scholar] [CrossRef] - Vigoureux, J.M.; Grossels, Ph. A relativistic-like presentation of optics in stratified planar media. Am. J. Phys.
**1993**, 61, 707–712. [Google Scholar] [CrossRef] - Monzón, J.J.; Sánchez-Soto, L.L. Fully relativisticlike formulation of multilayers optics. J. Opt. Soc. Am. A
**1999**, 16, 2013–2018. [Google Scholar] [CrossRef] - Monzón, J.J.; Sánchez-Soto, L.L. Fresnel formulas as Lorentz transformations. J. Opt. Soc. Am.
**2000**, 17, 1475–1481. [Google Scholar] [CrossRef] - Monzón, J.J.; Sánchez-Soto, L.L. Optical multilayers as a tool for visualizing special relativity. Eur. J. Phys.
**2001**, 22, 39–51. [Google Scholar] [CrossRef] - Giust, R.; Vigoureux, J.M. Hyperbolic representation of light propagation in a multilayer medium. J. Opt. Soc. Am. A
**2002**, 19, 378–384. [Google Scholar] [CrossRef] - Giust, R.; Vigoureux, J.M.; Lages, J. Generalized composition law from 2 × 2 matrices. Am. J. Phys.
**2009**, 77, 1068–1073. [Google Scholar] [CrossRef] - Başkal, S.; Georgieva, E.; Kim, Y.S.; Noz, M.E. Lorentz group in classical ray optics. J. Opt. B Quantum Semiclass. Opt.
**2004**, 6, 4554. [Google Scholar] - Başkal, S.; Kim, Y.S. Problems of measurement in Quantum Optics and Informatics, the Language of Einstein Spoken by Optical Instruments. Opt. Spectrosc.
**2005**, 99, 443–446. [Google Scholar] [CrossRef] - Başkal, S.; Kim, Y.S. Wigner rotations in laser cavites. Phys. Rev. E
**2002**, 66, 026604. [Google Scholar] [CrossRef] [PubMed] - Han, D.; Hardekopl, E.E.; Kim, Y.S. Thomas precession and squeezed states of light. Phys. Rev. A
**1989**, 39, 1269. [Google Scholar] [CrossRef] - Ungar, A.A. Analytic Hyperbolic Geometry. Mathematical Foundation and Applications; World Scientific: Hackensack, NJ, USA, 2005. [Google Scholar]
- Ungar, A.A. Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett.
**1988**, 1, 57–68. [Google Scholar] [CrossRef] - Feynman, R.P.; Leighton, R.; Sands, M. The Feynman Lectures on Physics; Addison-Wesley: London, UK, 1977; Volume I, Chapter 15. [Google Scholar]
- Tudor, T. On a quasi-relativistic formula in polarization theory. Opt. Lett.
**2015**, 40, 1–4. [Google Scholar] [CrossRef] [PubMed] - Williams, M.W. Depolarization and cross polarization in ellipsometry of rough surfaces. Appl. Opt.
**1986**, 25, 3616–3622. [Google Scholar] [CrossRef] [PubMed] - DeBoo, B.; Sasian, J.; Chipman, R. Degree of polarization surfaces and maps for analysis of depolarization. Opt. Express.
**2004**, 12, 4941–4958. [Google Scholar] [CrossRef] [PubMed] - Ferreira, C.; José, I.S.; Gil, J.J.; Correas, J.M. Geometric modeling of polarimetric transformations. Mono. Sem. Mat. G. de Galdeano
**2006**, 33, 115–119. [Google Scholar] - Tudor, T.; Manea, V. The ellipsoid of the polarization degree. A vectorial, pure operatorial Pauli algebraic approach. J. Opt. Soc. Am. B
**2011**, 28, 596–601. [Google Scholar] [CrossRef] - Ossikovski, R.; Gil, J.J.; San José, I. Poincaré sphere mapping by Mueller matrices. J. Opt. Soc. Am. A
**2013**, 30, 2291–2305. [Google Scholar] [CrossRef] [PubMed] - Gil, J.J.; Ossikovski, R.; San José, I. Singular Mueller matrices. J. Opt. Soc. Am. A
**2016**, 33, 600–609. [Google Scholar] [CrossRef] [PubMed] - Tudor, T.; Manea, V. Symmetry between partially polarized light and partial polarizers in the vectorial Pauli algebraic formalism. J. Mod. Opt.
**2011**, 58, 845–852. [Google Scholar] [CrossRef] - Poincaré, H. Theorie Mathématique de la Lumière; Carrè: Paris, France, 1892. [Google Scholar]
- Rindler, W. Relativity. Special, General and Cosmological; Oxford University Press: New York, NY, USA, 2006. [Google Scholar]
- Sard, R.D. Relativistic Mechanics; W.A. Benjamin Inc.: New York, NY, USA, 1970. [Google Scholar]

**Figure 3.**When the strength of the boost increases the ellipsoid corresponding to a given P-sphere (v = 0.40) is pushed farther and farther and becomes smaller and smaller: (a) u = 0.20; (b) u = 0.40; (c) u = 0.68; (d) u = 0.80.

**Figure 4.**Increasing the radius v of the P-sphere at a given u (u = 0.80), the corresponding ellipsoid becomes greater and greater and comes back to the origin of the Poincaré space: (

**a**) v = 0.45; (

**b**) v = 0.50; (

**c**) v = 0.80; (

**d**) v = 0.90.

**Figure 5.**Behavior of the ellipsoid when u increases at a higher level of v (v = 0.900): (

**a**) u = 0.850; (

**b**) u = 0.900; (

**c**) u = 0.994; (

**d**) u = 0.997.

**Figure 6.**Behavior of the ellipsoid when v increases at a higher level of u (u = 0.997): (

**a**) v = 0.925; (

**b**) v = 0.997; (

**c**) v = 0.999; (

**d**) v = 0.9998.

**Figure 7.**Δx and ${a}_{x}$ as function of u, with v as parameter, at low and moderate values of v. (

**a**) upper line v = 0.2, lower curve v = 0.8; (

**b**) lower curve v = 0.2, upper curve v = 0.8.

**Figure 8.**Δx and ${a}_{x}$ as function of u at high values of v. (

**a**) upper curve v = 0.95, lower curve v = 0.99; (

**b**) lower curve v = 0.95, upper curve v = 0.99.

**Figure 9.**Δx and ${a}_{x}$ as function of v with u as parameter; (

**a**) lower curve u = 0.95, upper curve u = 0.99; (

**b**) upper curve u = 0.95, lower curve u = 0.99.

**Figure 10.**Back and forth play of Δx when u and v increase alternatively toward their extreme limit.

**Figure 11.**The function Δx (u, v) in the region of physical interest: $u,\mathrm{v}\in [0,1].$ (

**a**,

**b**) mean two different perspectives.

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Tudor, T.
Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics. *Symmetry* **2018**, *10*, 52.
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Tudor T.
Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics. *Symmetry*. 2018; 10(3):52.
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**Chicago/Turabian Style**

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2018. "Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics" *Symmetry* 10, no. 3: 52.
https://doi.org/10.3390/sym10030052