Geometrical particle models on 3D null curves

Ferrández, Ángel; Giménez, Ángel; Lucas, Pascual

doi:10.1016/s0370-2693(02)02450-4

Cited by 47 publications

(63 citation statements)

References 11 publications

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“…The norm of x ∈ R 4 1 is defined by x = (sign(x) x, x ) 1/2 , where sign(x) denotes the signature of x which is given by sign(x)=1, 0 or -1 when x is spacelike, lightlike or timelike vector. For any two vectors x and y in R 4 1 , we say x is pseudo-perpendicular to y if x, y = 0. For vectors x = (x 1 , x 2 , x 3 , x 4 ), y = (y 1 , y 2 , y 3 , y 4 ) and z = (z 1 , z 2 , z 3 , z 4 ) in R 4 1 , we define a vector x ∧ y ∧ z by…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 99%

“…And Minkowski space form with the positive curvature is called de Sitter space. We know that de Sitter 3-space is a vacuum solution of the Einstein equation and an important cosmological model for physical universe [2,4,7]. The Euclidean rectifying curves have many interesting geometric properties.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, when the Frenet frame along a spacelike or a timelike curve contains a null vectors, such curve is said to be a pseudo null curve. The Frenet equations of a pseudo null curve, lying fully in R 4 1 , are given in [6,11]. However, most of papers and books are studying the geometrical properties of spacelike curves without any null Fernet frames in Minkowski 4-space.…”

Section: Introductionmentioning

confidence: 99%

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Singularity analysis of pseudo null hypersurfaces and pseudo hyperbolic hypersurfaces

Sun¹

2016

J. Nonlinear Sci. Appl.

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Section: Preliminaries and The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Singularity analysis of pseudo null hypersurfaces and pseudo hyperbolic hypersurfaces

Sun¹

2016

J. Nonlinear Sci. Appl.

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“…In addition, we can see other examples in [6] and [7] which show a particle model entirely based on geometry of null curves in physics.…”

Section: Introductionmentioning

confidence: 91%

“…Since α is of type AW (3), (10) holds on α. So substituting (6) and (8) into (10), we have (14). The converse statement is trivial.…”

Section: Curves Of Aw(k)-typementioning

confidence: 99%

On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space

Külahçı

Bektaş

Ergüt

2008

Zeitschrift Für Naturforschung A

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Lightlike surfaces of Darboux‐like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3‐space

Wang

2018

Math Methods in App Sciences

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In this paper, by introducing a new frame on spacelike curves lying in lightcone 3-space, we investigate the geometric properties of the lightlike surface of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix generated by spacelike curves in lightcone 3-space. As an extension of our previous work and an application of the singularity theory, the singularities of the lightlike surfaces of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix are classified, several new invariants of spacelike curves are discovered to be useful for characterizing these singularities, meanwhile, it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3-space and two-dimensional lightcone whose vertices are at the singularities of lightlike surfaces. One concrete example is provided to illustrate our results. KEYWORDS Darboux-like indicatrix, lightcone helix, lightlike surfacesMath Meth Appl Sci. 2020; :5-34.wileyonlinelibrary.com/journal/mmawhere (e 1 , e 2 , e 3 , e 4 ) is the canonical basis of R 4 1 . One can easily show that ⟨a, x ∧ y ∧ z⟩ = det(a, x, y, z). We say that a vector x ∈ R 4 1 ∖{0} is spacelike, lightlike, or timelike if ⟨x, x⟩ is positive, zero, or negative, respectively. The norm of a vector x ∈ R 4 1 is defined by || x|| = √ |⟨x, x⟩|, we call x the unit vector if || x|| = 1. Let ∶ I → R 4 1 be a regular curve in R 4 1 (ie, ′ (t) ≠ 0 for any t ∈ I), where I is an open interval. For any t ∈ I, the curve is called spacelike curve, lightlike curve, or timelike curve if all its velocities are ⟨ ′ (t), ′ (t)⟩ > 0, ⟨ ′ (t), ′ (t)⟩ = 0, or ⟨ ′ (t), ′ (t)⟩ < 0, respectively. We call the nonlightlike curve if is a timelike curve or a spacelike curve. The arc-length of a nonlightlike curve , measured from (t 0 ) (t 0 ∈ I), is s(t) = ∫ t t 0 || ′ (t)||dt. Then, the parameter s is determined such that || ′ (s)|| = 1 for the nonlightlike curve, where ′ (s) = d ds (s). Therefore, we say that a nonlightlike

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Geometrical particle models on 3D null curves

Cited by 47 publications

References 11 publications

Singularity analysis of pseudo null hypersurfaces and pseudo hyperbolic hypersurfaces

Singularity analysis of pseudo null hypersurfaces and pseudo hyperbolic hypersurfaces

On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space

Lightlike surfaces of Darboux‐like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3‐space

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