BIBECHANA Vol. 21, No. 2, August 2024, 103-112 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher:Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Geometrical Interpretation of Space Contraction in Two-dimensional Lorentz Transformation Chandra Bahadur Khadka Department of Physics, Tri-Chandra Multiple Campus, Tribhuvan University, Kathmandu-44600, Nepal ∗Corresponding author. Email: chandrabahadur9988@gmail.com Abstract This paper points out that the transformation equations for the spatial and temporal coordi- nates between two frames of reference in the existing generally accepted version of the Lorentz transformation are deficient, since transformation equations are based on one dimensional motion between inertial frames. Therefore, all possible space-time coordinate transformation equations between moving and stationary frames by prolonging Lorentz transformation in a two-dimensional plane are thoroughly proposed in this article. If v denotes the relative velocity between stationary frame (x, y) and moving frame (x ′ , y ′ ), then the transformation equations along X and Y-axis under two-dimensional Lorentz are given, respectively, by the formulas x ′ = ( x− vtx√ x2+y2 ) √ 1− v2 c2 and y ′ = ( y− vty√ x2+y2 ) √ 1− v2 c2 . In this work we conclude that length and breadth of a rectangle appears to be shortened to the observer when there is the relative velocity between the rectangle and observer along both X and Y-axis. In particular, we present a concise and carefully reasoned account of a new aspect of Lorentz transformation which decently allows for the determination of space-time coordinates transformation equations in two dimensions of space. Keywords Frame of reference, Lorentz transformation, Space contraction, Special relativity. Article information Manuscript received: September 5, 2023; Revised: January 10, 2024; Accepted: January 19, 2024 DOI https://doi.org/10.3126/bibechana.v21i2.62271 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction The theory of special relativity [1, 2] at the current time stands as a universal theory comprising the rel- ativistic nature of mass and the common space-time arena in which all fundamental phenomena occur. The well-known Lorentz transformation equations [3] form the basis of Einstein’s special relativity and give the transformation equations for spatial and temporal coordinates in two frames of reference on the basis of constancy of light and its independence with relative velocity between source and observer. 103 http://nepjol.info/index.php/BIBECHANA chandrabahadur9988@gmail.com https://doi.org/10.3126/bibechana.v21i2.62271 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 104 There are many well- known written texts on the formulation of relativistic mechanics. The relativis- tic space-time coordinate formulas were presented in 1904 by Lorentz [3], and later that the Lorentz transformation was simplified and clearly recorded on 5th June 1905 by Poincare [4], who gave it a form very close to its contemporary version: x ′ = x− vt√ 1− v2 c2 , t ′ = t− xv c2√ 1− v2 c2 y ′ = y, z ′ = z (1) where y, z,y ′ and z ′ are coordinates along the direction perpendicular to the direction of motion. On 30 June 1905, Einstein published his work [5] where based on the postulate of constant speed of light, he derived the correct transformation of co- ordinates, law for addition of velocities and mass energy equivalence principle. Lewis and Tolman [6] extended the relativity principle and introduced the theoretical dependence of mass on velocity. Deriva- tions such as that by Poincare for space-time coor- dinates transformation between inertial frames and by Tolman for relativistic mass are propagated in several excellent textbooks, including famous Feyn- man’s lectures [7]. There are some known research results about the extension of Lorentz transfor- mation for determination of space time coordinate transformation along various directions. There are research papers that provide a deeper sense of the concept of relativity with the universal frame of reference [8] and all possible experiments to falsify such theories were conducted in [9]. In the article [10], new mathematical formalism of special rela- tivity was developed. Research is also conducted on the practical aspects of relativity [11]. In Ref. [12], the trajectory of a relativistic particle in 1+1 dimensions was obtained in the representation of Lobachevsky geometry. The dynamics of a rela- tivistic particle that does not have an electric charge and is under the action of an external force has been analyzed on the basis of the special theory of relativity in work [13]. Pagano et al [14] have discussed different roles of Lorentz transformation in classical wave propagation theories and in rela- tivistic mechanics. Karplyuk et al [15] showed how the factorization of an arbitrary Lorentz transfor- mation is performed as a sequence of a spatial rota- tion and a boost. Using the Lorentz transformation, Alex-Amor et al [16] had shown that a particular class of space-time modulated gratings behave ef- fectively as moving media. Ref. [17] presents a concise and carefully reasoned account of a new as- pect of gravitational redshift theory revealing the change of mass with gravity. Furthermore, Ref. [18] points out the possible extension of special relativ- ity to derive the new mathematical formulas of lin- ear momentum, force and kinetic energy. A recently published research article [19] shows the simultane- ous contraction of length, breadth and height of cuboids by developing three-dimensional Lorentz transformation. There are many publications on special relativity with important theoretical results. The Lorentz transformation equations in most of the literature to date are formulated based on one- dimensional motion along X-axis between inertial frames governed by equation (1). Equation (1) gen- erates the space contraction along the X-axis only and can’t explain spatial behavior along Y and Z- directions. Therefore, it manifests the inadequacy of the current form of Lorentz transformation, es- pecially the need for the inclusion of relative mo- tion along both X and Y-directions between iner- tial frames of reference. Here, we present the cor- responding modified Lorentz transformation in two dimensional XY plane by introducing the relative motion between inertial frames along both X and Y-directions simultaneously. The modified trans- formation equations, in place of equation (1) are: x ′ = x− vxt√ x2+y2√ 1− v2 c2 , y ′ = y − vyt√ x2+y2√ 1− v2 c2 t ′ = t− v √ x2+y2 c2√ 1− v2 c2 , z ′ = z (2) Comparing equations (1) and (2), the space coordi- nates transformation take place along only X-axis in old transformation (1) and time depends only on x coordinate while the space coordinates take place along both X-axis and Y-axis in modified transfor- mation (2) and time depends on both x and y co- ordinates. With above motivation, the structure of this pa- per is systematically arranged as follows. In section 2, the transformation equations along both X and Y-axis have been displayed by introducing the rel- ative motion between inertial frames in the two- dimensional XY plane. In section 3, based on mod- ified transformation equations, formulas of length and breadth contraction of a rectangle in a moving frame are obtained for the first time. Some conclu- sions are summed up in the last section. 2 Methods 2.1 The equation of wave front of light Let two frame of reference s and s' such that s' frame of reference is moving with uniform veloc- ity v as shown in figure (1). Let origin O and O' of two co-ordinate system coincide at t = t′ = 0 and a source of light is flashed at the origin O at t = 0, when O and O' coincide. Then, in view of Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 105 constancy of speed of electromagnetic wave with re- spect to the motion of two frames of reference, each observer at O and O' claim to be at the center of the spreading spherical wave front of light pulse. When light is at point P, let space-time co-ordinate mea- sured by the observer O and O' be (x, y, z, t) and (x′, y′, z′, t′) respectively. Since both the observers are at the center of the same expanding wave front, the equation of wave front in frame s and frame s' must be equal. x2 + y2 + z2 − c2t2 = x ′2 + y ′2 + z ′2 − c2t ′2 (3) Since there is no relative motion along Z- direction then, z = z ′ = 0 Above equation (3) reduces to x2 + y2 − c2t2 = x ′2 + y ′2 − c2t ′2 r2 − c2t2 = r ′2 − c2t ′2 (4) 2.2 Geometrical calculations In figure (1), θ be the angle made by line OP with positive X-axis in s frame of reference. Draw PQ' and PQ perpendiculars on X'-axis and X-axis re- spectively. In right angled triangles PQ'O' and PQO, (i) angle OPQ = angle O'PQ' (same angle) (ii) angle PQO = angle PQ'O' = 90° (iii) angle QOP = angle Q’O’P (remaining angle) Therefore, angle QOP = angle Q'O'P' = θ. In right angled triangle O'PQ' PO′2 = O′Q′2 +Q′P 2 or, r ′2 = x ′2 + y ′2 (5) Also, sinθ = PQ ′ PO′ = y ′ r′ cos θ = Q ′ O ′ PO′ = x ′ r′ (6) Similarly, in right angled triangle OPQ PO2 = OQ2 +QP 2 , r2 = x2 + y2 (7) Also, sinθ = PQ OP = y r cos θ = OQ OP = x r (8) 2.3 Transformations Equations In two dimensions, there is relative motion between inertial frames along both X and Y- directions si- multaneously. Let r and r' be position of point P measured from frame of reference s and s' respec- tively. In figure (1), frame s' moves with velocity v in XY plane making an angle θ with positive X- axis of stationary frame. Therefore, velocity v has two components vx = v cos θ and vy = v sin θ along X-axis and Y-axis respectively. Thus, the transformation equation from frame s to s' are y ′ = γ(y − vyt) x ′ = γ(x− vxt) where vy = v sin θ and vx = v cos θ be com- ponent of velocity along Y and X-axis in s frame of reference respectively, then above transformation equation becomes, y ′ = γ(y − vtsinθ) x ′ = γ(x− vtcosθ) where γ denotes the Lorentz factor. Now, we have, y ′ = γ (y − vtsinθ) r (9) x ′ = γ (x− vtcosθ) r (10) Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 106 Figure 1: The inertial system s and s’ in two dimensional planes Now, radius vector r' in s' frame of reference from equation (5) is r ′2 = x ′2 + y ′2 Substituting value from transformation equa- tions (9) and (10), r ′2 = γ2 ( x− vxt r )2 + γ2 ( y − vyt r )2 or, r ′2 = γ2 ( x2 − 2vtx2 r + v2x2t2 r2 + y2 − 2vty2 r + v2y2t2 r2 ) or, r ′2 = γ2 ( x2 + y2 − 2tv ( x2+y2 r ) + t2v2 ( x2+y2 r2 )) Using equation (7) we get, or, r ′2 = γ2 ( r2 − 2vtr2 r + t2v2r2 r2 ) or, r ′2 = γ2 ( r2 − 2tvr + t2v2 ) or, r ′2 = γ2(r − vt)2 r ′ = γ(r − vt) (11) This is the transformation equation from frame s to frame s' in term of radius vector when there is relative motion along X and Y-axis simultaneously. Again, the transformation equations from frame s' to s are y = γ ′ (y′′ + v ′ yt ′ ) x = γ ′ (x′′ + x ′ yt ′ ) where v′y = vsinθ and v′x = vcosθ be compo- nent of velocity along Y and X- axis respectively, then above transformation equations becomes, y = γ ′ (y′′ + vsinθt ′ ) x = γ ′ (x′′ + vcosθt ′ ) Substituting value of sinθ and cosθ from equa- tion (6) we get, y = γ ′ ( y ′ + vy ′ t ′ r′ ) (12) x = γ ′ ( x ′ + vx ′ t ′ r′ ) (13) Now, radius vector in s frame of reference from equation (7) is r2 = x2 + y2 Substituting value from transformation equa- tion (12) and (13), r2 = γ ′2 ( x ′ + vx ′ t ′ r′ )2 + γ ′2 ( y ′ + vy ′ t ′ r′ )2 or, r2 = γ ′2 ( x2 + 2vt ′ x ′2 r′ + v2x ′2t ′2 r′2 + y ′2 + 2vt ′ y ′2 r′ + v2y ′2t ′2 r′2 ) or, r2 = γ ′2 ( x ′2 + y ′2 + 2vt ′ ( x ′2+y ′2 r′ ) + v2t ′2 ( x ′2+y ′2 r′2 )) Using equation (5) we get, or, r2 = γ ′2 ( r ′2 + 2vt ′ r ′2 r′ + v2t ′2r ′2 r′2 ) or, r2 = γ ′2 ( r ′2 + 2vt ′ r ′ + v2t ′2 ) or, r2 = γ ′2 ( r ′ + vt ′ )2 or, r = γ ′ (r ′ + vt ′ ) (14) This is the transformation equation from frame s' to frame s in term of radius vector when there is relative motion along X and Y-axis simultaneously. 2.4 The Lorentz transformation equations The transformation equation relating r' and r can be written from equation (11) and (14) as, r ′ = γ(r − vt) and r = γ ′ ( r ′ + vt ′ ) or, r = γ ′ ( γ(r − vt) + vt ′ ) or, r = γ ′ γr − γ ′ γvt+ γ ′ vt ′ or, t ′ = r γ′v − γ ′ γr γ′v + γ ′ γvt γ′v or, t ′ = γt+ r γ′v − γr v or, t ′ = γ [ t− r v ( 1− 1 γγ′ )] (15) Equation (3) gives equation of wave front, or, r2 − c2t2 = r ′2 − c ′2t ′2 Substituting value of r' and t' from (11) and (15), we get r2−c2t2 = γ2(r−vt)2−c2γ2 [ t− r v ( 1− 1 γγ′ )]2 Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 107 or, r2−c2t2 = γ2r2−2γ2rvt+γ2v2t2−c2γ2t2+ 2c2γ2t rv ( 1− 1 γγ′ ) −c2γ2 r2 v2 ( 1− 1 γγ′ )2 or, r2 − c2t2 = r2 [ γ2 − c2γ2 v2 ( 1− 1 γγ′ )2] + rt [ −2γ2v + 2c2γ2 v ( 1− 1 γγ′ )] + t2 ( γ2v2 − c2γ2 ) Equating the Coefficients of r2 ,rt and t2 on both sides, γ2 − c2γ2 v2 ( 1− 1 γγ′ )2 = 1 (16) −2γ2v + 2c2γ2 v ( 1− 1 γγ′ ) = 0 (17) γ2v2 − γ2c2 = −c2 (18) From equation (18) −γ2 ( c2 − v2 ) = −c2 γ2 = c2 (c2−v2) = 1 1− v2 c2 γ = 1√ 1− v2 c2 (19) Further, γ2 = 1 1− v2 c2 or, 1− v2 c2 = 1 γ2 or, v2 c2 = 1− 1 γ2 (20) From equation (17), we have, −v + c2 v ( 1− 1 γγ′ ) = 0 or, −v2+c2 ( 1− 1 γγ ′ ) v or, v2 = c2 ( 1− 1 γγ′ ) or, v2 c2 = ( 1− 1 γγ′ ) (21) Comparing equations (20) and (21) we get, 1− 1 γ2 = ( 1− 1 γγ′ ) or, 1 γ = 1 γ′ or, γ = γ ′ = 1√ 1− v2 c2 Therefore, the required transformation equation from (11) is r ′ = r−vt√ 1− v2 c2or, √ x′2 + y′2 = √ x2 + y2 − vt√ 1− v2 c2 (22) From (15) we get, t ′ = γ [ t− r v ( 1− 1 γγ′ )] Using equation (21) we get, or, t ′ 1√ 1− v2 c2 [ t− r v . v2 c2 ] or, t ′ = t− rv c2√ 1− v2 c2 or, t ′ = t−v √ x2+y2 c2√ 1− v2 c2 The inverse Lorentz transformation equations are obtained by interchanging the coordinates and changing v by −v in above transformation equa- tions as follows. r = r ′ +vt ′√ 1− v2 c2 or, √ x2 + y2 = √ x′2+y′2+vt ′√ 1− v2 c2 t = t ′ + v √ x′2+y′2 c2√ 1− v2 c2 (23) These equations convert measurement made in frame s' into those in frame s. From these equations it is thus amply clear that space time coordinates transformation equations in two-dimensional space (XY plane) depend upon both x and y coordinates. 3 Results and Discussions The transformation equation from moving frame to stationary frame of reference when there is relative motion along X and Y-axis can be written from equations (12) and (13) as follows. y = γ ′ ( y ′ + vy ′ t ′ r′ ) = y ′ + vy ′ t ′ r′√ 1− v2 c2 (24) x = γ ′ ( x ′ + vx ′ t ′ r′ ) = x ′ + vx ′ t ′ r′√ 1− v2 c2 (25) Above equations can be modified by using equa- tion (6) as follows, y = γ ′ ( y ′′ + vt′sinθ ) = y ′ +vt′sinθ√ 1− v2 c2 x = γ ′ ( x ′′ + vt′cosθ ) = x ′ +vt′cosθ√ 1− v2 c2 Now, radius vector in S frame of reference from equation (7) is r = √ x2 + y2 Using equations (24) and (25) we get, or, r = √√√√(x′+ vx ′ t ′ r ′√ 1− v2 c2 )2 + ( y′+ vy ′ t ′ r ′√ 1− v2 c2 )2 or, r = √√√√√( x′+ vx ′ t ′ r ′ )2 + ( y′+ vy ′ t ′ r ′ )2 (√ 1− v2 c2 )2 or, r = √ x2+ 2vt ′ x ′2 r ′ + v2x ′2t ′2 r ′2 +y′2+ 2vt ′ y ′2 r ′ + v2y ′2t ′2 r ′2√ 1− v2 c2 or, r = √ x′2+y′2+2vt′ ( x ′2+y ′2 r ′ ) +v2t′2 ( x ′2+y ′2 r ′2 ) √ 1− v2 c2 Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 108 Substituting the value r ′2 = x ′2 + y ′2 we get, or, r = √ r′2+2vt′ ( r ′2 r ′ ) +v2t′2 ( r ′2 r ′2 ) √ 1− v2 c2 or, r = √ r′2+2vt′r′+v2t′2√ 1− v2 c2 or, r = √ (r′+vt′) 2√ 1− v2 c2or, r = r ′ + vt ′√ 1− v2 c2 (26) This is the required transformation equation in terms of radius vector. This process of mathemat- ical calculation reveals that transformation equa- tions (24) and (25) in terms of x and y coordinates are separated form of transformation equation (26). Putting value of r andr ′ on equation (26) we get, √ x2 + y2 = √ x′2 + y′2 + vt ′√ 1− V 2 c2 (27) Case(I): when there is relative motion along sin- gle dimension say X-axis, then y = y ′ = 0. As a result, equation (27) becomes, √ x2 = √ x′2+vt ′√ 1−V 2 c2or, x = x ′ + vt ′√ 1− v2 c2 (28) Which is the desired ordinary Lorentz transforma- tion equation of space coordinate along X-axis. Case (II): when there is relative motion along single dimension say Y-axis, then x = x ′ = 0. As a result, equation (27) becomes,√ y2 = √ y′2+vt ′√ 1−V 2 c2or, y = y ′ + vt ′√ 1− v2 c2 (29) Which is the desired ordinary Lorentz transfor- mation equation of space coordinate along Y-axis. Consider two frames of references s and s ′ such that s ′ frame is moving with velocity v in two di- mensional XY plane as shown in figure (2). Con- sider a rectangle ABCD in stationary frame of ref- erence. Let (x1, y1) and (x2, y2) are coordinates of point A and C measured from s frame while( x1 ′ , y1 ′ ) and ( x2 ′ , y2 ′ ) are coordinates of point A and C measured from s ′ as shown in figure (2). Let length and breadth of rectangle are taken along X-axis and Y-axis respectively, then the length of the rectangle along AB in figure (2) is given by, L0 = x2 − x1 (30) This length L0 of the rectangle in stationary frame of reference is known as proper length. The inverse Lorentz transformation equation along X- axis can be written from equation (25) as follows. x1 = x1 ′ + vx1 ′ t ′ r1 ′√ 1− v2 c2 x2 = x2 ′ + vx2 ′ t ′ r2 ′√ 1− v2 c2 Putting these values in equation (30) we get, L0 = x2 ′ + vx2 ′ t ′ r2 ′√ 1− v2 c2 − x1 ′ + vx1 ′ t ′ r1 ′√ 1− v2 c2 or, L0 = x2 ′ −x1 ′ + vx2 ′ t ′ r2 ′ − vx1 ′ t ′ r1 ′√ 1− v2 c2 or, L0 √ 1− v2 c2 = x2 ′ − x1 ′ + vx2 ′ t ′ r2 ′ − vx1 ′ t ′ r1 ′ or, x2 ′ − x1 ′ = L0 √ 1− v2 c2 + vx1 ′ t ′ r1 ′ − vx2 ′ t ′ r2 ′ or, x2 ′ − x1 ′ = L0 √ 1− v2 c2 + vt ′ ( x1 ′ r1 ′ − x2 ′ r2 ′ ) or, L = L0 √ 1− v2 c2 +vt ′  x1 ′√ x′2 1 + y′21 − x2 ′√ x′2 2 + y′22  (31) Where L = x2 ′ − x1 ′ be the length of rect- angle along X-axis in frame s ′ called improper length, r1 ′ = √ x′2 1 + y′21 be radius vector measured froms ′ of point A having coordinate ( x1 ′ , y1 ′ ) and r2 ′ = √ x′2 2 + y′22 be radius vector measured froms ′ of point C having coordinate ( x2 ′ , y2 ′ ) . Similarly, breath of the rectangle along BC in figure (2) is given by, B0 = y2 − y1 (32) Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 109 Figure 2: Length and breadth contraction of the rectangle Figure 3: Relative geometrical shape of rectangle Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 110 This breadth B0 of the rectangle in stationary frame of reference is known as proper breadth. The inverse Lorentz transformation equation along Y- axis can be written from equation (24) as follows. y1 = y1 ′ + vy1 ′ t ′ r1 ′√ 1− v2 c2 y2 = y2 ′ + vy2 ′ t ′ r2 ′√ 1− v2 c2 Putting these values in equation (32) we get, B0 = y2 ′ + vy2 ′ t ′ r2 ′√ 1− v2 c2 − y1 ′ + vy1 ′ t ′ r1 ′√ 1− v2 c2 or, B0 = y2 ′ −y1 ′ + vy2 ′ t ′ r2 ′ − vy1 ′ t ′ r1 ′√ 1− v2 c2 or, B0 √ 1− v2 c2 = y2 ′ − y1 ′ + vy2 ′ t ′ r2 ′ − vy1 ′ t ′ r1 ′ or, y2 ′ − y1 ′ = B0 √ 1− v2 c2 + vy1 ′ t ′ r1 ′ − vy2 ′ t ′ r2 ′ or, y2 ′ − y1 ′ = B0 √ 1− v2 c2 + vt ′ ( y1 ′ r1 ′ − y2 ′ r2 ′ ) or, B = B0 √ 1− v2 c2 +vt ′  y1 ′√ x′2 1 + y′21 − y2 ′√ x′2 2 + y′22  (33) where B = y2 ′ − y1 ′ be the breadth of rectangle along Y-axis in frame s ′ called improper breadth, r1 ′ = √ x′2 1 + y′21 be radius vector measured froms ′ of point A having coordinate ( x1 ′ , y1 ′ ) and Chandra Bahadur Khadka/ BIBECHANA 21 (2024) 103-112 111 r2 ′ = √ x′2 2 + y′22 be radius vector measured froms ′ of point C having coordinate ( x2 ′ , y2 ′ ) . From equations (31) and (33), it is seen that both length and breadth of rectangle appear to be shortened to observer in frames ′ if there is relative motion between rectangle and observer along both X-axis and Y-axis. On the other hand, the length of the rectangle appears to be shortened keeping breadth fixed if there is relative motion along the X-axis only. Similarly, the breadth of the rectan- gle appears to be shortened keeping length fixed if there is relative motion along Y-axis only. A neat depict of space contraction in one and two- dimensional motion is shown in figure (3). Figure (3) shows that a square in one stationary frame appears to the observer in the other frame in one dimensional motion (say along X-axis) to be rectangle due to contraction along X-axis. A square gets shortened in both X and Y-directions if there is relative motion along both X and Y-axis simultaneously as shown in figure (3). The trans- formation equations along the X and Y-axis that give rise to such phenomena of shortening in spa- tial coordinates have been meticulously presented in table (1). From the table, it is clearly seen that the modi- fied Lorentz transformation represents the general- ization of ordinary transformation (one-dimension system). Therefore, the modified transformation is the description of the behavior of the space-time coordinate transformation happening in two dimen- sions of space. 4 Conclusions All possible space-time coordinate transformation equations by introducing relative motion between inertial frames in two dimensional XY-plane have been thoroughly outlined in this article. On the ba- sis of transformation equations, it is definitely pos- sible to obtain space contraction along the X and Y-axis simultaneously. The corresponding length and breadth contraction of rectangle when there is relative motion along both X and Y-axis can be written from equation (31) and (33) as follows. L = L0 √ 1− v2 c2 + vt ′ ( x1 ′ √ x′2 1+y′2 1 − x2 ′ √ x′2 2+y′2 2 ) B = B0 √ 1− v2 c2 + vt ′ ( y1 ′ √ x′2 1+y′2 1 − y2 ′ √ x′2 2+y′2 2 ) From these equations, it should be concluded that both length and breadth of a rectangle ap- pear to be shortened when there is relative motion along both X and Y-directions. Also, the trans- formation equation for temporal coordinate can be written from equation (23) as follows, t ′ = t−v √ x2+y2 c2√ 1− v2 c2 This equation reveals that time depends upon both x and y coordinate when there is relative motion along both X and Y- axis. When moving frames ′ in figure (1) moves along X-axis only, then y = y ′ = 0, hence above equation reduces to, t ′ = t− vx c2√ 1− v2 c2 Which is the required transformation equation of time in ordinary Lorentz transformation (one dimensional system along X-axis). Hence, modi- fied transformation is just an extended form of or- dinary transformation in a two-dimensional plane. An extensive analysis of transformation of temporal and spatial coordinates between inertial frames can be obtained via the natural extension of Lorentz transformation in two dimensional spaces. The mathematical relations illuminated here will have many applications for manifestation of structure of space-time coordinate transformation between iner- tial frames of reference and will play an important role in many other areas of theoretical physics, es- pecially those which are connected with relativistic mechanics. Conflict of interest The author declares no conflict of interest. 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