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Free particle geodesic affine parameter time-space coordinate systems

Bian Bao-Min Lai Xiao-Ming Yang Lin Li Zhen-Hua He An-Zhi

Free particle geodesic affine parameter time-space coordinate systems

Bian Bao-Min, Lai Xiao-Ming, Yang Lin, Li Zhen-Hua, He An-Zhi
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  • Taking the time-series t as independent variable, the parameter equations {Xi(t)} of free particle space geodesic can be given. By transforming affine parameter R(t) we achieve homogeneous geodesic differential equations, and derive the first-order differential equations which are satisfied by affine parameter R and the sequence of analytical solutions R marked by rational number Cu. In light of R we define the distance unit of flat four-dimensional coordinate system {t,r,θ,φ}, and then establish a free particle geodesic affine parameter time-space coordinate system {t,ξ,θ,φ}. By the study of the diagonalization process of special relativity time-space interval model metric tensor g in {t,ξ,θ,φ}, we find the spatial and temporal line characteristic quantities t1(t,ξ), τ1(τ,ξ),tt(t,τ,ξ) and ττ1(t,τ,ξ) corresponding to diagonal metric. Derived from these quantities, the dimension of time-space coordinate system is less than 4.
    [1]

    Weinberg S 1980 Gravitation and Cosmology (1st Ed.) (Beijing: Science Press) p168, 468 (in Chinese) [温伯格 S 1980 引力论和宇宙论 (第一版) (北京:科学出版社) 第168, 468页]

    [2]

    Liang C B, Zhou B 2006 Introduction to Differential Geometry and General Relativity (Beijing: Science Press) p69, 190, 192 (in Chinese) [梁灿彬, 周彬 2006 微分几何入门与广义相对论(北京:科学出版社) 第69, 190, 192页]

    [3]

    Pope A C, Matsubara T, Szalay A S, Blanton M R, Eisenstein D J, Gray J, Lain B, Bahcall N A, Brinkmann J, Budavari T, Connolly A J, Frieman J A, Gunn J E, Johnston D, KentS M, Lupton R H, Meiksin A, Nichol R C, Scranton R, Strauss M A, Szapudi I, Tegmark M, Vogeley M S, Weinberg D H, Zehavi I 2004 Astrophys J. 607 655

    [4]

    Bennett C L, Hill R S, Hinshaw G, Nolta M R, Odegard N, Page L, Spergel D N, Weiland J L, Wright E L, Halpem M, Larosik N, Kogut A, Limon M, Meyer S S, Tucker G S, Wollack E 2003 The Astrophysical Journal 148 97

    [5]

    Mather J C, Cheng E S, Cottingham D A, Eplee R E, Fixsen D J, Hewagama T 1994 The Amer. Astro. Soc. 420 439

    [6]

    Benítez N, Riess A, Nugent P, Dickingson M, Chornock E, Filippenko V 2002 The Astrophysical Journal 577 L1

    [7]

    Riess A G, Filippenko A V, Challis P, Clocchiattia A, Diereks A, Gamavich P M, Gilliland R L, Hogan C J, Jha S, Kirshner R P, Lebundgut B, Phillips M M, Reiss D, Schmidt B P, Schommer R A, Smith R C, Spyromilio J, Stubbs C, Suntzeff N B, Tonry J 1998 Astron. J. 116 1009

    [8]

    Schmidt B P, Suntzeff N B, Phillips M M, Schommer R A, Clocchiatti A, Kirshner R P, Garnavich P, Challis P, Leibundgut B, Spyromilio J, Riess A G, Filippenko A V, Hamuy M, Smith R C, Hogan C, Stubbs C, Diercks A, Reiss D, Gilliland R, Tonry J, Maza J, Dressler A, Walsh J, Ciardullo R 1998 The Astrophysical Journal 507 46

    [9]

    Cai R G 2007 Phys. Lett. B 657 228

    [10]

    Zhai X H, Zhao Y B 2006 Chin. Phys. 15 2465

    [11]

    Feng B, Wang X L, ZhangX M 2005 Phys. Lett. B 607 35

    [12]

    Li M 2004 Phys. Lett. B 603 1

    [13]

    Carroll S M 2001 Living Rev. Rel 4 1

    [14]

    Kamenshchik A Y, Moschella U, Pasquier V 2001 Phys. Lett. B 511 265

    [15]

    Nottale L 2010 Found. Sci. 15 101

    [16]

    Sorrell W H 2009 Astrophys. Space. Sci. 323 205

    [17]

    Avinash K, Rvachev V L 2000 Foundations of Physics 30 139

    [18]

    Hawking S W, Ellis G F R 2006 The Large Scale Structure of Space-Time (Changsha: Hunan Science and Technology Press) p30 (in Chinese) [霍金 S W, 埃利斯 G F R 2006 时空的大尺度结构(长沙:湖南科学技术出版社) 第30页]

    [19]

    Lai X M, Bian B M, Yang L, Yang J, Bian N, Li Z H, He A Z 2008 Acta. Phys. Sin. 57 7955 (in Chinese) [赖小明, 卞保民, 杨玲, 杨娟, 卞牛, 李振华, 贺安之 2008 物理学报 57 7955]

    [20]

    Bian B M, Lai X M, Yang L, Li Z H, He A Z 2012 Acta. Phys. Sin. 61 080401 (in Chinese) [卞保民, 赖小明, 杨玲, 李振华, 贺安之 2012 物理学报 61 080401]

    [21]

    Yu Y Q 1997 Introduction of general relativity (Beijing: Peking University Press) p18, 133, 158 (in Chinese) [愈允强 1997 广义相对论引论 (北京:北京大学出版社) 第18, 133, 158页]

    [22]

    Zhu H, Ji C C 2011 Fractal theory and its applications (Beijing: Science Press) p23 (in Chinese)[朱华, 姬翠翠 2011 分形理论及应用(北京:科学出版社) 第23页]

    [23]

    Yu Y Q 2002 Cosmophysics lectures (Beijing: Peking University Press) p104, 105, 214 (in Chinese)[俞允强 2002 物理宇宙学讲义(北京:北京大学出版社) 第104, 105, 214页]

  • [1]

    Weinberg S 1980 Gravitation and Cosmology (1st Ed.) (Beijing: Science Press) p168, 468 (in Chinese) [温伯格 S 1980 引力论和宇宙论 (第一版) (北京:科学出版社) 第168, 468页]

    [2]

    Liang C B, Zhou B 2006 Introduction to Differential Geometry and General Relativity (Beijing: Science Press) p69, 190, 192 (in Chinese) [梁灿彬, 周彬 2006 微分几何入门与广义相对论(北京:科学出版社) 第69, 190, 192页]

    [3]

    Pope A C, Matsubara T, Szalay A S, Blanton M R, Eisenstein D J, Gray J, Lain B, Bahcall N A, Brinkmann J, Budavari T, Connolly A J, Frieman J A, Gunn J E, Johnston D, KentS M, Lupton R H, Meiksin A, Nichol R C, Scranton R, Strauss M A, Szapudi I, Tegmark M, Vogeley M S, Weinberg D H, Zehavi I 2004 Astrophys J. 607 655

    [4]

    Bennett C L, Hill R S, Hinshaw G, Nolta M R, Odegard N, Page L, Spergel D N, Weiland J L, Wright E L, Halpem M, Larosik N, Kogut A, Limon M, Meyer S S, Tucker G S, Wollack E 2003 The Astrophysical Journal 148 97

    [5]

    Mather J C, Cheng E S, Cottingham D A, Eplee R E, Fixsen D J, Hewagama T 1994 The Amer. Astro. Soc. 420 439

    [6]

    Benítez N, Riess A, Nugent P, Dickingson M, Chornock E, Filippenko V 2002 The Astrophysical Journal 577 L1

    [7]

    Riess A G, Filippenko A V, Challis P, Clocchiattia A, Diereks A, Gamavich P M, Gilliland R L, Hogan C J, Jha S, Kirshner R P, Lebundgut B, Phillips M M, Reiss D, Schmidt B P, Schommer R A, Smith R C, Spyromilio J, Stubbs C, Suntzeff N B, Tonry J 1998 Astron. J. 116 1009

    [8]

    Schmidt B P, Suntzeff N B, Phillips M M, Schommer R A, Clocchiatti A, Kirshner R P, Garnavich P, Challis P, Leibundgut B, Spyromilio J, Riess A G, Filippenko A V, Hamuy M, Smith R C, Hogan C, Stubbs C, Diercks A, Reiss D, Gilliland R, Tonry J, Maza J, Dressler A, Walsh J, Ciardullo R 1998 The Astrophysical Journal 507 46

    [9]

    Cai R G 2007 Phys. Lett. B 657 228

    [10]

    Zhai X H, Zhao Y B 2006 Chin. Phys. 15 2465

    [11]

    Feng B, Wang X L, ZhangX M 2005 Phys. Lett. B 607 35

    [12]

    Li M 2004 Phys. Lett. B 603 1

    [13]

    Carroll S M 2001 Living Rev. Rel 4 1

    [14]

    Kamenshchik A Y, Moschella U, Pasquier V 2001 Phys. Lett. B 511 265

    [15]

    Nottale L 2010 Found. Sci. 15 101

    [16]

    Sorrell W H 2009 Astrophys. Space. Sci. 323 205

    [17]

    Avinash K, Rvachev V L 2000 Foundations of Physics 30 139

    [18]

    Hawking S W, Ellis G F R 2006 The Large Scale Structure of Space-Time (Changsha: Hunan Science and Technology Press) p30 (in Chinese) [霍金 S W, 埃利斯 G F R 2006 时空的大尺度结构(长沙:湖南科学技术出版社) 第30页]

    [19]

    Lai X M, Bian B M, Yang L, Yang J, Bian N, Li Z H, He A Z 2008 Acta. Phys. Sin. 57 7955 (in Chinese) [赖小明, 卞保民, 杨玲, 杨娟, 卞牛, 李振华, 贺安之 2008 物理学报 57 7955]

    [20]

    Bian B M, Lai X M, Yang L, Li Z H, He A Z 2012 Acta. Phys. Sin. 61 080401 (in Chinese) [卞保民, 赖小明, 杨玲, 李振华, 贺安之 2012 物理学报 61 080401]

    [21]

    Yu Y Q 1997 Introduction of general relativity (Beijing: Peking University Press) p18, 133, 158 (in Chinese) [愈允强 1997 广义相对论引论 (北京:北京大学出版社) 第18, 133, 158页]

    [22]

    Zhu H, Ji C C 2011 Fractal theory and its applications (Beijing: Science Press) p23 (in Chinese)[朱华, 姬翠翠 2011 分形理论及应用(北京:科学出版社) 第23页]

    [23]

    Yu Y Q 2002 Cosmophysics lectures (Beijing: Peking University Press) p104, 105, 214 (in Chinese)[俞允强 2002 物理宇宙学讲义(北京:北京大学出版社) 第104, 105, 214页]

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  • Received Date:  07 January 2012
  • Accepted Date:  23 February 2012
  • Published Online:  05 September 2012

Free particle geodesic affine parameter time-space coordinate systems

  • 1. Department of Information Physics and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

Abstract: Taking the time-series t as independent variable, the parameter equations {Xi(t)} of free particle space geodesic can be given. By transforming affine parameter R(t) we achieve homogeneous geodesic differential equations, and derive the first-order differential equations which are satisfied by affine parameter R and the sequence of analytical solutions R marked by rational number Cu. In light of R we define the distance unit of flat four-dimensional coordinate system {t,r,θ,φ}, and then establish a free particle geodesic affine parameter time-space coordinate system {t,ξ,θ,φ}. By the study of the diagonalization process of special relativity time-space interval model metric tensor g in {t,ξ,θ,φ}, we find the spatial and temporal line characteristic quantities t1(t,ξ), τ1(τ,ξ),tt(t,τ,ξ) and ττ1(t,τ,ξ) corresponding to diagonal metric. Derived from these quantities, the dimension of time-space coordinate system is less than 4.

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