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姓名 林永康(Yung-Kang Lin)  查詢紙本館藏   畢業系所 物理學系
論文名稱 幾何代數與微分形式間之轉換及其在重力之應用
(Geometric Algebra and Differential Forms: Translation and Gravitational Application)
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摘要(中) 幾何代數已成功應用在各物理領域, 我們希望能應用在重力上.
摘要(英) Geometric Algebra already shows its power on Classical Mechanics,Electrodynamics, Quantum Mechanics and Special Relativity. It is a very handy tool for understanding and developing physics. It handles flat spacetime physics fairly well which give us the motivation to test the Gauge Theory of Gravity based on Geometric Algebra. Here we show in detail how to translate between the popular differential form approach and the Gauge Theory of Gravity-Geometric Algebra expressions. We then test on an application: the energy-momentum pseudotensor. The new formulism can handle this application well.
關鍵字(中) ★ 幾何代數
★ 規範場論
★ 重力
★ 微分形式
關鍵字(英) ★ differential forms
★ pseudotensor
★ gauge theory
★ gravity
★ geometric algebra
論文目次 1. Geometric Algebra
1.1 Introduction and Outline
1.2 From Geometry to Algebra
1.3 Even Subalgebra
1.4 Differentiation
1.5 Integration
2. Gauge Field Theory of Gravity
2.1 Local Spacetime Transformations
2.2 Metric and Einstein Equation
2.3 Equivalence Principle and Geodesic Equation
3. Translation
3.1 Connection
3.2 Torsion and Curvature Tensor
3.3 Bianchi and Contracted Bianchi Identities
3.4 Volume Elements
3.5 Einstein Tensor
4. Application and Conclusion
4.1 Einstein Pseudotensor
4.2 Dual Relation
4.3 Conclusion
參考文獻 [1] D. Hestenes and G. Sobczyk, "Clifford Algebra to Geometric Calculus". Reidel, Dordrecht, 1984.
[2] D. Hestenes, "New Foundations for Mathematical Physics", available at [4].
[3] D. Hestenes, "Space-Time Algebra". Gordon and Breach, New York, 1966.
[4] http://modelingnts.la.asu.edu/GC_R&D.html.
[5] D. Hestenes, "Spacetime Calculus for Gravitation Theory", available at [4].
[6] D. Hestenes, "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics", available at [4].
[7] D. Hestenes, "New Foundations for Classical Mechanics", D. Reidel Publishing, 1986.
[8] http://www.mrao.cam.ac.uk/~clifford/.
[9] A. Lasenby, C. Doran, "A Lecture Course in Geometric Algebra", available at [8].
[10] C. Doran, A. Lasenby, "Physical Applications of Geometric Algebra", available at [8].
[11] A. Lasenby, C. Doran and S. Gull, "Gravity, gauge theories and geometric algebra", Phil. Trans. R. Soc. Lond. A, 356:487-582, 1999.
[12] A. Lasenby, C. Doran and S. Gull, "Astrophysical and Cosmological Consequences of a Gauge Theory of Gravity", N. Sanchez and A. Zichichi, eds. Advances in Astrofundamental Physics. Erice, 1994 (World Scientific Publishing Co., 1995), p. 359-401. [13] S. Gull, A. Lasenby, and C. Doran. "Imaginary Numbers are Not Real - the Geometric Algebra of Spacetime". Found. Phys., 23 (9):1175, 1993.
[14] A. Lasenby, C. Doran, and S. Gull. "A multivector derivative approach to Lagrangian field thoery". Found. Phys., 23(10):1295, 1993. [15] C. Doran, "Geometric Algebra and Its Application to Mathematical Physics", PhD thesis, University of Cambridge, 1994.
[16] A. Lewis, "Geometric Algebra and Covariant Methods in Physics and Cosmology", PhD thesis, University of Cambridge, 2000.
[17] J. Nester, "The Hamiltonian of Dynamic Geomtry", 1990, unpublished lecture notes.
[18] C.M. Chen, J. Nester, "Quasilocal Quantities for General Relativity and Other Gravity Theories", Class. Quantum Grav. 16 1279-304, 1999; gr-qc/9809020.
[19] C.M. Chen, J. Nester, "A symplectic Hamiltonian Derivation of Quasilocal Energy-Momentum for GR", Grav.Cosmol. 6 257-270, 2000; gr-qc/0001088.
[20] C.C. Chang, J. Nester and C.M. Chen, "Pesudotensor and Quasilocal Energy-Momentum", Phys. Rev. Lett. 83 1897-901, 1999; gr-qc/9809040.
[21] J. Nester, "Generalized Pesudotensor", 2002, unpublished lecture notes.
[22] J.Y. Lin, "Applications of Geometric Algebra to Gravity Theory", Msc thesis, National Central University, 1997.
[23] S.N. Liang "Geometric Algebra: Spinors and the Positivity of Gravitational Energy", Msc thesis, National Central University, 2003.
[24] C. W. Misner, K. S. Thorne and J. A. Wheeler, "Gravitation", W. H. Freeman and Company, 1973.
[25] T. Franhel, "The Geometry of physics: an introduction".
[26] C. von Westenholtz, "Differential Forms in Mathematical Physics".
[27] F. W. Hehl and Y. N. Obukhov, "Foundations of Classical Electrodynamics", 2003.
[28] R. M. Wald, "General Relativity", 1984.
指導教授 聶斯特(James M. Nester) 審核日期 2003-1-15
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