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贗張量

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物理學數學中,贗張量(英語:pseudotensor)為在座標轉換等情形下,行為類似張量的數量。但在空間反演瑕旋轉時會多出負號,張量則不會。

贗張量的另一個意義出現在廣義相對論中:張量遵守嚴格的轉換律,而贗張量不是。也因此,當轉換參考系時,贗張量的形式一般來說無法保持不變;一個含有贗張量的方程式在一個參考系成立,在另個參考系就不見得成立。細節參見:廣義協變性

定義

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數學中的贗張量

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在做鏡射時,贗張量會多出一個負號,而張量不會。根據定義,類型(p,q)的贗張量P是個幾何物體,其分量以任意基底展開,可以指標(p + q)來寫出,其遵守轉換規則:

式子兩邊採用不同基底。[1][2][3]

這裡是贗張量的分量,分別以新、舊基底寫出;逆變指標的轉換矩陣(transition matrix),協變指標的轉換矩陣。

;此轉換規則與尋常張量歧異處在(−1)A

廣義相對論中的贗張量

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廣義相對論中,重力場自身的能量動量無法以能量-動量張量來描述,而需引入一個物理量,其在受限的座標轉換中行為類似張量。此即贗張量的例子,其中較著名的為藍道-利夫希茨贗張量英语Landau-Lifshitz pseudotensor

相關條目

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參考文獻

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  1. ^ Sharipov, R.A. (1996). Course of Differential Geometry, Ufa:Bashkir State University, Russia, p. 34, eq. 6.15. ISBN 5-7477-0129-0 [arXiv:math/0412421v1]
  2. ^ Lawden, Derek F. (1982). An Introduction to Tensor Calculus, Relativity and Cosmology. Chichester:John Wiley & Sons Ltd., p. 29, eq. 13.1. ISBN 0-471-10082-X
  3. ^ Borisenko, A. I. and Tarapov, I. E. (1968). Vector and Tensor Analysis with Applications, New York:Dover Publications, Inc. , p. 124, eq. 3.34. ISBN 0-486-63833-2
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贗張量
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