Intro to Differential Geometry and General Relativity - S. Warner Episode 6

Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | column 1 column 1 k then column 4 column 4 -c2k . Similarly by choosing other photons we can replace column 1 by either column 2 or column 3 showing that if we take column 1 column 1 k we have column i column i -kc2 if 1 i 3 if i 4 k Let us now take another more interesting photon given by Path D x1 c 2 t x2 - c V2 t x3 0 x4 t with T c V2 -c yỊ2 0 1 . You can check to see that IITII2 0 so that it does indeed represent a photon. Since IITII2 0 we get 1 1 1 2 2 2 2x2 D1 c y2 - D2 c y2 D4 D1 c y2 - D2 c y2 D4 3 3 3x2 2 - 4 4 4x2 D1 c y2 - D2 c y2 D4 - c D1 c y2 - D2 c y2 D4 0 and looking at a similar photon traveling in the opposite x2-direction 1 . f 1 . _ _ Ọ . f _ z yx -1 _ z f X . yx 1 _ z. f x 7-x 1 . yx _ tr T-x _ - gr 7-x D1 c yị2 D2 c y2 D4 D1 c y2 D2 c y2 D4 _ 2 . r _ 2 . r _ 2. 9 9 . _ A . r _ A . r _ A Ọ _ . TX 3 _ I J-X 3 _ I J-X 3i2 4 _ I trs TX 4 _ I trs TX 4x 2 r D1 c y2 D2 c y2 D4 - c D1 c y2 D2 c y2 D4 0 Subtracting these gives X 1 T x 1 I T x 2 2 3 r- 3 2 4 4q 2c D1 D2 D1 D2 D1 D2 - c D1 D2 J I A r T 1 n 1 I 7 2 yx 2 yx 3 yx 3 2 yx 4 yx 4-1 y 4cẠ 2 D2 D4 D2 D4 D2 D4 - c D2 D4 J 0. But we already know that the second term vanishes so we are left with yx 1 yx 1 yx 2 yx 2 yx 3 yx 3 2 yx 4 yx 4 _ y D1 D2 D1 D2 D1 D2 - c D1 D2 0 showing that columns 1 and 2 are also orthogonal. 51 Choosing similar photons now shows us that columns 1 2 and 3 are mutually orthogonal. Therefore we have 0 if i j column i column I k if 1 i j 3 IV _-kc2 if i j 4 But what is k Let us invoke condition b of Defintion . To measure the length of a vector in the new frame we need to transform the metric tensor using this coordinate change. Recall that using matrix notation the metric G transforms to G PTGP where P is the matrix inverse of D above. In the exercise set you will see that the columns of P have the same property IV above but with k replaced by 1 k. But G PTGP Now since G is just a constant multiple of an elementary matrix all it does is multiply the last row .