Differential Geometry and Tensor analysis With applications to General Relativity This is an article from my home page: www.olewitthansen.dk Ole Witt-Hansen 2015 Contents 1. Linear algebra .........................................................................................................................1 2. Generalized coordinates...........................................................................................................5 3. Tensors...................................................................................................................................10 4. Covariant derivative...............................................................................................................12 4.1 Christoffel symbols as expansion coefficients.....................................................................14 4.2 Expressing the Christoffel symbol by the metric tensor......................................................15 4.3 Geodesics and covariant derivatives ....................................................................................16 4.4 Directional change under parallel transport of a vector.......................................................20 4.5 Angular excess and its correspondence to curvature ...........................................................21 4.7 Generalization of the Riemann tensor to n-dimensional space............................................24 4.8 Symmetries and contractions of the Riemann curvature tensor...........................................25 Covariant derivative and geodesics 1 1. Linear algebra The concept of a tensor is much easier to grasp, if you have a solid background in linear algebra. So we start out with, some fundamental issues in that field. A linear vector function is a vector function which obeys the following condition. (1.1) f (a b) f (a) f (b) , The relation being valid for arbitrary vectors a and b and arbitrary real numbers and . If we have a base on the vector space consisting of mutually orthogonal unit vectors: e1 ,e2 ,e3 ,.....,en The linear function is completely determined by the mapping of these vectors, since a vector x x1e1 x2e2 x3e3 .... xnen As a consequence of (1.1) will be mapped into (1.2) f (x) x1 f (e1) x2 f (e2 ) x3 f (e3 ) .... xn f (en ) Each of the vectors f (e1 ), f (e2 ), f (e3 ),...., f (en ) can then be written as a linear combination of the basic vectors. (1.3) f (ek ) a1k e1 a2k e2 a3k e3 .....an k en I follows (because the base vectors are mutually orthogonal), that a j k e j f (ek ) (1.3) is usually written in matrix form a a a . a 11 12 13 1n a21 a22 a23 . a2n (1.3) ( f (e ), f (e ), f (e ),...., f (e )) (e ,e ,e ,.....,e ) a a a . a 1 2 3 n 1 2 3 n 31 32 33 3n . an1 an 2 an3 . an n Or we can write the above relation in tensor form using Einstein’s summation convention: That whenever the same index appears twice, summation is implied. Covariant derivative and geodesics 2 n (1.4) f (ek ) a j k e j is written as j1 f (ek ) a j k e j (Summation j = 1…n is implied) If y f (x) we get from (1.2) and (1.3) using tensor notation: f (x) x1 f (e1 ) x2 f (e2 ) x3 f (e3 ) .... xn f (en ) x j f (e j ) And f (ek ) a1k e1 a2k e2 a3k e3 .....ank en ai k ei We get: y f (x) ai j x jei Which implies: yi ai j x j Or when written in matrix form: y a a a . a x 1 11 12 13 1n 1 y2 a21 a2 2 a23 . a2 n x2 (1.5) y a a a . a x 3 31 32 33 3n 3 . . . a a a . a yn n1 n 2 n3 n n xn And when written in symbolic matrix form. y = A x Where we use double underscore to mark a matrix symbol. The multiplication of two matrices A and B with elements ai j and bi j to give the matrix C with elements ci j goes as follows: One makes the “scalar product” of the i’th row in A and the j’th column in B . Written in tensor notation: (1.6) ci j ai k bk j (implied summation over k) The determinant of a matrix A is written det(A) or |A| The determinant of a 2x2 matrix: a a det 11 12 a a a a 11 2 2 21 12 a21 a2 2 Whereas the determinant of a 3x3 matrix is: Covariant derivative and geodesics 3 a a a 11 12 13 (1.7) deta21 a22 a23 a11a22a33 a11a32a23 a21a12a33 a21a32a23 a31a12a23 a31a22a13 a31 a32 a33 The rule is that each factor in the product a1i a2 j a3k belongs to a different row and a different column from the other factors. If [i, j, k] is an even permutation of [1, 2, 3] then the term is signed with a plus, if it is an uneven permutation, the term is signed with a minus. The determinant, can also be expressed with the help of the Levi-Civitas symbol 1 if i, j, k is an even permutation of 1,2,3 i j k 1 if i, j, k is an uneven permutation of 1,2,3 0 otherwise The determinant of the matrix shown above can then be written: det(A) = a1i a2 j a3k i j k Since there are six permutations of three elements, the number of terms is six. The extension to higher dimensions including the Levi-Civitas symbol is straightforward. The unit matrix E has the elements i j , where i j is the Koneke symbol (1.7) i j = 1 if i = j else 0 Any n x n matrix that has a non zero determinant, is called regular. Any regular matrix A has an inverse matrix A-1, defined by the equations: (1.8) A A-1 = A-1A = E If the elements in the inverse matrix are bi j , then (1.7) can be written ai kbk j i j 1.1 Coordinate transformations First we shall consider an ordinary coordinate system in the plane, which is rotated an angle θ. The base vectors (e ,e ) are rotated by an angle θ 1 2 into (e ',e ') . From the figure we see, that: 1 2 e1 ' e1 cos e2 sin and e2 ' e1 sin e2 cos When written in matrix form. Covariant derivative and geodesics 4 cos sin (e1 ',e2 ') (e1 ,e2 ) sin cos According to (1.3) and (1.5), we get the corresponding transformation for the coordinates (x1 , x2) and (x1’ , x2’). x1' cos sin x1 (1.9) x2 ' sin cos x2 In 3-dimensional space, this transformation corresponds to a rotation around the z-axis, an angle θ. The transformation matrix then becomes: x' cos sin 0 x (1.10) y' sin cos 0 y (Rotation about the z-axis) z' 0 0 1 z For the rotations around the y-axis or the x-axis, we have quite similar expressions x' cos 0 sin x y' 0 1 0 y (Rotation about the y-axis) z' sin 0 cos z x' 1 0 0 x y' 0 cos sin y (Rotation about the x-axis) z' 0 sin cos z To bring the coordinate system into an arbitrary angular position, we need three rotations. They are traditionally chosen as a rotation around the z-axis an angle α, followed by a rotation around the new y-axis (y’) an angle β, and finally an angle γ around the new z-axis z’’. The overall rotation is found by multiplying the three matrices. The angles α, β, γ are called the Euler angles. They are illustrated below: R(,, ) Rz ''( ) Ry '( ) Rz () cos sin 0cos 0 sin cos sin 0 sin cos 0 0 1 0 sin cos 0 0 0 1 sin 0 cos 0 0 1 cos cos cos sin sin cos sin cos cos sin sin cos cos cos sin sin cos cos sin sin cos cos sin sin sin cos sin sin cos Covariant derivative and geodesics 5 Showing the Euler angles. 2. Generalized coordinates Hitherto we have considered only Cartesian coordinates, with an orthonormal base e ,e ,e ,.....,e . 1 2 3 n This means that the base fulfils the condition ei e j i j . The basic vectors are the same for every point in space. Because of the orthogonality the square of the infinitesimal distance element is the same in every point. ds e1dx1 e2dx2 e3dx3 .... endxn 2 2 2 2 2 (2.1) ds ds ds dx1 dx2 dx3 ...dxn In a generalized coordinate system x (x1, x2 , x3 ,..., xn ) , the base vectors are not necessarily an orthogonal system, and the base may (and usually does) wary from point to point. If x (x , x , x ,..., x ) represents a point in the generalized coordinates, we can define a base: 1 2 3 n (e1 ,e2 ,e3 ,...en ) along the axis of the coordinate system x . A differential displacement is given by: dx (dx1,dx2 ,dx3 ,...,dxn ) And the displacement vector (2.2) ds dx1e1 dx2e2 dx3e3 ... dxnen The length of the distance element is: 2 2 ds ds ds ds (dx1e1 dx2e2 dx3e3 ... dxnen )(dx1e1 dx2e2 dx3e3 ... dxnen ) Covariant derivative and geodesics 6 2 (2.3) ds ei e j dxi dx j (summation over i and j is understood) There is a tradition, which we shall substantiate presently, letting the index on the coordinate functions go upstairs. From now on we write (2.3) as: 2 i j (2.3) ds ei e j dx dx Since the base vectors in the generalized coordinate system, are not (necessarily) orthogonal vectors, the scalar products e e does not vanish in general fori j . i j We define the metric (fundamental) form gi j gi j (x) gi j (x1, x2 , x3 ,..., xn ) as (2.4) gi j ei e j And the distance element becomes 2 i j ds gi j dx dx In the coordinate system x (x1, x2 , x3 ,..., xn ) we write of course: 2 i j (2.4) gi j ei e j and ds gi j dx dx In a Cartesian coordinate system gi j ei e j i j It is important to note that although the relation between generalized coordinate are (in general) not linear relations, the relations between the differentials dx (dx1,dx2 ,dx3 ,...,dxn ) are in fact linear.