MATHEMATICAL PHYSICS UNIT – 8 Metric Tensor & Chriostofell Symbols

PRESENTED BY: DR. RAJESH MATHPAL ACADEMIC CONSULTANT SCHOOL OF SCIENCES U.O.U. TEENPANI, HALDWANI UTTRAKHAND MOB:9758417736,7983713112 STRUCTURE OF UNIT

 8.1. INTRODUCTION

 8.2. RIEMANNIAN SPACE: METRIC TENSOR

jk 풋  8.3. FUNDAMENTAL TENSORS gjk, g AND 휹풌  8.4. CHRISTOFELL’S 3-INDEX SYMBOLS

 8.5. GEODESICS 8.1. INTRODUCTION

 In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.  A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.

 While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.  The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point. 8.2. RIEMANNIAN SPACE: METRIC TENSOR

An expression which express the distance between two adjacent point is called a metric or line element. In three dimensional space the line element, i.e., the distance between two adjacent points (x, y, z) and (x + dx, y + dy, z + dz) in Cartesian coordinates is given by

ds2 = dx2 + dy2 + dz2.

In terms of general curvilinear coordinates, the line element becomes

33 2 (Using summation convention) dsg== du dugjkjkjkjk du du jk==11

This idea was generalised by Riemann to n-dimensional space.

The distance between two neighbouring points with coordinates xj and xj + dxj is given by nn 2 j kj k dsg== dx  dx jkjk g dx dx …(8.1) jk==11 (Using summation convention)

j where the coefficients gjk are the functions of coordinates x , subject to the restriction g =

determinant of gjk, i.e, 푔푗푘 ≠ 0. j k The quadratic differential form gjk dx dx is independent of the coordinates system and is called the Riemannian metric for n dimensional space. The space which is characterised by Riemannian metric is called Riemannian space. Hence the quantities gjk are the components of a covariant symmetric tensor of rank two, called the metric tensor or fundamental tensor. (dx1)2 + (dx2)2 + (dx3)2 + … + (dxn)2 or dxjdxk, the space is called n-dimensional Euclidean space. It is now obvious that Euclidean spaces are the particular cases of Riemannian space. In general theory of relativity (four dimensional space), the line element is given by

2 j k Ds = gjkdx dx (j, k = 1, 2, 3, 4). In special theory of relativity, the line element is given by (dx1)2 + (dx2)2 + (dx3)2 + … + (dxn)2 or dxj dxk. the space is called n-dimensional Euclidean space. It is now obvious that Euclidean spaces are the particular cases of Riemannian space. In general theory of relativity (four dimensional space), the line element is given by

2 i j k ds = gjkdx dx dx (j, k = 1, 2, 3, 4). In special theory of relativity, the line element is given by ds2 = (dx1)2 + (dx2)2 + (dx3)2 [with x4 = ict, i = √ (-1)] = dxjdxj (j = 1, 2, 3, 4).

2 j k As ds = gjk dx dx has been defined in general space (i.e., Riemannian space), it 2 j k is independent of the coordinate system, i.e., dx = gjk dx dx is an invariant. jk 풋 8.3. FUNDAMENTAL TENSORS gjk, g AND 휹 (i) Covariant fundamental tensor gjk. The line element or interval ds in Riemannian풌 space is given by 2 j k ds = gjk dx dx . …(8.2) As dxj dxk are contravariant vectors and ds2 is invariant for arbitrary choice of vectors dxj and dxk, it follows from quotient law that gjk is a covariant tensor, we have 2 j k j ds = gjk dx dx in system of variables x 휇 푣 휇 = 푔휇푣 푑푥 푑푥 in system of variables 푥 휇 푣 j k i.e., = 푔휇푣 푑푥 푑푥 = gjk dx dx . …(8.3) Now applying inverse transformation law to dxj and dxj, i.e., 푗 푗 휕푥 휇 푑푥 = 휇 푑푥 etc. 휕푥 푗 푘 휇 푣 휕푥 휇 휕푥 푣 푔 푑푥 푑푥 = 푔푗푘 푑푥 푑푥 휇푣 휕푥휇 휕푥푣 푗 푘 휕푥 휕푥 휇 푣 = 푔푗푘 푑푥 푑푥 휕푥휇 휕푥푣 휕푥푗 휕푥푘 휇 푣 i.e., 푔 − 푔 휇 푣 푑푥 푑푥 = 0 …(8.4) 휇푣 푗푘 휕푥 휕푥 As 푑푥휇 and 푑푥푣 are arbitrary contravarient vectors, we must have 휕푥푗 휕푥푘 푔 − 푐 = 0 휇푣 휕푥휇 휕푥푣 휕푥푗 휕푥푘 푔 = 푔푗푘 휇푣 휕푥휇 휕푥푣

Hence 푔푗푘 is a covariant tensor of rank 2.

푔푗푘 may be expressed as 1 1 푔 = 푔 + 푔 + 푔 − 푔 푗푘 2 푗푘 푘푗 2 푗푘 푘푗

= 퐴푗푘 + 퐵푗푘 …(8.5) 1 where 퐴 = 푔 + 푔 is symmetric tensor 푗푘 2 푗푘 푘푗 1 ቑ …(8.6) and 퐵 = 푔 − 푔 is symmetric tensor 푗푘 2 푗푘 푘푗 2 j k j k ∴ ds = gjk dx dx = (Ajk + Bjk) dx dx . …(8.7) We have

j k k j Bjk dx dx = Bkj dx dx (interchanging dummy indices j and k) j k = – Bjk dx dx

(cince Bjk is antisymmetric i.e., Bjk = – Bkj) j k i.e., 2Bjk dx dx = 0. As dxj and dxk are arbitrary vectors, we have

Bjk = 0 1 i.e., 푔 + 푔 = 0 2 푗푘 푘푗 i.e., 푔푗푘 + 푔푘푗 i.e., 푔푗푘 is symmetric.

So, we can write 푔푗푘 as 1 푔 . = 푔 + 푔 휇푣 2 휇푣 푣휇

Thus we have proved that the metric tensor gjk is covariant symmetric tensor of rank 2. This is called covariant fundamental tensor of rank 2. (ii) Contravariant fundamental tensor gjk. Let us define gjk as 푐표푓푎푐푡표푟 표푓 푔 푖푛 푔 푔푗푘 푗푘 …(8.8) 푔 where g is the determinant of gjk, i.e., 푔11 푔12 푔13 … 푔1푛 푔 푔 푔23 … 푔2푛 …21 …22 … … 푔 = 푔 = … 푗푘 … … … … … … … … … 푔푛1 푔푛2 푔푛3 … 푔푛푛

Since gjk is symmetric, g is symmetric which implies cofactor of gjk in g is symmetric and so gjk is symmetric. Let Aj be an arbitrary contravariant vector, then by quotient law, j Ak = gjk A …(8.9) is an arbitrary covariant vector. Now multiplying eqn. (8.9) by gkl, we get kl kl j g Ak = gjk g A . …(8.10) But g gkl = g 푐표푓푎푐푡표푟 표푓 푔푘푙 푖푛 푔 jk jk 푔 푙 = 훿푗 (by theory of determinants). Therefore equation (8.10) yields

kl 푙 j l g Ak = 훿푗 A = A …(8.11) kl i.e., the inner product of g with an arbitrary covariant vector Ak yields a contravariant vector. Hence by quotient law gkl is a contravariant tensor of rank 2. Thus gjk is symmetric contravariant tensor of rank two. This tensor is reciprocal of gjk and is called conjugate metric tensor or contravariant fundamental tensor of rank 2. 풋 풍 (iii) Mixed fundamental tensor 품풌 or 휹풋. we have kl 푙 gjk g = 훿푗 …(8.12)

kl As gjk and g are covariant and contravariant tensors of rank 2 recpectively, 푙 therefore, from quotient law 훿푗 is also a tensor of rank 2; it is a mixed tensor, contravariant in l and covarian in j and is known as mixed fundamental tensor. As important property of mixed fundamental tensor is that its components have the same value in all coordinate system, i.e., mixed fundamental tensor is invariant.

jk 푗 The three tensors gjk, g and 훿푘 are called the fundamental tensors and are of basic importance in general theory of relativity. 8.4.CHRISTOFELL’S 3-INDEX SYMBOLS

We now introduce two expressions (not tensors) formed of the fundamental tensors, known as Christofell’s symbols of first and second kind, namely : Christofell’s symbol of first kind. 1 휕푔 휕푔 휕푔 푗푘, 푙 = Γ = 푙푗 + 푘푙 − 푗푘 …(8.13) l,jk 2 휕푥푘 휕푥푗 휕푥푙 Christofell’s symbol of second kind. 푙 1 휕푔 휕푔 휕푔 = Γl = glm 푗푚 + 푘푚 − 푗푘 …(8.14) 푗푘 .jk 2 휕푥푘 휕푥푗 휕푥푚

From the symmetry property of gjk it follows that

푗푘, 푙 = 푘푗, 푙 or Γl,jk = Γl.kj …(8.15) 푙 푙 and = or Γl = Γl …(8.16) 푗푘 푘푗 .jk kj l thereby indicating that Christofell’s symbols Γl.jk and Γ jk are symmetrical with respect to indices j and k. Relations between Christofell’s symbols of first and second kind. (i) Replacing l by m in eqn. (8.13), we get 1 휕푔 휕푔 휕푥 Γ = 푚푗 + 푘푚 − 푗푘 m,jk 2 휕푥푘 휕푥푗 휕푥푚 Multiplying both sides of above equation by glm, we get 1 휕푔 휕푔 휕푔 푔푙푚Γ = 푔푙푚 푗푚 + 푘푚 − 푗푘 (since g = g ) m,jk 2 휕푥푘 휕푥푗 휕푥푚 jm mj l = Γjk [Using (8.14)] l 푙푚 i.e., Γjk = 푔 Γm,jk …(8.17) (ii) Interchanging l and m in eqn. (8.14), we get 1 휕푔 휕푔 휕푔 Γm = 푔푙푚 푗푙 + 푘푙 − 푗푘 jk 2 휕푥푘 휕푥푗 휕푥푙

Multiplying above equation by glm, we get 1 휕푔 휕푔 휕푔 푔 Γm = 푔 푔푙푚 푗푙 + 푘푙 − 푗푘 푙푚 jk 2 푙푚 휕푥푘 휕푥푗 휕푥푙 1 휕푔 휕푔 휕푔 = 푗푙 + 푘푙 − 푗푘 (since g gml = 훿푙 = 1) 2 휕푥푘 휕푥푗 휕푥푙 lm 푙

= Γl,jk 8.5. GEODESICS

In Euclidean three dimensional space the path of shortest distance between two fixed points is a straight line. Here we shall generalise this fundamental concept to Riemannian space.

The path of extremum (maximum or minimum) distance between any two points in Riemannian space is called the geodesic. Thus a geodesic is determined by the condition that the path between two fixed points A and B given by be extremum, i.e., B  ds B extremum (or stationary), …(8.18) A  ds A B   ds = 0 i.e., A …(8.19) where 훿 represents the variation symbol.

In Riemannian space, we have

2 j k ds = gjk dx dx …(8.20) Keeping the end points A and B fixed, let the path be deformed by giving every intermediate point an arbitrary infinitesimal displacement 훿푥푚 , so that expression (8.20) yields

j k j k j k 2 ds 훿 (ds) = 훿 (gjk) dx dx + gjk 훿 (dx ) dx + gjk dx 훿 (dx )

j k 휕푔푗푘 푚 k j j k = dx dx 훿푥 + gjk dx 훿 (dx ) + gjk dx 훿 (dx ). 휕푥푚

Dividing both sides by 2 ds and using the relation

휕푥푗 푑 훿 = 훿푥푗 ; 푑푠 푑푠

1 푑푥푗 푑푥푘 휕푔 푑푥푘 푑 푑푥푗 푑 We get 푑푠 = + − 푗푘 훿푥푚 + 푔 훿푥푗 + 푔 훿푥푘 푑푠. 2 푑푠 푑푠 휕푥푚 푗푘 푑푠 푑푠 푗푘 푑푠 푑푠

…..(8.21)

Substituting the value of 훿 푑푠 from (8.21) in (8.19), we get

B j k j k 1 dx dxg jk m dx d k dx d j m x+ gjk(  x) + g jk (  x) ds = 0 2  ds ds x ds ds ds ds A  On changing the dummy indices in the last two terms, we get

B jkjk 1 dx dxdxdxdg jk mm m xggxds++=jmmk .0.( ) 2  ds dsxdsdsds A  Integrating the second term by parts and remembering that the variation 훿 is zero at the fixed end points A and B,

B j kjk 1 dx dxddxdxg jk m −+= ggxjmmk ds  0. 2  ds ds xdsdsdsm A  As the infinitesimal displacements 훿푥푚 are arbitrary, therefore for the integral to be stationary the coefficient of 훿푥푚 in the integrand must vanish at all points on the path, i.e.,

j kjk 1 dx dxdg jk dxdx −+= ggjmmk 0 2 ds ds xm ds dsds 

j kjk 2 111dx dxdxdgjkjm x dg m −−= gmk 2 0. i.e., 222ds ds xds dsds

kk2 11dgmk dx d x − −gmk 2 = 0. 22ds ds ds …(8.22) But we have

kj dgjm g jm dxdgmk g mk dx ==kjand ds x ds ds x ds

With these substitutions, equation (8.21) becomes

j kjk 22 11dx dxd xdggjkjm x gmk  −−−+= ggjmmk 0. 22ds ds xxxdsdsmkj 22 i.e., 

Replacing the dummy indices j and k and l in the second bracketed terms, we get

j kll 22 11dx dxd xggjk d jm x gmk  − − −+=ggjm mk 0. 22ds ds xm  x k  xds j ds 22 

Using symmetry property of glm (i.e., glm = gml) above equation mau be written as

j k2 l 1 dx dxggmjgkm jk d x − − +glm = 0. 2 ds ds xk  x j  x m ds2  Now multiplying throughout by gmp, we get

jkl 2 1 dxdxd x mpmpggmjjk gkm ggg+−+= lm 0 2 dsdsxxxds kjm 2 

jkl 2 1 dxdxd x mpp g +=m. jkl  2 0 or 2 dsdsds

2 pjk d xdxdx mp 2 += g m. jk 0 i.e., dsdsds …(8.23)

2 pjk d xdx dx p 2 + = jk 0. or dsds ds …(8.24)

Equation (8.24) represents the required condition to be satisfied in order that the integral be stationary. Hence equation (8.24) represents the differential equation of a geodesic. For p = 1, 2, 3, 4 this equation gives four differential equations which determine a geodesic. THANKS