By Jayant V. Narlikar

Common relativity is now a vital a part of undergraduate and graduate classes in physics, astrophysics and utilized arithmetic. this straightforward, effortless creation to relativity is perfect for a primary path within the topic. starting with a entire yet easy assessment of detailed relativity, the ebook creates a framework from which to release the tips of normal relativity. After describing the elemental thought, it strikes directly to describe very important purposes to astrophysics, black gap physics, and cosmology. a number of labored examples, and diverse figures and pictures, support scholars relish the underlying innovations. There also are one hundred eighty workouts which try out and advance students' figuring out of the topic. The textbook offers the entire invaluable info and dialogue for an undemanding method of relativity. Password-protected suggestions to the workouts can be found to teachers at www.cambridge.org/9780521735612.

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We use the result just derived, namely that gik transforms as a covariant tensor so that ∂xm ∂xn g ik = · gmn . ∂x i ∂x k 49 50 Vectors and tensors Now define nµ Fν B kl = g pq ∂x k ∂x l ∂x p ∂xq and consider the product P Fig. 5. An example of a second-rank tensor. In the example illustrated, the stress tensor at P relates F ν , the stress force at P, to the direction nµ of the local normal to the surface at P. In general the two directions are not parallel. g ik B kl = ∂ x m ∂ x n ∂ x k ∂ x l pq g gmn ∂x i ∂x k ∂x p ∂xq = gmn g pq δ np = gmp g pq ∂xm ∂x l ∂x i ∂xq ∂xm ∂x l ∂x i ∂xq = δil .

In dynamics we encounter the moment-of-inertia tensor Iµν of a massive extended body, which is a second-rank tensor in three-dimensional space. If ωµ is the angular velocity of the body then its angular momentum is given by the vector Iµν ωµ . 1 Contraction The operation of contraction consists of identifying a lower index with an upper index in a mixed tensor. 3 Tensors tensor by 2, since the repeated index implies a sum over all of its four values. Thus Ai Bk is a tensor of rank 2 if Ai and Bk are vectors.

1) by the coordinate transformation x = r cos θ, y = r sin θ cos φ, z = r sin θ sin φ. We do not expect that a fundamental change in the properties of spacetime, such as its geometry, should be brought about by such a change of coordinates. However, consider another example. Let us take the geometry on the surface of a sphere of radius a. If we consider the sphere as embedded in a three-dimensional space with the Cartesian coordinates x, y, z, we may write the equation of the surface of the sphere as x 2 + y2 + z2 = a2.