Question

2. Einstein Summation Convention Einstein Summation Convention: When an expression contains one index as a super script, and the same index as a subscript, a summation is implied over all values that index can take Examples: where the expression still represents four equations. An index that is summed over is called a dummy index, and can be replaced by another index if convenient. e.g The index in the above expression is not involved in the sum operation. Such an index is known as a free index. A free index must appear on both sides of an equation. Therefore, if we elect to change a free index, we must do so on both sides of the equation

Answer 1

Given that Einstein Summation convention (a) We have to Identify the free and dummy indices, following equations. (b) Rewrite the given expressions into equilent expressions but with different indices. \r\n (b) (c) If greek indices go from 0 rightarrow 3 and Latin indices go from 1 rightarrow 3 Greek 0 rightarrow 3 (alpha, beta, gamma, mu, sigma, delta) Latin 1 rightarrow 3 (i, j) \r\n A^alpha B_alpha = -12.4 equation (4) n_alpha gamma A^gamma = B alpha equation (4) A^mu B^gamma = C^mu^gamma equation (16) A^i B^ij = 13D^j equation (3) R_mu gamma - 1/2 g_mu gamma g^alpha beta R_alpha beta = 8 pi J_mu gamma equation (16) C_alpha beta 2s(R^alpha gamma + ug^alpha sigma n_sigma n^gamma) = G_B s equation (16) (d) equation (10) are actually the Einstein Field equations of General Relativity eqn (3) can be written as, R_mu gamma - 1/2 g_mu gamma R = 8 pi T_mu gamma g^alpha beta R_alpha beta = R \r\n < null >