Question

1. Consider the conjugate-metric tensor, whose components in inertial system S is element of the matrix y = diag(-1, +1, +1, +1). In another inertial system s', the components of the conjugate- metric tensor are given by = AA, where A', is element to of the matrix A= - n | - n1+ 6 -1) 1 - ny 7:6-1) -yon. 7:7(7-1) - ny (7-1) 1+n: 6-1) 7.7, 6-1) 1.6 - 1) 6-1) 1+ (-1) ) that is associated with the boost from S to S' along the direction n. Compute the tensor component 7/

Answer 1

. With the given matsex structure of A, one Dp33 = $, se poate = -(13.)* +(9*;)2+(48_92 +(99,5 [since y. only has dirond components] - x + x (7-1 + x (-) + [1 + m (-1)] = ? (7-3)*(*} + m) - ritmez +1+27 (V-1) + mai (r-1) the fact that ñ is a unit Now we me vector, namely 2 2 + n + n t 1 + - = n n y X Thes 9:33 - m (r-1) (1 - n ) - x?p?n ? + 1 + 2n? (Y -1) + neet (-1)2 = 1 + m (8-19%+2n? (T-1) - x?p2m

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