Question

1. Be explicit calculation verify that: $\left ( ds \right )^2=g_{\mu v}dx^{\mu}dx^{v} =g^{\mu v}dx_{\mu}dx_{v}=dx_{\mu}dx^{\mu}$

2. Express a 4-dimensional boost (Lorenz transformation) parallel to the z-direction at velocity fraction β using hyperbolic functions of an angle.

Answer 1

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To verify: (ds)^2 = g_mu nu dx^mu dx^nu = g^mu nu dx_mu dx_nu = dx_mu dx^mu Consider a length element, ds along mu^th coordinate i.e x^mu having a basis unit vector e_mu so the length in mu^th direction implies ds^mu = e_mu dx^mu Let us suppose reciprocal vector basis, e_mu such that e^mu e_nu = delta^nu_mu So contracting the length vector vector ds middot vector ds = (ds)^2 = dx_mu dx_nu e^mu e^nu = dx_mu dx_nu g^mu nu g^mu nu is the contravarient metric matrix. also e_mu e_nu = g_mu nu & e_mu = g_mu nu e^nu, e^mu_nu = g^mu nu e_nu \r\n So, the metric is used to raise or lower an index, in the tensor as shown. So metric can also be defined as ds^2 = g_mu nu dx^mu dx^nu also ds^2 = dx^mu dx_mu (e^mu nu e_mu) = dx^mu dx_mu \r\n When boost is parallel to z - direction, the coordinates transform as t� = r (t -