PHYSICS
Notes
Page 160: To derive Eq. (6.68) from Eq. (6.67), simply notice that \( g_{\alpha \lambda} g^{\lambda \sigma} = {\delta^\sigma}_\alpha \).
Page 160: In Eq. (6.72), it is important to understand that \( {V^\mu}_{; \beta} \) in \( \nabla_\alpha ({V^\mu}_{; \beta}) \) is not a number. The whole \( \nabla_\alpha ({V^\mu}_{; \beta}) \) denotes \( {(\boldsymbol{\nabla}(\boldsymbol{\nabla} \vec{V}))_{\alpha \beta}}^\mu \).
Page 176: For Eq. (7.14), an underlying assumption is that \( \phi_{, 0} = O(\phi) \), i.e. that \( \phi \) does not change too fast.
Page 180: In Eq. (7.33), we assume that \( g \) is diagonal, which is reasonable for a nearly flat metric.
Page 192: Eq. (8.25) deserves a bit more elaboration. We cannot directly use Eq. (6.68) since it is valid only in the locally inertial frame. We really have to start from Eq. (6.63). Notice that \( {\Gamma^\alpha}_{\sigma \mu} \) is to the first order of \( h_{\alpha \beta, \gamma} \) and therefore the last two terms in Eq. (6.63) are to the second order of \( h_{\alpha \beta, \gamma} \) and can be ignored. Further, Eq. (6.64) has to be replaced by
\[ {\Gamma^\alpha}_{\mu \nu, \sigma} = \frac{1}{2} g^{\alpha \beta} (g_{\beta \mu, \nu \sigma} + g_{\beta \nu, \mu \sigma} - g_{\mu \nu, \beta \sigma}) + \frac{1}{2} {g^{\alpha \beta}}_{, \sigma} (g_{\beta \mu, \nu} + g_{\beta \nu, \mu} - g_{\mu \nu, \beta}), \]
of which the second term is also to the second order of \( h_{\alpha \beta, \gamma} \) and can be ignored, reducing it back to Eq. (6.64). The rest can be derived in a similar fashion as from Eq. (6.65) up to Eq. (6.68).