Editing Einstein tensor (section)

== Explicit form ==
The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. However, this  expression is complex and rarely quoted in textbooks.  The complexity of this expression can be shown using the formula for the Ricci tensor in terms of [[Christoffel symbols]]:
<math display="block">\begin{align}
  G_{\alpha\beta}
    &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R \\
    &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} g^{\gamma\zeta} R_{\gamma\zeta} \\
    &= \left(\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}\right) R_{\gamma\zeta} \\
    &= \left(\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}\right)\left(\Gamma^\epsilon{}_{\gamma\zeta,\epsilon} - \Gamma^\epsilon{}_{\gamma\epsilon,\zeta} + \Gamma^\epsilon{}_{\epsilon\sigma} \Gamma^\sigma{}_{\gamma\zeta} - \Gamma^\epsilon{}_{\zeta\sigma} \Gamma^\sigma{}_{\epsilon\gamma}\right), \\[2pt]
  G^{\alpha\beta}
    &= \left(g^{\alpha\gamma} g^{\beta\zeta} - \frac{1}{2} g^{\alpha\beta}g^{\gamma\zeta}\right)\left(\Gamma^\epsilon{}_{\gamma\zeta,\epsilon} - \Gamma^\epsilon{}_{\gamma\epsilon,\zeta} + \Gamma^\epsilon{}_{\epsilon\sigma} \Gamma^\sigma{}_{\gamma\zeta} - \Gamma^\epsilon{}_{\zeta\sigma} \Gamma^\sigma{}_{\epsilon\gamma}\right),
\end{align}</math>
where <math>\delta^\alpha_\beta</math> is the [[Kronecker tensor]] and the Christoffel symbol <math>\Gamma^\alpha{}_{\beta\gamma}</math> is defined as
<math display="block">\Gamma^\alpha{}_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}\left(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon}\right).</math>
and terms of the form <math>\Gamma ^\alpha _{\beta \gamma, \mu}</math> or <math>g_{\beta\gamma,\mu}</math> represent partial derivatives in the ''μ''-direction, e.g.:
<math display="block">\Gamma^\alpha{}_{\beta\gamma, \mu} = \partial _\mu \Gamma^\alpha{}_{\beta\gamma} = 
\frac{\partial}{\partial x^\mu}
\Gamma^\alpha{}_{\beta\gamma}</math>

Before cancellations, this formula results in <math>2 \times (6 + 6 + 9 + 9) = 60</math> individual terms. Cancellations bring this number down somewhat.<!-- exactly how much? -->

In the special case of a locally [[inertial reference frame]] near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:

<math display="block">\begin{align}
  G_{\alpha\beta}
    & = g^{\gamma\mu}\left[ g_{\gamma[\beta,\mu]\alpha} + g_{\alpha[\mu,\beta]\gamma} - \frac{1}{2} g_{\alpha\beta} g^{\epsilon\sigma} \left(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}\right)\right] \\
    & = g^{\gamma\mu} \left(\delta^\epsilon_\alpha \delta^\sigma_\beta - \frac{1}{2} g^{\epsilon\sigma}g_{\alpha\beta}\right)\left(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}\right),
\end{align}</math>
where square brackets conventionally denote [[Antisymmetric tensor|antisymmetrization]] over bracketed indices, i.e.
<math display="block">g_{\alpha[\beta,\gamma]\epsilon} \, = \frac{1}{2} \left(g_{\alpha\beta,\gamma\epsilon} - g_{\alpha\gamma,\beta\epsilon}\right).</math>