6.2. Einstein’s equation

Idea:

1. Try with the simplest relation between momentum-energy tensor \(\mathsf{T}\) and a second order tensor representing the geometry of space, i.e. proportionality with Ricci’s curvature tensor \(\mathsf{R}\)

   \[\mathsf{R} = \kappa \mathsf{T} \ .\]

   Remark. This relationship doesn’t agree with local energy-momentum balance \(\nabla \cdot \mathsf{T} = \mathbf{0}\), as \(\nabla \cdot \mathsf{R} \ne \mathbf{0}\),

   \[\begin{split}\begin{aligned} \nabla_{\mu} R^{\mu}_{\ \ \nu} & = \frac{1}{2} \nabla_{\nu} R \\ \nabla_{\mu} R^{\mu \sigma} & = \nabla_{\mu} \left( g^{\sigma \nu} R^{\mu}_{\ \ \nu} \right) = g^{\sigma \nu} \nabla_{\mu} R^{\mu}_{\ \ \nu} = \frac{1}{2} g^{\sigma \nu }\nabla_{\nu} R = \nabla_{\mu} \left( \frac{1}{2} g^{\sigma \mu} R \right) , \\ \end{aligned}\end{split}\]

   as the covariant derivative of the metric components of the metric tensor are identically zero. Moving the extreme terms of this equality on the same side of the equal sign, \(\nabla_{\mu} \left( R^{\mu \sigma} - \frac{1}{2} g^{\mu \sigma} R \right) = 0\).

   Covariant derivative of the metric tensor

   \[\begin{aligned} \nabla_{\nu} g_{\mu \eta} & = \partial_{\nu} g_{\mu \eta} - \Gamma_{\nu \mu}^{\sigma} g_{\sigma \eta} - \Gamma_{\nu \eta}^{\sigma} g_{\mu \sigma} = 0 \ , \end{aligned}\]

   as \(\partial_{\nu} g_{\mu \eta} = \Gamma_{\nu \mu}^{\sigma} g_{\sigma \eta} + \Gamma_{\nu \eta}^{\sigma} g_{\mu \sigma}\) (see the derivative of the components of the metric tensor in Differential geometry for general relativity.

2. Correct the relation, replacing Ricci’s tensor with its divergence-free part

   \[R^{\mu \sigma} - \frac{1}{2} g^{\mu \sigma} R = \kappa T^{\mu \sigma} \ ,\]

   or

   \[\mathsf{R} - \frac{1}{2} \mathsf{g} R = \kappa \mathsf{T} \ .\]

3. Check against the classical limit, to find:

   • if Newton gravitation is the classical limit of general relativity

   • the values of the constants involved in the model, \(\kappa\)

   Under some assumptions:

   • weak gravitation for the linearization of the metric tensor \(g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}\) around Minkowski flat time-space (here \(\eta_{\mu \nu}\) describing Minkowski metrics).

   • pressure-less mass distribution at rest

   • mass distribution “at rest” (w.r.t. the “quasi inertial observer”)

6.2.1. Alternative expression of Einstein’s equations

Let’s evaluate the trace of \(\mathsf{R}\), i.e. \(R = R^{\sigma}_{\ \ \sigma}\) as a function of the trace of \(\mathsf{T}\).

\[\begin{split}\begin{aligned} R & := R^{\sigma}_{\ \ \sigma} = \\ & = g_{\sigma \varphi} R^{\varphi \sigma} = \\ & = g_{\sigma \varphi} \left( \frac{1}{2} g^{\varphi \sigma} R + \kappa T^{\varphi \sigma} \right) = 2 R + \kappa T \\ \end{aligned}\end{split}\]

and thus the relation \(R = - \kappa T\) holds between the trace of Ricci and energy-momentum tensors (here \(g_{\sigma \varphi} g^{\sigma \varphi} = \delta_{\sigma}^{\sigma} = 4\) in the 4-dimensional time-space). Thus, Einstein equation can be recast as

(6.2)\[\mathsf{R} = \kappa \left( \mathsf{T} - \frac{1}{2} \mathsf{g} \, T \right) \ .\]

6.2.2. Classical limit

1. Linearization of the metric tensor. \(g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}\), and the inverse (linearized) relation gives \(g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}\)

2. Linearization of Christoffel symbols.

\[\Gamma_{00}^{i} = - \frac{1}{2} \partial_i h_{00}\]

in the static limit, \(\partial_0 \equiv 0\), for the spatial components, \(i = 1:3\).

3. Linearization of the geodesics equation.

\[\ddot{q}^{\mu} + \Gamma_{\nu \sigma}^{\mu} \dot{q}^{\nu} \dot{q}^{\mu} = 0\]

For \(\mu = i = 1:3\), as the \(\tau \sim t\) in the slow-regime limit, and \(q^0 \sim c t\) and \(q^i = x_i\),

\[\ddot{x}_i = - c^2 \Gamma_{00}^i = \frac{c^2}{2} \partial_i h_{00} \ .\]

This equation must be compared with the dynamical equation of the Newtonian mechanics, \(\ddot{\vec{r}} = - \nabla \Phi\). In order to get the same equation, \(\partial_i h_{00} = - \frac{2}{c^2} \partial_i \Phi\), and thus - except for an arbitrary constant (irrelevant) - \(h_{00} = - \frac{2}{c^2} \Phi\).

4. Linearized Einstein equation, with energy-momentum tensor (of the pressure-less medium) \(T^{\mu \nu} = \rho c^2 \delta^{\mu}_0 \delta^{\nu}_0\), so that its trace reads \(T = \rho c^2\). Thus the linearized \(00\) component of Einstein equation reads

\[R_{00} = \kappa \left( \rho c^2 - \frac{1}{2} \rho c^2 \right) = \kappa \frac{1}{2} \rho c^2\]

5. Linearized relation between curvature tensor and the linearized metrics.

\[R_{00} = \partial_i \Gamma^{i}_{00} = - \frac{1}{2} \partial_{ii} h_{00} = \frac{1}{c^2} \nabla^2_{\vec{r}} \Phi\]

6. Comparison of the two expressions of \(R_{00}\).

\[\frac{\kappa}{2} \rho c^2 = \frac{1}{c^2} \nabla^2_{\vec{r}} \Phi \ , \]

or rearranging,

\[\nabla^2 \Phi = \frac{\kappa c^4}{2} \rho \ .\]

By direct comparison with the Poisson equation for Newtonian gravitation, \(\nabla^2 \Phi = 4 \pi G \rho\), it follows that

\[\kappa = \frac{8 \pi G}{c^4} \ .\]