Einstein’s Field Equations

Abstract

This chapter starts with a hydrodynamical description of energy–momentum conservation in a Newtonians context in order to give some intuition about the relativistic formulation of energy–momentum conservation as represented by a vanishing divergence of the energy–momentum tensor . Einstein demanded that energy–momentum conservation should follow from the field equations, and hence he needed a divergence-free curvature tensor. This is deduced from Bianchi’s 2. identity. It is shown that one need not postulate that free particles follow geodesic curves, but that it follows from the field equations.

Exercises

• 7.1. Newtonian approximation of perfect fluid

  Let

  $$ T^{\alpha \beta } = p\,\eta^{\alpha \beta } + \left( {\rho + p/c^{2} } \right)u^{\alpha } u^{\beta } $$

  be the components of the energy momentum tensor of a perfect fluid in flat spacetime with Minkowski metric $\eta_{\mu \nu }$. Here $p$ is the pressure and $\rho$ the mass density of the fluid, and $u^{\alpha }$ the components of its 4-velocity.

  • (a) Explain why the conservation law $T_{;\beta }^{\alpha \beta } = 0$ in this case reduces to $T_{,\beta }^{\alpha \beta } = 0$.

  • (b) We shall consider the Newtonian limit where $p/c^{2}$ can be neglected compared to $\rho$ in the second term of $T^{\alpha \beta }$, and the components of the 4-velocity of the fluid are $u^{\alpha } \approx \left( {c,\,\,\vec{v}} \right)$ where $\vec{v}$ is the ordinary velocity of a fluid element. Show that in this case the conservation law (a) implies mass conservation as represented by the equation of continuity ,

    $$ \frac{\partial \rho }{\partial t} + \nabla \cdot \left( {\rho \,\vec{v}} \right) = 0, $$

  • (c) And momentum conservation as represented by the Euler equation of motion ,

    $$ \rho \left( {\frac{{\partial \vec{v}}}{\partial t} + \left( {\vec{v} \cdot \nabla } \right)\vec{v}} \right) = - \nabla p. $$

• 7.2. The energy–momentum tensor of LIVE

  • (a) Show that the energy–momentum tensor of a Lorentz invariant medium is proportional to the metric tensor.

  • (b) Show that a Lorentz invariant perfect fluid has equation of state $p = - \rho c^{2}$.

  • (c) How is the density of Lorentz invariant vacuum energy, LIVE, related to the cosmological constant?