Next: The simplest pure gravity Up: Symmetry Breaking in General Previous: Symmetry Breaking in General

A glance at GR

Basically, the main idea of GR is to represent gravitational interaction not via action of forces in usual Minkowski space-time, but rather by changing the geometry of space-time, curving it in such a way that its geodesics become exactly the right paths for motion of the interacting particles.

The metric $g_{\mu\nu}(x)$ of the 4-dimensional space-time is thus taken as the gravitational field itself. It is not apriori given, but must be found from solving some equations, which are obtained in the standard way from the least action principle. The action for gravity coupled to some other fields $\phi^{I}(x)$ looks like this

$\begin{eqnarraystar}S &=& \int{\mathcal{L}\sqrt{-g}d^4x}, \quad g \equiv det(g_{... ...nu}, \quad R_{\mu\nu} = g^{\alpha\beta}R_{\mu\alpha\nu\beta}. \end{eqnarraystar}$

Lagrangian density for pure gravity is given there by scalar curvature

, which is the simplest curvature invariant one can construct out of curvature tensor $R_{\mu\nu\alpha\beta}$ .

Minimizing the action gives equations for $g_{\mu\nu}(x)$ and $\phi^{I}(x)$ . Unfortunately, they turn out to be very non-linear, so that in dealing with them one is forced to forget about superposition principle, which, for example, in electrodynamics allows us easily write down the field due to arbitrary distribution of charges summing over point charge contributions.

In order to obtain analytical solutions we need to simplify the problem from the very beginning by 1) taking fewer number of different fields and 2) assuming high symmetry of the solution, thus reducing the number of equations to solve, unknown functions to find and variables on which they are to depend.

One of the most important questions in analyzing gravity solutions is whether or not they are singular? It happens sometimes that what seems to be a (physical) singularity of a solution turns out to be just a consequence of a bad choice of coordinates for describing it (like at the origin of spherical coordinates). Therefore, one should look on the behavior of quantities, characterizing the solution and independent of a coordinate choice; i.e. on curvature scalars. Amongst those the commonly used are the following three:

$R \quad= g^{\mu\nu}R_{\mu\nu}$ ``Scalar curvature''

$RR \; = R^{\mu\nu}R_{\mu\nu}$ ( just ... )

$RK \,= R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$ ``Kretschmann curvature invariant''.

Next: The simplest pure gravity Up: Symmetry Breaking in General Previous: Symmetry Breaking in General

Dmitry Belyaev
2000-05-13

$R \quad= g^{\mu\nu}R_{\mu\nu}$	``Scalar curvature''
$RR \; = R^{\mu\nu}R_{\mu\nu}$	( just ... )
$RK \,= R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$	``Kretschmann curvature invariant''.