The basic quantities and equations of General Relativity

In order to formalize the ideas mentioned in the previous section, general relativity views spacetime as a 4-dimensional manifold equipped with a metric $ \hbox{\vec g}_{\alpha \beta}$ of Lorentzian signature where the Greek indices range from 0 to 3. At any given point in the manifold the signature enables one to distinguish between time-like, space-like and null directions. The metric further induces a whole range of higher level geometric concepts on the manifold. It defines a scalar product between vectors which leads to the measurement of length and the idea of orthogonality. From the metric and its derivatives one can derive a connection on the manifold which facilitates the definition of a covariant derivative. The notion of a derivative is more complicated in a curved manifold than in the common case of flat geometry and Cartesian coordinates because the basis vectors will in general vary from point to point in the manifold. It is therefore no longer possible to identify the derivative of a tensor with the derivative of its components. Instead one obtains extra terms involving the derivatives of the basis vectors. In terms of a covariant derivative these terms are represented by the connection. In general relativity one uses a metric-compatible connection defined by

$\displaystyle \Gamma^{\gamma}_{\alpha \beta}$ $\displaystyle = \frac{1}{2} \hbox{\vec g}^{\gamma \delta} (\partial_{\alp+ha} \h... ...hbox{\vec g}_{\alpha \delta} - \partial_{\delta} \hbox{\vec g}_{\alpha \beta}+),$    

where the Einstein summation convention, according to which one sums over repeated upper and lower indices, has been used. These connection coefficients are also known as the Christoffel symbols and define a covariant derivative of tensors of arbitrary rank by

$\displaystyle \nabla_{\delta} \hbox{\vec T}^{\alpha \beta}{}_{\gamma}$ $\displaystyle = \partial_{\delta} \hbox{\vec T}^{\alpha \beta}{}_{\gamma}+ + \Ga... ...{\gamma} - \Gamma^{\rho}_{\gamma \delta} \hbox{\vec T}^{\alpha \beta}{}_{\rho+},$    

where $ \partial_{\delta}$ represents the standard partial derivative with respect to the coordinate $ x^{\delta}$ . So for each upper index one adds a term containing the connection coefficients and for each lower index a corresponding term is subtracted. With the definition of a covariant derivative we can finally write down the exact definition of a "straight line" in a curved manifold. A geodesic is defined as the integral curve of a vector field $ \hbox{\vec v}$ which is parallel transported along itself
$\displaystyle \hbox{\vec v}^{\alpha} \nabla_{\alpha} \hbox{\vec v}^{\beta+}$ $\displaystyle = 0.$    

Based on the covariant derivative we can also give a precise definition of curvature. For this purpose the Riemann tensor is defined by
$\displaystyle \hbox{\vec R}^{\alpha}{}_{\beta \gamma \delta}$ $\displaystyle = \partial_{\gamma} \Gamma^{\alpha}_{\delta \beta} - \parti+al_{\d... ...o}_{\delta \beta} - \Gamma^{\alpha}_{\delta \rho} \Gamma^{\rho}_{\gamma \beta+}.$    

If we use a coordinate basis, i.e. $ \hbox{\vec e}_{\alpha} = \partial/\partial x^{\alpha}$ , this definition can be shown to imply that for any vector field $ \hbox{\vec v}^{\alpha}$
$\displaystyle \hbox{\vec R}^{\alpha}{}_{\beta \gamma \delta} \hbox{\vec v+}^{\beta}$ $\displaystyle = \nabla_{\gamma} \nabla_{\delta} \hbox{\vec v}^{\alpha} -\+nabla_{\delta} \nabla_{\gamma} \hbox{\vec v}^{\alpha},$    

which is commonly interpreted by saying that a vector $ \hbox{\vec v}$ is changed by being parallel transported around a closed loop unless the curvature vanishes. In order to describe the effect of the matter distribution on the geometry of spacetime one defines the Ricci tensor as the contraction of the Riemann tensor $ \hbox{\vec R}_{\beta \delta} = \hbox{\vec R}^{\alpha}{}_{\beta \alpha \delta}$ , where again the Einstein summation convention for repeated indices has been used. The geometry and the matter are then related by
$\displaystyle \hbox{\vec G}_{\alpha \beta} := \hbox{\vec R}_{\alpha \beta+} -1/2\,R\,\hbox{\vec g}_{\alpha \beta}\, = 8\pi \hbox{\vec T}_{\alpha \beta},$+    

where $ R=\hbox{\vec R}^{\alpha}{}_{\alpha}$ is the Ricci scalar and $ \hbox{\vec T}_{\alpha \beta}$ the energy momentum tensor. The interaction between the matter distribution and the geometry of spacetime can be summed up in the words of Misner, Thorn and Wheeler: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve".