58 General Relativity
to Einstein’s system of ten gravitational equations as we shall see later. Finally, we may
sum over the two indices of the Ricci tensor to obtain theRicci scalar푅:
푅=푔훽훾푅훽훾, (3.18)which we will need later. Actually we first met contraction in its simplest form, the
scalar product of two vectors [Equation (3.9)].
3.3 The Einstein Equation
Realizing that the space we live in was not flat, except locally and approximately, Ein-
stein proceeded to combine the equivalence principle with the requirement of gen-
eral covariance. The inhomogeneous gravitational field near a massive body being
equivalentto (a patchwork of flat frames describing) a curved space-time, the laws
of nature (such as the law of gravitation) have to be described bygenerally covariant
tensor equations. Thus the law of gravitation has to be a covariant relation between
mass density and curvature. Einstein searched for the simplest form such an equation
may take.
In the analysis of classical fields as carriers of forces one describes the dynamics by
equations of motion. A convenient way to derive the equations of motion is achieved
by introducing theLagrangianL which is the difference between kinetic and potential
energies, and theaction∫Ldt. Then theaction principleis an extremal of the action,
훿
∫
Ldt= 0 , (3.19)which delivers the equations of motion. This is also called theprinciple of least action.
But here we are interested in deriving the Einstein equation of gravitation. We
start with a Lagrangian defined in terms of aLagrangian density퐿, and an action of
the relativistic form
훿∫ 퐿d^4 푥휇= 0. (3.20)The Einstein–Hilbert Action. In general relativity, the action is usually assumed to
be a functional of the metric tensor푔휇휈and the affine connexions (3.13). This is the
Einstein–Hilbert actionproposed in 1915 byDavid Hilbert(1862–1943),
푆=
∫[
1
2 휅
푅+푀
]
√
−푔d^4 푥, (3.21)where휅= 8 휋퐺∕푐^4 ,Ris the Ricci scalar 3.18 and푔=det(푔휇휈).Theterm푀describes
any matter fields appearing in the theory.
Note that the integral푆is defined over all of spacetime d^4 푥, which is of course
a simplification. General relativity assumes the Copernican principle to be true, but
this is only accurate on small scales and modifications are needed at larger scales. The
scale at which the assumption breaks down is still debated but unknown.