The Einstein Equation 59The action principle then requires the variation of this action with respect to the
inverse metric푔휇휈to vanish,훿=0. Spelled out,
∫
[
1
2 휅
훿
√
−푔푅
훿푔휇휈
+
훿(
√
−푔푀)
훿푔휇휈
]
훿푔휇휈d^4 x. (3.22)Multiplying the integrand by 2휅∕
√
−푔the square bracket is a function that can be set
equal to zero for any variations훿푔휇휈, and we are left with an equation with three terms
for the motion of the metric field:
훿푅
훿푔휇휈+
푅훿
√
−푔
√
−푔훿푔휇휈
=−
2 휅
√
−푔
훿
(√
−푔푀
)
훿푔휇휈
. (3.23)
The right hand side of this equation is a tensor which can be chosen to be proportional
the stress-energy tensor푇휇휈since it contains all the matter terms푀.
The first term on the left is the variation of the Ricci scalar. From its definition in
Equation (3.18) we have
훿푅=푅휇휈훿푔휇휈+푔휇휈훿푅휇휈. (3.24)One can show that the second term does not contribute when integrated over the
whole space-time so we have the result
훿푅
훿푔휇휈=푅휇휈. (3.25)
The second term in Equation (3.23) requires the rule for the variation of a
determinant,
훿푔=푔푔휇휈훿푔휇휈. (3.26)Using this we get
훿√
−푔=−
훿푔
2
√
−푔
=−
1
2
√
−푔(푔휇휈훿푔휇휈). (3.27)
It follows that the left-side terms in Equation (3.22) are
퐺휇휈=푅휇휈−1
2
푅푔휇휈 (3.28)
Equation (3.28) expresses that the energy densities, pressures and shears embodied
by the stress-energy tensor determine the geometry of space-time, which, in turn,
determines the motion of matter. Thus we arrive at the covariant formula forEin-
stein gravity:
퐺휇휈=^8 휋퐺
푐^4푇휇휈. (3.29)
Stress–Energy Tensor. Let us now turn to the distribution of matter in the Universe.
Suppose that matter on some scale can be considered to be continuously distributed
as in anideal fluid. The energy density, pressure and shear of a fluid of nonrelativistic