The General Relativity version of 
must contain
rather than 
since we saw in Special Relativity that 
is just the 00 component of the energy- momentum tensor. This is expected anyway since in General Relativity all forms of energy [ not just rest mass ] should be a source of gravity.
To get the General Relativity version of the equations
involving we just replace the Minkowski metric
by 
and the partial derivative
(,) by the covariant derivative (;). For example the energy- momentum
tensor for a perfect fluid in curved space time is
and the conservation equations become
In the above we have just used the strong form of the Equivalence Principle , which says that any non- gravitational law expressible in tensor notation in Special Relativity has exactly the same form in a local inertial frame of curved spacetime.
We expect the full [ non- vacuum ] field equations to be of the form
where O is a second order differential operator which is a 0/2 tensor
[ since the stress energy tensor T is a 0/2 tensor] and 
is a
constant. The simplest operator that reduces to the vacuum field
equations when T=0 takes the form
Now since 
[ 
in Special Relativity ], we require 
.
Using 
gives
Comparing this with the double contracted Bianchi identities
we see that the constant 
has to be 
.
Thus we are led to the field equations of General Relativity:
or
In general we can add a constant 
so the field equations become
In a vacuum 
, so taking the trace of the field equations we get
Since 
and the Ricci scalar 
we find
that 
, and substituting this back into the field equations leads to
We recover the previous vacuum equations if 
. Sometimes
is called the vacuum energy density.
We have ten equations [ since 
is symmetric ] for the ten metric components.
Note that there are four degrees of freedom in choosing coordinates
so only six metric components are really determinable. This corresponds
to the four conditions
which reduces the effective number of equations to six.
It is very important to realize that although Einstein's field equations look very simple, they in fact correspond in general to six coupled non- linear partial differential equations.