The non- vacuum field equations
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 .
we see that the constant has to be .
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.