Tensors and Relativity: Chapter 7

The non- vacuum field equations

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The General Relativity version of tex2html_wrap_inline829 must contain tex2html_wrap_inline863 rather than tex2html_wrap_inline865 since we saw in Special Relativity that tex2html_wrap_inline867 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 tex2html_wrap_inline863 we just replace the Minkowski metric tex2html_wrap_inline873 by tex2html_wrap_inline875 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 tex2html_wrap_inline889 is a constant. The simplest operator that reduces to the vacuum field equations when T=0 takes the form


Now since tex2html_wrap_inline893 [ tex2html_wrap_inline895 in Special Relativity ], we require tex2html_wrap_inline897 .

Using tex2html_wrap_inline899 gives


Comparing this with the double contracted Bianchi identities 


we see that the constant tex2html_wrap_inline901 has to be tex2html_wrap_inline903 .

Thus we are led to the field equations of General Relativity: 




In general we can add a constant tex2html_wrap_inline905 so the field equations become


In a vacuum tex2html_wrap_inline907 , so taking the trace of the field equations we get


Since tex2html_wrap_inline909 and the Ricci scalar tex2html_wrap_inline911 we find that tex2html_wrap_inline913 , and substituting this back into the field equations leads to


We recover the previous vacuum equations if tex2html_wrap_inline915 . Sometimes tex2html_wrap_inline905 is called the vacuum energy density.

We have ten equations [ since tex2html_wrap_inline919 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.

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Next: The weak field approximation Up: Title page Previous: Introduction