Tensors and Relativity: Chapter 6

The Einstein Tensor

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Next: Index Up: The Bianchi identities; Ricci Previous: The Ricci tensor

Let us apply the Ricci contraction to the Bianchi identities

equation3215

Since tex2html_wrap_inline3559 and tex2html_wrap_inline3894 , we can take tex2html_wrap_inline3896 in and out of covariant derivatives at will. We get:

equation3217

Using the antisymmetry on the indices tex2html_wrap_inline3456 and tex2html_wrap_inline3612 we get

equation3219

so

equation3221

These equations are called the contracted Bianchi identities .

Let us now contract a second time on the indices tex2html_wrap_inline3434 and tex2html_wrap_inline3886 :

equation3223

This gives

equation3225

so

equation3227

or

equation3229

Since tex2html_wrap_inline3906 , we get

equation3231

Raising the index tex2html_wrap_inline3612 with tex2html_wrap_inline3910 we get

equation3233

Defining

equation3235

we get

equation3237

The tensor tex2html_wrap_inline3912 is constructed only from the Riemann tensor and the metric, and it is automatically divergence free as an identity. It is called the Einstein tensor , since its importance for gravity was first understood by Einstein. We will see in the next chapter that Einstein's field equations for General Relativity are

equation3239

where tex2html_wrap_inline3914 is the stress- energy tensor. The Bianchi Identities then imply

equation3241

which is the conservation of energy and momentum.