# I Energy-Momentum Tensor from the Action Principle

1. Feb 17, 2017

### davidge

I'm reading Hawking-Ellis 1973 book on General Relativity and I'm finding it very interesting. I like the way they present the theory. At the beggining of the book, they derive the energy-momentum tensor from the Lagrangian of the system, where the Lagrangian is a function of matter fields $\Psi_{(i)\ c...d}^{a...b}$ ($(i)$ label the fields), their first covariant derivatives, and the metric. He then goes through the construction of the integral, by imposing the usual condition that the action should not vary. He arrives at and that it's possible to write $\Psi_{(i)\ c...d; \ e}^{a...b}$ in terms of $\Delta(g_{bc})_{;d}$ and integrate by parts to get .

Can anyone show how to get this final integral from 1?

The another thing is that they show a variety of examples of energy-momentum tensor for different fields, but they don't show how to get the Lagrangian for that fields in first place. I would like to know how to derive the Lagrangian say, for Electromagnetism.