Abstract
The importance of the gauge invariance of the linearized Riemann tensor lies in the fact that the presence or absence of a real gravitational field is characterized by the presence or absence of a nonvanishing Riemann tensor. This tensor represents the gravitational field, and in the weak field approximation we have, in it, an invariant characterization of the field, i.e., an expression for the field that is independent of which quasi-canonical coordinate system we are using.
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Notes
- 1.
The term ‘matter’ will here include electromagnetic fields.
- 2.
The minus sign attached to the ? is conventional. ? can, in fact, have either sign.
- 3.
\(\updelta g_{\mu\nu}\) is assumed to have compact support in spacetime.
- 4.
In flat spacetime, the world line is of course straight, but we make no use of this at this point
- 5.
The analogy goes deeper than this. The electromagnetic field tensor is a curl. The linearized Riemann tensor is a double curl. It is obtained by antisymmeterizing the second derivative ?h ??,?? /2 in ? and ? and in ? and ?.
- 6.
Note that these equations allow one to determine the pressure and energy distribution in the gas directly from a knowledge of ?? and ?, and hence of the function Z m .
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© 2011 Springer-Verlag Berlin Heidelberg
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DeWitt, B., Christensen, S.M. (2011). Production of Gravitational Fields by Matter. In: Christensen, S. (eds) Bryce DeWitt's Lectures on Gravitation. Lecture Notes in Physics, vol 826. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36911-0_8
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DOI: https://doi.org/10.1007/978-3-540-36911-0_8
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