Abstract
The core of general relativity are the Einstein field equations, which relate curvature to energy and momentum. This chapter introduces curvature and the energy-momentum tensor to derive tensor equations between them that fulfill the correct Newtonian limit—the Einstein field equation. Finally, the small curvature limit in which the Einstein field equations are linear is worked out. This establishes the contact with the other classical field theory, Maxwell electrodynamics.
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- 1.
Multiplying (4.9) by \(g^{\lambda \nu }\), \(g^{\lambda \mu }\), and \(g^{\nu \kappa }\) yields \(\mathcal {R}_{\mu \kappa }=-g^{\lambda \nu } \mathcal {R}_{\mu \lambda \nu \kappa } = -g^{\lambda \nu } \mathcal {R}_{\lambda \mu \kappa \nu }=g^{\lambda \nu } \mathcal {R}_{\mu \lambda \kappa \nu }\) and \(g^{\lambda \mu } \mathcal {R}_{\lambda \mu \nu \kappa }=g^{\nu \kappa } \mathcal {R}_{\lambda \mu \nu \kappa }=0\).
- 2.
Note that we have tacitly toggled between \( \mathcal {T}^{00}\) and \(\mathcal {T}_{00}\), which here is OK. Generally, however, something like this will lead to the wrong result (cf. the example in Sect. 4.4)!
- 3.
As we shall see in the next section, the correct Newtonian limit is given by (4.51), which for \(\mathcal {T}_{\mu \nu } =0\) implies \(\vec \nabla ^2 \varphi =-\Lambda \). Our naive estimate is therefore off by a factor \(-2\) and the potential really is \(\varphi =-\frac{1}{6}\Lambda r^2\).
- 4.
Note: \(\bar{h} =\eta ^{\alpha \beta } \bar{h}_{\alpha \beta }= -h\).
- 5.
Lorenz, Ludvig Valentin, Danish physicist, *Helsingoer 18.1.1829, †Frederiksberg 9.5.1981.
- 6.
This is the proper generalisation of the Poisson equation (2.48) that is valid when \(\mathcal {T}_{00}=\rho \) is the only non-negligible element of the energy-momentum tensor. In the special case of a homogenous medium of density \(\rho \) and pressure P we have \(\mathcal {T}_{00}=\rho \) and \(\mathcal {T}_{ii}=P\) so that \(\Delta \varphi = 4 \pi G (\rho +3P)-\Lambda \). This will be a useful relation in cosmology.
- 7.
The following discussion from here to the next example may be omitted in a first reading.
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Hentschke, R., Hölbling, C. (2020). Field Equations of General Relativity. In: A Short Course in General Relativity and Cosmology. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-46384-7_4
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DOI: https://doi.org/10.1007/978-3-030-46384-7_4
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