The Weak-Field Approximation
Derivation of the linearized Einstein equations, and applications of their solutions (in the limit of weak and static fields) to discuss two classical tests of general relativity: the bending of light rays and the radar-echo delay. A special topic (not av
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The Weak-Field Approximation
The Einstein equations introduced in the previous chapter connect the space–time curvature to the energy-momentum density of the matter sources. Such equations will receive here a definitive gravitational interpretation, obtained by comparing their linearized version with the equations of Newton’s gravitational theory. In this way we will be able to fix the value of the constant χ which controls the coupling of matter and geometry, and which we have previously introduced as a free dimensional parameter. By solving the linearized Einstein equations we will obtain the space–time geometry associated to a weak and static gravitational field: we will find, in this way, interesting dynamic effects and new types of interaction between sources and geometry which were absent in the Newtonian limit discussed in Chap. 5. We will discuss, in particular, two effects: the bending of light rays and the radar-echo delay. Both effects concern the propagation of electromagnetic signals in the gravitational field of our solar system; their experimental verification, in both cases, has provided important support to the idea of a geometric description of gravity, in general, and to the Einstein gravitational theory, in particular.
8.1 Linearized Einstein Equations Consider a space–time geometry which is only slightly different from that of the Minkowski space–time so that, in a Cartesian chart, the metric gμν can be expanded (0) around the Minkowski metric. We will set, to zeroth order, gμν = ημν and, to first (1) order, gμν = hμν . Neglecting all terms of order higher than the first we have then the expansion gμν ημν + hμν ,
|hμν | 1,
(8.1)
where the symmetric tensor hμν describes small geometric fluctuations which can be treated perturbatively. By inserting this expansion into the Einstein equations, M. Gasperini, Theory of Gravitational Interactions, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-88-470-2691-9_8, © Springer-Verlag Italia 2013
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The Weak-Field Approximation
and neglecting all terms of order h2 and higher, we can obtain a system of linear differential equations for hμν determining, in this approximation, the dynamical evolution of the deviations from the Minkowski geometry. Let us first notice, to this aim, that the covariant and contravariant components of h are connected (to first order) by the Minkowski metric: hμ ν = g να hμα = ηνα hμα + O h2 , (8.2) h ≡ hμ μ = g μν hμν = ημν hμν + O h2 . The inverse metric is then given by g μν ημν − hμν ,
(8.3)
g μα gνα = δνμ + hν μ − hμ ν + O h2 = δνμ + O h2 .
(8.4)
and satisfies the condition
Let us now compute the connection. To zeroth order we have the Minkowski metric, (0) with a vanishing connection Γμν α = 0. To first order in h, using Eqs. (8.1) and (8.3), we have 1 (1) β Γνα = ηβρ (∂ν hαρ + ∂α hνρ − ∂ρ hνα ). (8.5) 2 Since the non-zero component of Γ are proportional to the gradients of h, we can then neglect all contributions of type Γ 2 when computing the first-order expression of the
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