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Weak Gravitational Fields

Most gravitational fields encountered in the physical universe are weak. Exceptions are the strong fields near compact objects (black holes and neutron stars) or in the very early universe. It is remarkable that Einstein investigated weak gravitational fields quite exhaustively only one month after his first systematic exposition of GR (see [74]). Because of a computational error in his derivation of the so-called quadrupole formula for the power, emitted by a material source in the form of gravitational waves,1 he took the subject up again somewhat later and added some important considerations (see [75]).

5.1 The Linearized Theory of Gravity In this chapter, we consider systems for which the metric field is nearly flat (at least in a certain region of spacetime). Then there exist coordinate systems for which gμν = ημν + hμν ,

|hμν |  1.

(5.1)

For example, in the solar system, we have |hμν | ∼ |φ|/c2  GM /c2 R ∼ 10−6 . However, the field can vary rapidly with time, as is the case for weak gravitational waves. For such fields, we expand the field equations in powers of hμν and keep only the linear terms. The Ricci tensor is given, up to quadratic terms, by Rμν = ∂λ Γ λνμ − ∂ν Γ λλμ .

(5.2)

Here, we can use the linearized approximation  1 1 Γ αμν = ηαβ (hμβ,ν + hβν,μ − hμν,β ) = hα μ,ν + hα ν,μ − hμν ,α . 2 2

(5.3)

1 The term ‘gravitational waves’ (onde gravifique) first appeared in 1905 in a paper by Poincaré, in which he proposed the first Lorentz-invariant equation for the gravitational field.

N. Straumann, General Relativity, Graduate Texts in Physics, DOI 10.1007/978-94-007-5410-2_5, © Springer Science+Business Media Dordrecht 2013

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5

Weak Gravitational Fields

We adopt the convention that indices are raised or lowered with ημν or ημν , respectively. Then the Ricci curvature can be written as Rμν =

 1 λ h μ,λν − hμν − hλ λ,μν + hλ ν,λμ , 2

(5.4)

and for the scalar Riemann curvature we obtain R = hλσ ,λσ − h,

(5.5)

h := hλλ = ηαβ hαβ .

(5.6)

where

In the linear approximation, the Einstein tensor is given by 2Gμν = −hμν − h,μν + hλ μ,λν + hλ ν,λμ + ημν h − ημν hλσ ,λσ .

(5.7)

The contracted Bianchi identity reduces to Gμν ,ν = 0,

(5.8)

as one also sees by direct computation. As a consequence of (5.8) and the linearized field equations hμν + h,μν − hλ μ,λν − hλ ν,λμ − ημν h + ημν hλσ ,λσ = −16πGTμν ,

(5.9)

we obtain T μν,ν = 0.

(5.10)

Thus in this approximation the gravitational field generated by Tμν does not react back on the source. For example, if we consider incoherent dust, Tμν = ρuμ uν , then μ Eq. (5.10) implies, beside (ρuμ ),μ = 0, also uν u ,ν = 0 or duμ /ds = 0, so that the μ integral curves of u are straight lines. As a first step of an iterative procedure for weak fields, one can determine hμν from the linearized field equations (5.9) with the “flat” Tμν and compute the reaction on physical systems by setting gμν = ημν + hμν in the basic equations of Sect. 2.4. This procedure makes sense as long as the reaction is small. We