2 t p 39 t p d1 m h x 35 which

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Unformatted text preview:  @ @ 1  þ ðD À 2Þ 2 @@T : À DÀ1 m (3.6) Thus we have seen that the equation of motion (3.2) implies the following three equations:    @ @ 1  À 2 T ðh À m2 Þh ¼ À T À DÀ1 m   þ 2 @ @ T þ @ @ T m    @ @ 1  þ ðD À 2Þ 2 @@T ; À DÀ1 m    @ T þ 2 @ T @ h ¼ À 2 m ð D À 1Þ m  DÀ2 @ @@T; À4 m DÀ1   DÀ2 @@T: (3.7) h¼À 2 TÀ 4 m ð D À 1Þ m DÀ1 Conversely, it is straightforward to see that these three equations imply the equation of motion (3.2). Taking the first equation of (3.7) and tracing, we find     DÀ2 ðh À m2 Þ h þ 2 @@T ¼ 0: Tþ 4 m ðD À 1Þ m DÀ1 Under the assumption that ð@2 À m2 Þf ¼ 0 ) f ¼ 0 for any function f, the third equation is implied. This will be the case Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 with good boundary conditions, such as the retarded boundary conditions we impose when we are interested in the classical response to sources. The second equation of (3.7) can also be shown to follow under this assumption, so that we may obtain the solution by Fourier transforming only the first equation of (3.7). This solution can also be obtained by applying the propagator (2.44) to the Fourier transform of the source. Despite the absence of gauge symmetry, we will often be interested in sources which are conserved anyway, @ T  ¼ 0. When the source is conserved, and under the assumptions in the paragraph above, we are left with just the equation    @ @ 1  À 2 T : ð@2 À m2 Þh ¼ À T À DÀ1 m (3.8) where T  ðpÞ is the Fourier transform of the source, R T  ðpÞ ¼ dD xeÀipx T  ðxÞ. To get the retarded field, we integrate above the poles in the p0 plane. B. Solution for a point source We now specialize to four dimensions so that  ¼ 1=MP , and we consider as a source the stress tensor of a mass M point particle at rest at the origin T  ðxÞ ¼ M  3 ðxÞ; 00 T  ðpÞ ¼ 2M  ðp0 Þ: 00 (3.10) Note that this source is conserved. For this source, the general solution (3.9) gives h00 ðxÞ ¼ 2M Z d3 p ipÁx 1 e ; 3MP ð2Þ3 p2 þ m2 h0i ðxÞ ¼ 0; hij ðxÞ ¼   pi pj M Z d3 p ipÁx 1 e ij þ 2 : 3MP ð2Þ3 m p2 þ m2 (3.11) Using the formulas Z d3 p 1 1 eÀmr ; eipÁx 2 ¼ ð2Þ3 p þ m 2 4 r Z d3 p Z d3 p pi pj 1 eipÁx 2 ¼ À@i @j e i p Áx 2 3 2 ð2Þ ð2Þ3 p þ m2 p þm  1 eÀmr 1 ¼ ð1 þ mrÞij 4 r r 2  1 À 4 ð3 þ 3mr þ m2 r2 Þxi xj ; r (3.12) Kurt Hinterbichler: Theoretical aspects of massive gravity pffiffiffiffiffiffiffiffi where r  xi xi , we have 2M 1 eÀmr ; h0 i ð x Þ ¼ 0 ; 3M P 4 r  M 1 eÀmr 1 þ mr þ m2 r2 hij ðxÞ ¼ ij 3M P 4 r m2 r 2  1 À 2 4 ð3 þ 3mr þ m2 r2 Þxi xj : mr (3.13) Note the Yukawa suppression factors eÀmr characteristic of a massive field. For future reference, it is convenient to record these expressions in spherical coordinates for the spatial variables. Using the formula ½FðrÞij þ GðrÞxi xj Šdxi dxj ¼ 2 Gðr...
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This document was uploaded on 09/28/2013.

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