RevModPhys.84.671

Gauge solutions to the original equations of motion

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Unformatted text preview: : 2 679 (2.51) Quantum mechanically, this results from the Fadeev-Popov gauge fixing procedure. We have L þ LGF ¼ 1h hh À 1hhh; 2 4 (2.52) whose equations of motion are (2.50). Note, however, that the classical gauge condition we have been using is not obtained as an equation of motion and must be imposed separately if solutions are to be compared. We can write the gauge fixed Lagrangian as L þ LGF ¼ 1 ~ h O; h , where 2 ~ O; ¼ h½1ð  þ   Þ À 1  Š: (2.53) 2 2 Going to momentum space and inverting, we obtain the propagator   Ài 1 1  ; D ;¼ 2 ð    þ    ÞÀ D À 2  p2 (2.54) ~ which satisfies the Eq. (2.43) with O in place of O. This propagator grows similar to $1=p2 at high energy. Comparing the massive and massless propagators, Eqs. (2.44) and (2.54), and ignoring for a second the terms in Eq. (2.44) which are singular as m ! 0, there is a difference in coefficient for the last term, even as m ! 0. For D ¼ 4, it is 1=2 vs 1=3. This is the first sign of a discontinuity in the m ! 0 limit. III. LINEAR RESPONSE TO SOURCES We now add a fixed external symmetric source T  ðxÞ to the action (2.1), Z 1 S ¼ dD x À @ h @ h þ @ h @ h À @ h @ h 2 1 1 þ @ h@ h À m2 ðh h À h2 Þ þ h T  : 2 2 (3.1) Àð Here  ¼ MP DÀ2Þ=2 is the coupling strength to the source.5 The equations of motion are now sourced by T , hh À @ @ h  À @ @ h  þ  @ @ h þ @ @ h À  hh À m2 ðh À  hÞ ¼ ÀT : (3.2) In the case m ¼ 0, acting on the left with @ gives identically zero, so we must have the conservation condition @ T ¼ 0 if there is to be a solution. For m Þ 0, there is no such condition. 5 The normalizations chosen here are in accord with the general pffiffiffiffiffiffiffiffi relativity definition T  ¼ ð2= ÀgÞL=g , as well as the normalization g ¼ 2h . Kurt Hinterbichler: Theoretical aspects of massive gravity 680 A. General solution to the sourced equations We now find the retarded solution of Eq. (3.2), to which the homogeneous solutions of Eq. (2.2) can be added to obtain the general solution. Acting on the equation of motion (3.2) with @ , we find  @ h À @ h ¼ 2 @ T : (3.3) m Plugging this back into Eq. (3.2), we find hh À @ @ h À m2 ðh À  hÞ  ¼ ÀT þ 2 ½@ @ T þ @ @ T À  @@T Š; m where @@T is short for the double divergence @ @ T  . Taking the trace of this we find h¼À   DÀ2 @@T: TÀ 4 m 2 ð D À 1Þ m DÀ1 Applying this to Eq. (3.3), we find   @ h ¼ À 2 @ T þ 2 @ T m ðD À 1Þ m  DÀ2 @ @@T; À4 m DÀ1 (3.4) The general solution for a conserved source is then Z dD p 1 eipx 2 ð2ÞD p þ m2     p p 1  þ 2 T ðpÞ ; (3.9)  T ðpÞ À DÀ1 m h ðxÞ ¼  (3.5) which when applied along with Eq. (3.4) to the equations of motion gives    @ @ 1  À 2 T ð@2 À m2 Þh ¼ À T À DÀ1 m   þ 2 @ @ T þ @ @ T m  ...
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