Now be invariant under the following gauge

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Unformatted text preview: hether it lies within the effective theory or can be consistently ignored. We now construct the full nonlinear gravitational ¨ Stuckelberg. This method was brought to our attention by Arkani-Hamed, Georgi, and Schwartz (2003) and Schwartz (2003), but was, in fact, known previously from work in string theory (Green and Thorn, 1991; Siegel, 1994). The full finite gauge transformation for gravity is @f @f g ðfðxÞÞ; @x @x 9 (6.1) Note that merely finding a ghost free interacting Lorentz invariant massive gravity theory is not hard totake; for instance, Uð; hÞ ¼ À2½detð  þ h  Þ À hŠ in Eq. (5.3), while letting the kinetic interactions be those of the linear graviton only. A Hamiltonian analysis just like that of Sec. II.A shows that h00 and h0i both remain Lagrange multipliers. The problem is that this theory does not go to GR in the m ! 0 limit, it goes to massless gravity. The real challenge is to construct a ghost free Lorentz invariant massive gravity that reduces to GR. 10 The objections of Alberte, Chamseddine, and Mukhanov (2011), Folkerts, Pritzel, and Wintergerst (2011), and Kluson (2011) are addressed by Hassan and Rosen (2011a), de Rham, Gabadadze, and Tolley (2011a, 2011c), respectively. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (6.3) with fðxÞ the gauge function. This is because the combination G is gauge invariant (not covariant). To see this, first transform11 g , @ Y @ Y g ðY ðxÞÞ ! @ Y @ Y @ f jY @ f jY g ðfðY ðxÞÞÞ (6.7) [here jY means the function is evaluated at Y ðxÞ] and then transform Y , ¨ A. Stuckelberg for gravity and the restoration of diffeomorphism invariance g ðxÞ ! (6.2) 11 The transformation of fields that depend on other fields is potentially tricky. To get it right, it is sometimes convenient to tease out the dependencies using delta functions. For example, suppose we have a scalar field ðxÞ, which we know transforms according to ðxÞ ! ðfðxÞÞ. How should ðY ðxÞÞ transform? To make it clear, write ðY ðxÞÞ ¼ Z dyðyÞðy À Y ðxÞÞ: (6.4) Now the field  appears with coordinate dependence, which we know how to deal with, ! Z dyðfðyÞÞðy À Y ðxÞÞ ¼ ðfðY ðxÞÞÞ: (6.5) Going through an identical trick for the metric, which we know transforms as g ðxÞ ! @f @f g ðfðxÞÞ; @x @x we find g ðY ðxÞÞ ! @ f jY @ f jY g ðfðY ðxÞÞÞ: (6.6) Kurt Hinterbichler: Theoretical aspects of massive gravity where gð0Þ ðxÞ is the absolute metric, which here is also the  background metric. We then expand G as in Eq. (6.10), and Y  as in Eq. (6.9). To linear order in h ¼ g À gð0Þ and  A , the expansion reads ! @ ½fÀ1 ðY ފ @ ½fÀ1 ðY ފ @ f jfÀ1 ðY Þ Â @ f jfÀ1 ðY Þ g ðY ðxÞÞ ¼ @ ½fÀ1 Š jY @ Y  @ ½fÀ1 Š jY  @ Y  @ f jfÀ1 ðY Þ @ f jfÀ1 ðY Þ g ðY ðxÞÞ H ¼ h þ rð0Þ A þ rð0Þ A ;   ¼   @ Y  @ Y  g ðY ðxÞÞ  ¼ @ Y  @ Y  g...
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