Takes the form 2h00 mp 2 2hij mp 2 c ij h0i

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Unformatted text preview: rameter, and if ðrÞ ¼ Àk=r for some constant k, then the angle for the bending of light at impact parameter b around the heavy source is given by ¼ 2ð1 þ Þ=b. Looking at Eq. (3.20), the massless graviton gives us the values ¼À GM ; r c ¼À GM ; massless graviton; r (3.22) 2 using 1=MP ¼ 8G. The PPN parameter is therefore ¼ 1 and the magnitude of the light bending angle for light incident at impact parameter b is ¼ 4GM ; b massless graviton: (3.23) For the massive case, the metric (3.13) is not quite in the right form to read off the Newtonian potential and light bending. To simplify things, we notice that while the massive gravity action is not gauge invariant, we assumed that the coupling to the test particle is that of GR, so this coupling is Kurt Hinterbichler: Theoretical aspects of massive gravity 682 gauge invariant. Thus we are free to make a gauge transformation on the solution h , and there will be no effect on the test particle. To simplify the metric (3.13), we go back to Eq. (3.11) and notice that the pi pj =m2 term in hij is pure gauge, so we can ignore this term. Thus our metric is gauge equivalent to the metric h00 ðxÞ ¼ 2M 1 eÀmr ; 3M P 4 r h0i ðxÞ ¼ 0; hij ðxÞ ¼ A. Vector example (3.24) M 1 eÀmr ij : 3M P 4 r We then have, in the small mass limit, 4 3 2 c ¼À 3 ¼À GM ; r GM ; r ij (3.25) massive graviton: These are the same values as obtained for the ! ¼ 0 BransDicke theory. The Newtonian potential is larger than for the massless case. The PPN parameter is ¼ 1 , and the magni2 tude of the light bending angle for light incident at impact parameter b is the same as in the massless case, ¼ 4GM ; b massive graviton: (3.26) If we want, we can make the Newtonian potential agree with GR by scaling G ! 3 G. Then the light bending would change 4 to ¼ 3GM=b, off by 25% from GR. What this all means is that linearized massive gravity, even in the limit of zero mass, gives predictions which are order 1 different from linearized GR. If nature were described by either one or the other of these theories, we would, by making a finite measurement, be able to tell whether the graviton mass is mathematically zero or not, in violation of our intuition that the physics of nature should be continuous in its parameters. This is the vDVZ discontinuity (van Dam and Veltman, 1970; Zakharov, 1970) [see also Iwasaki (1970) and Carrera and Giulini (2001)]. It is present in other physical predictions as well, such as the emission of gravitational radiation (van Nieuwenhuizen, 1973b). ¨ IV. THE STUCKELBERG TRICK We have seen that there is a discontinuity in the physical predictions of linear massless gravity and the massless limit of linear massive gravity, known as the vDVZ discontinuity. In this section, we expose the origin of this discontinuity. We see explicitly that the correct massless limit of massive gravity is not massless gravity, but rather massless gravity plus extra degrees of freedom, as expected since t...
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This document was uploaded on 09/28/2013.

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