*This preview shows
page 1. Sign up
to
view the full content.*

**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...

View
Full
Document