Physics Letters B 749 (2015) 144–148
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Propagation peculiarities of mean field massive gravity
S. Deser
a,b
, A. Waldron
c
, G. Zahariade
d
a
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
b
Physics Department, Brandeis University, Waltham, MA 02454, USA
c
Department of Mathematics, University of California, Davis, CA 95616, USA
d
Department of Physics, University of California, Davis, CA 95616, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
18 April 2015
Accepted
16 July 2015
Available
online 28 July 2015
Editor:
M. Cveti
ˇ
c
Massive gravity (mGR) describes a dynamical “metric” on a fiducial, background one. We investigate
fluctuations of the dynamics about mGR solutions, that is about its “mean field theory”. Analyzing mean
field massive gravity (mGR) propagation characteristics is not only equivalent to studying those of the full
non-linear theory, but also in direct correspondence with earlier analyses of charged higher spin systems,
the oldest example being the charged, massive spin 3/2 Rarita–Schwinger (RS) theory. The fiducial and
mGR mean field background metrics in the mGR model correspond to the RS Minkowski metric and
external EM field. The common implications in both systems are that hyperbolicity holds only in a weak
background-mean-field limit, immediately ruling both theories out as fundamental theories; a situation
in stark contrast with general relativity (GR) which is at least a consistent classical theory. Moreover,
even though both mGR and RS theories can still in principle be considered as predictive effective models
in the weak regime, their lower helicities then exhibit superluminal behavior: lower helicity gravitons
are superluminal as compared to photons propagating on either the fiducial or background metric. Thus
our approach has uncovered a novel, dispersive, “crystal-like” phenomenon of differing helicities having
differing propagation speeds. This applies both to mGR and mGR, and is a peculiar feature that is also
problematic for consistent coupling to matter.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Consistency is a powerful tool for studying field theories. Al-
ready
classically, there are stringent conditions that are extremely
difficult to fulfill for systems with spin s > 1, the most important
exception being (s = 2, m = 0) general relativity. Key consistency
requirements are
(i) Correct
degree of freedom (DoF) counts.
(ii) Non-ghost
kinetic terms.
(iii) Predictability.
(iv) (Sub)luminal
propagation.
Requirements
(i) and (ii) are closely related (as are (iii) and (iv)).
Models whose constraints do not single out the correct propagat-
ing
DoF suffer from relatively ghost kinetic terms: the relevant
E-mail addresses: deser@brandeis.edu (S. Deser), wally@math.ucdavis.edu
(A. Waldron),
zahariad@ucdavis.edu (G. Zahariade).
example here is the sixth ghost excitation that plagues generic
massive gravity (mGR) theories [1]. The discovery that a class of
mGR models satisfied requirements (i) and (ii) generated a revival
of interest in massive spin 2 theories [2–7] even though failure of
the propagation requirements (iii) and (iv) were long known to be-
devil
higher spin theories [8,9].
The
predictability requirement is that initial data can be prop-
agated
to the future of spacetime hypersurfaces. In PDE terms,
this means that the underlying equations must be hyperbolic [10].
The final requirement, that signals cannot propagate faster than
light, can be imposed once the hyperbolicity requirement is sat-
isfied.
The classic example of a model that obeys requirements
(i) and (ii) as well as (iii) but only in a weak field region, is the
charged, massive, s = 3/2RS theory. Curiously enough, the propa-
gation
problems of this model were first discovered in a quantum
setting by Johnson and Sudarshan [11] who studied the model’s
canonical field commutators (this is easy to understand in ret-
rospect,
because field commutators and propagators are directly
related [12]). The first detailed analysis of the model’s propagation
characteristics was carried out by Velo and Zwanziger; our aim is
http://dx.doi.org/10.1016/j.physletb.2015.07.055
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
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