Since 2010 there has been renewed interest in bigravity after the development by
Claudia de Rham,
Gregory Gabadadze, and
Andrew Tolley (dRGT) of a healthy theory of massive gravity. Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric g_{\mu\nu} can only be written down with the help of a second metric, as the only nonderivative term that can be written using one metric is a
cosmological constant. In the dRGT theory, a nondynamical "reference metric" f_{\mu\nu} is introduced, and the interaction terms are built out of the
matrix square root of g^{-1}f. In dRGT massive gravity, the reference metric must be specified by hand. One can give the reference metric an
Einstein–Hilbert term, in which case f_{\mu\nu} is not chosen but instead evolves dynamically in response to g_{\mu\nu} and possibly matter. This
massive bigravity was introduced by
Fawad Hassan and
Rachel Rosen as an extension of dRGT massive gravity. The dRGT theory is crucial to developing a theory with two dynamical metrics because general bimetric theories are plagued by the
Boulware–Deser ghost, a possible sixth polarization for a massive graviton. The dRGT potential is constructed specifically to render this ghost nondynamical, and as long as the
kinetic term for the second metric is of the Einstein–Hilbert form, the resulting theory remains ghost-free. :S = -\frac{M_g^2}{2}\int d^4x \sqrt{-g}R(g )-\frac{M_f^2}{2}\int d^4x \sqrt{-f}R(f) + m^2M_g^2\int d^4x\sqrt{-g}\displaystyle\sum_{n=0}^4\beta_ne_n(\mathbb{X}) + \int d^4x\sqrt{-g}\mathcal{L}_\mathrm{m}(g,\Phi_i). As in standard general relativity, the metric g_{\mu\nu} has an Einstein–Hilbert kinetic term proportional to the
Ricci scalar R(g) and a minimal coupling to the matter Lagrangian \mathcal{L}_\mathrm{m}, with \Phi_i representing all of the matter fields, such as those of the
Standard Model. An Einstein–Hilbert term is also given for f_{\mu\nu}. Each metric has its own
Planck mass, denoted M_g and M_f respectively. The interaction potential is the same as in dRGT massive gravity. The \beta_i are dimensionless coupling constants and m (or specifically \beta_i^{1/2}m) is related to the mass of the massive graviton. This theory propagates seven degrees of freedom, corresponding to a massless graviton and a massive graviton (although the massive and massless states do not align with either of the metrics). The interaction potential is built out of the
elementary symmetric polynomials e_n of the eigenvalues of the matrices \mathbb K = \mathbb I - \sqrt{g^{-1}f} or \mathbb X = \sqrt{g^{-1}f}, parametrized by dimensionless coupling constants \alpha_i or \beta_i, respectively. Here \sqrt{g^{-1}f} is the
matrix square root of the matrix g^{-1}f. Written in index notation, \mathbb X is defined by the relation :X^\mu{}_\alpha X^\alpha{}_\nu = g^{\mu\alpha}f_{\nu\alpha}. The e_n can be written directly in terms of \mathbb X as :\begin{align} e_0(\mathbb X)&=1,\\ e_1(\mathbb X)&=[\mathbb X], \\ e_2(\mathbb X)&=\frac12\left([\mathbb X]^2-[\mathbb X^2]\right), \\ e_3(\mathbb X)&=\frac16\left([\mathbb X]^3-3[\mathbb X][\mathbb X^2]+2[\mathbb X^3]\right), \\ e_4(\mathbb X)&=\operatorname{det}\mathbb X, \end{align} where brackets indicate a
trace, [\mathbb X] \equiv X^\mu{}_\mu. It is the particular antisymmetric combination of terms in each of the e_n which is responsible for rendering the Boulware–Deser ghost nondynamical. == See also ==