Acausal Propagation. Thought of the Day 62.1

Whereas the Proca theory is the unique local linear massive variant of Maxwell’s electromagnetism, the most famous massive gravity with 6∞³ degrees of freedom, the Freund-Maheshwari-Schonberg massive gravity, is just one member (albeit the best in some respects) of a 2-parameter family of massive theories of gravity, all of which satisfy universal coupling. Adding a mass term involves adding a term quadratic in the potential; higher-order (cubic, quartic, etc.) self-interaction terms might also be present. The nonlinearity of the Einstein tensor implies, in contrast to the electromagnetic case, that there is no obviously best choice for defining the gravitational potential. While any such definition requires a background metric η_μν in order that the potential vanish when gravity is turned off (typically flat space-time), thus making massive theories bimetric, one can still choose among g_μν − η_μν , √-gg^μν − √-ηη^μν, g^μν − η^μν and so on, as well as various nonlinear choices such as g_μαη^αβg_βν − η_μν. In some cases the availability of nonlinear field redefinitions might make some expressions that look like mass term + interaction term with one definition of the gravitational potential, appear as a pure quadratic mass term with another definition; nonetheless the Einstein tensor remains nonlinear, no matter what definition of the potential is used. By contrast, the linearity of the Maxwell field strength tensor makes it natural to have a mass term that is also linear in A_μ in the field equations (and hence quadratic in A_μ in the Lagrangian density). While one can explore introducing nonlinear algebraic terms in A_μ describing self-interactions in electromagnetism, such terms induce acausal propagation if not chosen carefully.

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