Non Metric Mass

Mass terms are often introduced into wave equations: for example introducing a mass term for a scalar field gives the Klein-Gordon equation (2 −m2)φ = 0. Proceeding similarly with the metric of general relativity one recovers a vanishing mass term because gab;c = 0. For non-metric theories gab;c = −Qcab, so that the wave equation associated with the metric (2−M)gab = 0 no longer entails vanishing mass. This equation can be rewritten in the form M(x) + ∇̃aQ. + (ǫ+ d2 − 2)QaQa. = 0, where ǫ = 0, 1, 2, or3 and d is the dimension of the spacetime. For any given non-metric theory it is possible to insert the metric into this wave equation and produce a non-metric mass. Alternatively one can choose this equation to be a priori, and then try to construct theories for which it is the primary equation. This can be achieved using a simple Lagrangian theory. More ambitiously it is possible to investigate whether the introduction of non-metric mass has similar consequences to having a mass term in the Klein-Gordon and Proca equations: namely whether there are wave-like solutions, and what the rate of decay of the fields are. In order to find out a more intricate theory than the simple theory is needed. Such a theory can be found by conformally rescaling the metric and then arranging that the conformal parameter cancels out the object of non-metricity in the Schouten connection. Once this has been achieved one can conformally rescale general relativity and then compare the properties of the wave equations. On the whole its consequences are similar to m in the Klein-Gordon equation, the main difference being M is position dependent. The Proca m breaks gauge invairiance, nothing similar happens for the non-metric mass M or for the Klein-Gordon m. The dynamics of the rescaled theory are not clearly defined; the best definition criterion is the initial value problem and this is taken to signify well-defined dynamics.