Dynamical tensors, gravity, and the SME

The following notes are based on a lecture given at the “Summer School on the CPTand Lorentz-violating Standard Model Extension”, organized by the IUCSS and held in Bloomington, IN from June 3–9, 2012. The goal of this talk is to address the following questions: what issues can arise when dynamical tensor fields are introduced into a theory? And how does the presence of gravity influence these issues? We know from Kostelecký’s “no-go” theorem that if our theory is to include both Lorentz-violating effects and gravity, the Lorentz violation must be due to a dynamical tensor field that spontaneously breaks Lorentz symmetry, rather than a fixed background tensor field. However, further features (some interesting, some problematic) can still arise if we are not careful about our choice of model. 1 Dynamics of the metric How do we describe the dynamics of the metric gμν in physics? The existence of an action principle is central to much of physics, so ideally we would like to define an action for the metric. To get to this point, though, we have to define a set of related notions leading up to this. I will merely sketch out the picture here; more details can be found in advanced texts in general relativity, such as Wald [1], Poisson [2], or Weinberg [3]. To describe the dynamics of anything on a curved manifold, we need to have a notion of what it means to take a derivative on a curved manifold. This is complicated by the fact that tensors defined at different points of the manifold essentially “live” in different spaces, i.e., the respective tangent spaces at the points in question. For our purposes, the most fruitful way to define the notion of such derivatives is to define a covariant derivative operator ∇μ, which acts on rank-n tensors to yield a rank-pn`1q tensor. From a pragmatic perspective, the covariant derivative is most easily defined in terms of the coordinate derivative “plus an extra piece” depending on the derivatives of the metric components; in other words, “∇μ “ Bμ ` Γμν.” For example, the covariant derivative of a vector field Aν will be ∇μA “ BμA ` ΓμρA.