Tensor-induced B modes with no temperature fluctuations

The recent indications for a tensor-to-scalar ratio $r \simeq 0.2$ from BICEP2 measurements of the cosmic microwave background (CMB) B-mode polarization present some tension with upper limits $r \lesssim 0.1$ from measurements of CMB temperature fluctuations. Here we point out that tensor perturbations can induce B modes in the CMB polarization without inducing any temperature fluctuations nor E-mode polarization whatsoever, but only, at the expense of violating the Copernican principle. We present this mathematical possibility as a new ingredient for the model-builder's toolkit in case the tension between B modes and temperature fluctuations cannot be resolved with more conventional ideas.