Tractography in Curvilinear Coordinates

Coordinate invariance of physical laws is central in physics, it grants us the freedom to express observations in coordinate systems that provide computational convenience. In the context of medical imaging there are numerous examples where departing from Cartesian to curvilinear coordinates leads to ease of visualization and simplicity, such as spherical coordinates in the brain's cortex, or universal ventricular coordinates in the heart. In this work we introduce tools that enhance the use of existing diffusion tractography approaches to utilize arbitrary coordinates. To test our method we perform simulations that gauge tractography performance by calculating the specificity and sensitivity of tracts generated from curvilinear coordinates in comparison with those generated from Cartesian coordinates, and we find that curvilinear coordinates generally show improved sensitivity and specificity compared to Cartesian. Also, as an application of our method, we show how harmonic coordinates can be used to enhance tractography for the hippocampus.

Implications of a frame dependent gravitational effective action for perturbations on the Robertson–Walker metric

In earlier work we showed that a frame dependent effective action motivated by the postulates of three-space general coordinate invariance and Weyl scaling invariance exactly mimics a cosmological constant in Robertson–Walker (RW) spacetimes. Here we study the implications of this effective action for small fluctuations around a spatially flat RW background geometry. The equations for the conserving extension of the modified stress-energy tensor can be integrated in closed form, and involve only the metric perturbation [Formula: see text]. Hence the equations for tensor and vector perturbations are unmodified, but there are Hubble scale additions to the scalar perturbation equations, which nonetheless admit no propagating wave solutions. Consequently, there are no modifications to standard gravitational wave propagation theory, but there may be observable implications for cosmology. We give a self-contained discussion, including an analysis of the restricted class of gauge transformations that act when a frame dependent effective action is present.

A frame-dependent gravitational effective action mimics a cosmological constant, but modifies the black hole horizon

A frame-dependent effective action motivated by the postulates of three-space general coordinate invariance and Weyl scaling invariance exactly mimics a cosmological constant in Robertson–Walker spacetimes. However, in a static spherically symmetric Schwarzschild-like geometry it modifies the black hole horizon structure within microscopic distances of the nominal horizon, in such a way that [Formula: see text] never vanishes. This could have important implications for the black hole “information paradox”.

The equivalence principle in a quantum world

We show how modern methods can be applied to quantum gravity at low energy. We test how quantum corrections challenge the classical framework behind the equivalence principle (EP), for instance through introduction of nonlocality from quantum physics, embodied in the uncertainty principle. When the energy is small, we now have the tools to address this conflict explicitly. Despite the violation of some classical concepts, the EP continues to provide the core of the quantum gravity framework through the symmetry — general coordinate invariance — that is used to organize the effective field theory (EFT).