We show that the field equations of new massive gravity (NMG) consist of a massive (tensorial) Klein-Gordon-type equation with a curvature-squared source term and a constraint equation. We also show that, for algebraic type D and N spacetimes, the field equations of topologically massive gravity (TMG) can be thought of as the "square root" of the massive Klein-Gordon-type equation. Using this fact, we establish a simple framework for mapping all types D and N solutions of TMG into NMG. Finally, we present new examples of types D and N solutions to NMG.

We find the most general algebraic type N solution with non-vanishing scalar curvature, which comprises all type N solutions of new massive gravity in three dimensions. We also give the special forms of this solution, which correspond to certain critical values of the topological mass. Finally, we show that at the special limit, the null Killing isometry of the spacetime is restored and the solution describes AdS pp-waves.

n th root of a Lie algebra and its dual (that is fractional supergroup ) based on the permutation group S n invariant forms is formulated in the Hopf algebra formalism. Detailed discussion of S 3 -graded sl(2) algebras is done.

We show that for four-dimensional spacetimes with a non-null hypersurface orthogonal Killing vector and for a Chern-Simons (CS) background (non-dynamical) scalar field, which is constant along the Killing vector, the source-free equations of CS modified gravity decouple into their Einstein and Cotton constituents. Thus, the model supports only general relativity solutions. We also show that, when the cosmological constant vanishes and the gradient of the CS scalar field is parallel to the non-null hypersurface orthogonal Killing vector of constant length, CS modified gravity reduces to topologically massive gravity in three dimensions. Meanwhile, with the cosmological constant such a reduction requires an appropriate source term for CS modified gravity.

The UME Kibble balance project was initiated at the second half of 2014. During this period we have studied the theoretical aspects of Kibble balances in which an oscillating magnet generates AC Faraday's voltage in a stationary coil and constructed a trial version to implement this idea. The remarkable feature of this approach is that it can establish the link between the Planck constant and a macroscopic mass by one single experiment in the most natural way. Weak dependences on variations of environmental and experimental conditions, small sizes and other useful features offered by this novel approach reduce the complexity of the experimental setup. This paper describes the principles of oscillating magnet Kibble balance and gives details of the preliminary Planck constant measurements. The value of the Planck constant determined with our apparatus is with a relative standard uncertainty of 6 ppm.