Multipole analysis for linearized $$f(R,{\mathcal {G}})$$ gravity with irreducible Cartesian tensors

Abstract

The field equations of f (R, G) gravity are rewritten in the form of obvious wave equations with the stressenergy pseudotensor of the matter fields and the gravitational field as its source under the de Donder condition. The linearized field equations of f (R, G) gravity are the same as those of linearized f (R) gravity, and thus, their multipole expansions under the de Donder condition are also the same. It is also shown that the Gauss-Bonnet curvature scalar G does not contribute to the effective stress-energy tensor of gravitational waves in linearized f (R, G) gravity, though G plays an important role in the nonlinear effects in general. Further, by applying the 1/r expansion in the distance to the source to the linearized f (R, G) gravity, the energy, momentum, and angular momentum carried by gravitational waves in linearized f (R, G) gravity are provided, which shows that G, unlike the nonlinear term R 2 in the gravitational Lagrangian, does not contribute to them either.

Individual Detection and Characterization of Non‐Electrocatalytic, Redox‐Inactive Particles in Solution by using Electrochemistry

“…The distortion of the electrode-solution charge balance at the electrical double layer (EDL) at the working electrode j solution by particles in solution has been demonstrated to give rise to measurable current-time events recorded with mercury electrodes and solid microelectrodes. [84][85][86][87][88][89][90][91][92] The events are spikeshaped, similar to those observed in some Faradaic impact experiments, though they show a 'peculiar' potential-dependence that enables identification. Thus, while Faradaic impacts due to a single redox reaction appear at potentials more positive (if it is an electro-oxidation) or more negative (in the case of an electro-reduction) than a 'threshold potential' (related to the formal potential and rate of the redox process) and they always have the same sign, the sign of capacitive spikes switches at a certain inversion potential that have been related to the potential of zero charge [87] and to the electronic properties/capacitance of the particles [86,88] (see Scheme 4).…”

Synthesis of Furo[3,2‐c]coumarins via Lewis Acid‐Mediated Multicomponent Tandem Reactions

“…Based on the aforementioned results and some related research, [9][10][11][12][13] a plausible mechanistic pathway is depicted in Scheme 3. Initially, the condensation of phenylglyoxal 2a with Full Paper…”

Cu(I)‐Catalyzed Cross‐Coupling of Diazo Compounds with Terminal Alkynes: An Efficient Access to Allenes

“…[27] In view of the importance of allenylphosphonates, in 2016 we developed the application of Cu(I)-catalyzed reaction for the synthesis of allenylphosphonates that involves the Cu(I)catalyzed cross-coupling of terminal alkynes with diazophosphonates ( Figure 20). [28] The reaction is operationally simple providing good yields over a wide range of substrates.…”