Corresponding to the similarity between the Lorentz gauge ∂ µ A µ = 0 in electrodynamics and g µν Γ ρ µν = 0 in gravity, we show that the counterpart of the radiation gauge ∂ i A i = 0 is g ij Γ ρ ij = 0, in stead of other forms as discussed before. Particularly: 1) at least for a weak field, g ij Γ ρ ij = 0 fixes the gauge completely and picks out exactly the two physical components of the gravitational field;2) like A 0 , the non-dynamical components h 0µ are solved instantaneously; 3) gravitational radiation is generated by the "transverse" part of the energy-momentum tensor, similar to the transverse current J ⊥ . This "true" radiation gauge g ij Γ ρ ij = 0 is especially pertinent for studying gravitational energy, such as the energy flow in gravitational radiation. It agrees with the transverse-traceless (TT) gauge for a pure wave, and reveals remarkably how the TT gauge can be adapted in the presence of source.

Using the density functional theory (DFT) method at the B3LYP /6−311G (D) level, we studied how silicon doping affects the geometrical structure, stability, and electronic and spectral properties of magnesium clusters. The stable isomers of SiMg n (n = 1‐12) clusters were calculated by searching numerous initial configurations using the CALYPSO program. The geometrical structure optimization shows that most stable SiMg n (n = 3‐12) clusters are three‐dimensional. In addition, geometrical structure growth patterns show that some structures of SiMg n clusters can be directly formed by replacing one Mg atom in the corresponding Mg n + 1 cluster with one silicon atom, such as SiMg8 and Mg9 clusters. The stability of SiMg n clusters is analyzed by calculating the average binding energy, fragmentation energy, and second‐order energy difference. The results show that SiMg n clusters with n = 5 and 8 are more stable than others. MO contents analysis show that the Si 3p‐orbitals and Mg 3s‐orbital are mainly responsible for the stability of these two clusters. The results of the natural charge population (NCP) and natural electronic configure (NEC) analysis of the electronic properties reveal that the charges in SiMgn (n = 1‐12) clusters transfer from magnesium atoms to silicon frame, and electronic charge distributions are primarily governed by s‐ and p‐orbital interactions. In addition, the Vertical ionization potential (VIP), vertical electron affinity (VEA), and chemical hardness of ground sates of SiMg n (n = 1‐12) clusters were studied in detail and compared with the experimental results. The conclusions show that the chemical hardness of most SiMg n clusters are lower than that of pure Mg n + 1 (n = 1‐12) clusters, except for n = 1 and 8. This indicates that the doping of silicon atom can always reduce the chemical hardness of pure magnesium clusters. Finally, the infrared and Raman spectral properties of SiMg5 and SiMg8 clusters were calculated and discussed in detail.

No abstract

We discuss various proposals of separating a tensor field into pure-gauge and gauge-invariant components. Such tensor field decomposition is intimately related to the effort of identifying the real gravitational degrees of freedom out of the metric tensor in Einstein's general relativity. We show that, as for a vector field, the tensor field decomposition has exact correspondence to, and can be derived from, the gauge-fixing approach. The complication for the tensor field, however, is that there are infinitely many complete gauge conditions, in contrast to the uniqueness of Coulomb gauge for a vector field. The cause of such complication, as we reveal, is the emergence of a peculiar gauge-invariant pure-gauge construction for any gauge field of spin ≥ 2. We make an extensive exploration of the complete tensor gauge conditions and their corresponding tensor field decompositions, regarding mathematical structures, equations of motion for the fields, and nonlinear properties. Apparently, no single choice is superior in all aspects, due to an awkward fact that no gauge-fixing can reduce a tensor field to be purely dynamical (i.e., transverse and traceless), as can the Coulomb gauge in a vector case.

No abstract