We obtain the Misner-Sharp mass in the massive gravity for a four dimensional spacetime with a two dimensional maximally symmetric subspace via the inverse unified first law method. Significantly, the stress energy is conserved in this case with a wi
dely used reference metric. Based on this property we confirm the derived Misner-Sharp mass by the conserved charge method. We find that the existence of the Misner-sharp mass in this case does not lead to extra constraint for the massive gravity, which is notable in modified gravities. In addition, as a special case, we also investigate the Misner-Sharp mass in the static spacetime. Especially, we take the FRW universe into account for investigating the thermodynamics of the massive gravity. The result shows that the massive gravity can be in thermodynamic equilibrium, which fills in the gap in the previous studies of thermodynamics in the massive gravity.
Relations between radio surface brightness ($Sigma$) and diameter ($D$) of supernova remnants (SNRs) are important in astronomy. In this paper, following the work Duric & Seaquist (1986) at adiabatic phase, we carefully investigate shell-type superno
va remnants at radiative phase, and obtain theoretical $Sigma$-$D$ relation at radiative phase of shell-type supernova remnants at 1 GHz. By using these theoretical $Sigma$-$D$ relations at adiabatic phase and radiative phase, we also roughly determine phases of some supernova remnant from observation data.
Gravity/fluid correspondence becomes an important tool to investigate the strongly correlated fluids. We carefully investigate the holographic fluids at the finite cutoff surface by considering different boundary conditions in the scenario of gravity
/fluid correspondence. We find that the sonic velocity of the boundary fluids at the finite cutoff surface is critical to clarify the superficial similarity between bulk viscosity and perturbation of the pressure for the holographic fluid, where we set a special boundary condition at the finite cutoff surface to explicitly express this superficial similarity. Moreover, we further take the sonic velocity into account to investigate a case with more general boundary condition. In this more general case, two parameters in the first order stress tensor of holographic fluid cannot be fixed, one can still extract the information of transport coefficients by considering the sonic velocity seriously.
We derive the $Sigma$-$D$ relation of Galactic supernova remnants of shell-type separately at adiabatic-phase and at radiative-phase through two sets of different formulas, considering the different physical processes of shell-type remnants at both s
tages. Also statistics on Galactic shell-type remnants about 57 was made. Then we do some comparison with other results obtained before. It shows that all the best fit lines in the $Sigma$-$D$ relation plots newly are to some extent flatter than those derived by some authors at early time. Our theoretical and statistical outcomes are in somewhat good consistency.
