Generalized Vaidya Solutions and Misner-Sharp mass for nn-dimensional massive gravity

Dynamical solutions are always of interest to people in gravity theories. We derive a series of generalized Vaidya solutions in the $n$-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity with a singular reference metric. Similar to the case of the Einstein gravity, the generalized Vaidya solution can describe shining/absorbing stars. Moreover, we also find a more general Vaidya-like solution by introducing a more generic matter field than the pure radiation in the original Vaidya spacetime. As a result, the above generalized Vaidya solution is naturally included in this Vaidya-like solution as a special case. We investigate the thermodynamics for this Vaidya-like spacetime by using the unified first law, and present the generalized Misner-Sharp mass. Our results show that the generalized Minser-Sharp mass does exist in this spacetime. In addition, the usual Clausius relation $\delta Q= TdS$ holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state. We find that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution. Furthermore, the covariant generalized Minser-Sharp mass in the $n$-dimensional de Rham-Gabadadze-Tolley massive gravity is also derived by taking a general metric ansatz into account.Comment: 10 pages, no figure, version published in PR

The effects of massive graviton on the equilibrium between the black hole and radiation gas in an isolated box

It is well known that the black hole can has temperature and radiate the particles with black body spectrum, i.e. Hawking radiation. Therefore, if the black hole is surrounded by an isolated box, there is a thermal equilibrium between the black hole and radiation gas. A simple case considering the thermal equilibrium between the Schwarzschild black hole and radiation gas in an isolated box has been well investigated previously in detail, i.e. taking the conservation of energy and principle of maximal entropy for the isolated system into account. In this paper, following the above spirit, the effects of massive graviton on the thermal equilibrium will be investigated. For the gravity with massive graviton, we will use the de Rham-Gabadadze-Tolley (dRGT) massive gravity which has been proven to be ghost free. Because the graviton mass depends on two parameters in the dRGT massive gravity, here we just investigate two simple cases related to the two parameters, respectively. Our results show that in the first case the massive graviton can suppress or increase the condensation of black hole in the radiation gas although the $T-E$ diagram is similar like the Schwarzschild black hole case. For the second case, a new $T-E$ diagram has been obtained. Moreover, an interesting and important prediction is that the condensation of black hole just increases from the zero radius of horizon in this case, which is very different from the Schwarzschild black hole case.Comment: 9 pages, 4 figure

Research on Innovation Ecosystem of the Crowd Innovation Space Based on System Dynamics

The crowd innovation space is a new innovation business service platform, developed from the traditional incubator, including makerspace, entrepreneurship café and so on. It is essential for mass entrepreneurship and innovation to get full use of the creativity of the crowed innovation space and explore new impetus in New Economic Norm. Based on the traditional enterprise innovation ecosystem theory, this paper explores the connotation of innovation ecosystem of the crowd innovation space. And then, we use the knowledge to model the innovation ecosystem of the crowd innovation space with system dynamics. Based on the model, we explore the operational mechanism of innovation ecosystem of the crowd innovation space. Through the analysis of the operational mechanism and connotation, we put forward to some proposal to improve the effectiveness of the service ability in innovation of the crowd innovation space

Value of superb microvascular imaging ultrasonography in the diagnosis of carpal tunnel syndrome: Compared with color Doppler and power Doppler.

The aim of this study was to compare the value of superb microvascular imaging (SMI) in carpal tunnel syndrome (CTS) with that of color Doppler ultrasonography (CDUS) and power Doppler ultrasonography (PDUS).Fifty patients with symptomatic CTS and 25 healthy volunteers were enrolled. The cross-sectional area (CSA), CDUS score, PDUS score, and SMI score of the median nerve (MN) at the carpal tunnel were recorded. The value of different ultrasonography (US) diagnostic strategies was calculated.The blood flow display ratio in the MN of the healthy volunteers had no statistical difference between CDUS, PDUS, and SMI (20%, 32%, and 48%, respectively, P \u3e.05). The blood flow display ratio for SMI in patients was significantly higher than that of CDUS and PDUS (90%, 52%, and 60%, respectively,