Schlömilch's formula is generalized and applied to the thermal Casimir effect of a fermionic field confined a three-dimensional rectangular box. The analytic expressions of the Casimir energy and Casimir force are derived for arbitrary temperature and edge sizes. The low and high temperature limits and finite temperature cases are considered for the entire parameter space spanned by edge sizes and/or temperature. In the low temperature limit, it is found that for typical rectangular box, the effective 2-dimensional parameter space spanned by the two edge size ratios can be split into four regions. In one region, all three forces between three pairs of faces are attractive, and in another two regions, the force along the longest edge becomes repulsive and in the last region the force along both the longest and medium sized edge becomes repulsive. Three forces cannot be made simultaneously repulsive. For the waveguide under low temperature, the Casimir force along the longer side of the waveguide cross-section transforms from attractive to repulsive when the aspect ratio of the cross-section exceed a critical value. For the parallel plate scenario under low temperature, our results agrees with previous works. For high tempera limit, it is shown that both the Casimir energy and force approach zero due to the high temperature suppression of the quantum fluctuation responsible for the Casimir energy. For the finite temperature case, we separate the parameter space into four subcases (C1 to C4) and various edge size and temperature effects are analyzed. In general, we found that in all cases the Casimir energy is always negative, while the Casimir force at any finite or low temperature can be either repulsive or attractive depending on the sizes of the edges. For the case (C1) that is similar to parallel plates with relatively high temperature, it is found that the Casimir force is always attractive, regardless the change of the plate separation. At the given temperature, The Casimir energy/force densities approach the infinite parallel plate limit even when the plate edge size is 2 times the plate separation. For the case (C2) that is similar to a waveguide with relatively high temperature, the Casimir force along the longer side of the waveguide cross-section transform from attractive to repulsive when this side exceed a critical value. This critical point forms a boundary in the parameter space when the shorter edge of the waveguide cross-section changes and the boundary values decreases with respect to temperature increase. Case (C3) covers the low temperature parallel plate, typical rectangular box and waveguide geometries. For the waveguide case, the force along the waveguide longitude also transform from attractive to repulsive when the waveguide length exceed certain critical value. These critical values changes with respect to temperature in a nontrivial way. For the typical waveguide case (C4) at low temperature, the Casimir energy density along the longitudinal direction is a constant while force density dec...
Abstract. The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large r will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that an SU(2) Yang-Mills-Einstein SSS spacetime whose metric function is not known, will allow the existence of timelike COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics.
As a follow up of the seminal work by Guiot, Borquez, Deur, and Werner on “Graviballs and Dark Matter”, we explicitly show that in string theory, local and nonlocal higher derivative theories, as well as general asymptotically-free or finite theories, gravitationally interacting bound states can form when the energy is larger than the Planck energy. On the other hand, in higher derivative or nonlocal theories with interaction governed by a dimensionless or a dimensionful coupling constant, the bound states form when the energy is smaller than the Planck energy. Such bound states are allowed because of the softness of the scattering amplitudes in the ultraviolet region. Indeed, in such theories, the potential is finite while the force is zero or constant in r = 0. Finally, since the bound states that form in the early Universe may have an energy that ranges from the Planck mass to any arbitrarily large or small value, we argue that they can serve as dark matter candidates and/or as the seeds for the structure’s formation at large scale in the Cosmos.
As a follow up of the seminal work by Guiot, Borquez, Deur, and Werner on "Graviballs and Dark Matter", we explicitly show that contrary to Einstein's gravity, in string theory, local and nonlocal higher derivative theories, as well as general asymptotically-free or finite theories, gravitationally interacting bound states can form when the energy is larger than the Planck energy. On the other hand, in higher derivative or nonlocal theories with interaction governed by a dimensionless or a dimensionful coupling constant, the bound states form when the energy is smaller than the Planck energy. Such bound states are allowed because of the softness of the scattering amplitudes in the ultraviolet region. Indeed, in such theories, the potential is finite while the force is zero or constant in r = 0. Finally, since the bound states that form in the early Universe may have an energy that ranges from the Planck mass to any arbitrarily large or small value, we argue that they can serve as dark matter candidates and/or as the seeds for the structure's formation at large scale in the Cosmos.
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