The Schwarzschild-de Sitter solution in the Einstein theory with a positive cosmological constant Λ = m 2 /α becomes an exact solution to the dRGT non-linear massive gravity theory with the mass parameter m when the theory parameters α and β satisfy the relation β = α 2 . We study the perturbative behaviour of this black hole solution in the non-linear dRGT theory with β = α 2 . We find that the linear perturbation equations become identical to those for the vacuum Einstein theory when they are expressed in terms of the gauge-invariant variables. This implies that this black hole is stable in the dRGT theory as far as the spacetime structure is concerned in contrast to the case of the bi-Schwarzschild solution in the bi-metric theory. However, we have also found a pathological feature that the general solution to the perturbation equations contain a single arbitrary function of spacetime coordinates. This implies a degeneracy of dynamics in the Stückelberg field sector at the linear perturbation level in this background. Physical significance of this degenercy depends on how the Stückelberg fields couple observable fields. 2/26Conversely, if a solution to the field equations (6) satisfies this relation, it must be a solution to the vacuum Einstein equations with Λ. Hence, if it is spherically symmetric, the solution must be diffeomorphic to the Schwarzschild-de Sitter solution. Note that this does not implies the uniqueness of the solution because although the matrices of two solutions are related 4/26

We analyze the phenomena of spontaneous symmetry breaking in Quantum Finance by using as a starting point the Black-Scholes (BS) and the Merton-Garman (MG) equations expressed in the Hamiltonian form. In this scenario the martingale condition (state) corresponds to the vacuum state which becomes degenerate when the symmetry of the system is spontaneously broken. We then analyze the broken symmetries of the system and we interpret from the perspective of Financial markets the possible appearance of the Nambu-Goldstone bosons.

I derive general conditions in order to explain the origin of the Vainshtein radius inside dRGT. The set of equations, which I have called "Vainshtein" conditions are extremal conditions of the dynamical metric (gμν) containing all the degrees of freedom of the theory. The Vainshtein conditions are able to explain the coincidence between the Vainshtein radius in dRGT and the scale [Formula: see text], obtained naturally from the Schwarzschild de-Sitter (S-dS) space inside general relativity (GR). In GR, this scale was interpreted as the maximum distance in order to get bound orbits. The same scale corresponds to the static observer position if we want to define the black hole temperature in an asymptotically de-Sitter space. In dRGT, the scale marks a limit after which the extra degrees of freedom of the theory become relevant.

We demonstrate that the spontaneous symmetry breaking condition, when applied to any system, have a correspondence with the Quantum Yang-Baxter equations. This correspondence guarantees the appropriate dispersion relation and the appropriate counting for the Nambu-Goldstones bosons.