The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.

We introduce a symmetry principle that forbids a bulk cosmological constant in six and ten dimensions. Then the symmetry is extended in six dimensions so that it insures absence of 4-dimensional cosmological constant induced by the six-dimensional curvature scalar, at least, for a class of metrics. A small cosmological constant may be induced in this scheme by breaking of the symmetry by a small amount. Cosmological constant problem is a long standing problem [1]. The problem can be stated as the huge discrepancy between the observational and the theoretically expected values of the cosmological constant [2] and the lack of understanding of its extremely small value [3]. Numerous schemes, to solve this problem, range from the models which employ supersymmetry, supergravity, superstrings, anthropic principles, modified general relativity, self-tuning mechanisms, quantum cosmology, extra dimensions, and combinations of these ideas [2,[4][5][6][7]. Although they shed some light on the direction of the solution of this problem, they have not given a wholly satisfactory, widely accepted answer to this question. Among these attempts extra-dimensional models become more popular because they give model builders more flexibility [5][6][7]. This is mainly due to the fact the no-go theorem of Weinberg [2] is intrinsically four-dimensional; for example, the equations of motion for a field constant in 4 dimensions may contain a contribution from extra-dimensional kinetic term in the Lagrangian hence making the Weinberg's argument non-applicable in higher dimensions. Moreover, the models where a four-dimensional space is embedded in a higher-dimensional space may have striking differences. For example, four-dimensional world may be embedded in extra dimensions in such a way that the 4-dimensional brane remains flat under energy density changes on the brane through the counter balance of the curvature due to the extra dimensions and the brane tension [7]. However, these models, although appealing, at present have some technical problems such as need for large extra

A more conventional realization of a symmetry which had been proposed towards the solution of cosmological constant problem is considered. In this study the multiplication of the coordinates by the imaginary number i in the literature is replaced by the multiplication of the metric tensor by minus one. This realization of the symmetry as well forbids a bulk cosmological constant and selects out 2(2n + 1)-dimensional spaces. On contrary to its previous realization the symmetry, without any need for its extension, also forbids a possible cosmological constant term which may arise from the extra-dimensional curvature scalar provided that the space is taken as the union of two 2(2n + 1)-dimensional spaces where the usual 4-dimensional space lies at the intersection of these spaces. It is shown that this symmetry may be realized through space-time reflections that change the sign of the volume element. A possible relation of this symmetry to the E-parity symmetry of Linde is also pointed out. Recently a symmetry [1-3] which may give insight to the origin of the extremely small value [4] of the cosmological constant compared to its theoretical value [2,5] was proposed. As in the usual symmetry arguments the symmetry forces the cosmological constant vanish and the small value of the cosmological constant is attributed to the breaking of the symmetry by a small amount. In [1] the symmetry is realized by imposing the invariance of action functional under a transformation where all coordinates are multiplied by the imaginary number i. It was found that this symmetry select out the dimensions D obeying D = 2(2n + 1) n = 0, 1, . . . , that is, D = 2, 6, 10, . . . and it gives some constraints on the form of the possible Lagrangian terms as well. Moreover that symmetry has more chance to survive in quantum field theory when compared to the usual scaling symmetry because the n-point functions are invariant under this symmetry. In this Letter we study a symmetry transformation where the coordinates remain the same while the metric tensor is multiplied by minus one. We show that this symmetry is equivalent to the one given in [1]. Although its results are mainly the same as [1] it is more conventional in its form, in the sense that the space-time coordinates remain real. On contrary to [1] we use the same symmetry to forbid 4-dimensional cosmological constant as well as to forbid a bulk cosmological constant. Moreover we show that the multiplication of the metric tensor by minus one may be related to a parity-like symmetry in the extra dimensions. We also discuss the relation of this symmetry to the anti-podal symmetry of Linde [6][7][8], whose relation to the previous realization of the present symmetry is discussed also in [3] for the 4-dimensional case.The symmetry principle given in [1] may be summarized as follows: the transformation

Two different realizations of a symmetry principle that impose a zero cosmological constant in an extra-dimensional set-up are studied. The symmetry is identified by multiplication of the metric by minus one. In the fist realization of the symmetry this is provided by a symmetry transformation that multiplies the coordinates by the imaginary number i. In the second realization this is accomplished by a symmetry transformation that multiplies the metric tensor by minus one. In both realizations of the symmetry the requirement of the invariance of the gravitational action under the symmetry selects out the dimensions given by D = 2(2n + 1), n = 0, 1, 2 . . . , and forbids a bulk cosmological constant. Another attractive aspect of the symmetry is that it seems to be more promising for quantization when compared to the usual scale symmetry. The second realization of the symmetry principle is more attractive in that it is possible to make a possible brane cosmological constant zero in a simple way by using the same symmetry, and the symmetry may be identified by reflection symmetry in extra dimensions.

In this study we reconsider the phenomenological problems related to tachyonic modes in the context of extra time-like dimensions. First we reconsider a lower bound on the size of extra time-like dimensions and improve the conclusion in the literature. Next we discuss the issues of spontaneous decay of stable fermions through tachyonic decays and disappearance of fermions due to tachyonic contributions to their self-energies. We find that the tachyonic modes due to extra time-like dimensions are less problematic than the tachyonic modes in the usual 4-dimensional setting because the most troublesome Feynman diagrams are forbidden once the conservation of momentum in the extra time-like dimensions is imposed.