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This chapter starts with a description of the intellectual investigations in worlds with extra spatial dimensions. It briefly introduces the idea of Kaluza–Klein theories and space compactification. Recent research has shown that extra dimensions may provide a solution to the naturalness problem. The most promising proposals are those of large extra dimensions and warped extra dimensions. The conceptual foundations of these theories are discussed, explaining the role of branes as a way of confining matter and forces. Finally, the chapter describes how LHC experiments can obtain empirical evidence in favour of the existence of extra spatial dimensions.

This Chapter contains a review of many important aspects of modern black hole theory and its applications. It begins with a general definition of a (not‐necessary stationary) black hole and formulation of the most important results on generic properties of black holes, including the Penrose theorem on the structure of the event horizon, the Hawking theorems on the topology and area of the event horizon and black hole uniqueness theorems. Gravitational radiation from black holes in a binary system and modern status and perspectives of the gravitation waves search from black holes and other compact sources are discussed. We also describe black hole models proposed for the explanation of the gamma‐ray bursts. Modeling of black hole properties, in particular their Hawking radiation, in the laboratory experiments is reviewed. We also discuss recent models with large extra dimensions and possibility of micro black hole creation in the collider experiments. This subject is directly connected with the problem of the higher dimensional black holes. Higher dimensional generalization of the Kerr metric, and a variety of new exact solutions for higher dimensional black objects with the non‐spherical topology of the horizon are discussed. The Chapter ends with remarks on two closely related problems on the wormhole and ‘time machine’ existence. It is shown that in order to create and support macroscopic objects of this type a new exotic form of the matter is requires. It seams that this and possible instabilities make the existence of such objects questionable at least at the present state of our knowledge. These and other fascinating open problems are still wait for their solution.

Gravity is the weakest interaction known in the Physics. Nevertheless it plays the leading role in astrophysics and cosmology. We discuss specific properties of gravity, explaining why it happens, and introduce the notion of a black hole. We describe final states of a star evolution and conditions, when a massive star collapses and forms a black hole. The Chapter contains also a brief review of the astrophysical evidences of the black hole existence, and describes methods used for identification of stellar mass and supermassive black holes. At the end of this Chapter we review the status of black holes in the modern theoretical physics, unsolved problems of black hole physics, and new ideas how to use black holes as probes of extra dimensions.

Many extensions to the Standard Model of elementary particles predict the existence of long-range interactions between neutral macrobodies in addition to Newtonian gravity. This chapter summarizes the constraints on these interactions obtained from the Casimir effect and compares them with the parallel progress in gravitational measurements. The availability of new precise measurements of the Casimir force, considered in Chapter 19, has provided further impetus for rapid progress in this direction. As a result, in the last few years, the previously known constraints on Yukawa interactions in the submicrometer range have been strengthened by up to ten thousand times. As shown in the chapter, the strongest constraints at the shortest separations follow from measurements of the Casimir force.

The Kaluza–Klein theory is a shining example of Einstein’s unification program. It not only unifies gravitation with electromagnetism in a GR theory with a 5D spacetime but also suggests a possible interpretation of the charge space and gauge symmetry as reflecting the existence of a compactified extra dimension. The ‘Kaluza–Klein miracle’ is the discovery that the GR field equation in the Kaluza–Klein spacetime is composed of 4D Einstein equations and the Maxwell equations. This is partially explained by the realization that a displacement in the extra dimension coordinate can be identified with the electromagnetic U(1) gauge transformation. From quantum mechanics one also learns that the signature of an extra dimension is the existence of a tower of particles, having identical spin and gauge quantum numbers, with increasing masses that are controlled by the compactification scale. Brief comments are offered of the more recent efforts in the construction of unified theories with extra dimensions.

Dimensional considerations are a powerful, integral part of physics. Starting with the familiar linear dimension, physics expands the idea in different ways. The essence of even complicated problems is often captured through simple analysis of the dimensions of relevant variables. In particular, invoking an extra fictitious dimension is a powerful method for solving or gaining insight into complex problems. Many examples are given, from all areas of physics, and from ones accessible to a high-school student to advanced ones of current research physics. These include a technique of mathematical analysis called Lagrange multipliers, electromagnetism, space-time, the symmetry structure of the hydrogen atom, and atomic and nuclear structure.

This chapter examines how it is possible to deduce consequences of particle physics for cosmology in light of the recent systematic exploration of the electroweak scale by experiments at the Large Hadron Collider (LHC). The two main results from the first phase of LHC, operation, namely, the discovery of a Higgs-like particle and the absence so far of new particles predicted by “natural” theories beyond the Standard Model (supersymmetry, extra dimensions and composite Higgs), are put in a historical context to highlight their importance, and are then presented in detail. For completeness, a short review of neutrino physics, which cannot be probed at the LHC, is also given. The ability of all these results to resolve the three fundamental questions of cosmology—the nature of dark energy and dark matter and the origin of matter–antimatter asymmetry—is discussed in each case.