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General Relativity is today the best theory of gravity addressing a wide range of phenomena. Our understanding of physical laws, from cosmology to local scales, cannot be properly formulated without taking into account it. It is based on one of the most fundamental principles of Nature, the Equivalence Principle, which represents the core of the Einstein theory of gravity. The confirmation of its validity at different scales and in different contexts represents one of the main challenges of modern physics both from the theoretical and the experimental points of view. A major issue related to this principle is the fact that we actually do not know if it is valid at quantum level. Furthermore, recent progress on relativistic theories of gravity have to take into account new issues like Dark Matter and Dark Energy, as well as the validity of fundamental principles like local Lorentz and position invariance. Experiments allow to set stringent constraints on well established symmetry laws, on the physics beyond the Standard Model of particles and interactions, and on General Relativity and its possible extensions. In this review, we discuss precision tests of gravity in General Relativity and alternative theories and their relation with the Equivalence Principle. In the first part, we discuss the Einstein Equivalence Principle according to its weak and strong formulation. We recall some basic topics of General Relativity and the necessity of its extension. Some models of modified gravity are presented in some details. The second part of the paper is devoted to the experimental tests of the Equivalence Principle in its weak formulation. We present the results and methods used in high-precision experiments, and discuss the potential and prospects for future experimental tests.
We investigate leading order deviations from general relativity that violate the Einstein equivalence principle in the gravitational standard model extension. We show that redshift experiments based on matter waves and clock comparisons are equivalen
We present in detail the scientific objectives in fundamental physics of the Space-Time Explorer and QUantum Equivalence Space Test (STE-QUEST) space mission. STE-QUEST was pre-selected by the European Space Agency together with four other missions f
We review matter wave and clock comparison tests of the gravitational redshift. To elucidate their relationship to tests of the universality of free fall (UFF), we define scenarios wherein redshift violations are coupled to violations of UFF (type II
We propose an experiment to test the Weak Equivalence Principle (WEP) with a test mass consisting of two entangled atoms of different species. In the proposed experiment, a coherent measurement of the differential gravity acceleration between the two
We briefly summarize motivations for testing the weak equivalence principle and then review recent torsion-balance results that compare the differential accelerations of beryllium-aluminum and beryllium-titanium test body pairs with precisions at the