From Special Relativity to Feynman Diagrams.pdf

Reference for further reading see refs 1 11 12

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Reference For further reading see Refs. [1, 11, 12]
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Chapter 3 The Equivalence Principle 3.1 Inertial and Gravitational Masses In this section we discuss the principle of equivalence . We shall see that, besides allowing the extension of the principle of relativity to a generic , not necessarily inertial, frame of reference, it allows to define gravity, in a relativistic framework, as a property of the four-dimensional space–time geometry. The principle of equivalence , in the so-called weak form, asserts the exact equiv- alence between the inertial mass m I and the gravitational one m G . We recall that the inertial mass m I is defined through Newton’s second law of dynamics: F = m I a . (3.1) Its physical meaning is, as is well known, that of inertia of a body, that is its reluctance to be set in motion or, more generally, to change its velocity. The gravitational mass m G , on the other hand, enters the definition of Newton’s universal law of gravitation according to: F = − G m G M G r 2 u r (3.2) where G = 6 . 6732 × 10 11 Nm 2 / Kg 2 is the gravitational constant, m G and M G are the gravitational masses of the two attracting bodies. 1 For definiteness we refer to the situation where the mass M G attracts the mass m G . We see that the gravitational mass sets the strength of the gravitational force that, ceteris paribus, a given body exerts on another one. In this sense it would better deserve the name of gravitational charge , in analogy with Coulomb’s law of the electrostatic interaction, where the electric charge sets the strength of the electric interaction, the mathematical structures of the two laws being exactly the same. 1 It goes without saying that we are referring to two spherical bodies or to bodies whose dimensions are negligible with respect to their distance r . R. D’Auria and M. Trigiante, From Special Relativity to Feynman Diagrams , 63 UNITEXT, DOI: 10.1007/978-88-470-1504-3_3, © Springer-Verlag Italia 2012
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64 3 The Equivalence Principle However, besides the enormous quantitative difference between the strengths of gravitational and electrostatic forces 2 there is a major qualitative difference between them: While any two bodies have mass, they do not necessarily have charge. As a consequence the gravitational force is universal , while the electric force is not. It is important to note that the equality between inertial and gravitational mass, expressed by the principle of equivalence, is in some sense surprising, given the sub- stantial difference between these two physical concepts, which reflects into different operational definitions of their respective measures. It is, however, one of the best established results from the experimental point of view. Indeed a large number of experiments were devised, since Newton’s times, to ascertain the validity of this unexpected coincidence; among them, of particular importance from a historical point of view is the Eötvos experiment , of which we give a short description in Appendix A. The precision reached in this experiment is such that m m I | m G m I | m I , is less than 2 × 10
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