From Special Relativity to Feynman Diagrams.pdf

3 thanks to the advanced tech nological features of

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3 . Thanks to the advanced tech- nological features of modern experimental physics, the above ratio has been pushed to less than 10 15 . This justifies the theoretical assumption of the exact equality between inertial and gravitational mass: m I = m G , (3.3) that is the principle of equivalence in its weak form , can be safely assumed as one of the experimentally best established principles of theoretical physics. The great intuition Einstein had at the beginning of last century, was to realize the considerable importance of this seemingly curious “coincidence”, since it implies that locally, it is impossible to distinguish between the effects of a gravitational field and those of an accelerated frame of reference. To justify such a conclusion we illustrate a so-called “Gedankenexperiment”, that is a conceptual experiment, originally formulated by Einstein to be performed in a frame of reference attached to an elevator; in our era of space journeys, it seems however more appropriate to update this experiment by replacing Einstein’s elevator with a spaceship. Note that the time duration of the experiment inside the spaceship, together with its spatial extension, define a four-dimensional region of space–time. In the following we shall moreover restrict ourselves to the framework of the Newtonian theory of gravitation. Let us now describe this conceptual experiment, in the following four steps: (i) Suppose the spaceship, to be simply referred to by A , is initially placed, with the engines turned off, in a region of space which is far enough from any celestial body for the net gravitational force acting on it to be negligible. By definition this is an inertial frame of reference. If a physicist performs experiments within the spaceship, he will find that all bodies (if not subject to other kind of forces) move in a rectilinear uniform motion, according to Galileo’s principle of inertia. 2 We recall that the ratio between the electric and gravitational forces between two protons is of the order of 10 38 .
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3.1 Inertial and Gravitational Masses 65 (ii) Let us assume next that, at some stage, the spaceship reaches the proximity of a planet, and thus becomes subject to a gravitational acceleration of the form: g ( r ) = − G M r 2 u r , (3.4) M being the (gravitational) mass of the planet. In the presence of such an attraction A starts orbiting around the planet, thus becoming an accelerated frame of reference . The accelerated frame of reference defined by any massive body which is not subject to forces other than gravity is usually referred to as a free falling frame . Our spaceship A orbiting around the planet, is an example of a free falling frame. In this situation an observer inside A still finds that all bodies execute an inertial motion : Indeed, the same acceleration g ( r ), acting on the frame A , also acts on each body inside of it. It follows that the motion of bodies in A is inertial with respect to the spaceship itself, since their relative acceleration with respect to A is zero.
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