From Special Relativity to Feynman Diagrams.pdf

Of the tidal field acting on a given mass m is the

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of the tidal field, acting on a given mass m , is the tidal force . We stress that all these considerations have been obtained using the classical Newtonian formula for the gravitational field which is both static and non-relativistic. In order to show what the effect of tidal forces on an extended body is, let us consider two bodies, say A and B , subject to their mutual gravitational interaction, and let us assume that A is free falling in the gravitational field of B , like for instance an orbiting satellite. We also assume, for the sake of simplicity, that the free falling body A is spherical. We call S the frame of reference attached to A and S the one attached to B . Let the origin of S coincide with the barycenter r 0 of A , the z -axis coincide with direction joining r 0 to the center of mass of the attracting body B and the x and y -axes lie, as usual, in the plane orthogonal to the z -axis (see Fig. 3.2 ). With reference to this configuration we observe that x 3 0 z 0 r 0 , so that ( 3.17 ) gives: f z = 2 G Mm r 3 0 h z , (3.18) f x = − G Mm r 3 0 h x , (3.19)
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3.2 Tidal Forces 71 f y = − G Mm r 3 0 h y , (3.20) that is, in matrix notation: f x f y f z = m G M r 3 0 0 0 0 G M r 3 0 0 0 0 2 G M r 3 0 h x h y h z , (3.21) where the entries of the matrix on the right hand side are g i /∂ x k computed in r 0 . From ( 3.18 ) it follows that f z is attractive or repulsive according to the sign of h z ; that is, referring to Fig. 3.1 , it points downwards on the part of the body facing B ( z > 0 ), and upwards on the opposite side ( z < 0 ). The two horizontal compo- nents f x e f y , instead, are always attractive, that is they are directed towards the origin of S . The net result is an outward stress acting along the line joining A and B and an inward stress on the horizontal planes z = const . Tidal forces can be very strong in the astrophysical phenomena; for example the tidal forces exerted by the greatest planets of the solar system on their satellites induce a tidal heating due to the consequent internal friction; in the case of the Jupiter satellite Io this results in dramatic volcanic eruptions. 6 In the case of a deformable body, tidal forces deform a spherical body to the shape of an ellipsoid, the major axis lying along the A-B direction. This is in fact what happens in the case of the earth where the corresponding phenomenon, induced by the moon, gives rise to the ordinary oceanic tides, whence the denomination tidal forces has its origin. We may indeed think of the earth as the body A in free fall on the gravitational field of the moon, the body B . The “thin” layer represented by the oceans covering the earth’s surface is indeed deformable. It follows that on the side facing the moon and on the opposite side tidal forces give rise to high tides, while in the directions perpendicular tothelineearth-moontidal forces producelow tides (Fig. 3.1 ). Because of the bipolar character of these bulges and compressions, the periodicity of tides is of 12 h. We may make a crude estimation of the tidal size on the earth.
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