From Special Relativity to Feynman Diagrams.pdf

Under the linear lorentz transformations relating

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under the linear Lorentz transformations, relating inertial frames, in the presence of gravity the implementation of the principle of relativity requires that: The laws of the Physics be covariant under general coordinate transformations ( 3.13 ). Note that covariance under general coordinate transformations means, as we discussed in the case of the Lorentz transformations, that the equations describing the physical laws have exactly the same form, albeit in the transformed variables, in every coordinate system. In other words: A theory including a treatment of the gravitational field must be generally covariant . It is really amazing that this conclusion, assumed by Einstein as the starting point for a relativistic theory of gravitation, can be drawn simply from the principle of equivalence, that is the equality between inertial and gravitational masses. Because of its general covariance the relativistic theory of gravitation is called general theory of relativity. 5 Linearity was then a consequence of the requirement that the principle of inertia holds in both the old and the transformed frames: A motion which is uniform with respect to one of them cannot be seen as accelerated with respect to the other.
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3.2 Tidal Forces 69 Fig.3.1 Tidal forces 3.2 Tidal Forces Wehaveseenthatitisalwayspossibleto locally eliminatetheeffectsofagravitational field by using a free falling frame . It is then extremely important to examine those effects of a gravitational field which cannot be eliminated in the free falling frame, that is which manifest themselves when we go beyond the crude equivalence implied by the approximate relation ( 3.5 ). In the present section we shall work purely in the classical limit, that is with no reference to the corrections implied by special relativity, and show that what remains of a gravitational force in the free falling frame are the tidal forces , see Fig. 3.1 . Let us start indeed from equation ( 3.8 ) which, in the classical case, assuming m I = m G , contains no approximations: g ( r ) g ( r 0 ) = a , (3.14) the position vector r defining a generic point in a neighborhood of r 0 . We shall also denote by h = r r 0 = ( h i ) the relative position vector between the two points, with components h k x k x k 0 , where k = 1, 2, 3 and ( x k ) ( x , y , z ), ( x k 0 ) ( x 0 , y 0 , z 0 ) (see Fig. 3.2 ). Let us compute, to the first order in h k , the i -th component of the gravitational acceleration in the free falling frame. From ( 3.14 ) it follows: a i = 3 k = 1 g i x k h = 0 h k ≡ − 3 k = 1 2 φ x k x i h = 0 h k + O ( | h | 2 ). (3.15) where i = 1, 2, 3 and φ denotes the gravitational potential. Taking into account that g i = − ∂φ x i = − G Mx i r 3 , (3.16)
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70 3 The Equivalence Principle Fig.3.2 Coordinates equation ( 3.15 ) becomes: a i = − G M r 3 0 3 k = 1 δ ik 3 x i 0 x k 0 r 2 0 h k . (3.17) The acceleration field a ( a i ), defined in ( 3.17 ), is the remnant, to order O ( | h | ), of the gravitational field in the free falling frame. We see that it is essentially given by the gradient of the gravitational field and is called the tidal field . Correspondingly, the force f = m a
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