From Special Relativity to Feynman Diagrams.pdf

Fig a1 the eotvös experiment 538 appendix a the

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Fig. A.1 The Eotvös’ experiment 538 Appendix A: The Eotvös’ Experiment
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Appendix B The Newtonian Limit of the Geodesic Equation In this section, we show that in the non-relativistic limit v ( c , by further assuming the gravitational field to be weak and stationary , the geodesic equation ( 3.56 ) reduces to the Newton equation of a particle in a gravitational field. As previously pointed out, the metric field g lm ð x Þ is the generalization of the Newtonian potential, and the statement that the gravitational field be weak and stationary is expressed by condi- tions ( 3.61 ) and ( 3.62 ), computing all quantities to first order in v = c and h . We first rewrite ( 3.56 ) by splitting the coordinate index l into l ¼ 0 and l ¼ i ð i ¼ 1 ; 2 ; 3 Þ : d 2 ð ct Þ d s 2 þ C 0 00 d ð ct Þ d s µ 2 þ 2 C 0 0 i d ð ct Þ d s dx i d s þ C 0 ij dx i d s dx j d s ¼ 0 ; ð B : 1 Þ d 2 x i d s 2 þ C i 00 d ð ct Þ d s µ 2 þ C i jk dx k d s dx j d s þ 2 C i 0 j dct d s dx j d s ¼ 0 : ð B : 2 Þ Since dx i d s µ ¶· dx 0 d s µ ¼ v i c ; ð B : 3 Þ one recognizes that the condition v = c ( 1 makes the last two terms of both equations negligible, so that ( B.1 ) and ( B.2 ) become: 1 c d 2 t d s 2 þ C 0 00 dt d s µ 2 ¼ 0 ð B : 4 Þ 1 c 2 d 2 x i d s 2 þ C i 00 dt d s µ 2 ¼ 0 : ð B : 5 Þ R. D’Auria and M. Trigiante, From Special Relativity to Feynman Diagrams , UNITEXT, DOI: 10.1007/978-88-470-1504-3, Ó Springer-Verlag Italia 2012 539
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Taking into account that the time derivative of g lm is zero for a stationary field, from (3.59) we find: C 0 00 ¼ 1 2 g 0 q ð± o q g 00 þ 2 o 0 g q 0 Þ ¼ ± 1 2 ð g 0 q ± h 0 q Þ o q g 00 þ O ð h 2 Þ ¼ ± g 00 o 0 h 00 þ O ð h 2 Þ ¼ O ð h 2 Þ ’ 0 ; ð B : 6 Þ C i 00 ¼ ± 1 2 g ij o j g 00 ¼ ± 1 2 ð g ij ± h ij Þ o j h 00 þ O ð h 2 Þ ¼ 1 2 o j h 00 þ O ð h 2 Þ ; ð B : 7 Þ where we have taken into account ( 3.61 ), ( 3.62 ), the fact that g ij ¼ ± d ij , and the inverse of relation ( 3.61 ), namely: g lm ¼ g lm ± h lm þ O ð h 2 Þ : ð B : 8 Þ Equation ( B.4 ) implies dt d s ¼ const :; ð B : 9 Þ so that d 2 x i d s 2 ¼ dt d s ± ² 2 d 2 x i dt 2 : Taking into account ( B.9 ), and ( B.7 ), ( B.5 ) becomes: d 2 x i dt 2 ¼ ± c 2 2 o i h 00 ; ð B : 10 Þ where the minus sign on the right hand side originates from the metric. This is exactly Newton’s equation of a particle in a gravitational field if we identify the Newtonian potential / ð x Þ with h 00 as follows: / c 2 ¼ 1 2 h 00 : ð B : 11 Þ Indeed, with such identification, (3.64) can be rewritten as: d 2 x i dt 2 ¼ ± o i / : ð B : 12 Þ Furthermore, from the previous equations, we also see that in the limit of non- relativistic, weak and static field we can write: g 00 ¼ 1 þ h 00 ¼ 1 þ 2 / c 2 : ð B : 13 Þ 540 Appendix B: The Newtonian Limit of the Geodesic Equation
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Appendix C The Twin Paradox The so called twin paradox is the seemingly contradictory situation arising from a naive application of the time dilation phenomenon discussed in Chap. 1 to the following conceptual experiment.
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