From Special Relativity to Feynman Diagrams.pdf

On the metric this implies g μν ? μν h μν o h 2

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require the gravitational field to be weak and static. On the metric this implies: g μν = η μν + h μν + O ( h 2 ), (3.61) g μν t = c g μν x 0 = 0 , (3.62) where h μν ( x ) is the first order deviation from the flat Minkowski space corresponding to the absence of gravitational field. In Appendix B we show that, in this case, the only non-vanishing component of the affine connection is: i 00 = 1 2 g i μ ( μ g 00 ) 1 2 η i j j h 00 = 1 2 i h 00 , (3.63) where i is a three-dimensional space index and we have used η i j = − δ i j . With these approximations the geodesic equation for the index μ = 0 gives dt d τ 1 , while for the spatial index μ = i we have: 1 c 2 d 2 x i dt 2 = − 1 2 i h 00 . (3.64) Equation ( 3.64 ) then coincides with the Newton equation if we set: φ c 2 = 1 2 h 00 . (3.65) where φ is the classical gravitational potential. With this identification, ( 3.64 ) becomes: d 2 x i dt 2 = − i φ. (3.66) In particular from ( 3.61 ) we find: g 00 = 1 + h 00 = 1 + 2 φ c 2 . (3.67)
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86 3 The Equivalence Principle 3.5.2 Time Intervals in a Gravitational Field Equation 3.67 allows us to evaluate how time intervals are affected by the presence of a gravitational field. Let us suppose that we are in a free falling frame, where the Minkowskian coordinates ξ α can be used. According to the principe of equivalence, a clock at rest in such a system measures a time interval which coincides with the proper time in the absence of gravity: d τ 2 = 1 c 2 η αβ d ξ α d ξ β = η 00 dt 2 = dt 2 (α, β = 0 , 1 , 2 , 3 ), (3.68) since, for a clock at rest, d ξ i dt = 0 . In any other frame of reference with coordinates x μ , like our laboratory, the gravitational field is present and the proper time interval will take the following form: d τ 2 = 1 c 2 g μν dx μ dx ν . (3.69) If in this frame the clock has four-velocity dx μ / dt = v μ , then the time interval dt between two consecutive (infinitely close) ticks satisfies the relation: d τ dt 2 = 1 c 2 g μν dx μ dt dx ν dt . (3.70) In particular, if the clock is at rest in the laboratory frame, that is if v i = 0 , we obtain: dt d τ = ( g 00 ) 1 2 . (3.71) The dilation factor on the right hand side of ( 3.71 ), however, cannot be observed, since the gravitational field affects in the same way the ticks of the standard clock and those of the clock being studied. However, the difference between dt 1 e dt 2 in two different points x 1 , x 2 can be observed; indeed ( 3.71 ) implies: dt 1 = d τ( g 00 ( x 1 )) 1 2 , (3.72) dt 2 = d τ( g 00 ( x 2 )) 1 2 , (3.73) so that: dt 2 dt 1 = g 00 ( x 1 ) g 00 ( x 2 ) 1 2 . (3.74)
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3.5 Motion of a Particle in Curved Space–Time 87 In particular, in the classical Newtonian limit, we may use ( 3.67 ) and, recalling ( 3.67 ), we obtain dt 2 dt 1 = 1 + 2 φ 1 c 2 1 + 2 φ 2 c 2 1 2 1 + 2 φ 1 c 2 1 2 1 2 φ 2 c 2 1 2 1 φ 2 φ 1 c 2 = 1 φ c 2 , (3.75) where we have defined φ 1 = φ( x 1 ) and φ 2 = φ( x 2 ) and used that φ c 2 is very small in most situations. 25 For example, if a clock is placed at the point x 1 far away from other bodies, so that no gravitational field is present, we have φ( x 1 ) = 0 , and therefore: dt 1 = d τ, (3.76) since g 00 ( x 1 ) = η 00 = 1 . The same clock placed at a point x 2 , e.g., on the earth’s surface, will tick time intervals dt 2 such that: dt 2 dt 1 = dt
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