The Path Integral, Perturbation Theory and Complex Actions

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d we have the time delay, on top of the ous notation and to be taken first order it the integration from 0 to L Shapiro 2 and then multiply by 2, to get the total lig travel in ✏) Z time. We assume r(t 0 time L )=r 0 (1 + ✏ sin !t delay 0 ), and we take ✏ ! GM 0. Then w get t 0 ⇡ t (x 2 + r0) 2 1/2 v Z g T /v L = L g + o(✏), and we have the time delay, on top of t 0 vg(x 2 2 + r Euclidean value v • v the time delay g T L = L, (in obvious notation GM 0(1 2 + ✏ sin !(t (x 2 + and to be taken first ord g T L = given by Z L ✓ in ✏) vg(x 2 2 + r0(1 2 + ✏ sin GM!(t (x 2 1+ r0) 2 1/2 /v g )) 2 ) d 1/2 GM 0 Z Z L ✓ ⇡ n !(t (x 2 + r L 2 v g T L = 0) 1/2 GM dx /v g )) 2 ) 1/21 0 vg 2 ⇡ 0 (r0✏ 2 0 ◆ sin !(t Z L (x 2 + r0) 2 1/2 /v g )) ⇡ (x 2 + r0) 2 3/2 GM dx vg(x 2 2 + r0(1 2 + ✏ sin !(t (x 2 + r0) 2 1/2 /v g )) 2 ) 1/2 The vg 2 t appearing in Equation (27) (x 2 is + ar 0) 2 function 3/2 ✓ ◆ GM already first 1 order, (r we 0✏ 2 sin can dx (27) !(t use the (x 2 + zero r0) 2 1/2 order /v g )) expr dx (2 no point (x 2 + inr 0) 2 doing 1/2 this further (x 2 + calculation. r0) 2 3/2 To get already first order, 0 vg 2 we can use the zero order expression t(x) =x/v s a function of x, but since the delay is ro order expression t(x) =x/v g . There is ation. To get an estimate of the order of ) we replace the sin ⇡ 1, assume r 0 ⌧ L s replacing the integration by a simple The t appearing inand Equation use simple (27) is dimensional a function of x, analysis but since replacing the delayt already • but first here order, we factor can use L to the get zero order expression t(x) =x/v g . There no and point usein simple doing this dimensional further calculation. analysis To replacing get an estimate the integration of the order by the factor size Lof to theget RHS in Equation (27) we replace the sin 1, assume r 0 ⌧ and use simple dimensional analysis replacing the integration by a simp • approximately factor L to get v g T L ⇡ GM ✓v g T L ◆⇡ GM ✓ r 1 0✏ 2 r 2 vg 1 0✏ 2 L 2 The length L is v g T L ⇡ GM almost ✓v g 2 11km◆ for L 2 the case of Ligo [2 r 2 1 0✏ The ✓length L is almost in actual 11km fact for the the time case of dependent (2 L 2 Ligo [2], thus part) the is second incom M v 2 g The length L is almost 11km for the case of Ligo [2], thus the second term(an v 2 Euclidean value v g T L = L, (in obvious notation and to be taken ◆ r0✏ 2 L 2 (r0✏ 2 sin ! (x (r0✏ 2 + r sin !(t 0) 2 1/2 (x 2 + r0) 2 1/2 / (x 2 + r0) 2 1/2 The t appearing in Equation (27) is a function of x, but since th the size of the RHS in Equation (27) we replace no point in doing this further calculation. To get an estimate of th the size of the RHS in Equation (27) we replace the sin ⇡ 1, assum in actual 1 fact the find time v g dependent T L ⇡ MG/v part) g 2 ⇠ (28) is 10 incompletely 28 m which is negligible simply Nevertheless, it may still be practical to us Tuesday, 2 April, 13 2 28
Conclusions • We have proposed a simple method for measuring the speed of propagation of gravity. • The null condition is perhaps the most problematic, it must be enforced to a higher precision than the effect we want to measure. • We believe there is a greater chance of seeing the effect using the potential rather than the force. • It would be an important validation to experimentally verify that gravity does indeed travel at a finite propagation velocity, and to measure that velocity. Tuesday, 2 April, 13
Page 1 and 2: A proposal to measure the speed of
Page 3 and 4: Motivation • Tests of Newton’s
Page 5 and 6: • The cylinder is about 60 cm lon
Page 7 and 8: • Here the lower disc is rotated
Page 9 and 10: Tuesday, 2 April, 13
Page 11 and 12: • The masses generate equal and o
Page 13 and 14: Tuesday, 2 April, 13 In a greatly e
Page 15 and 16: • Which can be written as: F = Gm
Page 17 and 18: • The denominators should be: F =
Page 19 and 20: • For a path difference of the or
Page 21 and 22: ulated from the gravitational analo
Page 23 and 24: • Then the solution is given by t
Page 25 and 26: • expanding then to lowest order
Page 27 and 28: • Hence we get e assumption that
Page 29 and 30: Perspectives for measurement • Th
Page 31 and 32: Negative mass and the gravitational
Page 33 and 34: • A sphere with a spherical cavit
Page 35 and 36: • Thus the potential is to lowest
Page 37 and 38: Q Bounce • This experiment places
Page 39: Interferometric gravitational wave

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