Global Journal of Researches in Engineering, A: Mechanical & Mechanics, Volume 22 Issue 1

Figure 2 g) Shapiro Time Delay When v = c equation (2a) gives t c c v r K r d d 2 2 = =⋅ ta Therefore dt dr dr dc c r r K =  2 2 or dc dr cr K = 2 2 Integration leads to [ ] ∞ ∞     −= i i rc K c 2 , cr K c c i i 2 0 += − where c is now the speed of light where r tends to infinity. Or         − = 2 2 1 rc GM c c i (16) Consider the case of light grazing the Sun at a radius R s and calculate the journey time. As an approximation assume the path to be a straight line. Then, since dt dx c i = ( ) 2 / 21 rc GM dx dt c − = where 2 2 s R x r + = also as 2 2 rc GM << 1 dx R x c GM dt c s ∫ ∫         + + = 2 2 2 / 2 1           + + += x R x c GM x s 2 2 2 2 ln 2 Between the limits x = 0 to x and t = 0 to t we have the total time ( ) ( ) s s R c GM x R x c GM c x t 2 ln 2 2 2ln 2 3 2 2 3 − + + + = so the additional time due to the gravitational effect is         + + =∆ s s R x R x c GM t 2 2 3 ln 2 (17) This gives a time delay of 232 µ s for the double transit time from Earth to Venus. This is exactly the same as quoted by Bertotti, reference made to C. M. Will. The value for Mars is 247 µ s which is as quoted by Reasenberg, Shapiro et. al., is also given by the above equation. This shows that the speed of light is reduced by gravity. In the case of light grazing the Sun the reduction in speed/c o is 6 2 10 4 2 − ×≈ s s Rc GM , that is, only four parts per million. The variable speed of light has been incorporated into the basic equation however this would not have any major effect on the motion of material bodies. The valuation of the deflection of light is not changed. Because the light path is slightly curved the increase in path length will add to the delay but as the R s r x y n v=ds/dt ψ c n Global Journal of Researches in Engineering (A ) Volume XxXII Issue I Version I 7 Year 2022 © 2022 Global Journals Gravitomagnetics a Simpler Approach Applied to Dynamics within the Solar System