1. (60 Points) Consider synchronization of clocks recording the time coordinate between two infinites- imally close 'sta

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1 60 Points Consider Synchronization Of Clocks Recording The Time Coordinate Between Two Infinites Imally Close Sta 1 (76.74 KiB) Viewed 30 times

1. (60 Points) Consider synchronization of clocks recording the time coordinate between two infinites- imally close 'stationary observers Alice and Bob located at constant spatial locations A (r = const) and B (r-dx = const) respectively in a general spacetime with line element ds? = 900 (dez)+ 2go dz" dx' + gudr'dr6.1=1,2,3) The synchronization is carried out using a light signal emitted by Bob at B that is received at A by Alice and who then reflects it back to Bob. (a) From the line element, compute the coordinate time differences d.º(1) and drº(2) between the detection event at A and the two events of emission and absorption at B. Hint: Remember that for the light signal, ds? = 0. (b) The two clocks are said to be synchronized if the A-clock time zº corresponds to the B-clock time +0.70 where 6.rº = (1/2)[0r0(1) + d.10(2)). Show that: 820 = d. 900 (c) Now consider measurement of the spatial distance between Alice and Bob using data from clocks synchronized as in part (b). If dr is Alice's proper time that elapses between the emission and detection of the light signal by Bob, then the instantaneous spatial separation between Alice and Bob is defined as dl=ds/2 (in uits c=1). described by the line Show that the geometry of the instantaneous space defined in this way element: 90 ( 90.90% dard: 900 Note that the metric of the instantaneous 3-space is not the metric induced on the surface z=const. Hint: By observing the diagram, we can quickly see that Bob's proper time is given by drbo = [d.xº 2) – dz"(!)?. Consider finding Alice's proper time by using the appropriate com- ponent of the given metric 1 Alice Bob *° + dx012) x(0) dr dr dl A xº + dx 0(1) Bx' + dx' - const x = const