Question

7. Consider a set of particles in a ring in flat space-time. The coordinates of any particle in the ring is given by x, y or equivalently by r, where I = r cos 0 and y=rsin. During the passage of a gravitational wave the proper distance between the origin (t=y=0) and any particle in the ring is given by: As = R [1+34, cos(wt) cos(20) (9) a) Show that if we change to a coordinate system X, Y where X = 1(1+5Acos(wt) and Y = yſ1 - A cos(wt)] then As? = x² + y2 for all particles in the ring. [3 marks] b) Show that the deformed shape of the ring is an ellipse (at least to first order in A1). HINT: The equation of an ellipse is 2+ =const. (10) q2 Also consider that t is constant when attempting this. You want to show that the ring forms an ellipse at any given point in time [2 marks] 8. Imagine that some cattoli

Answer 1

Ans where 8sin o ye A Gravitotional wave for proper distance betwcen the orgin (xay-o) As= R) I+A+ COs (wt) cos 20 X = x [i++ A + cs Cwt) ] Y= Y [i+% coswt)] then + COS x-+ y? an par ticles in the Agt + cos wt cos20)n (I+ cos wt. ) eyt (1-A cos w EjZ ost o(I+ R? +5into (1-ntcoswete cos" we ]} 2 TI+At os wt Co820) upkook)

J x?ag-R (1+A t cos wG cos 20) (i+ At Cos ot Cos 20) =45 9ien Hence prove d. b. The Deformed Shape of the rìng 9s an clipse const. -a (it tAtCaswt) 1- A+ losw t . 1+ At cos wt that when know (1+ t cas uot) k²(\+AL coswt)L p?l1-At cos w| f an eiripge. This is equation R (1+ coscwt), b = R6- cos at)