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Unformatted text preview: Chapter 13 Uniform Acceleration 13.1 Events at the same proper distance from some event Consider the set of events that are at a fixed proper distance from some event. Locating the origin of space-time at this event, the equation for this set of events is: x 2- c 2 t 2 = d 2 (13.1) The parameter, d , is the proper distance of these events from the origin event. The origin event and the events on the curve are related by this distance d and thus for the set of events on the curve the origin is called the magic point and d is the distance from the magic point to the curve. In space-time, this is a two branch hyperbola with light cones emanating from the origin as the asymptotes. If we now consider only the branch that has x > 0, x = d 2 + c 2 t 2 , we have a single curve. In Figure 13.2, We plot several of them for different d . Since this equation is a form invariant under the Lorentz transformations, all inertial observers will have the same curve and Lorentz transformations will map points on the curve to points on the curve. By locating a light cone on the event at ( d, 0), we can see that all the events on the curve at later times are in the future; the curve is monotonically asymptotic to a light cone that is later in space-time. Thus all the events at later times on the curve are in the future of ( d, 0). Similarly all the events that are before t = 0 are in the past of ( d, 0). Thus the curve is time-like and is therefore a candidate for the motion of a material particle. In the next section, we will see that this is the trajectory of the uniformly accelerated object. 305 306 CHAPTER 13. UNIFORM ACCELERATION Figure 13.1: The locus of events that are at the same proper distance from the origin. 13.2 Uniformly accelerated motion Since this curve is time-like, it is a possible state of motion for a material particle. It is certainly a case of motion that is not uniform, not a straight line in space-time. For any observer in uniform motion, an object following this trajectory will appear to be approaching at a very rapid rate, almost- c , and slowing down until at some event it is as close as it will ever get and at rest with respect to the observer and then moving away so that at long times later it is receding at almost c . Since the Lorentz transformations are homogeneous and linear, lines through the origin are transformed into lines through the origin and space- like lines are transformed into space-like lines and similarly for time-like lines. Thus if you pick an event, say ( x , t ), on this curve, the line through it and the origin which is space-like can be transformed to the space-like line through ( d, 0) and (0 , 0) by the Lorentz transformation with v = c 2 t x ....
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- Spring '07