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Unformatted text preview: AST 204 13 February 2008 Lecture 4 A. More Spacetime and Geometry I. Like Time or Distance? Timelike and Spacelike Vectors The square of the interval (or modulus, as we shall call it for 4vectors not connecting two events) associated with a 4vector vectoru is, as we have seen, s 2 = u 2 t u 2 x u 2 y u 2 z = u 2 t u 2 , where u is an ordinary 3vector, the space part of vectoru . We should emphasize here that this separation is not a real geometric notionit is associated with a particular reference frame (coordinate system), and is no more natural than artificially separating a 3vector into a zcomponent and a 2vector in the x y plane. It is nevertheless often useful to do, just as similar artificial constructs in 3space are. It is clear that s 2 can have either sign, depending on the relative sizes of the time component and the length of the space part, and can, of course, also be zero . In the last case the 4vector is called null , and is, as we have seen, parallel to the path of a photon in spacetime. If the sign of s 2 is positive , the vector is called timelike ; the simplest example is a vector with no space part at all: the vector connecting two events occurring at the same place in some reference frame. If, on the other hand, s 2 is negative , the vector is called spacelike ; again the simplest example is a 3vector with no time componentthis is a vector connecting two simultaneous events in some frame, for example. In this case the length of the 3vector is ( s 2 ) 1 / 2 . The path of a particle in spacetime is called its world line . Now in a frame in which it is instantaneously at rest, however complex its motion, its velocity is zero, and so the 4vector tangent to its world line has only a time component, and is hence timelike . Furthermore, if we let be the time kept by a clock carried by this particle (the particles proper time , then the 4vector vectoru = dvectorx d = parenleftbigg dt d , d x d parenrightbigg clearly has modulus unity, because again in the frame in which the particle is instan taneously at rest the time component is 1 and the space part vanishes. vectoru is called the 4velocity of the particle; for very slowly moving particles it is simply (1 , v ). Newtonian mechanics is that regime in which all the particles satisfy the approximation that the veloc ity is much, much less than unity (c), so their 4velocities have time components negligibly different from 1 . Notice that if a particle starts from rest in some frame, the 4velocity has modulus unity and always does so, which means that it remains timelike, which means that  dx  < dt , or  dx /dt  < 1 Particles accelerating from rest in any frame can NEVER exceed the velocity of light. We will see physically what the reason for this is shortly....
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This note was uploaded on 02/26/2012 for the course AST 204 taught by Professor Knapp during the Spring '08 term at Princeton.
 Spring '08
 Knapp
 Astronomy, Space

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