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lecture4 - AST 204 13 February 2008 Lecture 4 A More...

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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 4-vectors not connecting two events) associated with a 4-vector 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 3-vector, the space part of vectoru . We should emphasize here that this separation is not a real geometric notion–it is associated with a particular reference frame (coordinate system), and is no more ‘natural’ than artificially separating a 3-vector into a z-component and a 2-vector in the x y plane. It is nevertheless often useful to do, just as similar artificial constructs in 3-space 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 4-vector 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 3-vector with no time component–this is a vector connecting two simultaneous events in some frame, for example. In this case the length of the 3-vector 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 4-vector tangent to its world line has only a time component, and is hence timelike . Furthermore, if we let τ be the time kept by a clock carried by this particle (the particle’s proper time , then the 4-vector vectoru = dvectorx = parenleftbigg dt , d x 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 4-velocity 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 4-velocities have time components negligibly different from 1 . Notice that if a particle starts from rest in some frame, the 4-velocity 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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