Geometric Representation of Space Time

Geometric Representation of Space Time - " :r; '"

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- Space. Time Diagrams 189 World line of particle OIL-------------_% Section A.I and waxes of frame S orthogonal (perpendicular) to one another (Fig. A.I). If we wanted to represent the motion of a particle in this frame, we would draw a curve, called a world line, which gives the loci of space- time points corresponding 10the motion. * The tangent to the world line at any point, being dxldw = 1.. (dx/dt), is always inclined at an angle c . less than 45 0 with the time axis. For this angle (see Fig. A·I) is given by tan () = = uf c and we must have u < c for a material particle. The world line of a light wave, for which u = c, is a straight line making a 45 0 angle with the axes. Consider now the primed frame (8') which moves relative to S with a-velocity v along the common x-x' axis. The equation of motion of 5' relative to S can be obtained by selling x' = 0 (which locates the origin of S'); from Eq. A·I, we see that this corresponds to x = f3w (= vt). We draw the line x' = 0 (that IS, x = f3w) on our diagram (Fig. A.2) and note that, smce v < c and f3 < 1, the angle which this line makes with the w.axis, <!>(= tan- 1 {3), is less than 45°. Just as the w-axis corresponds to x = 0 and is the time axis in frame 5, so the line = 0 gives the time axis * Minkowski referred to space-time as "the world." Hence, events are world points and a collection , / of events giving the history of_a particle is a laws on the Interaction of par- ticles can be thought of as the geometri.c relations between their world lines. In this sense, Minkowski may be said to have geometrized phYSICS. I I i I I ,I 1- " ::r;'" <.~ +0 12c.l,'t: v , 't-r 1\ Supplementary Topic A The Geometric Repvesentation of Space. . Time 188 Notice the symmetry in this form of the equations. To represent the situation geometrically, we begin by drawing the x A-I Space.Time Diagrams In classical p'hysics~ the time coordinate ie unaffected by a trans- fO:J:111ClHon from one inertialframe to another. The time coordinate, e, one inertial 'system does not depend on.the space coordinates, x. y, ,z of another inertial system. .thetransformation equation being t; = t. In relativity.ihowever.repace alldtimear~.::interdependent. The time coordi- nate of one inertial system depends on both the time and the space coor- dinates of another inertial system, the transformation equation being e = [t - (vlc 2 )xJl VI v 21c2 , Hence, instead of treatmg space and time separately, as is quite properly done in classical theory, it isnatural in relativity to treat them together. J:I. Minkowski [1] was first to show clearly how this could be done. In what follows, we shall consider only one Bpace axis, the x-axis,and shall ignore the y and z axes. We lose 1'10 generality by this algebraic simplification and this procedure willenable us to focus more clearly on the interdependence of Bpac,:: and time and its geometric representation.
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This note was uploaded on 10/25/2007 for the course PHYS 1116 taught by Professor Elser, v during the Fall '05 term at Cornell University (Engineering School).

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Geometric Representation of Space Time - " :r; '"

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