Geometric Representation of Space Time

Geometric Representation of Space Time - :r fro.tv <.~ 0...

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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 10 the motion. * The tangent to the world line at any point, being dxl dw = 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 () = dxl dw = 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 x' = 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 worldlin~~pn:ysl.cal 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; '" fro.tv <.~ +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 inertial frame to another. The time coordinate, e, of 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 is natural 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 will enable us to focus more clearly on the interdependence of Bpac,:: and time and its geometric representation.
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