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The task at hand is to describe the motion of a free (relativistic) point particle in space-time. One way to approach the problem is by using an action principle. Before we do that, let s set up the arena in which the particle moves. Let its motion be de ned with respect to space-time coordinates X where X 0 is the timelike coordinate (i.e., X 0 = ct) and X i where i 0 are the spacelike coordinates (say x, y, and z). While you are probably used to lowercase letters like x to represent coordinates, in string theory uppercase letters are used, so we will stick to that convention. Anticipating the fact that string theory takes place in a higher-dimensional arena, rather than the usual one time dimension and three spatial dimensions we are used to, we consider motion in a D-dimensional space-time. There is one time dimension but now we allow for the possibility of d = D 1 spatial dimensions. We reserve 0 to index the time dimension hence our coordinates range over = 0, . . . , d . Now, the motion or trajectory of a particle is described such that the coordinates are parameterized by , which parameterizes the world-line of the particle. That is, this is the time given by a clock that is moving or carried along with the particle itself. We can emphasize this parameterization by writing the coordinates as functions of the proper time: X = X ( ) (2.1)



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To describe distance measurements, we are going to need a metric, that is, a function which allows us to de ne the distance between two points. Here we will stick with special relativity and use the at space Minkowski metric which is usually denoted by . You may recall that the time and spatial components of the metric have different sign; the choice used is referred to as the signature of the metric. In string theory, it is convenient to place the negative sign with the time component, so in the case of d = 3 spatial dimensions we can write the Minkowski metric as a matrix 1 0 = 0 0 0 1 0 0 0 0 1 0 0 0 0 1

Figure 7.4 shows the path of Tarde s S-curve. Note that the curve levels off at the end. It does not curve down. Tarde saw that





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Feb 5, 2016 · C#, Silverlight, ASP.NET, Win32, Windows Forms, Microsoft Azure, .NET Framework, .NET Framework 4.0, Graphics Functions.​ Controls, Graphics, C#, ASP.NET, User Interface, Graphics and 3D, Microsoft Azure, Image, How to, BitmapImage.

(2.2)

More compactly, we can write = ( , + , + , + ). Generalizing to D-dimensional Minkowski space-time, we simply associate a plus sign with products of spatial coordinates. So the Lorentz invariant length squared of a vector is ( X )2 = X X = ( X 0 )2 + ( X 1 )2 + In nitesimal lengths or distances are described by ds 2 = dX dX = ( dX 0 )2 ( dX 1 )2 ( dX d )2 (2.4) + ( X d )2 (2.3)

Figure 7.4 The S-Curve (adapted from Tarde, Laws of Imitation)

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We include the minus sign out in front of the metric in Eq. (2.4) to ensure that ds = dX dX is real for timelike trajectories. With these notations in hand, we are ready to describe the trajectory of a free relativistic particle using the action principle. The action principle tells us that the relativistic motion of a free particle is proportional to the invariant length of the particles trajectory. That is, S = ds First let s gure out what the constant of proportionality is. EXAMPLE 2.1 Given that the action of a free, non-relativistic particle is S0 = dt (1/2)mv 2, where m is the mass of the particle and v is the particle velocity, determine the nature of the constant in Eq. (2.5). (2.5)

SOLUTION For simplicity, we consider motion in one spatial dimension. Now S = ds = dt 2 dx 2 = dt 1 dx 2 dt 2

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