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World line
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==World lines in special relativity== So far a world line (and the concept of tangent vectors) has been described without a means of quantifying the interval between events. The basic mathematics is as follows: The theory of [[special relativity]] puts some constraints on possible world lines. In special relativity the description of [[spacetime]] is limited to ''special'' coordinate systems that do not accelerate (and so do not rotate either), termed [[inertial frame of reference|inertial coordinate system]]s. In such coordinate systems, the [[speed of light]] is a constant. The structure of spacetime is determined by a [[bilinear form]] η, which gives a [[real number]] for each pair of events. The bilinear form is sometimes termed a ''spacetime metric'', but since distinct events sometimes result in a zero value, unlike metrics in [[metric space]]s of mathematics, the bilinear form is ''not'' a mathematical metric on spacetime. World lines of freely falling particles/objects are called [[geodesic]]s. In special relativity these are straight lines in [[Minkowski space]]. Often the time units are chosen such that the speed of light is represented by lines at a fixed angle, usually at 45 degrees, forming a cone with the vertical (time) axis. In general, useful curves in spacetime can be of three types (the other types would be partly one, partly another type): * '''light-like''' curves, having at each point the speed of light. They form a cone in spacetime, dividing it into two parts. The cone is three-dimensional in spacetime, appears as a line in drawings with two dimensions suppressed, and as a cone in drawings with one spatial dimension suppressed. [[Image:World line2.svg|thumb|upright=1.2|An example of a [[light cone]], the three-dimensional surface of all possible light rays arriving at and departing from a point in spacetime. Here, it is depicted with one spatial dimension suppressed.]] [[File:Lorentz transform of world line.gif|thumb|upright=1.2|The momentarily co-moving inertial frames along the trajectory ("world line") of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the [[spacetime]] of the observer. The small dots are specific events in spacetime. Note how the momentarily co-moving inertial frame changes when the observer accelerates.]] * '''time-like''' curves, with a speed less than the speed of light. These curves must fall within a cone defined by light-like curves. In our definition above: '''world lines are time-like curves in spacetime'''. * '''space-like''' curves falling outside the light cone. Such curves may describe, for example, the length of a physical object. The circumference of a cylinder and the length of a rod are space-like curves. At a given event on a world line, spacetime ([[Minkowski space]]) is divided into three parts. * The '''future''' of the given event is formed by all events that can be reached through time-like curves lying within the future light cone. * The '''past''' of the given event is formed by all events that can influence the event (that is, that can be connected by world lines within the past [[light cone]] to the given event). ** The '''lightcone''' at the given event is formed by all events that can be connected through light rays with the event. When we observe the sky at night, we basically see only the past [[light cone]] within the entire spacetime. * '''Elsewhere''' is the region between the two light cones. Points in an observer's '''elsewhere''' are inaccessible to them; only points in the past can send signals to the observer. In ordinary laboratory experience, using common units and methods of measurement, it may seem that we look at the present, but in fact there is always a delay time for light to propagate. For example, we see the [[Sun]] as it was about 8 minutes ago, not as it is "right now". Unlike the '''present''' in Galilean/Newtonian theory, the '''elsewhere''' is thick; it is not a 3-dimensional volume but is instead a 4-dimensional spacetime region. ** Included in "elsewhere" is the '''simultaneous hyperplane''', which is defined for a given observer by a [[space]] that is [[hyperbolic-orthogonal]] to their world line. It is really three-dimensional, though it would be a 2-plane in the diagram because we had to throw away one dimension to make an intelligible picture. Although the light cones are the same for all observers at a given spacetime event, different observers, with differing velocities but coincident at the event (point) in the spacetime, have world lines that cross each other at an angle determined by their relative velocities, and thus they have different simultaneous hyperplanes. ** The '''present''' often means the single spacetime event being considered. ===Simultaneous hyperplane=== Since a world line <math> w(\tau) \isin R^4</math> determines a velocity 4-vector <math> v = \frac {dw}{d\tau}</math> that is time-like, the Minkowski form <math> \eta(v,x)</math> determines a linear function <math> R^4 \rarr R</math> by <math> x \mapsto \eta( v , x ) .</math> Let ''N'' be the [[kernel (linear algebra)|null space]] of this linear functional. Then ''N'' is called the '''simultaneous hyperplane''' with respect to ''v''. The [[relativity of simultaneity]] is a statement that ''N'' depends on ''v''. Indeed, ''N'' is the [[orthogonal complement]] of ''v'' with respect to η. When two world lines ''u'' and ''w'' are related by <math> \frac {du}{d\tau} = \frac {dw}{d\tau}, </math> then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve the movement of information by light. For instance, the traditional electro-static force described by [[Coulomb's law]] may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve [[retarded potential]]s.
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