World line
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A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. Humans have a world line, starting at the time and place of their birth. The autobiography of George Gamow is entitled My world line. The log book of a ship is a description of the ship's world line, as long as it contains a time tag attached to every position. The world line allows one to calculate the speed of the ship, given a measure of distance (a so called metric), appropriate for the curved surface of the Earth. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" (such as an orbit in space or a trajectory of a truck on a road map) by the element of time.
The idea of world lines originates in physics and was pioneered by Einstein. The term is now most often used in relativity theories, (special relativity and general relativity). However, world lines are a general way of representing the course of events. The use of it is not bound to any specific theory.
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Usage in physics
In physics, a world line of an object (approximated as a point in space, e.g. a particle or observer) is the sequence of spacetime events corresponding to the history of the object. A world line is a special type of curve in spacetime. Below an equivalent definition will be explained: A world line is a time-like curve in spacetime. Each point of a world line is an event that can be labeled with the time and the spatial position of the object at that time.
For example, the orbit of the Earth in space is approximately a circle, a three dimensional (closed) curve in space: the Earth returns every year to the same point in space. However, it arrives there at a different (later) time. The world line of the Earth is a helix in spacetime (a curve in a four-dimensional space) and does not return to the same point.
Spacetime is the collection of points called events, together with a continuous and smooth coordinate system identifying the events. Each event can be labeled by four numbers: a time coordinate and three space coordinates and thus spacetime is a four-dimensional space. The mathematical term for spacetime is a four-dimensional manifold. The concept may be applied as well to a higher dimensional space. For easy visualisations of four dimensions, two space coordinates are often suppressed. The event is then represented by a point in a two-dimensional spacetime, a plane usually plotted with the time coordinate, say <math>t<math>, upwards and the space coordinate, say <math>x<math> horizontally.
A world line traces out the path of a single point in spacetime. A world sheet is the analogous two-dimensional surface traced out by a one dimensional line (like a string) traveling through spacetime. The worldsheet of an open string (with loose ends) is a strip; that of a closed string (a loop) is a cylinder.
World lines as a tool to describe events
A one-dimensional line or curve can be represented by the coordinates as a function of one parameter. Each value of the parameter corresponds to a point in spacetime and varying the parameter traces out a line. So in mathematical terms a curve is defined by four coordinate functions <math>x^a(\tau),\; a=0,1,2,3<math> (where <math>x^{0}<math> usually denotes the time coordinate) depending on one parameter <math>\tau<math>. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant.
Sometimes, the term world line is loosely used for any curve in spacetime. This terminology causes confusions. More properly, a world line is a curve in spacetime which traces out the (time)history of a particle, observer or small object. One usually takes the proper time of an object or an observer as the curve parameter <math>\tau<math> along the world line.
Trivial examples of spacetime curves
Worldlines1.jpg
A curve that consists of a horizontal line segment (a line at constant coordinate time), may represent a rod in spacetime and would not be a world line in the proper sense. The parameter traces the length of the rod.
A line at constant space coordinate (a vertical line in the convention adopted above) may represent a particle at rest (or a stationary observer). A tilted line represents a particle with a constant coordinate speed (constant change in space coordinate with increasing time coordinate). The more the line is tilted from the vertical, the larger the speed.
Two world lines that start out separately and then intersect, signify a collision or encounter. Two world lines starting at the same event in spacetime, each following their own path afterwards, may represent the decay of a particle in two others or the emission of one particle by another.
World lines of a particle and an observer may be interconnected with the world line of a photon (the path of light) and form a diagram which depict the emission of a photon by a particle which is subsequently observed by the observer (or absorbed by another particle).
Tangent vector to a world line, four-velocity
The four coordinate functions <math>x^a(\tau),\; a=0,1,2,3<math> defining a world line, are real functions of a real variable <math>\tau<math> and can simply be differentiated in the usual calculus. Without the existence of a metric (this is important to realize) one can speak of the difference between a point <math>p<math> on the curve at the parameter value <math>\tau_0<math> and a point on the curve a little (parameter <math>\tau_0+\Delta\tau<math>) further away. In the limit <math>\Delta\tau\rightarrow 0<math>, this difference divided by <math>\Delta\tau<math> defines a vector, the tangent vector of the world line at the point <math>p<math>. It is a four-dimensional vector, defined in the point <math>p<math>. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore called four-velocity <math>\vec{v}<math>, or in components:
- <math>\vec{v} = (v^0,v^1,v^2,v^3) =
\left( \frac{dx^0}{d\tau}\;,\frac{dx^1}{d\tau}\;, \frac{dx^2}{d\tau}\;, \frac{dx^3}{d\tau} \right)<math>
where the derivatives are taken at the point <math>p<math>, so at <math>\tau=\tau_0<math>
All curves through point p have a tangent vector, not only world lines. The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar. Therefore all tangent vectors in a point p span a linear space, called the tangent space at point p. For example, taking a 2-dimensional space, like the (curved) surface of the Earth, its tangent space at a specific point would be the flat approximation of the curved space.
Imagine a pendulum clock floating in space. We see in our mind in four stages of time; NOW, THEN, BEFORE, and THE PAST. Imagine the pendulum swinging and also the “Tick Tock” of the internal mechanism. Each swing from right to left represents a movement in space, and the period between a “Tick” to a “Tock” represents a period of time.
Now, if we image a wavy line between the different locations of the pendulum at the time intervals of: NOW, THEN, BEFORE and THE PAST. The line is a World line and is a representation of where the pendulum was in space-time at any point between the intervals. Time flows from The Past to Now.
World lines in special relativity
So far a worldline (and the concept of tangent vectors) is defined in spacetime even without a definition of a metric. We now discuss theories in which, in addition, a metric is defined.
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), called inertial coordinate systems. In such coordinate systems, the velocity of light is a constant. Spacetime now has a special type of metric imposed on it, the Lorentz metric and is called a Minkowski space, which allows for example a description of the path of light.
World lines of particles/objects at constant speed are called geodesics. 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 verical (time) axis. In general curves in spacetime with a given metric can be of three types:
- light-like curves, having at each point the speed of light. They form a cone in spacetime dividing it in two parts. The cone is a three dimensional hyperplane in spacetime (which shows as a line in drawings with two dimensions suppressed and as a cone in drawings with one spatial dimension suppressed.
World_line.png
> example of a light cone
Example of a light cone, the (3-dim) surface of all possible light rays arriving and departing from a point in spacetime, here depicted with one spatial dimension suppressed
- time-like curves, with a speed less than the speed of light. These curves must fall within a cone defines 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, 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 (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.
- The present is the region between the two light cones. Points in an observer's present are inacessible to her/him; 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, "Now you see it, now you don't," 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 Galilean/Newtonian theory, the present is thick; it is not a sheet but a volume.
- The present instant is defined for a given observer by a plane normal to her/his world line. It is the locus of simultaneous events, and is really three-dimensional, though it would be a 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, different observers, with differing velocities but coincident at an event or point in the spacetime, have world lines that cross each other at an angle determined by their relative velocities, and thus the present instant is different for them. This fact, that simultaneity depends on relative velocity, made problems for many scientists and laymen to accept relativity in the early days. The illustration with the light cones may make it appear that they can't be at 45 degrees to two lines that intersect, but it is true and can be demonstrated with the Lorentz transformation. The geometry is Minkowskian, not Euclidean.
World lines in general relativity
The use of world lines in general relativity is basically the same as in special relativity. However, now all coordinates systems are allowed. A metric exists and is determined by the mass distribution in spacetime. Again the metric defines light-like, space-like and time-like curves. Also in general relativity, world lines are time-like curves in spacetime, where time-like curves fall within the lightcone. However, lightcones are not necessarily inclined to 45 degrees.
World lines of free falling particles or objects (such as planets around the Sun or an astronaut in space) are called geodesics.
See also
Some specific type of world lines,