World line

The world line (or worldline) of an object is the path that object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from concepts such as an "orbit" or a "trajectory" (e.g., a planet's "orbit in space" or the "trajectory" of a car on a road) by the "time" dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their (relatively) more absolute position states—to reveal the nature of special relativity or gravitational interactions. 
The idea of world lines originates in physics and was pioneered by Hermann Minkowski. The term is now most often used in relativity theories (i.e., special relativity and general relativity).
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 relative to the sun. However, it arrives there at a different (later) time. The "world line" of the Earth is helical 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; 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 visualizations of four dimensions, two space coordinates are often suppressed. The event is then represented by a point in a Minkowski diagram, which is a plane usually plotted with the time coordinate, say formula_1, upwards and the space coordinate, say formula_2 horizontally.
As expressed by F.R. Harvey
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 world sheet of an open string (with loose ends) is a strip; that of a closed string (a loop) resembles a tube.
Once the object is not approximated as a mere point but has extended volume, it traces out not a "world line" but rather a world tube.
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 formula_3 (where formula_4 usually denotes the time coordinate) depending on one parameter formula_5. 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 that 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 formula_5 along the world line.
Trivial examples of spacetime curves.
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 its own path afterwards, may represent the decay of a particle into 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 depicting the emission of a photon by a particle that is subsequently observed by the observer (or absorbed by another particle).
Tangent vector to a world line: four-velocity.
The four coordinate functions formula_3
defining a world line, are real functions of a real variable formula_5 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 formula_9 on the curve at the parameter value formula_10 and a point on the curve a little (parameter formula_11) farther away. In the limit formula_12, this difference divided by formula_13 defines a vector, the tangent vector of the world line at the point formula_9. It is a four-dimensional vector, defined in the point formula_9. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore called four-velocity formula_16, or in components:
where the derivatives are taken at the point formula_9, so at formula_19.
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.
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), called inertial coordinate systems. 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 called a "spacetime metric", but since distinct events sometimes result in a zero value, unlike metrics in metric spaces of mathematics, the bilinear form is "not" a mathematical metric on spacetime.
World lines of freely falling particles/objects 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 vertical (time) axis. In general, useful curves in spacetime can be of three types (the other types would be partly one, partly another type):
At a given event on a world line, spacetime (Minkowski space) is divided into three parts.
Simultaneous hyperplane.
Since a world line formula_20 determines a velocity 4-vector formula_21 that is time-like, the Minkowski form formula_22 determines a linear function formula_23 by formula_24 Let "N" be the 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 formula_25 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 potentials.
World lines in general relativity.
The use of world lines in general relativity is basically the same as in special relativity, with the difference that spacetime can be curved. A metric exists and its dynamics are determined by the Einstein field equations and are dependent on the mass-energy distribution in spacetime. Again the metric defines lightlike (null), spacelike and timelike curves. Also, in general relativity, world lines are timelike curves in spacetime, where timelike curves fall within the lightcone. However, a lightcone is not necessarily inclined at 45 degrees to the time axis. However, this is an artifact of the chosen coordinate system, and reflects the coordinate freedom (diffeomorphism invariance) of general relativity. Any timelike curve admits a comoving observer whose "time axis" corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis. See also for example Eddington-Finkelstein coordinates.
World lines of free-falling particles or objects (such as planets around the Sun or an astronaut in space) are called geodesics.
World lines in quantum field theory.
Quantum field theory, the framework in which all of modern particle physics is described, is usually described as a theory of quantized fields. However, although not widely appreciated, it has been known since Feynman that many quantum field theories may equivalently be described in terms of world lines. The world line formulation of quantum field theory has proved particularly fruitful for various calculations in gauge theories and in describing nonlinear effects of electromagnetic fields.
World lines in literature.
In 1884 C. H. Hinton wrote an essay "What is the fourth dimension ?", which he published as a scientific romance. He wrote
A popular description of human world lines was given by J. C. Fields at the University of Toronto in the early days of relativity. As described by Toronto lawyer Norman Robertson:
Almost all science-fiction stories which use this concept actively, such as to enable time travel, oversimplify this concept to a one-dimensional timeline to fit a linear structure, which does not fit models of reality. Such time machines are often portrayed as being instantaneous, with its contents departing one time and arriving in another—but at the same literal geographic point in space. This is often carried out without note of a reference frame, or with the implicit assumption that the reference frame is local; as such, this would require either accurate teleportation, as a rotating planet, being under acceleration, is not an inertial frame, or for the time machine to remain in the same place, its contents 'frozen'.
Author Oliver Franklin published a science fiction work in 2008 entitled "World Lines" in which he related a simplified explanation of the hypothesis for laymen.
In the short story "Life-Line", author Robert A. Heinlein describes the world line of a person:
Heinlein's "Methuselah's Children" uses the term, as does James Blish's "The Quincunx of Time" (expanded from "Beep").
A visual novel named Steins;Gate, produced by 5pb., tells a story based on the shifting of world lines. Steins;Gate is a part of the "Science Adventure" series. World lines and other physical concepts like the Dirac Sea are also used throughout the series.
Neal Stephenson's novel Anathem involves a long discussion of worldlines over dinner in the midst of a philosophical debate between Platonic realism and nominalism.
Absolute Choice depicts different world lines as a sub-plot and setting device.
A space armada trying to complete a (nearly) closed time-like path as a strategic maneuver forms the backdrop and a main plot device of "Singularity Sky" by Charles Stross.