World line


The world line 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" by the time dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their 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.

Usage in physics

In physics, a world line of an object 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 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 time. The world line of the Earth is helical in spacetime 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, upwards and the space coordinate, say 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 traveling through spacetime. The world sheet of an open string is a strip; that of a closed string 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 depending on one parameter. 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 history of a particle, observer or small object. One usually takes the proper time of an object or an observer as the curve parameter along the world line.

Trivial examples of spacetime curves

A curve that consists of a horizontal line segment, 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 may represent a particle at rest. A tilted line represents a particle with a constant coordinate speed. 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 and form a diagram depicting the emission of a photon by a particle that is subsequently observed by the observer.

Tangent vector to a world line: four-velocity

The four coordinate functions
defining a world line, are real functions of a real variable and can simply be differentiated in the usual calculus. Without the existence of a metric one can speak of the difference between a point on the curve at the parameter value and a point on the curve a little farther away. In the limit, this difference divided by defines a vector, the tangent vector of the world line at the point. It is a four-dimensional vector, defined in the point. It is associated with the normal 3-dimensional velocity of the object and therefore called four-velocity, or in components:
where the derivatives are taken at the point, so at.
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 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 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, 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 axis. In general, useful curves in spacetime can be of three types :
, 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.
of the observer. The small dots are specific events in spacetime. Note how the momentarily co-moving inertial frame changes when the observer accelerates.
At a given event on a world line, spacetime is divided into three parts.
Since a world line determines a velocity 4-vector that is time-like, the Minkowski form determines a linear function by 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 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, 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 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 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:
Because they oversimplify world lines, which traverse four-dimensional spacetime, into one-dimensional timelines, almost all purported science-fiction stories about time travel would not be possible in reality. Some device or superpowered person is generally portrayed as departing from one point in time, and with little or no subjective lag, arriving at some other point in time—but at the same literally geographic point in space, typically inside a workshop or near some historic site. However, in reality the planet, its solar system, and its galaxy would all be at vastly different spatial positions on arrival. Thus, the time travel mechanism would also have to provide instantaneous teleportation, with infinitely accurate and simultaneous adjustment of final 3D location, linear momentum, and angular momentum.
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.
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 closed time-like path as a strategic maneuver forms the backdrop and a main plot device of "Singularity Sky" by Charles Stross.