In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. 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 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 Einstein. The term is now most often used in relativity theories (i.e., general relativity and special relativity). 2-dimensional renderings (ie. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In psychology and the cognitive sciences, perception is the process of acquiring, interpreting, selecting, and organizing sensory information. ... Look up position in Wiktionary, the free dictionary. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... “Gravity” redirects here. ... This article needs additional references or sources for verification. ... “Einstein” redirects here. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

However, world lines are a general way of representing the course of events. The use of it is not bound to any specific theory. Thus in general usage, a world line is the sequential path of personal human events (with time and place as dimensions) that marks the history of a person — perhaps starting at the time and place of one's birth until their death. 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.

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. This article needs additional references or sources for verification. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ...

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. A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ...

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 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 t, upwards and the space coordinate, say x horizontally. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

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. 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. ...

World lines as a tool to describe events

Worldline, worldsheet, and world volume, as they are derived from particles, strings, and branes.

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 (where x⁰ usually denotes the time coordinate) depending on one parameter τ. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant. Image File history File links Download high resolution version (811x735, 114 KB)I made this with Inkscape. ... Image File history File links Download high resolution version (811x735, 114 KB)I made this with Inkscape. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... This article or section is in need of attention from an expert on the subject. ...

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 τ along the world line. In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ...

Trivial examples of spacetime curves

Three different world lines representing travel at different constant speeds. t is time and x distance.

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. Image File history File links Worldlines1. ...

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 in to 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 depicts 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 defining a world line, are real functions of a real variable τ 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 p on the curve at the parameter value τ₀ and a point on the curve a little (parameter τ₀ + Δτ) farther away. In the limit , this difference divided by Δτ defines a vector, the tangent vector of the world line at the point p. It is a four-dimensional vector, defined in the point p. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore called four-velocity , or in components:

where the derivatives are taken at the point p, so at τ = τ₀.

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. The fundamental concept in linear algebra is that of a vector space or linear space. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

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 speed 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. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... An inertial reference frame is one in which Newtons first and second laws of motion are valid. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

World lines of particles/objects at constant speed are called geodesics. In special relativity these are straight lines in Minkowski space. In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the 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, 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 into two parts. The cone is a three-dimensional hyperplane in spacetime, which appears as a line in drawings with two dimensions suppressed and as a cone in drawings with one spatial dimension suppressed.

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.

• 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ...

• 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, which 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 inaccessible 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. The fact that simultaneity depends on relative velocity caused problems for many scientists and laymen trying to accept relativity in the early days. The illustration with the light cones may make it appear that they cannot 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.

In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ... The Sun (Latin: ) is the star at the center of the Solar System. ... A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ...

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 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. An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... In physics, the adjective light-like refers to a contour in spacetime in the context of special relativity whose proper length vanishes. ... In the context of special relativity, space-like separated points (or events) in spacetime have a spacetime interval less than 0 (see sign convention). ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In theoretical physics, general covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Proper frame is the reference frame co-moving with an object. ... In general relativity, Eddington-Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry which are adapted to radial null geodesics (i. ...

World lines of free-falling particles or objects (such as planets around the Sun or an astronaut in space) are called geodesics. In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...

See also

Some specific type of world lines:

• geodesics
• closed timelike curves

In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is closed. ...

External links

• World lines article on h2g2.