Кез-келген дененің уақыт-кеңістіктегі жолы

Unique path of an object as it travels through spacetime

The world line (or worldline) of an object is the path that an object traces in 4 dimensional spacetime. It is an important concept of 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 inclusion of the dimension time, and typically encompasses a large area of spacetime wherein paths which are straight perceptually are rendered as curves in space time to show their (relatively) more absolute position states—to reveal the nature of special relativity or gravitational interactions. The idea of world lines was originated by physicists and was pioneered by Hermann Minkowski. The term is now used most often in the context of relativity theories (i. e., special relativity and general relativity).

A world line of an object (generally 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 either a time like or a null 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 therefore helical in spacetime (a curve in a four dimensional space) and does not return to the same point. Spacetime is the collection of 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 (a topological space that locally resembles Euclidean space near each point). The concept may be applied as well to a higher dimensional space. For easy visualizations of four dimensions, two space coordinates are often suppressed. An event is then represented by a point in a Minkowski diagram, which is a plane usually plotted with the time coordinate, say , vertically, and the space coordinate, say , horizontally. As expressed by F. R. Harvey
A curve M in [spacetime] is called a worldline of a particle if its tangent is future timelike at each point. The arclength parameter is called proper time and usually denoted τ. The length of M is called the proper time of the particle. If the worldline M is a line segment, then the particle is said to be in free fall. 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 not a world line but rather a world tube.

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 usually denotes the time coordinate) 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 used informally 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 uses the proper time of an object or an observer as the curve parameter along the world line.

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 simply traces the length of the rod. A line at constant space coordinate (a vertical line using 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 e. g. 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).

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 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 termed 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):

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. 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.

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.

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 include timelike curves and null 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.

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. This preceded much of his work on the formulation which later became more standard. 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.