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The world line (or worldline) of an object is the
path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...
that an object traces in 4-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. It is an important concept in modern
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and particularly
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
. The concept of a "world line" is distinguished from concepts such as an "
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
" or a "
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
" (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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
or
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
al interactions. The idea of world lines originates in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and was pioneered by
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
. The term is now most often used in relativity theories (i.e.,
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
).


Usage in physics

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a world line of an object (approximated as a point in space, e.g., a particle or observer) is the sequence of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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 Helical may refer to: * Helix, the mathematical concept for the shape * Helical engine, a proposed spacecraft propulsion drive * Helical spring, a coilspring * Helical plc, a British property company, once a maker of steel bar stock * Helicoil A t ...
in spacetime (a curve in a four-dimensional space) and does not return to the same point. Spacetime is the collection of
events Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of ev ...
, together with a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
and
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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 A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contractio ...
, which is a plane usually plotted with the time coordinate, say t, upwards and the space coordinate, say x horizontally. As expressed by F.R. Harvey :A curve M in pacetimeis called a ''worldline of a particle'' if its tangent is future timelike at each point. The arclength parameter is called
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
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 In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
. A world line traces out the path of a single point in spacetime. A
world sheet In its most general sense, the term "world" refers to the totality of entities, to the whole of reality or to everything that is. The nature of the world has been conceptualized differently in different fields. Some conceptions see the worl ...
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 x^a(\tau),\; a=0,1,2,3 (where x^ usually denotes the time coordinate) depending on one parameter \tau. 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 In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
of an object or an observer as the curve parameter \tau 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 x^a(\tau),\; a = 0, 1, 2, 3 defining a world line, are real functions of a real variable \tau 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 \tau_0 and a point on the curve a little (parameter \tau_0 + \Delta\tau) farther away. In the limit \Delta\tau \to 0, this difference divided by \Delta\tau 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 In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
\vec, or in components: \vec = \left(v^0, v^1, v^2, v^3\right) = \left( \frac\;,\frac\;, \frac\;, \frac \right) where the derivatives are taken at the point p, so at \tau = \tau_0. 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, called the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
puts some constraints on possible world lines. In special relativity the description of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
is limited to ''special'' coordinate systems that do not accelerate (and so do not rotate either), called
inertial coordinate system In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
s. In such coordinate systems, the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
is a constant. The structure of spacetime is determined by a
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
η, which gives a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s of mathematics, the bilinear form is ''not'' a mathematical metric on spacetime. World lines of freely falling particles/objects are called
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s. In special relativity these are straight lines in
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
. 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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
) 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 In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
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 In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
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 The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
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 Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
that is
hyperbolic-orthogonal In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyperb ...
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.


Simultaneous hyperplane

Since a world line w(\tau) \isin R^4 determines a velocity 4-vector v = \frac that is time-like, the Minkowski form \eta(v,x) determines a linear function R^4 \rarr R by x \mapsto \eta( v , x ) . Let ''N'' be the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...
of this linear functional. Then ''N'' is called the simultaneous hyperplane with respect to ''v''. The
relativity of simultaneity In physics, the relativity of simultaneity is the concept that ''distant simultaneity'' – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possi ...
is a statement that ''N'' depends on ''v''. Indeed, ''N'' is the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of ''v'' with respect to η. When two world lines ''u'' and ''w'' are related by \frac = \frac , 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 Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve
retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so th ...
s.


World lines in general relativity

The use of world lines in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
is basically the same as in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, with the difference that
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
can be
curved In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
. A
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
exists and its dynamics are determined by the
Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
and are dependent on the mass-energy distribution in spacetime. Again the metric defines
lightlike In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
(null),
spacelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
and
timelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
curves. Also, in general relativity, world lines are
timelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
curves in spacetime, where
timelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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 In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the Invariant (physics), invariance of the ''form'' of physical laws under arbitrary Derivative, differentiable coordinate transf ...
) of general relativity. Any
timelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
curve admits a
comoving observer In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a spec ...
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
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s.


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 Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction works titled ''Scientific Romances''. He was interested in n-dimensional space, higher dimensions, particularly the Four-dimensional space, ...
wrote an essay "What is the fourth dimension ?", which he published as a
scientific romance Scientific romance is an archaic, mainly British term for the genre of fiction now commonly known as science fiction. The term originated in the 1850s to describe both fiction and elements of scientific writing, but it has since come to refer to ...
. He wrote :Why, then, should not the four-dimensional beings be ourselves, and our successive states the passing of them through the three-dimensional space to which our consciousness is confined. A popular description of human world lines was given by J. C. Fields at the
University of Toronto The University of Toronto (UToronto or U of T) is a public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution ...
in the early days of relativity. As described by Toronto lawyer Norman Robertson: :I remember ieldslecturing at one of the Saturday evening lectures at the
Royal Canadian Institute The Royal Canadian Institute for Science (RCIScience), known also as the Royal Canadian Institute, is a Canadian nonprofit organization dedicated to connecting the public with Canadian science. History The organization was formed in Toronto as t ...
. It was advertised to be a "Mathematical Fantasy"—and it was! The substance of the exercise was as follows: He postulated that, commencing with his birth, every human being had some kind of spiritual aura with a long filament or thread attached, that traveled behind him throughout his life. He then proceeded in imagination to describe the complicated entanglement every individual became involved in his relationship to other individuals, comparing the simple entanglements of youth to those complicated knots that develop in later life. Kurt Vonnegut, in his novel ''
Slaughterhouse-Five ''Slaughterhouse-Five, or, The Children's Crusade: A Duty-Dance with Death'' is a 1969 semi-autobiographic science fiction-infused anti-war novel by Kurt Vonnegut. It follows the life and experiences of Billy Pilgrim, from his early years, to h ...
'', describes the worldlines of stars and people: :“Billy Pilgrim says that the Universe does not look like a lot of bright little dots to the creatures from Tralfamadore. The creatures can see where each star has been and where it is going, so that the heavens are filled with rarefied, luminous spaghetti. And Tralfamadorians don't see human beings as two-legged creatures, either. They see them as great millepedes - "with babies' legs at one end and old people's legs at the other," says Billy Pilgrim.” Almost all science-fiction stories which use this concept actively, such as to enable
time travel Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a w ...
, 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 Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and futuristic concepts such as advanced science and technology, space exploration, time travel, parallel unive ...
work in 2008 entitled ''World Lines'' in which he related a simplified explanation of the hypothesis for laymen. In the short story ''
Life-Line "Life-Line" is a short story by American author Robert A. Heinlein. Published in the August 1939 edition of ''Astounding'', it was Heinlein's first published short story. The protagonist, Professor Hugo Pinero, builds a machine that will predict ...
'', author
Robert A. Heinlein Robert Anson Heinlein (; July 7, 1907 – May 8, 1988) was an American science fiction author, aeronautical engineer, and naval officer. Sometimes called the "dean of science fiction writers", he was among the first to emphasize scientific accu ...
describes the world line of a person: :He stepped up to one of the reporters. "Suppose we take you as an example. Your name is Rogers, is it not? Very well, Rogers, you are a space-time event having duration four ways. You are not quite six feet tall, you are about twenty inches wide and perhaps ten inches thick. In time, there stretches behind you more of this space-time event, reaching to perhaps nineteen-sixteen, of which we see a cross-section here at right angles to the time axis, and as thick as the present. At the far end is a baby, smelling of sour milk and drooling its breakfast on its bib. At the other end lies, perhaps, an old man someplace in the nineteen-eighties. :"Imagine this space-time event that we call Rogers as a long pink worm, continuous through the years, one end in his mother's womb, and the other at the grave..." Heinlein's ''
Methuselah's Children ''Methuselah's Children'' is a science fiction novel by American writer Robert A. Heinlein. Originally serialized in ''Astounding Science Fiction'' in the July, August, and September 1941 issues, it was expanded into a full-length novel in 1958. ...
'' uses the term, as does
James Blish James Benjamin Blish () was an American science fiction and fantasy writer. He is best known for his ''Cities in Flight'' novels and his series of ''Star Trek'' novelizations written with his wife, J. A. Lawrence. His novel ''A Case of Conscienc ...
's ''
The Quincunx of Time ''The Quincunx of Time'' is a short science fiction novel by American writer James Blish. It is an extended version of a short story entitled "Beep", published by ''Galaxy Science Fiction'' magazine in 1954. The novel form was first published ...
'' (expanded from "Beep"). A
visual novel A , often abbreviated as VN, is a form of digital semi-interactive fiction. Visual novels are often associated with and used in the medium of video games, but are not always labeled as such themselves. They combine a textual narrative with sta ...
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 ''Science Adventure'' is a multimedia series consisting of interconnected science fiction stories, created mainly by 5pb., Mages, Nitroplus, and Chiyomaru Studio. The main entries mostly take the form of visual novel video games, but side entries s ...
" series. World lines and other physical concepts like the
Dirac Sea The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by th ...
are also used throughout the series.
Neal Stephenson Neal Town Stephenson (born October 31, 1959) is an American writer known for his works of speculative fiction. His novels have been categorized as science fiction, historical fiction, cyberpunk, postcyberpunk, and baroque. Stephenson's work exp ...
's novel
Anathem ''Anathem'' is a science fiction novel by American writer Neal Stephenson, published in 2008. Major themes include the many-worlds interpretation of quantum mechanics and the philosophical debate between Platonic realism and nominalism. Plot sum ...
involves a long discussion of worldlines over dinner in the midst of a philosophical debate between
Platonic realism Platonic realism is the philosophical position that universals or abstract objects exist objectively and outside of human minds. It is named after the Greek philosopher Plato who applied realism to such universals, which he considered ideal ...
and
nominalism In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings t ...
. 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 Charles David George "Charlie" Stross (born 18 October 1964) is a British writer of science fiction and fantasy. Stross specialises in hard science fiction and space opera. Between 1994 and 2004, he was also an active writer for the magazine '' ...
.


See also

* Specific types of world lines **
Geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s **
Closed timelike curve In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Sto ...
s **
Causal structure In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian ma ...
, curves that represent a variety of different types of world line **
Isotropic line In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, an ...
*
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
*
Time geography Time geography or time-space geography is an evolving transdisciplinary perspective on spatial and temporal processes and events such as social interaction, ecological interaction, social and environmental change, and biographies of individuals. T ...


References

* :*Various English translations on Wikisource:
Space and Time Space and Time or Time and Space, or ''variation'', may refer to: * '' Space and time'' or ''time and space'' or ''spacetime'', any mathematical model that combines space and time into a single interwoven continuum * Philosophy of space and time S ...
*
Ludwik Silberstein Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a se ...
(1914) ''Theory of Relativity'', p 130,
Macmillan and Company Macmillan Publishers (occasionally known as the Macmillan Group; formally Macmillan Publishers Ltd and Macmillan Publishing Group, LLC) is a British publishing company traditionally considered to be one of the 'Big Five' English language publi ...
.


External links


World lines
article on
h2g2 The h2g2 website is a British-based collaborative online encyclopedia project. It describes itself as "an unconventional guide to life, the universe, and everything", in the spirit of the fictional publication '' The Hitchhiker's Guide to ...
.
in depth text on world lines and special relativity
{{DEFAULTSORT:World Line Theory of relativity Minkowski spacetime Time in science