World line

# World line

Discussion

Encyclopedia
In physics, the world line of an object is the unique path of that object as it travels through 4-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. 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
Perception
Perception is the process of attaining awareness or understanding of the environment by organizing and interpreting sensory information. All perception involves signals in the nervous system, which in turn result from physical stimulation of the sense organs...

straight paths are recalculated to show their (relatively) more absolute position states — to reveal the nature of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

or gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

al interactions. The idea of world lines originates in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and was pioneered by Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

. The term is now most often used in relativity theories (i.e., general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

and special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

).

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
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

.

## Usage in physics

In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a world line of an object (approximated as a point in space, e.g., a particle or observer) is the sequence of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

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. However, it arrives there at a different (later) time. The world line of the Earth is helical
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

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
Event (relativity)
In physics, and in particular relativity, an event indicates a physical situation or occurrence, located at a specific point in space and time. For example, a glass breaking on the floor is an event; it occurs at a unique place and a unique time, in a given frame of reference.Strictly speaking, the...

, 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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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
Minkowski diagram
The Minkowski diagram was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical...

, which is a plane usually plotted with the time coordinate, say , upwards and the space coordinate, say horizontally.

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) is a volume.

## 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 (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 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
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

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 (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 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 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 on the curve at the parameter value and a point on the curve a little (parameter ) 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 (but it is not the same) 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
Tangent space
In mathematics, the tangent space of a manifold 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....

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.

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
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

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 system
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

s. 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
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

and is called a Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

, which allows for example a description of the path of light.

World lines of particles/objects at constant speed are called geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s. 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, 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.

• 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, which can be connected by world lines within the past light cone
Light cone
A light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime...

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
Light cone
A light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime...

within the entire spacetime.
• Elsewhere is the region between the two light cones. Points in an observer's elsewhere 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, but in fact there is always a delay time for light to propagate. For example, we see the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

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 present hypersurface, which is defined for a given observer by a 3-plane normal to her/his world line. It is the locus of simultaneous events for the chosen observer, and 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 the present hypersurface 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 this is indeed the case in Minkowski spacetime
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

.
• The present (without the specification of a hypersurface or "section") often means the single spacetime event being considered.

## World lines in general relativity

The use of world lines in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

is basically the same as in special relativity, with the difference that spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

can be curved
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

. A metric
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

exists and its dynamics are determined by the Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

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
Proper frame
A proper frame, or comoving frame, is a frame of reference that is attached to an object. The object in this frame is stationary within the frame, which is useful for many types of calculations....

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
Eddington-Finkelstein coordinates
In general relativity Eddington–Finkelstein coordinates, named for Arthur Stanley Eddington and David Finkelstein, are a pair of coordinate systems for a Schwarzschild geometry which are adapted to radial null geodesics...

.

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

s.

## World lines in literature

Because they oversimplify worldlines, which traverse four-dimesional spacetime, into one-dimensional timelines, almost all purported science-fiction stories about time travel
Time travel
Time travel is the concept of moving between different points in time in a manner analogous to moving between different points in space. Time travel could hypothetically involve moving backward in time to a moment earlier than the starting point, or forward to the future of that point without the...

are actually wishful fantasy stories. 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.

World lines appeared in Jeffrey Rowland's webcomic Wigu Adventures as part of the "Magical Adventures in Space" side story line, in which Topato Potato and Sheriff Pony accidentally delete a world line relating to the initial creation of Earth from asteroids, causing the Earth to never have existed. According to this webcomic, calculating the exact coordinates of a world line is "embarrassingly simple", and the deletion of the world line specified is executed by making a call and entering the coordinates of the world line, and pressing 3.

Author Oliver Franklin published a science fiction
Science fiction
Science fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...

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
"Life-Line" is a short story by American author Robert A. Heinlein. Published in 1939, it was Heinlein's first published short story.The protagonist, Professor Pinero, builds a machine that will predict how long a person will live. It does this by sending a signal along the world line of a...

, author Robert A. Heinlein
Robert A. Heinlein
Robert Anson Heinlein was an American science fiction writer. Often called the "dean of science fiction writers", he was one of the most influential and controversial authors of the genre. He set a standard for science and engineering plausibility and helped to raise the genre's standards of...

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..."

• Specific types of world lines
• Geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s
• Closed timelike curve
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...

s
• Causal structure, curves that represent a variety of different types of world line

• World lines article on h2g2
H2g2
h2g2 is a British-based collaborative online encyclopedia project engaged in the construction of, in its own words, "an unconventional guide to life, the universe, and everything", in the spirit of the fictional publication The Hitchhiker's Guide to the Galaxy from the science fiction comedy series...

.