---
title: "World line"
chunk: 1/4
source: "https://en.wikipedia.org/wiki/World_line"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:51:27.089519+00:00"
instance: "kb-cron"
---

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 spacetime 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).

== Usage in physics ==
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 
  
    
      
        t
      
    
    {\displaystyle t}
  
, vertically, and the space coordinate, say 
  
    
      
        x
      
    
    {\displaystyle x}
  
, 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.

== World lines as a method of describing 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
          
        
        (
        τ
        )
        ,
        
        a
        =
        0
        ,
        1
        ,
        2
        ,
        3
      
    
    {\displaystyle x^{a}(\tau ),\;a=0,1,2,3}
  
 (where 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x^{0}}
  
 usually denotes the time coordinate) depending on one parameter 
  
    
      
        τ
      
    
    {\displaystyle \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 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 
  
    
      
        τ
      
    
    {\displaystyle \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 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).

=== Tangent vector to a world line: four-velocity ===
The four coordinate functions 
  
    
      
        
          x
          
            a
          
        
        (
        τ
        )
        ,
        
        a
        =
        0
        ,
        1
        ,
        2
        ,
        3
      
    
    {\displaystyle x^{a}(\tau ),\;a=0,1,2,3}
  