World line: Difference between revisions

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The '''world line''' (or '''worldline''') of an object is the [[path (topology)|path]] that an object traces in 4-[[dimension]]al [[spacetime]]. It is an important concept inof 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'' dimension, and typically encompasses a large area of spacetime wherein paths which are straight [[perception|perceptually]] straightare pathsrendered areas recalculatedcurves in spacetime to show their ([[Principle of relativity|relatively]]) more absolute [[position states]]—to reveal the nature of [[special relativity]] or [[gravitation]]al interactions.

The idea of world lines originateswas inoriginated by [[physics|physicists]] and was pioneered by [[Hermann Minkowski]]. The term is now used most often used in the context of relativity theories (i.e., [[special relativity]] and [[general relativity]]).

In [[physics]], aA 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 [[helix|helical]] in spacetime (a curve in a four-dimensional space) and does not return to the same point.

Spacetime is the collection of points called [[event (relativity)|events]], together with a [[continuous function|continuous]] and [[smooth function|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. TheAn event is then represented by a point in a [[Minkowski diagram]], which is a plane usually plotted with the time coordinate, say <math>t</math>, upwardsvertically, and the space coordinate, say <math>x</math>, horizontally.

: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 &tau;. The length of M is called the ''proper time'' of the worldline or particle. If the worldline M is a line segment, then the particle is said to be in [[free fall]].<ref>{{cite book|first = F. Reese|last = Harvey|year = 1990|chapter-url = https://books.google.com/books?id=6HnNCgAAQBAJ&pg=PA62|chapter = Special Relativity" section of chapter "EuclidieanEuclidean / Lorentzian Vector Spaces|title = Spinors and Calibrations|pages = 62–67|publisher = [[Academic Press]]|isbn = 9780080918631}}</ref>{{rp|62-6362–63}}

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 toolmethod toof describedescribing events==

Sometimes, the term '''world line''' is loosely 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 takesuses the [[proper time]] of an object or an observer as the curve parameter <math>\tau</math> 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 inusing 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.

The four coordinate functions <math>x^a(\tau),\; a = 0, 1, 2, 3</math>

defining a world line, are real number functions of a real variable <math>\tau</math> and can simply be differentiated inby the usual calculus. Without the existence of a metric (this is important to realize) one can speak ofimagine the difference between a point <math>p</math> on the curve at the parameter value <math>\tau_0</math> and a point on the curve a little (parameter <math>\tau_0 + \Delta\tau</math>) farther away. In the limit <math>\Delta\tau \rightarrowto 0</math>, this difference divided by <math>\Delta\tau</math> defines a vector, the '''tangent vector''' of the world line at the point <math>p</math>. It is a four-dimensional vector, defined in the point <math>p</math>. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore calledtermed [[four-velocity]] <math>\vec{v}</math>, or in components:

:<math display="block">\vec{v} = \left(v^0, v^1, v^2, v^3\right) = \left( \frac{dx^0}{d\tau}\;,\frac{dx^1}{d\tau}\;, \frac{dx^2}{d\tau}\;, \frac{dx^3}{d\tau} \right)</math>

wheresuch that the derivatives are taken at the point <math>p</math>, so at <math>\tau = \tau_0</math>.

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 infor a point p span a [[linear space]], calledtermed 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.

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), calledtermed [[inertial frame of reference|inertial coordinate system]]s. 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 calledtermed a ''spacetime metric'', but since distinct events sometimes result in a zero value, unlike metrics in [[metric space]]s of mathematics, the bilinear form is ''not'' a mathematical metric on spacetime.

The use of world lines in [[general relativity]] is basically the same as in [[special relativity]], with the difference that [[spacetime]] can be [[curvature|curved]]. A [[metric tensor|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 areinclude [[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 [[Proper frame|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]].

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<ref>{{cite journal|last = Feynman|first = Richard P.|author-link = Richard Feynman|title = Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction|journal = [[Physical Review]]|year = 1950|volume = 80|issue = 1|pages = 108–128|doi = 10.1103/PhysRev.80.440|url = https://journals.aps.org/pr/abstract/10.1103/PhysRev.80.440}}</ref> that many quantum field theories may equivalently be described in terms of world lines. This preceded much of his work<ref>{{cite journal|last = Feynman|first = Richard P.|author-link = Richard Feynman|title = An operator calculus having applications in quantum electrodynamics|journal = [[Physical Review]]|year = 1951|volume = 84|issue = 13|pages = 108–128440-457|doi = 10.1103/PhysRev.84.108|url = https://authors.library.caltech.edu/3530/1/FEYpr51.pdf|bibcode = 1951PhRv...84..108F}}</ref> thaton manythe quantumformulation fieldwhich theorieslater maybecame equivalentlymore be described in terms of world linesstandard. The [[Path integral formulation#Quantum field theory|world line formulation of quantum field theory]] has proved particularly fruitful for various calculations in gauge theories<ref>{{cite journal|last1 = Bern|first1 = Zvi|author-link1 = Zvi Bern|first2 = David A.|last2 = Kosower|title = Efficient calculation of one-loop QCD amplitudes|journal = [[Physical Review Letters]]|volume = 66|issue = 13|year = 1991|pages = 1669–1672|pmid = 10043277|doi = 10.1103/PhysRevLett.66.1669|bibcode = 1991PhRvL..66.1669B}}</ref><ref>{{cite journal|last1 = Bern|first1 = Zvi|author-link1 = Zvi Bern|first2 = Lance|last2 = Dixon|author-link2 = Lance J. Dixon|first3 = David A.|last3 = Kosower|title = Progress in one-loop QCD computations|journal = [[Annual Review of Nuclear and Particle Science]]|volume = 46|year = 1996|pages = 109–148|url = http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-7111.pdf|arxiv = hep-ph/9602280|doi = 10.1146/annurev.nucl.46.1.109| doi-access=free|bibcode = 1996ARNPS..46..109B}}</ref><ref>{{cite journal|last = Schubert|first = Christian|title = Perturbative quantum field theory in the string-inspired formalism|journal = [[Physics Reports]]|volume = 355|issue = 2–3|year = 2001|pages = 73–234|doi = 10.1016/S0370-1573(01)00013-8|arxiv = hep-th/0101036|bibcode = 2001PhR...355...73S|s2cid = 118891361}}</ref> and in describing nonlinear effects of electromagnetic fields.<ref>{{cite journal|last1 = Affleck|first1 = Ian K.|author-link1 = Ian Affleck|first2 = Orlando|last2 = Alvarez|first3 = Nicholas S.|last3 = Manton|author-link3 = Nicholas S. Manton|title = Pair production at strong coupling in weak external fields|journal = [[Nuclear Physics B]]|volume = 197|issue = 3|year = 1982|pages = 509–519|doi = 10.1016/0550-3213(82)90455-2|bibcode = 1982NuPhB.197..509A}}</ref><ref>{{cite journal|last1 = Dunne|first1 = Gerald V.|first2 = Christian|last2 = Schubert|title = Worldline instantons and pair production in inhomogenous fields|journal = [[Physical Review D]]|volume = 72|issue = 10|year = 2005|page = 105004|doi = 10.1103/PhysRevD.72.105004|arxiv = hep-th/0507174|url = httphttps://cds.cern.ch/record/855960/files/0507174.pdf?version=1|bibcode = 2005PhRvD..72j5004D|s2cid = 119357180}}</ref>