World line

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Theworld line(orworldline) of an object is thepaththat an object traces in 4-dimensionalspacetime.It is an important concept of modernphysics,and particularlytheoretical physics.

The concept of a "world line" is distinguished from concepts such as an "orbit"or a"trajectory"(e.g., a planet'sorbit in spaceor thetrajectoryof a car on a road) by inclusion of the dimensiontime,and typically encompasses a large area of spacetime wherein paths which are straightperceptuallyare rendered as curves in spacetime to show their (relatively) more absoluteposition states—to reveal the nature ofspecial relativityorgravitationalinteractions.

The idea of world lines was originated byphysicistsand was pioneered byHermann Minkowski.The term is now used most often in the context of relativity theories (i.e.,special relativityandgeneral 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 ofspacetimeevents 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, theorbitof 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. Theworld lineof the Earth is thereforehelicalin spacetime (a curve in a four-dimensional space) and does not return to the same point.

Spacetime is the collection ofevents,together with acontinuousandsmoothcoordinate systemidentifying 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-dimensionalmanifold(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 aMinkowski diagram,which is a plane usually plotted with the time coordinate, say$t$,vertically, and the space coordinate, say$x$,horizontally. As expressed by F.R. Harvey

A curve M in [spacetime] is called aworldline of a particleif its tangent is future timelike at each point. The arclength parameter is calledproper timeand usually denoted τ. The length of M is called theproper timeof the particle. If the worldline M is a line segment, then the particle is said to be infree fall.

A world line traces out the path of a single point in spacetime. Aworld sheetis 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 aworld linebut rather a world tube.

World lines as a method of describing events

World line, worldsheet, and world volume, as they are derived fromparticles,strings,andbranes.

A one-dimensionallineorcurvecan 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$x^{0}$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 termworld lineis used informally foranycurve in spacetime. This terminology causes confusions. More properly, aworld lineis a curve in spacetime that traces out the(time) historyof a particle, observer or small object. One usually uses theproper timeof an object or an observer as the curve parameter$\tau$along the world line.

Trivial examples of spacetime curves

Three different world lines representing travel at different constant four-velocities.tis time andxdistance.

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 acollisionor "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$x^{a}(\tau ),\;a=0,1,2,3$ defining a world line, are real number functions of a real variable$\tau$and can simply be differentiated by the usual calculus. Without the existence of a metric (this is important to realize) one can imagine 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, thetangent vectorof 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 termedfour-velocity${\vec {v}}$,or in components: $${\vec {v}}=\left(v^{0},v^{1},v^{2},v^{3}\right)=\left({\frac {dx^{0}}{d\tau }}\;,{\frac {dx^{1}}{d\tau }}\;,{\frac {dx^{2}}{d\tau }}\;,{\frac {dx^{3}}{d\tau }}\right)$$

such that 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 for a point p span alinear space,termed thetangent spaceat 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 ofspecial relativityputs some constraints on possible world lines. In special relativity the description ofspacetimeis limited tospecialcoordinate systems that do not accelerate (and so do not rotate either), termedinertial coordinate systems.In such coordinate systems, thespeed of lightis a constant. The structure of spacetime is determined by abilinear formη, which gives areal numberfor each pair of events. The bilinear form is sometimes termed aspacetime metric,but since distinct events sometimes result in a zero value, unlike metrics inmetric spacesof mathematics, the bilinear form isnota mathematical metric on spacetime.

World lines of freely falling particles/objects are calledgeodesics.In special relativity these are straight lines inMinkowski 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-likecurves, 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.

An example of alight cone,the three-dimensional surface of all possible light rays arriving at and departing from a point in spacetime. Here, it is depicted with one spatial dimension suppressed.

The momentarily co-moving inertial frames along the trajectory ( "world line" ) of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is thespacetimeof the observer. The small dots are specific events in spacetime. Note how the momentarily co-moving inertial frame changes when the observer accelerates.

• time-likecurves, 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-likecurves 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.

• Thefutureof the given event is formed by all events that can be reached through time-like curves lying within the future light cone.
• Thepastof the given event is formed by all events that can influence the event (that is, that can be connected by world lines within the pastlight coneto the given event).
  • Thelightconeat 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 pastlight conewithin the entire spacetime.
• Elsewhereis the region between the two light cones. Points in an observer'selsewhereare 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 theSunas it was about 8 minutes ago, not as it is "right now". Unlike thepresentin Galilean/Newtonian theory, theelsewhereis thick; it is not a 3-dimensional volume but is instead a 4-dimensional spacetime region.
  • Included in "elsewhere" is thesimultaneous hyperplane,which is defined for a given observer by aspacethat ishyperbolic-orthogonalto 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.
  • Thepresentoften means the single spacetime event being considered.

Simultaneous hyperplane

Since a world line$w(\tau )\in R^{4}$determines a velocity 4-vector$v={\frac {dw}{d\tau }}$that is time-like, the Minkowski form$\eta (v,x)$determines a linear function$R^{4}\rightarrow R$by$x\mapsto \eta (v,x).$LetNbe thenull spaceof this linear functional. ThenNis called thesimultaneous hyperplanewith respect tov.Therelativity of simultaneityis a statement thatNdepends onv.Indeed,Nis theorthogonal complementofvwith respect to η. When two world linesuandware related by${\frac {du}{d\tau }}={\frac {dw}{d\tau }},$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 byCoulomb's lawmay be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involveretarded potentials.

World lines in general relativity

The use of world lines ingeneral relativityis basically the same as inspecial relativity,with the difference thatspacetimecan becurved.Ametricexists and its dynamics are determined by theEinstein field equationsand are dependent on the mass-energy distribution in spacetime. Again the metric defineslightlike(null),spacelike,andtimelikecurves. Also, in general relativity, world lines includetimelikecurves and null curves in spacetime, wheretimelikecurves 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. Anytimelikecurve admits acomoving observerwhose "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 exampleEddington-Finkelstein coordinates.

World lines of free-falling particles or objects (such as planets around the Sun or an astronaut in space) are calledgeodesics.

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 Feynmanthat many quantum field theories may equivalently be described in terms of world lines. This preceded much of his workon the formulation which later became more standard. Theworld line formulation of quantum field theoryhas proved particularly fruitful for various calculations in gauge theoriesand in describing nonlinear effects of electromagnetic fields.

World lines in literature

In 1884C. H. Hintonwrote an essay "What is the fourth dimension?", which he published as ascientific romance.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 byJ. C. Fieldsat theUniversity of Torontoin the early days of relativity. As described by Toronto lawyer Norman Robertson:

I remember [Fields] lecturing at one of the Saturday evening lectures at theRoyal Canadian Institute.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 novelSlaughterhouse-Five,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 enabletime travel,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'.

AuthorOliver Franklinpublished ascience fictionwork in 2008 entitledWorld Linesin which he related a simplified explanation of the hypothesis for laymen.

In the short storyLife-Line,authorRobert A. Heinleindescribes 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'sMethuselah's Childrenuses the term, as doesJames Blish'sThe Quincunx of Time(expanded from "Beep" ).

Avisual novelnamedSteins;Gate,produced by5pb.,tells a story based on the shifting of world lines. Steins;Gate is a part of the "Science Adventure"series. World lines and other physical concepts like theDirac Seaare also used throughout the series.

Neal Stephenson's novelAnatheminvolves a long discussion of worldlines over dinner in the midst of a philosophical debate betweenPlatonic realismandnominalism.

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" byCharles Stross.

See also

• Specific types of world lines
  • Geodesics
  • Closed timelike curves
  • Causal structure,curves that represent a variety of different types of world line
  • Isotropic line
• Feynman diagram
• Time geography

References

• Minkowski, Hermann (1909),"Raum und Zeit",Physikalische Zeitschrift,10:75–88

• Various English translations on Wikisource:Space and Time

• Ludwik Silberstein(1914)Theory of Relativity,p 130,Macmillan and Company.

External links

• World linesarticle onh2g2.

• in depth text on world lines and special relativity