# world line

• 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 space-time to show their (relatively) more absolute position states—to reveal the
nature of special relativity or gravitational interactions.

• 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.

• 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.

• One usually uses the proper time of an object or an observer as the curve parameter along the world line.

• 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.

• Below an equivalent definition will be explained: A world line is a time-like curve in spacetime.

• 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.

• More properly, a world line is a curve in spacetime that traces out the (time) history of a particle, observer or small object.

• Tangent vector to a world line: four-velocity The four coordinate functions defining a world line, are real number functions of a real variable and can simply be differentiated
by the usual calculus.

• 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.

• A world sheet is the analogous two-dimensional surface traced out by a one-dimensional line (like a string) traveling through spacetime.

• In our definition above: world lines are time-like curves in spacetime.

• 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.

• A line at constant space coordinate (a vertical line using the convention adopted above) may represent a particle at rest (or a stationary observer).

• Each event can be labeled by four numbers: a time coordinate and three space coordinates; thus spacetime is a four-dimensional space.

• 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.

• 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.

• Trivial examples of spacetime curves Three different world lines representing travel at different constant four-velocities.

• 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, oversimplify this concept to a one-dimensional timeline to fit a linear structure, which does not fit models of reality.

• A world line is a special type of curve in spacetime.

• 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’.

• 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.

• • 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.

• 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.

• Sometimes, the term world line is used informally for any curve in spacetime.

• The concept may be applied as well to a higher-dimensional space.

• World lines and other physical concepts like the Dirac Sea are also used throughout the series.

• 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.

• o The present often means the single spacetime event being considered.

• Absolute Choice depicts different world lines as a sub-plot and setting device.

• Two world lines starting at the same event in spacetime, each following its own path afterwards, may represent e.g.

• 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.

• 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.

• 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.

• However, although not widely appreciated, it has been known since Feynman[2] that many quantum field theories may equivalently be described in terms of world lines.

• At a given event on a world line, spacetime (Minkowski space) is divided into three parts.

• An event is then represented by a point in a Minkowski diagram, which is a plane usually plotted with the time coordinate, say , vertically, and the space coordinate, say
, horizontally.

• 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.

• 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.

• The basic mathematics is as follows: The theory of special relativity puts some constraints on possible world lines.

• It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore termed four-velocity , or in components: such that the derivatives
are taken at the point , so at .

Works Cited

[‘Harvey, F. Reese (1990). “Special Relativity” section of chapter “Euclidean / Lorentzian Vector Spaces”. Spinors and Calibrations. Academic Press. pp. 62–67. ISBN 9780080918631.
2. ^ Feynman, Richard P. (1951). “An operator calculus having applications
in quantum electrodynamics” (PDF). Physical Review. 84 (1): 108–128. Bibcode:1951PhRv…84..108F. doi:10.1103/PhysRev.84.108.
3. ^ Bern, Zvi; Kosower, David A. (1991). “Efficient calculation of one-loop QCD amplitudes”. Physical Review Letters.
66 (13): 1669–1672. Bibcode:1991PhRvL..66.1669B. doi:10.1103/PhysRevLett.66.1669. PMID 10043277.
4. ^ Bern, Zvi; Dixon, Lance; Kosower, David A. (1996). “Progress in one-loop QCD computations” (PDF). Annual Review of Nuclear and Particle Science.
46: 109–148. arXiv:hep-ph/9602280. Bibcode:1996ARNPS..46..109B. doi:10.1146/annurev.nucl.46.1.109.
5. ^ Schubert, Christian (2001). “Perturbative quantum field theory in the string-inspired formalism”. Physics Reports. 355 (2–3): 73–234. arXiv:hep-th/0101036.
Bibcode:2001PhR…355…73S. doi:10.1016/S0370-1573(01)00013-8. S2CID 118891361.
6. ^ Affleck, Ian K.; Alvarez, Orlando; Manton, Nicholas S. (1982). “Pair production at strong coupling in weak external fields”. Nuclear Physics B. 197 (3): 509–519.
Bibcode:1982NuPhB.197..509A. doi:10.1016/0550-3213(82)90455-2.
7. ^ Dunne, Gerald V.; Schubert, Christian (2005). “Worldline instantons and pair production in inhomogenous fields” (PDF). Physical Review D. 72 (10): 105004. arXiv:hep-th/0507174.
Bibcode:2005PhRvD..72j5004D. doi:10.1103/PhysRevD.72.105004. S2CID 119357180.
8. ^ Hinton, C. H. (1884). “What is the fourth dimension?”. Scientific Romances: First Series. S. Sonnenschein. pp. 1–32.
9. ^ Robinson, Gilbert de Beauregard (1979).
The Mathematics Department in the University of Toronto, 1827–1978. University of Toronto Press. p. 19. ISBN 0-7727-1600-5.
10. ^ Oliver Franklin (2008). World Lines. Epic Press. ISBN 978-1-906557-00-3.
11. ^ “Technovelgy: Chronovitameter”. Retrieved
8 September 2010.
• 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.

Photo credit: https://www.flickr.com/photos/perspective/8695372372/’]