6.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| World line | 3/4 | https://en.wikipedia.org/wiki/World_line | reference | science, encyclopedia | 2026-05-05T03:51:27.089519+00:00 | kb-cron |
=== Simultaneous hyperplane === Since a world line
w
(
τ
)
∈
R
4
{\displaystyle w(\tau )\in R^{4}}
determines a velocity 4-vector
v
=
d
w
d
τ
{\displaystyle v={\frac {dw}{d\tau }}}
that is time-like, the Minkowski form
η
(
v
,
x
)
{\displaystyle \eta (v,x)}
determines a linear function
R
4
→
R
{\displaystyle R^{4}\rightarrow R}
by
x
↦
η
(
v
,
x
)
.
{\displaystyle x\mapsto \eta (v,x).}
Let N be the null space of this linear functional. Then N is called the simultaneous hyperplane with respect to v. The relativity of simultaneity is a statement that N depends on v. Indeed, N is the orthogonal complement of v with respect to η. When two world lines u and w are related by
d
u
d
τ
=
d
w
d
τ
,
{\displaystyle {\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 by Coulomb's law may be pictured in a simultaneous hyperplane, but relativistic relations of charge and force involve retarded potentials.
== World lines in general relativity == The use of world lines in general relativity is basically the same as in special relativity, with the difference that spacetime can be curved. A 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 include 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 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. World lines of free-falling particles or objects (such as planets around the Sun or an astronaut in space) are called geodesics.
== 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 Feynman that many quantum field theories may equivalently be described in terms of world lines. This preceded much of his work on the formulation which later became more standard. The world line formulation of quantum field theory has proved particularly fruitful for various calculations in gauge theories and in describing nonlinear effects of electromagnetic fields.
== World lines in literature == In 1884 C. H. Hinton wrote an essay "What is the fourth dimension ?", which he published as a scientific 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 by J. C. Fields at the University of Toronto in the early days of relativity. As described by Toronto lawyer Norman Robertson:
I remember [Fields] lecturing at one of the Saturday evening lectures at the Royal 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 novel Slaughterhouse-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 enable time 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'. Author Oliver Franklin published a science fiction work in 2008 entitled World Lines in which he related a simplified explanation of the hypothesis for laymen. In the short story Life-Line, author Robert A. Heinlein describes the world line of a person: