8.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| World line | 2/4 | https://en.wikipedia.org/wiki/World_line | reference | science, encyclopedia | 2026-05-05T03:51:27.089519+00:00 | kb-cron |
defining a world line, are real number functions of a real variable
τ
{\displaystyle \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
{\displaystyle p}
on the curve at the parameter value
τ
0
{\displaystyle \tau _{0}}
and a point on the curve a little (parameter
τ
0
+
Δ
τ
{\displaystyle \tau _{0}+\Delta \tau }
) farther away. In the limit
Δ
τ
→
0
{\displaystyle \Delta \tau \to 0}
, this difference divided by
Δ
τ
{\displaystyle \Delta \tau }
defines a vector, the tangent vector of the world line at the point
p
{\displaystyle p}
. It is a four-dimensional vector, defined in the point
p
{\displaystyle p}
. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore termed four-velocity
v
→
{\displaystyle {\vec {v}}}
, or in components:
v
→
=
(
v
0
,
v
1
,
v
2
,
v
3
)
=
(
d
x
0
d
τ
,
d
x
1
d
τ
,
d
x
2
d
τ
,
d
x
3
d
τ
)
{\displaystyle {\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
{\displaystyle p}
, so at
τ
=
τ
0
{\displaystyle \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 a linear space, termed 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.
== 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 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), termed inertial coordinate systems. 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 termed a spacetime metric, but since distinct events sometimes result in a zero value, unlike metrics in metric spaces of mathematics, the bilinear form is not a mathematical metric on spacetime. World lines of freely falling particles/objects are called geodesics. In special relativity these are straight lines in Minkowski 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-like curves, 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.
time-like curves, 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-like curves 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.
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. The past of the given event is formed by all events that can influence the event (that is, that can be connected by world lines within the past light cone to the given event). The lightcone at 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 past light cone within the entire spacetime. Elsewhere is the region between the two light cones. Points in an observer's elsewhere are 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 the Sun as it was about 8 minutes ago, not as it is "right now". Unlike the present in Galilean/Newtonian theory, the elsewhere is thick; it is not a 3-dimensional volume but is instead a 4-dimensional spacetime region. Included in "elsewhere" is the simultaneous hyperplane, which is defined for a given observer by a space that is hyperbolic-orthogonal to 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. The present often means the single spacetime event being considered.