time space

Spacetimes are the arenas in which all physical events take place—an event is a point in spacetime specified by its time and place. For example, the motion of planets around the sun may be described in a particular type of spacetime, or the motion of light around a rotating star may be described in another type of spacetime. The basic elements of spacetime are events. In any given spacetime, an event is a unique position at a unique time. Examples of events include the explosion of a star or the single beat of a drum.

A spacetime is independent of any observer.[3] However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient coordinate system. Events are specified by four real numbers in any coordinate system. The worldline of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The worldline of the orbit of the earth is depicted in two spatial dimensions x and y (the plane of the earth's orbit) and a time dimension orthogonal to x and y. The orbit of the earth is an ellipse in space alone, but its worldline is a helix in spacetime.

The unification of space and time is exemplified by the common practice of expressing distance in units of time, by dividing the distance measurement by the speed of light.

[edit] Space-time intervals

Spacetime entails a new concept of distance. Whereas distances in Euclidean spaces are entirely spatial and always positive, in special relativity, the concept of distance is quantified in terms of the space-time interval between two events, which occur in two locations at two times:
s^2 = c^2\Delta t^2 - \Delta r^2\, (spacetime interval),

where:

c is the speed of light,
Δt and Δr denote differences of the time and space coordinates, respectively, between the events.

(Note that the choice of signs for s2 above follows the Landau-Lifshitz spacelike convention. Other treatments reverse the sign of s2.)

Space-time intervals may be classified into three distinct types based on whether the temporal separation (c2Δt2) or the spatial separation (Δr2) of the two events is greater.

Certain types of worldlines (called geodesics of the spacetime) are the shortest paths between any two events, with distance being defined in terms of spacetime intervals. The concept of geodesics becomes critical in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.

[edit] Time-like interval

\begin{align} \\ c^2\Delta t^2 &> \Delta r^2 \\ s^2 &> 0 \\ \end{align}

For two events separated by a time-like interval, enough time passes between them for there to be a cause-effect relationship between the two events. For a particle traveling at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval. Event pairs with time-like separation define a positive squared spacetime interval (s2 > 0) and may be said to occur in each other's future or past.

The measure of a time-like spacetime interval is described by the proper time:
\Delta\tau = \sqrt{\Delta t^2 - \frac{\Delta r^2}{c^2}} (proper time).

The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs. (The proper time defines a real number, since the interior of the square root is positive.)
[edit] Light-like interval

\begin{align} c^2\Delta t^2 &= \Delta r^2 \\ s^2 &= 0 \\ \end{align}

In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a squared spacetime interval of zero (s2 = 0).

Events which occur to or by a photon along its path (i.e., while travelling at c, the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a light cone, and all the events which preceded from a light-like interval define a second light cone.

[edit] Space-like interval

\begin{align} \\ c^2\Delta t^2 &< \Delta r^2 \\ s^2 &< 0 \\ \end{align}

When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a causal relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a reference frame such that the two events are observed to occur at the same time.

For these space-like event pairs with a negative squared spacetime interval (s2 < 0), the measurement of space-like separation is the proper distance:
\Delta\sigma = \sqrt{\Delta r^2 - c^2\Delta t^2} (proper distance).

Like the proper time of time-like intervals, the proper distance (Δσ) of space-like spacetime intervals is a real number value.