Introduction
In ordinary geometry, points are labeled by their coordinates
P1 = (x1,y1,z1,...) and P2 = (x2,y2,z2,...).
In spacetime, points are called "Events"
and are represented with a mix of space and time labels such as
E1 = (x1,y1,z1,t1) and E2 = (x2,y2,z2,t2).
We can define the distance between two points in ordinary space, L,
using the Pythagorean Theorem as:
L2 = (x2-x1)2 + (y2-y1)2w
e can also define the distance between two nearby events in spacetime, ds. d
s2 = c2(t2-t1)2 - (x2-x1)2
And like ordinary spatial distance, ds (which is called the spacetime interval) is invariant; that is, it is measured to be the same by all observers and its value does not depend on how we choose to label the events.
Spacetime Interval
The spacetime interval is defined , more compactly, to be:
ds2 = c2(dt)2 - dL2
where c is the speed of light and dt is the difference between the time labels at the two nearby events. The difference dt is called the coordinate time difference to remind us of the fact that its value depends on how we choose to label the events, in contrast to the interval ds which does not. You should note that, although in a purely spatial sense the L in the spacetime interval ds can be found using Pythagorean Theorem and was an invariant in 3-dimensional space, this number is no longer an invariant in the combined spacetime. This leads to many interesting "problems" and paradoxes because time and space are blended together, and their magnitudes are not the same for all observers viewing the two events from different coordinate frames. Notice something peculiar about the interval. Its square can be positive, zero or negative! It turns out that this is the only way we can get an invariant length in spacetime.
Time-like Interval
If ds2 > 0 the two nearby events are said to separated by a time-like interval. This means that a messenger could travel from one event to the other at less than the speed of light. In other words, a messenger could set off from one spatial point, at a given time, and arrive at the nearby spatial point at the time which together with the spatial point define the second event. It also means that you can find a coordinate frame in which the interval is something you can measure on a clock but not a meter stick. This is called the Proper Frame.
Space-like Interval
If ds2 < 0 the two nearby events are said to be separated by a space-like interval. This means that no messenger could travel from one event to the other because to do so would require the messenger to travel at superluminal speeds (speeds greater than that of light). It also means that there is some coordinate system in which this interval can be measured only on a meter stick and not a clock. This is called the Proper Frame.
Light-like Interval
Finally, if ds2 = 0 the two events are said to be separated by a light-like, or null, interval. That is, a messenger could connect these two events if the messenger were to travel at exactly the speed of light. That is, you could start from one point, at a given time, and arrive at the other just in time, if your speed were exactly that of light. It follows that two null separated events can be connected by a light ray. Indeed, worldlines that are null intervals are precisely the paths followed by light rays in spacetime.
Related EoC Articles
- Spacetime
- Spacetime: Foundations
- Spacetime: Gravity
- Spacetime: Light Cones
- Spacetime: Special Relativity
Preview Image
An artist's concept of twisted space-time around Earth. (Source: Spacetime Vortex - NASA.)
Citation
Odenwald, Sten, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2009. "Spacetime: Geometry." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published February 13, 2008].
<http://www.cosmosportal.org/articles/view/138091/>
Contact Us
Send to a Friend
Comments
There are no comments.