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Since the rod is stationary in K 0, the distance between the marks is the proper length of the rod regardless of the time lapse between the two markings. On the other hand, it is not the proper distance between the marking events if the marks are not made simultaneously in K 0."
On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted , the proper distance () at an arbitrary time is simply given by = where () is the scale factor (e.g. Davis & Lineweaver 2004). [2]
The proper time interval between two events on a world line is the change in proper time, which is independent of coordinates, and is a Lorentz scalar. [1] The interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line.
Length U upon the x-axis represents the rest length or proper length of a rod resting in S. The same interpretation can also be applied to distance U ′ upon the ct ′ - and x ′ -axes for clocks and rods resting in S ′ .
A time interval measured using a single clock that is motionless in a particular reference frame is called a proper time interval. [41] Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of . Instead of making straight up-and-down motions, observer B ...
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. [1] It is also known as Lorentz contraction or Lorentz–FitzGerald contraction (after Hendrik Lorentz and George Francis FitzGerald ) and is usually only noticeable ...
In the following, the rest length [3] or proper length [4] of an object is its length measured in the object's rest frame. (This length corresponds to the proper distance between two events in the special case, when these events are measured simultaneously at the endpoints in the object's rest frame. [4])
A fuller explanation of the concept of coordinate time arises from its relations with proper time and with clock synchronization. Synchronization, along with the related concept of simultaneity, has to receive careful definition in the framework of general relativity theory, because many of the assumptions inherent in classical mechanics and classical accounts of space and time had to be removed.