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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.
Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation. [3]: 220–221 The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O ...
Proper length [1] or rest length [2] is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics . In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously.
Fig 4–2. Relativistic time dilation, as depicted in a single Loedel spacetime diagram. Both observers consider the clock of the other as running slower. Relativistic time dilation refers to the fact that a clock (indicating its proper time in its rest frame) that moves relative to an observer is observed to run slower. The situation is ...
In terms of the synchronized clocks and rulers associated with a particular slice of flat spacetime, the three spacelike components of four-velocity define a traveling object's proper velocity = / i.e. the rate at which distance is covered in the reference map frame per unit proper time elapsed on clocks traveling with the object.
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.
In this frame, the separation in time, / , is called the proper time. Δs 2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated.
The arclength parameter is called proper time and usually denoted τ. The length of M is called the proper time of the particle. If the worldline M is a line segment, then the particle is said to be in free fall. [1]: 62–63 A world line traces out the path of a single point in spacetime.