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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]
A different term, proper distance, provides an invariant measure whose value is the same for all observers. Proper distance is analogous to proper time. The difference is that the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events ...
In relativity, proper time (from Latin, meaning own time) along a timelike world line is defined as the time as measured by a clock following that line. 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 ]
Another method is to use a clock indicating its proper time, which is traveling from one endpoint of the rod to the other in time as measured by clocks in the rod's rest frame. The length of the rod can be computed by multiplying its travel time by its velocity, thus L 0 = T ⋅ v {\displaystyle L_{0}=T\cdot v} in the rod's rest frame or L = T ...
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.
Any curve that differs from the geodesic purely spatially (i.e. does not change the time coordinate) in any inertial frame of reference will have a longer proper length than the geodesic, but a curve that differs from the geodesic purely temporally (i.e. does not change the space coordinates) in such a frame of reference will have a shorter ...
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Hsu and Hsu claimed that measuring time in units of distance allowed them to develop a theory of relativity without using the second postulate in their derivation. It is the principle of relativity, that Hsu & Hsu say, when applied to 4D spacetime , implies the invariance of the 4D-spacetime interval s 2 = w 2 − r 2 {\displaystyle s^{2}=w^{2 ...