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The kinematic origin of the cosmological redshift
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Image of Fig. 1.
Fig. 1.

Spacetime in different coordinate systems. (a) Plot of an expanding Robertson–Walker universe in the usual comoving coordinates . Coordinates are expressed in units of the Hubble length and time. The observer is located at . The dotted curves show the world lines of a particular galaxy and of a light signal from that galaxy reaching the observer at the present time . The dashed and solid curves are contours of constant Riemann normal coordinates ; that is, the coordinates that approximate flat spacetime as well as possible near the observer. (b) The roles of the two coordinate systems are reversed: the coordinate axes are the Riemann normal coordinates , and the dashed and solid curves are contours of constant comoving coordinates . As time passes, the galaxy moves to larger values of ; that is, it is not at rest in this coordinate system.

Image of Fig. 2.
Fig. 2.

The scale factor of a hypothetical “loitering” universe as a function of time (measured in units of the present time). At the time indicated by the dot, a galaxy emits radiation, which is observed at the present time. At the times of both emission and observation, the expansion is very slow, yet the galaxy’s observed redshift is large.

Image of Fig. 3.
Fig. 3.

Parallel transport of the velocity four-vector. The solid and dashed lines show the world line of a galaxy and the path of radiation from the galaxy to the observer in comoving coordinates. The short solid lines show the galaxy’s four-velocity being parallel transported to the observer along the path of the light.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The kinematic origin of the cosmological redshift