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- Thread starter Einstein's Cat
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Simon Bridge

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Chronos

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How does proper distance change with time? Is there an equation to describe that? I looked on Wikipedia and found no such thing

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Chronos

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I see, so proper distance changes with the expansion of the universe and thus with time because theoretical observers along it would appear to have relative velocity to a relative stationary observers? Therefore, meaning that the relatively moving observers would have a different interpretations of cosmological time and thus the proper distance would appear to be longer? Is this correct please?

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George Jones

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How does proper distance change with time?

If ##D## is the (now) proper distance between us and a distant galaxy, then

$$D \left( t \right) = R \left( t \right) \chi,$$

where ##R \left( t \right)## is the scale factor for the universe (a solution to Friedmann's equation), ##\chi## is the constant comoving coordinate difference between us and the galaxy, and ##t## is cosmological time.

Einstein's Cat, are you familiar with calculus? Assuming you are, the rate of change of proper distance is given by

$$\frac{dD}{dt} = \frac{dR}{dt} \chi.$$

If the scale factor is known, then so is its rate of change ##dR/dt## , thus giving the rate change of proper distance, ##dD/dt##.

Multiplying the left side of the above equation by one in the form ##1=R/R##, and using ##D = R \chi## gives the Hubble relation

$$\frac{dD}{dt} = \frac{dR}{dt} \frac{R}{R} \chi = H D,$$

where the Hubble parameter (a function of time) is give by

$$H = R \frac{dR}{dt}. $$

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