High School Hubble equation function of time

[Einstein's Cat]

Is there an equation which explains how the recessional velcoity of a galaxy changes with time? Furthermore, does an equation exist that describes the universe's accelerating expansion; is it the Friedman equation?

 

[Simon Bridge]

Have you seen: https://en.wikipedia.org/wiki/Hubble's_law#Idealized_Hubble.27s_Law

 

[Chronos]

The concept of 'absolute' time has plagued cosmologists and philosophers since the 'beginning' of time. Einstein helped clear up the mystery with GR, but, the notion of absolute time continues to befuddle both mathematicians and philosophers.

 

[Einstein's Cat]

The concept of 'absolute' time has plagued cosmologists and philosophers since the 'beginning' of time. Einstein helped clear up the mystery with GR, but, the notion of absolute time continues to befuddle both mathematicians and philosophers.

Have you seen: https://en.wikipedia.org/wiki/Hubble's_law#Idealized_Hubble.27s_Law

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

 

[Chronos]

From the perspective of physicists, the concepts of distance and time have no absolute meaning. Neither can exist independendent of the other. Hermann Minkowski is credited as the first to understand this odd state of affairs between space and time and coined the term spacetime to describe the interchangeability of these four fundamental dimensions of the universe back in 1908. Needless to say this greatly influenced the thinking of Einstein. See http://www.physicsoftheuniverse.com/topics_relativity_spacetime.html for a brief discussion.

 

[Einstein's Cat]

From the perspective of physicists, the concepts of distance and time have no absolute meaning. Neither can exist independendent of the other. Hermann Minkowski is credited as the first to understand this odd state of affairs between space and time and coined the term spacetime to describe the interchangeability of these four fundamental dimensions of the universe back in 1908. Needless to say this greatly influenced the thinking of Einstein. See http://www.physicsoftheuniverse.com/topics_relativity_spacetime.html for a brief discussion.

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?

 

[George Jones]

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

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