# Hubble law in co-moving frame

We normally assume that the recession of distant galaxies is due to the expansion of the space between the galaxies and us.

In a co-moving frame the expansion of space is factored out so that all objects remain at a fixed distance away from us in cosmological time. Thus the co-moving frame is equivalent to our local inertial frame extrapolated out to large distances.

In this co-moving frame, at the present cosmological time, the Hubble law defines a velocity field that increases linearly with distance away from us according to the expression:

v(t) = H_0 r(t).

where H_0 is the present value of the Hubble parameter and t is our local time.

Can these velocities be taken to be "true" velocities relative to us such that the proper time for a galaxy, at distance r moving with velocity v, is relativistically dilated compared to our local time?

John

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Chalnoth
It is fundamentally impossible to speak of a true velocity of a far-away object.

It is fundamentally impossible to speak of a true velocity of a far-away object.

The following article linked from the Wikipedia entry on the Hubble Law seems to argue that the Hubble linear velocity law:

V = H(t) L

where V is recession velocity and L is proper distance is generally true for all distances.

Thus this author implies that one can speak of the velocity of a far-away object.

However the author says that recession velocities can be greater than the speed of light according to this linear formula. I question that statement. I think relativistic time dilation would occur between the distant galaxy's local time and our own. This would imply that the distant galaxy cannot receed faster from us than the speed of light (or else the Lorentz factor would cease to be real valued).

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cepheid
Staff Emeritus
Gold Member
The following article linked from the Wikipedia entry on the Hubble Law seems to argue that the Hubble linear velocity law:

V = H(t) L

where V is recession velocity and L is proper distance is generally true for all distances.

Thus this author implies that one can speak of the velocity of a far-away object.

However the author says that recession velocities can be greater than the speed of light according to this linear formula. I question that statement. I think relativistic time dilation would occur between the distant galaxy's local time and our own. This would imply that the distant galaxy cannot receed faster from us than the speed of light (or else the Lorentz factor would cease to be real valued).

Your problem seems to stem from adopting a special relativistic view. In cosmology, you have to apply general relativity to explain the situation. It is my understanding that in general relativity, there is no unique way of defining the velocities and distances of far away objects. This FAQ posting addresses the question and may clear things up for you:

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Chalnoth
The following article linked from the Wikipedia entry on the Hubble Law seems to argue that the Hubble linear velocity law:

V = H(t) L

where V is recession velocity and L is proper distance is generally true for all distances.