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Correct interpretation of terms in proper velocity expression?

  1. Nov 10, 2012 #1
    The "cosmological" proper distance from the origin, [itex]D(t)[/itex], to an object at radial co-ordinate [itex]r[/itex] at cosmological time [itex]t[/itex] is given by

    [itex]D(t) = a(t) r(t) [/itex]

    The corresponding "cosmological" proper velocity [itex]v[/itex] of the object is given by

    [itex] v = \frac{dD}{dt} = \frac{da}{dt} r(t) + a(t) \frac{dr}{dt} [/itex]

    Using the definition of the Hubble parameter [itex]H(t) = \dot{a} / a[/itex] and the above equation [itex]D = a r[/itex] we find

    [itex] v(t) = H(t) D(t) + a(t) \frac{dr}{dt} [/itex]

    The first term is Hubble's law for the recessional velocity of a co-moving object whereas the second is the peculiar velocity term.

    I would like to further understand the meaning of the peculiar velocity term.

    To do so I use the relationship between an interval of co-moving time τ and cosmic time t

    [itex] d\tau = \frac{dt}{a(t)}[/itex]

    to rewrite the peculiar velocity term so that we have

    [itex] v(t) = H(t) D(t) + \frac{dr}{d\tau} [/itex]

    Thus the cosmological proper velocity of an object is the recessional velocity of its co-moving inertial frame plus the velocity of the object within this inertial frame. The co-ordinates of the object within its inertial frame are [itex](r,\tau)[/itex].

    Is this the right interpretation of the above equation?
    Last edited: Nov 10, 2012
  2. jcsd
  3. Nov 10, 2012 #2


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    Instead of "co-moving time" can I interpret that as "conformal time"?
    I don't mean to be picky about terminology but sometimes I get slowed down just by unfamiliar words.

    According to paragraph 6 of this essay in John Baez physics FAQ
    "comoving time" is actually just the SAME AS COSMIC TIME.
    So it is not the same as conformal time.
    An interval of conformal could differ by a factor of 1000 from an interval of cosmic (ie. "comoving").

    ==quote from Baez FAQ==
    ... However, this explanation glosses over one crucial point: the time coordinate. FRW spacetimes come fully equipped with a specially distinguished time coordinate (called the comoving or cosmological time). For example, a comoving observer could set her clock by the average density of surrounding speckles, or by the temperature of the Cosmic Background Radiation. (From a purely mathematical standpoint, the comoving time coordinate is singled out by a certain symmetry property.)...

    I haven't heard "comoving time" used much--maybe others have and I just didn't notice. If it is as uncommon as I think, it could cause confusion.

    IMHO better to say cosmic time t, or FRW time t.
    the tau as you define it would be conformal time

    I think your interpretation is perfectly fine, though. Good handling of the equations. Straightforward derivation. You clearly indicate that r is the CO-MOVING radial distance, so it doesn't change except due to the objects own peculiar motion.
    The objects own peculiar radial velocity is then, as you say, dr/dτ

    Maybe someone else will find something wrong. I don't
    Last edited: Nov 10, 2012
  4. Nov 10, 2012 #3
    Sorry - yes I meant conformal time [itex]\tau[/itex].
  5. Nov 10, 2012 #4


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    Hey great!
    That sets my mind at rest. So AFAICS everything is OK.
  6. Nov 10, 2012 #5
    So do inertial observers measure conformal time [itex]\tau[/itex] rather than cosmological time [itex]t[/itex]?
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