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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

    marcus

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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
    http://math.ucr.edu/home/baez/physics/Relativity/GR/hubble.html
    "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.)...
    ==endquote==

    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

    marcus

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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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