David_Bowman@georgetowncollege.edu wrote:
>
> Regarding Dick Fischer's question:
>
> >Okay, so how can two material objects - that quasar and this earth - in a
> >14 billion year old universe (give or take two) get 26 billion light years
> >apart? Neither of us can travel at anything like light speed! Am I
> >missing something?
>
> Dick, your question is a common one. The answer lies in the realization
> of the nature of the cosmic expansion of the universe as understood by
> general relativity. *If* spacetime was the nondynamical static flat
> spacetime of special relativity and the matter (and radiation) in the
> universe expanded from some fixed spatial center by exploding outwards
> into the preexisting empty space (as happens in a conventional explosion)
> then your observation would be correct that the distance between any two
> particles carried by the explosion cannot increase with time any faster
> than c.
>
> *But* this is *not* how the Hubble expansion of the universe is
> envisioned. The various galaxies (quasars and other large-scale
> structures) are *not* moving very fast through a static space. In fact,
> to a reasonable approximation their own local motions (w.r.t. a local
> frame of reference in which the expansion proceeds isotropically) can be
> *ignored altogether* on the larger length scales of the universe, and
> they can be thought of as being *at rest*. The picture is that there is
> no outer domain of empty space for the matter to expand into. Rather
> *all* the space of the universe is (and always has been since the Big
> Bang) more or less uniformly filled (on sufficiently large length scales
> much larger than typical intergalactic distances) with matter which is
> locally at rest. The universe expands because the space between the
> particles (i.e. galaxies and galactic clusters) is *stretching*. The
> distance between the galaxies is growing because more space is
> continuously being created between them as the space between them
> stretches.
>
> General relativity places no prior speed limit on how fast the cumulative
> effect of this stretching may occur.
>
> The Hubble law is that the rate of increase of separation between any two
> distant galaxies (i.e. rate of increase in the proper geodesic distance
> between these galaxies w.r.t. cosmic standard time) is directly
> proportional to the (proper) distance between them, v = H_0*d (where v is
> the rate of increase in the separating distance between the galaxies, H_0
> is the Hubble 'constant' and d is their current separation distance).
> The separation distance between two very mutually distant galaxies
> increases with time faster than it does for two galaxies that happen to
> be close together. Thus, for any two galaxies that happen to be farther
> apart than c/H_0 (i.e. d > c/H_0) the distance between them increases
> faster than c (i.e. v > c).
>
> So the reason that when light left the quasar 10 billion years ago when
> it was only 4 billion light years from Earth (or, more properly, from
> the part of the universe from which the Earth would later be formed) and
> it is just arriving here now with the current distance to the Quasar (or,
> more properly, to the remnant left behind by the quasar that it has since
> become) being 26 billion light years is simply that the quasar and Earth
> *have* been separating from each other *faster* than c. The average
> 'speed' of this increasing separation during the light travel time has
> apparently been 2.2*c.
>
> Like special relativity, general relativity *does* put a restriction on
> how fast one object can move with respect to another one at *the same
> place* (or nearly the same place). IOW, when object A emits object B,
> when object A absorbs or is struck by object B, and/or when object B
> passes by object A in a 'near miss' the magnitude of the velocity of
> object B relative to object A must never exceed c, and it is *only*
> exactly c when object B is massless (such as a photon of light).
> Otherwise this relative velocity has a speed which is less than c. Here
> we have implicitly assumed that object A had a positive mass so we could
> consider the velocity of B respect to it, i.e. w.r.t. a frame in which
> object A is at rest. The main restriction imposed by relativity
> regarding speeds is that no *causally informative influence* can travel
> *locally* any faster than c. To the extent that the dynamical nature
> of the geometry of spacetime itself can be ignored (such as in
> special relativity or in general relativity on a length/time scale
> much shorter than billions of (light)years), then this speed restriction
> also extends to relative velocities between objects that are not at the
> same place. But if the space/time separations between the objects
> are comparable to that of the horizon size and age of the universe itself
> then the v < c speed restriction does not apply for motions between such
> widely separated objects. But even in this case the Hubble expansion
> cannot be used to propagate causally significant influences between
> different objects any faster than c. If the cosmic expansion carries two
> objects apart faster than c then these objects can't exert any causal
> influence on each other unless the expansion either slows down enough
> in the future or if it was slow enough in the past for a light ray
> from one object to eventually make it to the other one.
>
> I hope this explanation helps.
Just a footnote to David's explanation. In the simplest cosmological models
of general relativity the scale factor which determines the separation of two
representative objects like galaxies (after matter dominates radiation but before
cosmological repulsion becomes significant) will vary as the 2/3 power of the time
since the expansion began. Its derivative, which will determine the relative speed
of separation of those two objects will thus vary as the -1/3 power & will get bigger
and bigger as we look back in time. For radiation dominated early epochs the scale
factor varies as t^(1/2) so the speeed goes as t^(-1/2) and can be many times greater
than c as t -> 0.
Shalom,
George
George L. Murphy
gmurphy@raex.com
http://web.raex.com/~gmurphy/
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