Cosmological
Horizons vs. Standard TheoryThe basic
idea of a cosmological horizon seems to be straightforward: if the
universe is expanding, then the further away an object is, the faster
it'll be receding from us. Beyond a certain distance, objects might be
expected to have a recession speed greater than lightspeed, which would
lead us to expect that we wouldn't be able to see them. The critical
distance from an observer would therefore seem to mark out an
observational horizon. However,
when we get
into this subject more deeply, things don't turn out to be quite so
straightforward.
Argument #1:
"Cosmological
horizons must
exist "In an
evenlyexpanding universe, the mutual recession between two chosen
points on the surface depends on how far they are apart. For a given
distance, the cosmological
recession velocity will be half lightspeed ... at double that distance,
it'll be lightspeed .... at double that distance,
the nominal recession
will be twice lightspeed. In a
spatiallyclosed expanding universe,
it'd seem that
we can always find objects at the critical distance, even if our
universe is very small, and so that we have to "loop the loop" around
the universe to get there.
Argument
#2:
"Cosmological horizons cannot exist"In
a spatiallyclosed universe with a reasonablydense, reasonablyeven
distribution of matter, any two adjacent nearby stars ought to be
capable of
seeing each other and being in contact with each other. But if
cosmological horizons exist, then any point in space could be said to
be on a cosmological horizon with respect to a suitablychosen viewing
position.
For any given pair of stars, there'll be a viewing position somewhere
whose nominal cosmological horizon passes between the two
stars, with one star is
reckoned to be inside the horizon, and the other one outside. And
yet ... signals still manage to pass between them, in both directions Mainstream
response #1: "Special relativity's velocityshift relationships don't
apply to cosmological shifts" A
distant object on our side of a cosmological horizon is quite capable
of accelerating to a location on the other side, which sounds as if it
might cause problems for special relativity's idea that an object can't
increase in velocity past someone else's speed of light. Cosmologists
counter that special relativity's arguments (and its shift
relationships) don't apply when cosmological curvature is in play,
because cosmology isn't a flat spacetime problem. But it spoils general
relativity's efficiency to have two different effects that look the
same, and associate similar effects with recession velocities, but
which apply those effect according to two different laws. It would seem
that a distant receding redshifted galaxy can have two different
velocities, a Hubble recession velocity and a local velocity compared
to its immediate environment, with the two types of recession
("velocity" and "Hubble velocity") each generating different Doppler
characteristics. This doesn't seem to be a very satisfactory situation.
Mainstream
response #2: "Cosmological horizons don't exist"
We're
sometimes told that the recession velocityshift relationship for
cosmological effects is freq'/freq = c / (c + v_{HUBBLE}),
which doesn't produce a zero observed frequency for any finite
value
of v_{HUBBLE}.
This would allow signals to reach us even from objects that
are nominally "Hubblereceding" at more than the speed of
light. But if we accept this answer, it'd suggest that
mainstream work that does
involve
cosmological horizons was wrong. There'd then seem to be some
disagreement between experts as to the correct way to tackle these
problems.
Argument
#3: "Cosmological horizons must exist"If
the age of the universe (measured using "insidertime") is finite,
then as we look further and further away, and see events that
took place progressively longer ago, we should eventually
reach a limit that corresponds to the initial event(s) at the
universe's origin. We should be able to plot a radius around us at
which the events that we'd be seeing would correspond to the Big
Bang. We shouldn't be able to see any further back or
any
further away than this, because there shouldn't be any events inside
our universe before this, and there also shouldn't be any space before
this for events to take place in. If
observerspace
coordinates seem to stop dead at or before this limit,
then for this to happen in a wellbehaved way, we seem to need the
observational limit to represent a coordinates where timeflow appears
frozen ... that is, we need our spatial limit to observation to behave
like an event horizon. A variation on
this argument describes the hypothetical initial originpoint of the
universe as a singularity, which then
triggers the cosmic
censorship hypothesis. According to
the c.c.h.,
singularities (which represent the limit at which physics appears to
break down) aren't allowed to be "visible". If they exist, they must
always be hidden from us by an intervening horizon ("no
naked singularities"). Considering
the problem in terms of the censorship hypothesis, and considering the
viewable universe as being topologically equivalent to a black
hole turned insideout, leads us to the idea that
perhaps effects due to the cosmological curvature
of spacetime might be expressed (with a topological twist)
using language and ideas that were originally developed for
the gravitational curvature of spacetime.
GravitationequivalenceAs
the universe expands, it's massdensity reduces. This means that as we
look at more and more distant objects, we're effectively looking back
in time to a period in which the background gravitational field
strength was higher. In literal observerspace
descriptions, we should see what appears to be a gravitational field
strength that increases with distance, and we should see the more
distant stars seeming more redshifted, apparently as a
function of this gravitational field, and/or as a function of their
recession. We can also argue that (in the observerspace description)
the apparent existence of the field should be associated with a
freefall motion of stars embedded within it, so that we should see
distant galaxies receding from us, and receding faster the further away
they are. This is simply an alternative geometrical
description of the same expandinguniverse model that we started out
with. It's the same physics looked at from a different perspective. At
a certain distance, the field strength difference between
"hereandnow" and "thereandthen" will be expected to have an escape
velocity equal to or greater than lightspeed, and at that
point we should expect the redshift to become total. Within
the observerspace description, this horizon then looks like a
gravitational horizon surrounding a highgravity object. If
we take our descriptions of the redshift distribution due to a black
hole, and due to spatiallyclosed cosmological model, we can combine
them to produce a single descriptions. This is a powerful method, as if
means that cosmological effects have to be bound by gravitational
theory, and vice versa While
it is very satisfying to be able to predict the same effects in
different ways, this trick doesn't seem to work with our
current implementation of general relativity. While we might like to
model the same horizon using cosmological arguments and gravitational
arguments, current GR can't manage it. Cosmological horizons are
surrounded by radiation that the observer can;t see directly, but which
appears when they accelerate through a region (see Unruh
radiation) allow indirect radiation. Cosmological
horizons
leak. GR1915's gravitational
horizons are
totally sealed, and completely cut a region off from the
outside world. If we want to say that current general
relativity is the correct classical description,
then gravitational horizons would have to operate according to
different rules to cosmological horizons. If we wanted to unify both
descriptions, then since cosmological horizons have
to leak, gravitational horizons would need to leak, too, and
according to 1960's relativity theory, this was supposed to be provably
impossible. The situation shifted slightly
in the 1970's, with the idea of Hawking
radiation ...
Cosmological Hawking
radiationAccording to modern quantum
mechanics, spacetime around an eventhorizonbounded region should
radiate, and according to 21stCentury quantum
mechanics, the patterns embedded in that radiation are now
supposed to belong to information that would be reckoned to have
previously been somewhere behind the horizon. As a result of
Stephen Hawking's 1970's arguments, we now believe that gravitational
horizons should be "leaky" after all, and we might think that this
solves the incompatibility problem mentioned earlier. Hawking
radiation was originally considered to be a strictly
"quantum" effect, but as we studied the relationships in more detail,
we found that the same basic statistical patterns also happened in
other
situations due to entirely classical mechanisms, and one of the
situations involving classical Hawking radiation
was radiation through a cosmological horizon.
The
idea of cosmological Hawking radiation seems
to dissolve the apparent contradiction between our earlier
arguments that cosmological horizons have to exist, but that
information has to be capable of passing across them. If we look at the
old classical indirectradiation mechanism, and compare them to the
behaviour of cosmological horizons, we find that a cosmological horizon
actually counts as a category of acoustic
horizon
... it's a mathematical surface that theoretically describes
observational limits, but the surface fluctuates and fizzes in response
to local events, some of which are behind the horizon surface. An
acoustic horizon is descriptive rather than prescriptive
 it doesn't dictate the physics, or the events that are allowed to
happen, it instead jumps about mathematically in response to
them. The "statistical mechanics" of acoustic horizons
involves fluctuation and radiation, just like the "quantum mechanics"
description of Hawking radiation, and acoustic horizon radiation is now
considered a legitimate instance of Hawking radiation.
If
we now believe that cosmological horizons and gravitational horizons both
radiate, does this mean that we can now finally unite the two
descriptions and solve our problem?
Not
quite, because while the cosmological radiation effects are "mundane"
and operate according to straightforward classical physics, current GR
has no way of replicating the effect classically. To get a GR black
hole to radiate we have to invoke Hawking radiation effects separately,
and put the effect in "by hand" using quantum mechanics. In the
"cosmological horizon" case, QM is describing the statistical results
of classical behaviour, but in the GR+QM black hole case,
there's nothing within GR for the QM statistics to relate to, we simply
say that since the QM result has to be correct, the
radiation must be due to some of those spooky counterintuitive effects
that QM descriptions sometimes produce. The difference
between these two descriptions seems to be due to the choice of
underlying metric, the classical
indirectradiation effect happens in the context of acoustic
metrics, while general relativity instead
reduces over small
regions to the Minkowski metric of
special
relativity. We can imagine a different general theory that
reduces to an acoustic metric (and which would therefore seem to allow
indirect radiation and solve the black hole information
paradox) but since the relationships of the metrics are
different, this would seem to mean stepping away from special
relativity to a different sort of relativistic model, and if we've been
trained to see relativity theory through they eyes of current
physics textbooks, we'll have no idea what such a "nonSR" theory of
relativity would look like. Also, if we were to
combine the cosmological and gravitational descriptions, we'd really
need the resulting unified description to apply the same velocityshift
relationship to both situations, and if the cosmological case can;t
use the shift relationships of SR, that'd mean that gravitational
theory couldn't use them either. Since we'd need a set of
relationships that could reproduce gravitational horizons, but which
wouldn;t be special relativity, parts of our gravitational theory and
parts of our cosmological theory would be wrong.
ConclusionsAlthough
Einstein's general theory was eventually applied to the case of a
spatiallyclosed, expanding universe, it was not originally devised
with this sort of universe in mind. The theory was not
designed to make use of relativistic and geometrical arguments that
only make sense in a spatially closed universe, such as the
gravitationequivalence idea mentioned above. The
"cosmological horizon" problem is an interesting one, in that it seems
to break current theory, and also seems to suggest how a replacement
theory might be constructed. To solve the "horizon problem", we seem to
be required to switch from special
relativity's Minkowski metric to an acoustic metric, which in turn
would seem to require us to implement the idea of local
lightspeed
constancy, and the principle of relativity in inertial physics, with
gravitomagnetic
arguments rather than with special
relativity's flat spacetime. If the mathematical horizon at a
cosmological threshold fluctuates and jumps about in response to
mediumscale and smallscale events there, then "cosmological
horizon physics" must allow lightspeeds to be affected in a
fundamental
way by the motion of bodies, even when those bodies wouldn't normally
be said to have significant gravitational fields. And since any
point
in the universe could be considered as being at a cosmological horizon
for someone, these
cosmological arguments for the variability of lightspeeds should also
apply to physics here on Earth. "Academic"
questions that at first sight would seem to be remote from
everyday experience and testing ("What is
our view of ancient events at the other side of the universe,
and of cosmologicalscale features?") can
end up telling us important things about our own physics ("If
I throw this baseball, does it affect the speed of light in the adjacent region?").
References Michael
A. Seeds, Foundations of Astronomy, Third Edition
(1992)
Box 6.1 The Hubble Law " ... the
more distant a galaxy, the faster it recedes from us. A
galaxy's radial velocity in kilometers per second equals the Hubble
constant times its distance in megaparsecs. ... V_{r}
= HD ... The best measurements of distance and velocity
suggest that H is approximately equal to 70 km/sec/Mpc."  Eric
Baird, Relativity in Curved Spacetime
(2007) section 12 "What's wrong with general
relativity?", especially 12.11 "Do
cosmological horizons count as 'acoustic'?" and
12.13 "Grand unification?"
 For
the reasons given above, trying to find a definitive statement on the
status of cosmological horizons under standard theory can be difficult.
For a critical overview of some apparently conflicting views, see:
 Tamara
M. Davis and Charles H.
Lineweaver, "Expanding Confusion: Common
Misconceptions of Cosmological Horizons and the Superluminal Expansion
of the Universe" Publications of the
Astronomical Society of Australia (PASA) 21 97109
(2003) astroph/0310808"We
use standard general relativity to illustrate and clarify several
common misconceptions about the expansion of the Universe. To show the
abundance of these misconceptions we cite numerous misleading, or
easily misinterpreted, statements in the literature. In the context of
the new standard LambdaCDM cosmology we point out confusions regarding
the particle horizon, the event horizon, the ``observable universe''
and the Hubble sphere (distance at which recession velocity = c). ... "
 Early
work by Zel'dovich on particleproduction across a field gradient was
cited in "MTW"'s Gravitation
chapter 30, p.816 (1973) as suggesting a possible solution to
problems involving causality and cosmological horizons. The subject of
quantum mechanics and indirectradiation effects seems to have been
tentatively considered for the case of cosmological horizons shortly before
Hawking's work applied similar arguments to
gravitationally horizonbounded bodies.
 For
references on Hawking radiation (quantum and classical) see the
references section at hawking_radiation.html .
