I retracted this for a while, before reinstating it. There might be some edits required.
---
In Observable Events Curve - Hang On! Aren't I Double Dipping? (SPOILER - No, I am not), I addressed a self-evident concern that I have had to revisit. I wrote:
“It could be argued that I used
the expansion twice, first to retard the advancement of the photon and then to
stretch the racetrack, pushing a sufficiently distant photon even further away
than it started.
“Well, yes, I did.”
Then I referred to this as “apparent double dipping”.
From there, I went on my merry way, until I ran into a
conceptual roadblock. That roadblock
made me reconsider the content of Observable Events Curve - World Line,
in which I point out that the OE curve could be thought of as the world line of
a photon.
The precise nature of the roadblock is difficult to explain,
but I will have a go. Let’s start with
the event horizon that I have subsequently described as the causal horizon, a
point sufficiently distant from the observer that it expands away at the speed
of light. The underlying concept is that
any photon emitted at the causal horizon will never interact with the observer
(assuming no reduction to the rate of expansion). It certainly While it seems obvious that anything recessing
from us at the speed of light will never be observed, this is not necessarily the case - see Magic Paving Stones and the articles that follow.
(Edit: What we can say is that ...) Anything closer to us than the causal horizon can,
eventually, be observed because its location is recessing at a subluminal rate. All well and good so far.
Now recall that a FUGE
universe is, by definition, granular.
In such a universe, the distance to the causal horizon expands by one
unit of Planck length every unit of Planck time. We can imagine it like this, for any given
direction:
The red line at the bottom is our notional observer’s position.
The green square is the notional additional “grain” of space
(Planck length) for every additional “grain” of time (Planck time).
Say that soon in the process we have an occurrence (avoiding
the term “event”) that generates a photon that has a velocity (speed and direction)
towards the observer, at the farthest possible location. The farthest possible location is moving away
from the observer at the speed of light and the photon is travelling towards
the observer at the speed of light (noting that photons can travel from one grain
of space to an adjoining grain of space in one grain of time).
So, what will happen is that there will be an additional
grain of space that appears, expanding the distance, and in the same grain of
time in which that happens, the photon moves closer towards the observer by one
grain of space:
This seems to mean that (edit: simplistically) a photon that is generated at that distance
would never reach the observer. It is,
of course, not that simple. There is no
reason why the additional grain of space should always appear right next to the
observer. Instead, it makes more sense
that if an additional grain of space were to appear, it would do so at a random
location across the distance. And that
could include locations on the other side of the separation between the photon
and the observer, meaning that eventually the photon would be
observed.
Showing this graphically is going to take more time than I
can justify, but I can show it in a chart. Consider when the distance is three grains. In
that case, there are four possible locations for an additional grain:
We can use RANDBETWEEN(0,<max before expansion>) to
select the location for the additional grain.
Say the universe radius is 100 grains at the time that the photon is generated
at the 100th grain. This is
what the chart looks like:
That seems to be a problem. And
it’s not a problem that is limited to a granular universe. It’s simply that this curve would be smoother
without granularity, which also happens when the grains are relatively smaller. Here is what it looks like with grains that
are about 30 times smaller (orange curve) versus the non-granular option (green
curve):
---
If we “double dip” the problem goes away. Say that we add two randomly located grains between
the observer and the causal horizon and look at a photon on that causal
horizon, we get this:
It doesn’t matter how often I recalculate, the chart remains
the same.
The next step is to consider a situation similar to that of
the OE curve. An observer at some
nominal time (t0) observes a photon that was generated a
distance x' away at
time t ago. We can set the time by
selecting a causal horizon distance (in my example that is 1500 grains of time
after expansion started) and the initial separation as an offset from that
causal horizon distance (in my example that is 1/6th, so 250 grains
of space). Two grains of space are
added, with a random distribution across the distance to the causal horizon. This is what we get:
This may look similar to the calculated OE curve, and indeed
it is. We can calculate it because we
can work out what the value of t0 is from the intercept with
the vertical axis. Overlaying the calculated
OE curve, we get:
The correlation is remarkable.
For this reason, I must walk back my walking back of the
claim that the OE curve represents the world line of the photon, or in other
words, the path of the photon between spacetime between generation and
observation.
That said, the apparent essentiality of the “double dipping” does need some more thought. I am not convinced that it is real and suspect that it might be an artefact of the modelling, especially since the OE curve was arrived at using the assumption of a Hubble parameter that matches that of a FUGE universe. (Edit: some additional faffing about that led to the Magic Paving Stones series has made me more convinced that it is real, but the mechanism still eludes me at this time, or rather, I have an idea as to a mechanism, but I don't really like it.)
I would certainly welcome someone’s ideas on this conundrum.
