I’ve been quiet for a while, partly because of so many other things going on in the world, but also partly because I’ve been struggling with a feature of the OE curve, specifically the apparent necessity for double dipping which I even retracted for a while because I was unhappy with it (and even once reinstated it continued to bother me). Since then, I have wracked my brain trying to come up with some way for the OE curve to work in such a way that double dipping was not necessary, but the underlying problem remained (see Observable Events Curve - Is Double Dipping Essential? for details).
Recently it struck
me that there was another way to think about it. Everything, including photons, travels
through spacetime at the same rate. I discussed
this a very long time ago in the post On Time, where it was expressed as “our
speed through spacetime is conserved (or invariant)”.
When we look at the
universe around us, we see two types of motion – kinetic motion and recession. By kinetic motion I mean the normal everyday motion
through space that we are used to.
Recession is the apparent motion of distant objects due to the expansion
of space. It is not actually true that objects
need to be distant to undergo recession, it is just that more local objects
undergo so little recession that it can be safely ignored.
We can only
perceive this recession though because we are relatively stationary and the vast
majority of our speed through spacetime is due to the temporal component. It is different with photons. For photons, the universe is basically a
block universe, with no time elapsing in their frame. Which means that any expansion that happens is
simultaneous (in the photon’s frame) with the photon’s transition from one part
of space to another.
This is simple to
think about for a photon that was emitted at the very beginning of the universe
and is observed today (about 14 billion years later). The entire universe was scrunched together at
emission and (in a FUGE universe) its radius has expanded by 14 billion light
years since, so photon has had a total speed through space time of 14 billion
light years (due to expansion) divided by 14 billion years, or the speed of
light. It is equally simple to think
about for a photon that was emitted very recently, such that the expansion
during the period between source and observation is negligible. The entirety of the travel was through space
and the speed will be the speed of light.
It gets a little
more complicated when we consider events that occurred between, but the principle
is the same. There are two comoving locations
(at rest with respect to the Hubble flow), one where the photon was emitted and
the other where the photon is observed. At
the time of emission, there is a separation between those two locations (according
to the observer [atto]). There is also
a temporal separation between emission and observation (atto). During the time elapsed between emission and
observation (atto), the comoving distance at emission will expand by an amount
that is relative to the initial separation (atto).
In terms of the OE curve, where the equations are
explained, we have:
In this example, a
photon that reaches an observer after t=4 billion years (noting that x=ct and t0=14
billion years) will have travelled x'=x(ct0-x)/ct0=2.85 billion
light years across space and experienced an amount of expansion of ct-x'=x2/ct0=x.t/t0=1.14
billion light years, for a total of x=ct=4 billion light years.
---
This does not, in
itself, explain how the double dipping issue is resolved. To do that, we have to realise that the concept
used in the “single dip” was in error.
My thinking was that when a FUGE universe expands, it does so by adding
one unit of Planck length to the radius every unit of Planck time, but where
the radius is incremented is stochastic – see Observable Events Curve - Not Quite a
Drunkard's Walk, but note that I am double dipping all the way in that
post.
I reasoned that if the
expansion is between the observer and the photon, then for each unit of Planck
time, the space would expand by one unit of Planck length and the photon would
travel one unit of Planck length, so the distance would remain constant. If the expansion was not between the observer
and the photon, then the photon would get one unit of Planck length closer to
the observer. There’s no way in this
conception that photons could move away from the observer’s location – which must
be the case for photons that reach us from the early universe. And, therefore, I seriously considered double
dipping.
But double dipping
is not necessary. If the photon is affected
by expansion of space that it is in the process of moving through, then it only
moves forward when the expansion happens behind it. Plugging that concept into the VBA script results
in this chart (each dot indicates where the expansion occurs, the red curve is
the simulation, the underlying green curve is the OE curve):
Precisely matching
the OE curve with no double dipping. So,
to answer the question – again – no, double dipping is not essential.
It maybe difficult to see at this resolution, but the simulation curve is wandering somewhat. This is an artefact of using one million year units, rather than units of Planck time. The same wandering would exist at the Planck scale but it would be imperceptible at the human scale.
