Observed Event Curve - A Photon Through Spacetime

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.

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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.