Observable Events Curve - Simplified

This is an attempt to describe the observable events (OE) curve concept as simply as I can manage, without the complications that are introduced when trying to address predictable objections.  If you can’t suspend judgement long enough to grasp the concept without raising objections, the complicated version is discussed in:

Taking Another Look at the Universe,

Mathematics for Taking Another Look at the Universe,

Concepts in Taking Another Look at the Universe,

Observable Events Curve - Fast and Slow Expansion,

Observable Events Curve - Addressing a Point of Potential Confusion,

Observable Events Curve - Hang On!  Aren't I Double Dipping? (SPOILER - No, I am not.),

Observable Events Curve - Shifting About Redshift (Including an Alternate FUGE-like Universe), and

A Very Fast Rocket and a Rather Distant Planet. 

It’s possible that your objection has already been anticipated and addressed.  Some other objections may be raised in response to the FUGE universe.  There are currently 17 additional articles that refer to FUGE (with that label) and many earlier articles that lead to the concept, so I won’t list them here.

If you have additional objections that I have not already addressed, please let me know.

---

If a photon hits your eye and is thus “observed”, then it must have travelled from where it originated via some path through spacetime.  To simplify, we will consider its path through empty space such there is no deviation in that path due to gravitational lensing.

The photon could be thought of travelling in a straight line from origin to observer, like this:

A complicating factor is that the universe through which the photon is travelling is expanding.  This means that the photon not only needs to cover the distance between the origin and observer, but also the additional space that is introduced due to expansion during its travel time.  So it looks a little like this:

Currently, the universal expansion rate is given by the inverse of the age of the universe (ꬱ).  For a FUGE universe, the expansion rate is always the inverse of the age of the universe.  This means the expansion rate starts off very fast (approaching infinity as ꬱ→0) and decreases as ꬱ increases.

In the second animation, I have shown a situation in which the origin and observer are being moved apart by expansion at a rate of 0.5c.  While it might seem that this means that the photon is approaching the observer at c-0.5c=0.5c, it’s not that simple.  Some keen-eyed readers will have noticed that the photon covers a greater distance in the second half of its travel than in the first.

Let’s call the distance from the observer to the photon x' and say they were initially 5 billion light years apart (in a universe that is ꬱ=10 billion years old [at that time]).  Saying that the photon is observed t=0 (where t is an indication of how long ago something happened). Tabling the results, we have:

x' (Gyr)	t (Gyr)
0.00	0
0.95	1
1.80	2
2.55	3
3.20	4
3.75	5
4.20	6
4.55	7
4.80	8
4.95	9
5.00	10

From this we can deduce that the universe is ꬱ=20 billion years old when the photon is observed.  Then we can chart this, with t as the horizontal axis and x' as the vertical axis:

This is what I call an observable events (OE) curve, because all of the points on that curve are spacetime coordinates (also known as “events”) from which a photon observed at (t,x)=(0,0) may have originated.  The equation associated with this curve is x'=(ct₀-x).x/ct₀ for any time t ago, where x=ct and t₀=ꬱ (the age of the universe at the time of observation).  Noting that all the events are relative to the observer, at (t,x)=(0,0), here is a final animation which may help cement understanding of the concept:

If we plug the value t₀=ꬱ=14 billion years into the formula (as a rough approximation of the age of the universe) and plot across the period 0 to 14 billion years, we get the observable events curve for our universe (if it is a FUGE universe):

In GIF form similar to the above, this becomes (noting that an observed photon cannot start at (t,x')=(14,0) precisely, otherwise the observer and the photon are already collocated, it is just at a time [t→0] when [x'→0]).  Note that in this version, we no longer have the origin, but rather the causal horizon relative to the observer, which can be thought of as a location that is receding away from the observer at c:

Note that this should not necessarily be considered to be representing the path of an individual photon since it implies that the photon is being dragged away in the earlier travel.  I'm still mulling over this aspect.

In summary:

• any photon which we could possibly observe must have had a path through spacetime on their way to being observed,
• the universe is expanding, so some element of the travelling distance covered by any photon on its way to us before being observed must have been due to expansion, and
• the longer ago a photon started on its way to us, the greater proportion of its travel must have been due to expansion – approaching 100% as the departure time approaches now minus the age of the universe.

A Very Fast Rocket and a Rather Distant Planet

The below is retained to note a probable misstep.

The OE curve should not be thought of as simply plotting the path of a photon.  It is instead the array of potential events from which photons that are observed might have originated.  Therefore the notion that the concept of the OE curve can be extended to describing the path of a rocket is questionable.

The reason why it probably does not work to consider the OE curve as a path in spacetime is a little complicated and has been troubling me for weeks, so far with no satisfactory resolution.  When I have sorted it out to an at least preliminary level of satisfaction, I will try to explain in a follow-up post.

---

I was trying to have a conversation with an old school friend about the OE curve and we went off on a tangent about a planet that is so far away that it is receding at 0.5c and an effort to get there in a rocket travelling at 0.75c.  There was a slight misunderstanding about the rocket passing the planet at 0.25c (that’s simply not how it works) but, because I was trying to talk about the OE curve, it got me wondering about how the situation could be charted in a similar way.

I started by considering the comoving distance to the distant planet (in FUGE universe) and the causal horizon.

The next step was to transform those to cosmological proper distances.

Note that my old school friend and I talked freely about the notion of a causal horizon, but the term does not appear to be particularly well defined.  I believe that we were both using a definition such that the causal horizon is expanding away from us at c, irrespective of how large the universe is.  In a FUGE universe, this is a straight line in terms of cosmological proper distances but not in a Standard Model universe (see The Problem(s) with the Standard Cosmological Model, noting that the charts there refer to radius of the universe, not the causal horizon, but the effect is similar).   In the chart above, the speed of the rocket is 0.75 times the recession speed of the causal horizon which would be, with the definition we were using, 0.75c.

There was also a certain lack of clarity when talking about the speed of the rocket and the relative speed of the planets (ours and the distant one).  To clarify, both planets were considered to be at rest relative to the CMB and the rocket was considered to be travelling at 0.75c relative to us (with the confusion arising from the question as to what the speed of the rocket was relative to the distant planet).  To make things simple, I referred to the speed of the rocket relative to the CMB as being a “kinetic speed” – as opposed to a recession speed, or a closing speed.  There’s also the opposite of a closing speed, which we could call an “opening speed”, which is the combination of kinetic speed away and recession speed.  Closing speed in this context is the combination of kinetic speed towards and recession speed.

After transformation from comoving to cosmological proper distances, I got this:

It didn’t intuitively make sense to me and I didn’t think it was particularly useful.  Plus there was something that seemed wrong.  I added another line, being the 0.75c line:

It makes sense, but it wasn’t quite what I was expecting and … well, it seems too perfect.  Is it a coincidence (noting that I don’t like coincidences)?

In thinking about it, I realised that travel from our planet to the distant planet is equivalent to travel from the distant planet to our planet, so I could use a variation of the OE curve to check the results.  Initially, I just considered a rocket that reaches us now (at t₀=13.8Gyr) and looked at when it must have passed a distant planet that is receding away from us at 0.5c, if it were travelling at 0.75c:

Then I changed the timing to make the time the rocket passed the distant planet 13.8Gyr:

Note that the intercept is 13.8Gyr from where causal horizon line (etc) crosses the horizontal axis, so it’s equivalent to now.  It seems to line up with what I got above.  So, given that I now had a spreadsheet into which I could reliably adjust the parameters, what would happen if I was thinking about a distant planet at a distance at which is receding at 0.3c and the rocket was going at 0.8c?

This indicates that the intercept of the red line with the orange curve and blue line is not a coincidence.  In retrospect, this is clearly not a coincidence.  What it represents is, according to the observer at (0,0), the effective speed of the rocket required to travel from the event of being collocated with the distant planet.

The remaining question is why is the red line 0.25c in one and 0.75c in the other?

It’s because they are different things.  For both the period of travel is the same, 27.6Gyr.  But in the first chart, the distance being indicated is the distance between us and the distant planet after travel, whereas in the third it’s the distance before travel.

Here’s the third chart marked up to illustrate:

Therefore, using this, we have a mechanism for plotting the path through spacetime of any object travelling at a constant velocity in a FUGE universe.

Out of interest, and because it was a point of contention, note that the path of the rocket as it approaches its destination (us) rises in gradient to 0.75c, or, in other words, its closing speed rises to 0.75c, which is as expected since that is its proper speed.