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.

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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.