In Mathematics for Taking Another Look at the Universe, I discussed the equation that I introduced but not fully explained in Taking Another Look at the Universe. In that second post, however, I still sorted of skated over a couple of elements and I didn’t specifically explain how I came to the equation that I was notionally explaining. That was not entirely unintentional.
I usually like doing things from first principles but, in this
instance, it was more a case of inspiration.
And I was trying to solve a different problem. Back in post, right at the end, I mention an
internal struggle that I have (which also triggered A 4D Black Hole?) related to
considering the universe as akin to both an expanding sphere and an expanding
glome. In it, I threatened to give this some
more thought and that was what I was doing.
So, because it is what I have been doing for years, I eliminated
one dimension from consideration and thought about a (spatial) circle moving
towards me (in time), but one which was expanding (from nothing) as it
went. Then I was inspired to think about
it shrinking back to nothing as it reached me and realised that this was equivalent
to me being stationary and having a sphere move past me (so long as I could
only see slices that were perpendicular to its motion). Add a dimension and I had a way to consider a(n
expanding) sphere that traced out a glome.
This is very similar to what I showed in the chart first
shared in Taking Another Look at the Universe:
The major difference is that I have eliminated yet another dimension. I am only considering one spatial dimension there. And time.
Which brings me to an element that I skated over. The blue curve represents potential events that
we could see if we look in precisely one direction – at one specific moment in
time. Naturally, we can’t see multiple events
in that way. We would just see photons,
and any that arrived simultaneously would be blended together.
While in Mathematics for Taking Another Look at the
Universe, I suggested that events on the blue curve are maximally
distant observed events (MDOEs) and all less distant events are below the curve,
this should not be taken as meaning that we could see those less distant events
simultaneously with MDOEs – photons from less distant events will have already
past us by. To work out when, you can
draw a line from 13,787 million years ago, through the distance of the event
from our location (at that time) and take the intercept with the y-axis. Divide that by the speed of light and you
have how long ago photons passed our location.
For observers at that time, the event being considered would
have been an MDOE.
I actually got to the equation x'=(ct0-x).x/ct0
via making a mistake. I was specifically
after a circle, which got me to the x2 element but, because
the image in Mathematics for Taking Another Look at the
Universe came later, I kept getting
stuck on the notion that there was a temporal component (inflation) and a
spatial component (distance away from us at the time), which meant that I added
them together. This is part of the
reason that I called the left-hand term x' which, I accept, is confusing. I knew, at some level, that the x2
value had to be negative but, in the intermediate stage, I just had to make it
negative, chart it and see how it turned out – and it immediately turned out
perfectly, which gave me some not inconsiderable concern.
Why did the x2 value have to be negative? In retrospect, it is bleeding obvious. To turn a normal parabola (y=x2)
upside-down, you need to make the right-hand term negative. To raise it up, you need to add a positive
value to the right (y=x2+c). To shift it to the right, you
need to add a negative term to the squared value (y=(x-b)2). Once I realised that I was looking for a(n
inverted and offset) parabola rather than a circle, things fell into place
quite quickly.
It would have all been simpler to have considered a photon
reaching us from the approximate era of instanton, when the universe was at a Planck
scale. The vast majority of the transit of
such a photon would have been due to expansion.
Photons generated from significantly later events would have a significant
element of their transit due to the distance by which our location and the
location of the event was separated (at that time). Photons that reach us from very recent events
are near enough to simultaneous and therefore their transit time is almost entirely
due to the spatial distance the photons have traversed.
The key element is that unlocked the equation is the realisation
that the events that reach us can all be laid out, moment by moment, into a sequence
that is 13.787 billion years long.
Therefore, photons from an event that reaches us after, say, 12 billion
years, cannot have travelled 12 billion light years, because the universe was
only about 2 billion years in radius at the time of the event. And this led to the chart below:
This chart locked in the equation and extinguished my doubts.
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The charts are actually reduced more than by two dimensions. I have only considered expansion of the radius – which in reality expands in all directions, so the charts are only half of what they should be. Imagine that the observer implied in the chart, is not limited to seeing photons from the region of space that expanded upwards, but also from the region that expanded downwards. That would imply a surface in the shape of an eye. Go a step further and imagine that the observer can look around, swivelling around the y-axis (out of the screen to the left, and eventually back again). This would imply a surface in a shape somewhat like a torus. Finally, imagine that the observer can look up and down. The resultant shape implied would be four dimensional and it is not possible for me to describe it, other than as a torus rotated around an additional dimension. It is easy, but I suspect wrong, to think of it as tracing out a sphere. However, as observers, we would naturally interpret what we see of the universe as the inside of an enormous sphere the centre of which we occupy.
