[paul cotter (OP)]

It is postulated that a horizon exists or will exist in the future whereby remote galaxies are receding so fast that the light from them will never be seen at some arbitrary distant(from the distant receding galaxies) point in space. A photon of light leaving such a galaxy will travel any distance in zero time in it's own frame of reference. So the question is will the photon reach the the distant observation post or as the horizon theory would suggest will it be lost forever in the permanently stretching fabric of space?

[Origin]

A photon of light leaving such a galaxy will travel any distance in zero time in it's own frame of reference.

A photon does not have a valid inertial rest frame in relativity. 

So the question is will the photon reach the the distant observation post or as the horizon theory would suggest will it be lost forever in the permanently stretching fabric of space?

The photon won't be 'lost' it just will never reach us.  Can a photon just continue on if it never hits something?  Yes.

[Halc]

It is postulated that a horizon exists or will exist in the future whereby remote galaxies are receding so fast that the light from them will never be seen at some arbitrary distant(from the distant receding galaxies) point in space.

This horizon is due to distance, not due to the recession speed. Light travels at the same pace regardless of the speed of the emitting object. This event horizon is due to acceleration (due to dark energy), similar to the event horizon that forms in Rindler coordinates. Without dark energy, light from a galaxy however distant will eventually reach here.

A photon of light leaving such a galaxy will travel any distance in zero time in it's own frame of reference.

As Origin points out, light has no valid frame of reference. Speaking of distance relative to light is meaningless.

So the question is will the photon reach the the distant observation post or as the horizon theory would suggest will it be lost forever in the permanently stretching fabric of space?

It will not reach the distant observer, but also will continue to move at light speed, not being 'lost' at all.
Similarly, if I accelerate forever to the left at 1g, light emitted from over a light year behind me will never reach me in any amount of time.

FYI, no paradox is suggested anywhere in the OP, except the title. Perhaps you want to reword the title question.

[paul cotter (OP)]

Could you elaborate on "not having a valid inertial rest frame", please. Physics is not a subject I have studied in any depth. I always assumed when the v squared/c squared in the Lorentz equation reached unity the universe would shrink to a singularity and time would cease to exist from the photon's frame of reference. Einstein was fond of thought experiments and he mused on what travel with a photon would be like but I don't know what his conclusions were. I did not mean to suggest that the photon would actually be lost and I do realise that light travels at c, regardless of the speed of the emitter. I just wanted to raise the topic of the horizon without going into too much detail. An explanation for "dummies" would be ideal but I doubt such exists. PS I don't buy dark matter' dark energy , dark flow. I realise some explanation is needed for observed effect but these concepts are too nebulous(for me, hope I don't start a row!)

[Halc]

Could you elaborate on "not having a valid inertial rest frame", please.

Per Galilean relativity (proposed 400+ years ago), the laws of physics are the same in any inertial reference frame. One of the properties of such a frame is that light moves at c, but in a hypothetical frame of the light itself, light would be defined as stationary, a contradiction.

I always assumed when the v squared/c squared in the Lorentz equation reached unity the universe would shrink to a singularity and time would cease to exist from the photon's frame of reference.

Except for the work 'shrink', yes. The physics of the universe would be singular, meaning time, space, speed, direction, and anything dependent of these things would be meaningless (which is different from zero or infinite). Being singular means the laws don't apply. They're meaningless. No conclusion can be drawn from them. So you can't say 'light would get from the distant galaxy to here' because there is no meaning to 'here' or 'there'.

So as for the horizon, in intertial coordinates in Minkkowskian spacetime, like will get from any location to any other location. The fact that it doesn't in our universe shows that spacetime isn't Minkowskian. Nobody uses inertial coordinates for really distant things. For one, nothing can travel faster than c relative to an inertial frame, but relative to cosmological coordinates, anything beyond the Hubble sphere recedes from us faster than c. Yes, we can very much see galaxies that recede faster than c and are currently beyond both the Hubble radius and also the event horizon, which is not far beyond it.

(for me, hope I don't start a row!)

... or even a column. 

[Eternal Student]

Hi.

Looks like a good discussion, I hope you won't mind if I join.

1.     Got to support  @Origin  and then @Halc.    There isn't an inertial rest frame for a photon.   Light can never be at rest in any inertial frame, instead it must move with velocity c.    So, you can construct a rest frame for light (yes, that can be done) but there is no way it is an inertial frame of reference - just straight from the basic properties of inertial frames and light.  As it hapens, in an accelerated (and thus non-inertial) frame you can have light travelling at any velocity you want, including being stationary.

2. 

Einstein was fond of thought experiments and he mused on what travel with a photon would be like but I don't know what his conclusions were.

   Yes, that seems correct.   To the best of my knowledge, no formal results were published in anything like a journal or research paper.  It was apparently an interesting or useful thing to consider but not easily expressed or useful to what was finally distilled and ultimately became Special Relativity.

3.   

It is postulated that a horizon exists or will exist in the future whereby remote galaxies are receding so fast that the light from them will never be seen....

    Yes.   It is often stated something like this.  It's not your ( @paul cotter ) fault, this just is how it is usually stated.
    The "recession" and it being "so fast" is not to be understood as if they have an ordinary sort of velocity through flat space.   On average they don't, they're stationary, it's just that space is behaving strangely.

4.  We have already suggested that you don't push the limits of imagining an inertial rest frame for a photon too far (because, and it's worth saying again, there isn't one).  However, if you must do this, then note that the Lorentz transformations only tell you what happens in flat Minkowski space.  So considering limits of the Lorentz transformation as  v/c ---> 1   is barely relevant.  We already have good reason to believe that the universe does not exhibit Minkowski geometry on large scales.   Even if the Lorentz transformations suggests that a flat Minkowski space becomes degenerate and reduces to single point when v/c -->1, the real universe does not have to do that and there can still be regions of space that the photon cannot reach.

Best Wishes.

[paul cotter (OP)]

Thank you all for your contributions. I won't say I fully understand all that has supplied so i'll have to digest your efforts.And halc, forget the row and column and we should just settle for a square matrix.

[paul cotter (OP)]

I have a further question related to the original one. When light passes through a region with high relative permittivity it no longer travels at C and the denominator in the Lorenz expression will no longer be zero and the photon will now have an inertial frame of reference, albeit for a very short time. Or does one use a reduced value of C such that the Lorenz expression still is indeterminate ie 1/0 ? 

[Halc]

When light passes through a region with high relative permittivity it no longer travels at C and the denominator in the Lorenz expression will no longer be zero and the photon will now have an inertial frame of reference, albeit for a very short time.

First of all, you're mixing quantum and classic terms. So to keep it classic, we can send a pulse of light through rapidly flowing water.  Make a pipe with a straight section and flat ends and move water in a loop through it fast enough and light entering one end will in principle stand still, yes. So yes, in a medium like water, there is a frame where light moving against the current will stand still. But time doesn't 'stand still' in that frame for anything any more than it does in any other valid inertial frame.

[paul cotter (OP)]

Thank you very much halc, you have answered my question. I didn't mean to suggest time stands still in that scenario, I probably could have expressed the question better. I have had no formal education in relativity and i'm still finding my way(obviously).

[Eternal Student]

Hi.

   This is complicated   BUT  an excellent question to ask and think about.

When light passes through a region with high relative permittivity it no longer travels at C...

    That would be correct considering light as a classical electromagnetic wave.  It's more conventional to describe the slowing of light by referring to the "refractive index" of the medium instead of the "permittivity" of the medium - but the basic idea is right.    Using classical e-m theory, a change in either the permittivity (a thing affecting Electric fields)   OR  the permeability  (a thing affecting magnetic fields)  should change the speed of propagation of an e-m wave.
   The typical equation connecting these quantities is:
   v = velocity of wave propagation  =       with    ε = permittivity.  μ = permeability.

- - - - - - - -
   However, as @Halc stated,   that's the speed of a classical wave.   A photon is a quantum mechanical particle and is quite a different thing.   A photon should always travel at the speed c, in any reference frame and through any medium.

    How can the wave propagate slower than c, when a photon is always moving at c?   Probably too complicated for me to explain,   I'd only get bits of it wrong anyway.   Most texts start by carefully defining and examining the difference between "group velocity",  "phase velocity" and the speed of a wavefront.

Here's someone else trying to explain it.  It's still a simplification and therefore quite accessible but equally it just jumps over and avoids discussing some complications.
"Why does light slow down in water",  FermiLab ,  available on YouTube.

Or does one use a reduced value of C such that the Lorenz expression still is indeterminate ie 1/0 ? 

   Definitely not.   c = the speed of light in a vacuum ≈ 3 x 10⁸ m/s   is always the constant that appears in the Lorentz transformations,  regardless of the medium in which a particle or thing you are considering is located.    To say that another way,  this value, c, has importance in special relativity.   Changing frames of reference won't be any different in some medium even if light has a different phase velocity in that medium.

Best Wishes.

[Rodneyhhernandez]

Probably misunderstanding, although any theory has a right to life. I believe that later with discoveries, this theory will be clear too.