What would a closed timelike curve look like? What exactly are closed timelike curves. In a metric in which they would exist, what would they look like. What would it be like travelling through them? It obviously wouldn't look like a door. Would it be a region of space that if you wonder into, it can happen that see your past self?
 A: A closed timelike curve wouldn't actually "look" like anything because it's an abstract thing. You can't actually see any lightcones or worldlines. A metric is an abstract thing too, to do with your measurements of distance and time, typically made using the motion of light. And the crucial point is this: you don't travel along your worldline. You move through space whilst light moves, along with pendulums and cogs and piezo-electric vibrating crystals. Your worldline is a plot of this in a static 3+1 dimensional "block universe" called spacetime. For an analogy, imagine I throw a red ball across the room, and you film it with your old-style cine-camera. Then you develop the film, cut it up into individual frames, and form them into a block. There's a red streak in the block. That's like the ball's world-line in the block universe. But note that there is no motion in there. The ball is not moving up through the block, and in similar vein you aren't moving up or along or around your worldline. You move through space, not through spacetime.       
As for what a closed timelike curve would be like, some say it would constitute time travel, but it isn't true. Others would say it would be like Groundhog Day, but that isn't true either. Because if your closed timelike curve was 24 hours long, it would be more like Mayfly Day. Your life lasts for 24 hours. And it is causeless. You are born from an egg, you live for a day, you lay the egg, you die. You live and die, but you don't live die repeat. Because you don't travel along your worldline. Hence a closed timelike curve does not offer any possibility of time travel. 
Don't think this is just something I've made up. Check out A World Without Time: The Forgotten Legacy of Gödel and Einstein. See this page in it, where author Palle Yourgrau says Wheeler conflated a circle with a cycle, precisely missing the force of Gödel's conclusion: 

He's right. You don't move round that closed timelike curve. And there is no way you can move such that everything else in this universe not only moved back to where it was, but never moved at all. Time travel is a fantasy I'm afraid. And so are closed timelike curves. 
A: To know what a closed timelike curve looks like, you just do like every spacetime metric. You compute geodesics and field equations and all of that. Unfortunately, things start getting complicated.
Closed timelike curves have a lot of weird behaviours, especially when it comes to matter fields upon them. They may not have a properly defined Cauchy problem, that is, knowing the field at a point in time may not help you to know it at all points in time. Physically speaking, this corresponds to when geodesics form loops, or in the case of naked singularities, come out of nowhere. The particle creates itself, in a way. There may therefore be an arbitrary amount of particles popping out of nowhere, making the analysis a bit difficult. The problem seems to be mostly worse with interacting fields, although some free fields also suffer from the problem.
Another problem is joining different regions of spacetime. It's always possible to solve fields locally, since spacetime always looks locally like Minkowski space, but if there are CTCs present, extending those solution to the entire spacetime becomes difficult, since it is assumed that they have to be consistent. For instance, consider the 2 dimensional torus spacetime, which is Minkowski spacetime with the following identifications : 
$(x,t) = (x,t+T) = (x+d,t)$
It's not too difficult to show that the geodesic with 4-velocity $(1,\vec{0})$ is consistent, it just loops once around the torus. A geodesic with a slightly askew velocity $(1, \vec{v})$ will loop back on itself eventually if the velocity is a rational value of the period of time and space. Otherwise, it will go on looping forever. That is another common problem of CTCs : EXPLODING. The same particle geodesic exists an infinite number of time at the same moment. This is usually a bad sign for the existence of CTCs because then you cannot pretend that the matter field is only a small perturbation on the spacetime.
Misner space is a CTC spacetime that starts out like a reasonable spacetime, until t = 0 where CTCs suddenly appear. If you check out the stress energy tensor of a free scalar field upon it, it will explode at t = 0, unless some conditions on the field are met, because the field get infinitely blueshifted. The same applies to the Gott spacetime. 
A third effect is that when quantum effects are included, the energy of the vacuum might diverge, due to geodesics being able to wrap around an arbitrary amount of time through a region.
All in all, that is a lot of effects where fields explode without provocation. I'm not exactly sure what things would look like inside a CTC, but odds are good they would not be very good. It's a common theme in CTC spacetimes : Particles radiate highly blueshifted particles when falling towards the CTC region of the Kerr black hole, wormholes collapse from vacuum radiation, Misner space explodes when CTCs try to form and so on. To answer your question, I guess that if they survive, they might be VERY BRIGHT.
Here's a few papers on the various topics : 
The Cauchy problem in spacetimes with CTCs and various field solutions : 


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*http://www.johngoodwin.me.uk/thesis/john-goodwin-thesis.pdf

*http://arxiv.org/pdf/gr-qc/0401004.pdf

*http://thelifeofpsi.com/wp-content/uploads/2014/09/Friedman-et-al.-1990.pdf

*http://arxiv.org/pdf/gr-qc/9411033.pdf

*http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/22/023/22023908.pdf
You can of course use much simpler models to check out geometrical optics. In which case it is usually done simply using point travelling along geodesics. Here's a few papers on it as well : 


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*http://arxiv.org/pdf/1102.0907.pdf

*http://authors.library.caltech.edu/6469/1/ECHprd91.pdf

*http://arxiv.org/pdf/gr-qc/0007064v2.pdf

*http://arxiv.org/pdf/gr-qc/0007063v1.pdf
That's about all I can think about for general optics and fields (including EM fields) phenomenon involving CTCs. more specific CTC spacetimes of course have their own optical effects, such as redshifting, blueshifting, gravitational lensing and such. Wormholes in particular have had some work done on them, although not in their CTC configuration. Here's a sample : 


*

*http://scitation.aip.org/content/aapt/journal/ajp/72/8/10.1119/1.1758220

*http://arxiv.org/pdf/astro-ph/9409051.pdf

*http://arxiv.org/pdf/0807.2774.pdf
A: Instead of piling up abstractions, one should define start coordinates in space-time $[x_0,y_0,z_0,ct_0]$ relative to solar baryentre, then write down the celebrated trajectory solution in terms of consecutive $x,y,z,ct$ leading to initial condition (? which would move with Milky Way rotation and possibly other unknown velocity) at some stage we should notice coordinate of time achieving maximum then going down while the pocket clock of the traveler is incrementing. You then see yourself passing by to time travel. The atoms of your body and the other body marginally doubled violated conservation of mass and energy but that's OK as long as it proves the theory of time-like loops.
A: Consciousness, by definition, cannot traverse through the whole loop. So there is no way to define "what it would look like".
You cannot just assume "what if consciousness could exist in a closed timelike loop". It is a self contradictory assumption. Consciousness must work by gaining knowledge of the universe along its worldline. So, by definition, consciousness cannot have a loop worldline, because a loop worldline would mean coming back to the original state.
All that we can know about closed timelike solutions is what abstract information they contain. They do not look like anything.
