enceladus_ said:
I heard Neil deGrasse Tyson talk about the life of a photon that was emitted from the Sun and hit oneself. He said that its entire life was an instant. I hope I'm not misquoting or misunderstanding what he said.
I don't think you're misquoting or misunderstanding; but I do think Tyson is stating things in a way that is likely to cause confusion. (He's not the only one, btw: we often get threads here on topics like this after one of Brian Greene's specials airs on PBS.)
The correct way to say what Tyson was saying is that a photon's worldline has zero length. In the simplest case, where gravity is negligible so we can use the formulas of special relativity, the length of a worldline is given by the Minkowski metric:
\tau^2 = c^2 t^2 - x^2
which is basically a spacetime version of the Pythagorean theorem: t is the travel time of the photon in some inertial frame, and x is the distance the photon covers in that time in the same frame. Zero length for a photon's worldline means \tau^2 = 0, which just means that ct = x: the photon moves at the speed of light.
Really what this is saying is that objects that move at the speed of light, like photons, are fundamentally different, physically, from objects like us, that move slower than light, and for which \tau^2 > 0. In that case, of objects moving slower than light, \tau is just the time elapsed on a clock that moves with the object. By analogy, then, pop science discussions often say that "no time elapses for a photon" because \tau = 0 for a photon.
However, the analogy is flawed because \tau has another meaning for objects that move slower than light (we call the worldlines these objects move on "timelike", by contrast with "null" worldlines that photons move on): each event on a timelike worldline can be labeled by a unique value of \tau (which we normally refer to as the "proper time" of the event according to the observer following the worldline). Interpreting \tau = 0 for a photon as meaning the photon "only takes an instant" to go from, say, the Sun to the Earth, would imply that the photon's worldline is not a line but a single point--a single event. Obviously it's not, because we could put a photon detector anywhere between the Sun and the Earth and detect the photon, and each possible location for the detector must be a distinct point on the photon's worldline. So \tau for the photon simply doesn't mean "time" the way it does for a timelike object.
enceladus_ said:
With that in mind, does that instant take into account the thousand of years (relative to us) it spends trapped in the mantle and core of the Sun, trying to get out?
No; that's a whole separate discussion. The "instant" only refers to the photon traveling from the surface of the Sun to Earth, through vacuum.
enceladus_ said:
Also, how would one describe the life of a photon that is emitted from the Sun, or any Star for that matter, and travels indefinitely throughout the Universe, never coming into contact with anything? Is this an indefinite instant?
Not really. The condition \tau = 0 for the photon's worldline still holds, but that doesn't make it an "instant". See above.
