It take 8 minutes for light from the sun to reach Earth. One day, it just disappears Goes dark suddenly, but not a nova.
What would it be like for those 8 minutes? Or would we know sooner? Anything the instant the sun goes out?
If it “just disappears”; the same as usual, no and no.
No information can travel faster than the speed of light, so there would be no physical effect at the Earth until the light hits us, along with anything else (like gravity) that is traveling at or near that speed.
To know sooner than 8 minutes, information would have to be communicated to us at a speed faster than that of light. So far, no method for that has been found.
I’d like to join in the no way of knowing before eight minutes club.
I have an question if I can hijack just a tad. The Sun goes dark. How long does life on Earth go on? We’d have to radiate away our heat before the atmosphere pooled into puddles at our feet, but how fast do we lose heat to the cosmos?
Also, how deep are those puddles? 
Here’s an answer based as much on epistemology as physics:
The situation described in the OP is only possible if several immutable laws of nature are violated. Once that happens, ***anything ***is possible . . . including information traveling faster than light.
As many times as I’ve seen this question answered, it never occurred me what an independent observer would see. Suppose you are located a long distance from our solar system at right angles to the plane, somewhere near the line projecting out of the sun perpendicular to the plane. Your distance is large compared to Earth’s distance from the sun, but close enough to be able to see both Earth and the sun, and light from concurrent events on each reaches you at more or less the same time (let’s not get into relativity of concurrent events for the purpose of this idea). You see the sun blink out, and continue to watch for 8 minutes when suddenly the Earth shoots off its orbit on a tangent. That would be a weird thing to see.
I don’t know the complete answer but overnight I know that temps can drop dramatically during a very clear night, like in the desert. So I would imagine things would get damn cold in a few days. The food chain would take a while to collapse but it would happen faster and more thoroughly than the mass extinction 65M years ago (I don’t know how fast that happened).
So any shockwaves or jarring would be slower than the speed of light? Interesting.
By “just disappears” do you mean it clicks off, like a light bulb, leaving a big, dark ball ? Or do you mean poof, and there’s a big empty space where the sun used to be?
And if, poof, suddenly there’s a big empty space, would the earth suddenly shoot out of orbit and continue in a straight line at +/- 67,000 mph? Would it happen instantly, or would it take 8 minutes for gravity to stop, as well as light?
Gravitational waves are thought to travel at the speed of light. It would take a bit over 8 minutes for the effect of the loss of the sun’s gravity to reach the earth.
This answer comes up a lot, and it always seems a bit of a cop-out. What the OP is asking is pretty clearly “…and everything else stays the same.” Otherwise, you’re almost saying you can’t ever answer a hypothetical (“but it didn’t!”), which might be epistemologically true, but isn’t very useful as a producer of knowledge.
Assume a race of aliens with much better technology than us come along and wrap the sun in a radiation-proof (maybe gravity-wave/graviton-proof) bag before hauling it off to replace a light bulb in some other galaxy. No laws violated, except for an intersteller misdemeanor about stealing suns from planets without leaving the standard crop-circle warning ahead of time.
The critical words here are “are thought”.
Unless I’m mistaken, this is a presumption based on the oft-quoted “information can’t move faster than light”. But it is only a presumption. The truth is that the scientists don’t yet really understand how gravity works, or whether it is limited to the speed of light.
If I am mistaken, please enlighten me.
Well, somewhat true, but we can only answer a question using the best facts and theories available at the time. Otherwise the answer to the OP would be, “Who the fuck knows?”
But as far as we know there is nothing that is gravity-wave/graviton proof. That is we know of no way to create a gravity shield. So either that law is violated or we Earthlings don’t know enough to answer the question of what happens.
If you ask me, it oughta be a felony.
Millions of years I expect, at least for bacteria. Some of them live deep in the Earth’s crust, eat rock and don’t depend on the Sun for energy. Tidal and radioactive heating will keep the interior of the Earth from freezing, I think.
That’s not the truth at all. Not having experimental evidence of a graviton particle does not translate to “not understanding how gravity works”. General relativity is a theory that describes gravity very accurately. Also it’s well established that faster than light travel breaks causality, and most people believe in a causal universe.
Not me. I believe in a formal universe.
General relativity has been a remarkably successful theory. It predicts the speed of gravity to be precisely the same as the speed of light. There are no compelling alternative theories and all viable alternatives also predict the same speed for gravity. There is still no quantum field theory that properly unifies quantum mechanics and relativity in the strong gravity limit, but this is irrelevant to the OP’s hypothetical situation, which involves weak gravity.
As far as I’ve been able to find, we haven’t observed any ‘gravity particles’ or ‘gravity waves’ or whatever, but we have observed interactions between bodies in space that are consistent with gravity propagating at the speed of light.
In particular, measuring the orbital decay rate of a pair of pulsars orbiting each other will tell you that gravity propagates at a non-instantaneous speed. (The exact speed you calculate depends on the theory you’re using, so it doesn’t prove GR by itself.)
In 2002, some guys indirectly measured the speed of gravity by watching Jupiter pass in front of a quasar, but the interpretation of their results is disputed. I don’t understand the math of that argument, so you’re on your own there.
The fun thing about “slow” gravity is that the earth experiences a force toward where the sun is <i>now</i>, not where it was 8 minutes ago. Gravity has error prediction.