If you bought an analogue clock to space, would it change according to the gravity?

A fantastic question! Really made me think for a second about how to explain this, thanks for asking. Best way in my opinion to approach this is to consider light. Light is one of the fundamental parts of how we understand the universe. We are generally taught that photons (little balls of energy that we see as light as well as other phenomena but these aren’t important right now) are massless. However, they can still be affected by gravity - not by the usual means of F = ma (specifically G = ma in terms of gravity) - but by the sheer warping of space-time by a heavy object. When I say heavy object, I’m talking on a planetary scale. As the fabric of space-time becomes warped, photons must follow that misshape. This means their path length increases. Think of a straight road from A - B. Now if you put a hill in between A - B, the path length has increased. However, in reality, the distance between A and B is the same as before, because nothing has changed except the area of space-time they are in. This is hard to get your head around - we as humans would have no idea without advanced technology that there has been a change in space-time - it would just be like that. For us, A - B would be the same, but not for a photon. Now, it becomes a question of speed = distance / time. Well, light has a fixed speed in a vacuum like space, which is 3 x 10^8 m/s, so speed is the same. Distance is a variable which goes the same way to time - if one goes up the other must go up as well. Therefore, if higher gravitational field strength increases the path length of a photon, the time taken for that interaction must be longer. If the time for a standard process (a photon travelling from A - B) becomes slower in a higher gravity field without being apparent to an onlooker, the speed of time itself must fundamentally change to satisfy the constant speed of light. Therefore, a high gravitational field strength makes time slow down. An easier way to look at this is to create a rudimentary rule in the form of an equation from our assumptions above: gravitational field strength (gfs) = k x path length of photon, where k is an unknown constant just to make things mathematically correct. Again, this is very touch and go, and the relationship may not be linear. In any case, working with this assumption leads us to gfs ∝ photon path length. Similarly, distance = speed x time gives us distance ∝ time. If photon path length = distance, gfs ∝ time taken. So time slows down in a high gfs. In answer to your questions, the actual clock wouldn’t change speed as far as you could tell in space - the time between ticks wouldn’t be different to you, but take it back down to earth after a good long few years and another clock previously synchronised will be behind your one because it has been in a higher gravity location. For astronauts, without getting into astrobiology, they wouldn’t change the speed of ageing - they would still live between 70-100 years by their reckoning. But to someone on Earth, they would have lived longer. If you haven’t already seen it, watch Interstellar! It deals with this in a really cool, enjoyable and (best of all) wholly accurate way, by which I mean the physics is 100% correct! Let me know if you have any more questions.