What is the difference between time and space in general relativity?

I know that similar questions have been asked before, I will try to be specific.

In special relativity time is the coordinate with minus sign in metric tensor. In general relativity the components of metric tensor can change value and sign. Questions:

1. In special relativity how the different sign is related to physical difference between time and space.
2. In general relativity what happens if the time (or space) component change sign? What is the physical meaning of this?
3. I know that the change of sign of my point 2. is allowed (e.g. in BHs). A more general question is: is any value/sign allowed for the components of metric tensor in GR or Einstein equations somehow constrain their value/sign? For example is it possible that all the diagonal components get the same sign? What would be the physical meaning?
• For question 1, you could think about ds^2, and how it needs to be invariant, rather than any physical difference, that's where the minus sign comes in (or plus, depending on the convention used). – user108787 May 26 '16 at 10:56

1)The first is how to think of causality and locality. Locality (sometimes people use locality to talk about microcausality but that's not very important) is the statement that two events cannot communicate with each other if they are separated by spacelike distances. So what does this mean? Suppose you have a metric $\eta_{\mu\nu} = diag(-1,1,1,1)$ which is the Minkowski metric and the proper distance is given as $$ds^2 = - dt^2 + \sum_i dx_i^2.$$ If the proper distance is less than 0, then you have a timelike or causal theory. Why is this and what does this mean? This means that if you have two events at $(t_1, X_1)$ and $(t_2, X_2)$, then the proper distance is essentially an interesting way to quantify how much of a spacetime interval one has to endure to travel between these two events. If the proper distance is less than 0, then it means that you need a positive amount of coordinate time to move between the two events. Think of the first event as you leaving a message at a stationary point $X$ at time $t_1$ and then your friend picks up the message at time $t_2$ from the same point $X$. The fact that there must be atleast a small amount of time to pass between the two events gives you the correct spacetime norm. These are timelike events. Following the logic that you cannot possibly expect your friend to decode the message you left for him/her before you even did so, you see that if the proper distance is greater than 0, this is what happens. This is why spacelike theories are a sign of something going wrong. And you will see that the only way to quantify this logically is to have the spatial and temporal indices with opposite sign in the metric. Having the minus sign for time and plus for space (or vice-versa if you grew up reading old GR papers or Steve Weinberg's book) allows you to read off very easily the properties of your theory. It doesn't really matter if both spatial and temporal entries in the metric change signs, it just means that you will have to redefine what you mean by spacelike and timelike intervals and hover around (not) worrying about minus signs.
3) You need to be very careful with the way you phrase this. What happens in a black hole is that the spacelike and timelike symmetries get interchanged. It does not mean that the space and time indices change signs. It is possible that all components get the same sign and this is usually called Wick rotation when you set $t \rightarrow i t$ and go to Euclidean spacetime but this is not to be interpreted physically and is usually used for things like determining the temperature and analytic continuation.
The true meaning of all the trouble is that coordinates are just tools for calculations and don't have any intrinsic meaning in GR. You should never interpret the coordinates alone. These can be very misleading and in particular become singular in some regions of spacetime (because coordinates are local). And never trust the labels of coordinates: Fact that some coordinate is called $t$ doesn't mean that it is time measures by some observer!