How can the spacetime interval be negative? It’s my understand that the spacetime interval is analogous to the distance squared between two points in euclidean space. If the spacetime interval is the distance squared between two events in spacetime, how could that distance squared ever be negative? That would mean that the distance is an imaginary number, and i would expect it to only be a positive number or zero. How is this possible if spacetime is an actual surface? Sorry if i’m thinking too physically here.
 A: The answer is quite simple, the reason is because you are measuring distances differently. In an Euclidean space your metric is diagonal with signature $(+,+,+,+)$ which means if you have vectors in such space ${\bf x} = (x_1,x_2,x_3,x_4)$ and ${\bf y}=(y_1,y_2,y_3,y_4)$, the distances square between them is
$$({\bf x-y})\cdot({\bf x-y})_{\rm{Euclidean}} =({\bf x-y})^2 = (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2+(x_4-y_4)^2\ge 0$$
However Minkowski metric is diagonal and has signature $(+,-,-,-)$ or $(-,+,+,+)$, so time is a bit special and is labeled usually with the index 0, so for 4-vectors, ${\bf x} = (x_0,x_1,x_2,x_3)$ and ${\bf y}=(y_0,y_1,y_2,y_3)$, you compute distances in the following way
$$({\bf x-y})\cdot({\bf x-y})_{\rm{Minkowski}} =({\bf x-y})^2 = (x_0-y_0)^2-(x_1-y_1)^2-(x_2-y_2)^2-(x_3-y_3)^2$$
as you can see in this case, the norm of a vector is not always positive, because the metric is Lorentzian meaning not positive-definite.
A: Concerning the spacetime intervals, there are 4 contradicting concepts. Depending on the convention you chose you could adopt the signature (-, +, +, +) or (+, -, -, -), and you could chose to extract the root or not. But that is not enough. We might presume that for lightlike phenomena it should be clear that the corresponding spacetime interval is zero, but even here there is no unanimous opinion, in particular with respect to the question if we can say that the proper time of photons in vacuum is zero or not.
Very little effort is done by theoretical physics to resolve these differences of conventions. It is thought that this question is of no particular importance. 
However, our current spacetime concept has a big problem with quantum mechanics. Up to now, all attempts of quantization of spacetime failed, our spacetime concept seems not to be compatible with quantum mechanics.
The following information may lead to a solution of these problems:


*

*First we should decide for one of the above-mentioned contradicting concepts. The solution of a signature (-, +, +, +) and with extraction of the root seems to correspond to the physical circumstances. ds = $d\tau$. Proper time is a physical object-related notion for distances (the time according to a clock following a given object).

*After having decided for the spacetime interval concept of proper time, we realize that only timelike intervals are defined, while spacelike intervals are not defined, they are negative squares or imaginary numbers. At first sight, this seems to be inadmissible, but if we check we will realize that the continuity of spacetime in space direction is a mere assumption only, that means, we assume that spacetime is a manifold without ever having proved it, and even though continuity of spacetime is not compatible with quantum mechanics. At my knowledge, nobody ever tried to prove that spacelike spacetime intervals exist.

*Historically, the mathematics of spacetime were first described by Minkowski in 1908. It was a fascinating idea that by Lorentz symmetry, space and time were linked by the equation of the spacetime interval, and rapidly ideas of a fourdimensional manifold emerged. A fourdimensional manifold would have been possible in Newton's spacetime (and there were some theories in this direction), but not in a Lorentzian spacetime where spacelike spacetime intervals are imaginary (according to the explanations above). After the famous lecture of Minkowski, the spacetime manifold was generally adopted as mathematical model for spacetime, a simple equation was considered to be a manifold. 20 years later, when the question of quantum gravity came up, nobody mistrusted the manifold character of spacetime, even if such manifold is not compatible with quantum mechanics.

*Physics today: Please be aware that today the concept of a continuous spacetime manifold is still mainstream. But according to the information above 1-3, it is impossible that you get a coherent answer on the basis of the mainstream model, because it is an intrinsic contradiction of the model, in the core of quantum gravity. 
