I agree that these terms —especially 'locality'— are used for different concepts and this is annoying. I will list several notions of causality and locality.
Causality (or Einsteinian locality): Results of experiments carried out at a space-like distance are not correlated. This assumes that there are not previous correlations before making the experiment. In a quantum theory, this implies that observables must commute at a space-like distance. A violation of this property in a Special Relativistic theory may give rise to time travels paradoxes.
Micro-causality (only used in field theory): A theory whose fundamental variables or the dynamical degrees of freedom commute or anticommute at space-like distance is sometimes said to be local. Note that there are theories which violate this property without violating causality.
Lagrangian density (only used in field theory): A theory whose action functional is expressed as a space-time integral of a local Lagrangian density is sometimes said to be local. Note that there are theories which violate this property without violating causality.
Non-local character of gravity: Here I will mention a few points, which are not independent of each other:
Equivalence principle: Free-fall observers cannot detect the gravitational field by making local experiments. All they feel are (non-local) tidal forces. This implies that we cannot define a gravitational energy-momentum tensor.
Gravitational entropy is not an extensive magnitude: The entropy of a Schwarzschild black hole does not scale with the volume, but with the horizon area. This means that the degrees of freedom are not local, but live on the boundary.
Usually, as we increase energy in a theory, we explore lower and lower distances and find out substructure and new particles/degrees-of-freedom (atoms, nuclei, nucleons, quarks). However, when we take into account gravity, when we increase the energy sufficiently, we inevitably form a black hole, which are not substructure o new, more-fundamental degrees of freedom, but a solution of the theory.
Horizons: A free-fall observer feels nothing when they cross the horizon (if the horizon is large enough, otherwise they would get spaghettificated due to tidal forces (non-local effect)). The space-time curvature at the horizon can be arbitrarily small. However, this is a physical boundary, in the sense of being a non-return point.
Cluster decomposition principle: Results of experiments carried out at the same time (in a given reference-frame) but in different spatial regions are not correlated. This assumes that there are not previous correlations before making the experiment. This property together with Poincare invariance implies causality. Relativistic QFT and non-relativistic condensed matter QFT verify this notion of locality. This property imposes certain smooth dependence of the Hamiltonian density on the creation/annihilation operators.
Non-local wave-function's collapse (quantum notion): People sometimes say that quantum mechanics is non-local because the collapse is a non-local process. In my opinion, the wave-function collapse is not a physical process, but something that affects our mathematical description of the physical system. A sort of algorithm to incorporate new information to the theory. So, in my opinion, this non-local collapse is not a signal of physical non-locality.
Non-local states and observables (quantum notion): In quantum mechanics there exist non-localized states and observables and I have heard people to call this non-locality. A particle whose linear momentum is very well defined may be an example of a non-localized state and the scattering operator may be an example of a non-local observable given that relate states at far past and future. Entangled states could be listed here.
Entanglement (or non-local correlations, quantum notion): In quantum mechanics there are quantum correlations which are non-local such as the spin correlations of the singlet state. One needs to measure the z component of both particles which can be very separated. A singlet state may also be called non-local. Since one cannot use these correlations to send information (a parallel classical channel is required, and this cannot be superluminal), therefore this property does not imply violations of causality.
Incompatibility of QM with local (causal) realism: QM and experiments violate Bell's inequalities. This leads to the incompatibility of QM and nature with either local (causal) realism (local hidden variables) or free will. People are currently discussing this on this site.
Only the first notion of locality (causality) must be required in a Poincare invariant theory. Semantic issue: Some people call a theory "special-relativistic" if the theory is Poincare invariant and causal, while other people by "special-relativistic" just mean Poincare invariant.
Examples:
- Non-relativistic quantum mechanics and non-relativistic QFT verify the cluster decomposition principle. However, there are non-local observables, entanglement, etc.
- Relativistic QFT verify causality and the cluster decomposition principle.
- Quantum Electrodynamics in Coulomb gauge is a relativistic QFT, but it does not have a Lagrangian or Hamiltonian density.
- Para-statistics theories verify the cluster decomposition principle, but they do not verify micro-causality.
- Classical General Relativity has a local Lagrangian density and it is locally Lorentz invariant, but it allows time machines in some topologies (and in these cases is not a causal theory) and horizons have non-local properties.
One way to define spacelike separation in special relativity is that any two events are spacelike separated if and only if there exists a reference frame in which the two events have the same time coordinate. So yes, if $x^0 = y^0$ the separation is spacelike.
Alternatively you can work from the definition where two events are spacelike separated if (and only if) the interval between them has the same sign as the spatial components of the metric. In other words, if your metric convention is $(-1,1,1,1)$, a spacelike interval has $\Delta s^2 > 0$, or if you use $(1,-1,-1,-1)$, it has $\Delta s^2 < 0$. If $x^0 = y^0$, then clearly the interval is determined only by the spatial components, and will necessarily have the same sign.
Best Answer
I agree with your definition of locality (probably not surprising :)).
Causality I would say is the statement that an event in the future should not affect an event in the past. We can formulate this in classical physics terms. Causality is necessary in order for there to be a well defined initial value problem: I should be able to choose an initial time slice, specify the field values and derivatives on that slice, and evolve the system forward from there unambiguously. Acausality would allow an event from the future to come back and affect what's going on in the past--in principle that would allow the field evolution to change the initial conditions you started with.
If you like, causality is the requirement that there should be no time machines that allow me to send information into my past--I should not be allowed to kill my own grandfather.
If you don't demand Lorentz invariance, locality and causality are distinct concepts. I can certainly imagine non-local theories that are causal--Newton's action at a distance version of gravity is certainly causal, but it is nonlocal. Similarly, I can imagine a universe where I can press a button and reverse the flow of time for me (ie, my clock runs in the opposite direction of the rest of the universe), where I can only interact with things locally but I now have clearly violated causality.
These notions however become related once you demand Lorentz invariance. The reason is that the notion of simultaneity is relative. In particular, the time ordering of spacelike separated events becomes observer dependent. So if two spacelike separated events can affect each other (which is definitely non-local), there is a frame where I am using this spacelike communication to talk to someone in my past. She can then (provided that she can also perform spacelike communication) talk to someone in my past but also inside my past light cone. So you can create a loop of communication that ends up in my past light cone. In this example, no one is moving faster than light (or, maybe more accurately, the non-local communication allows for superluminal transfer of information), but the nonlocal transfer of information has allowed something I said now to end up in my past light cone.
So if we don't want to allow spacelike transfer of information, what can we do? Well at a fixed time the only event that is not spacelike separated from me is the event where I am located. So I can only affect the fields and their derivatives at my location.
As a warning, in gravity when the spacetime metric becomes dynamical, all of this becomes more complicated! In special relativity when the metric is fixed, things are more clear.