Is causality synonymous with continuity?

Question

In general relativity, we use the term "time-like" to state that two events can influence one another. In fact, in order for an event to physically interact with another one, they have to be inifnitely close both in time and space.

As far as I know (correct me if I'm wrong) this principal of "near action/causality" is conserved in all branches of modern physics and that is one of the reasons people are looking for "force carriers".

If this is the case, then would it be accurate to say that causality is simply a measure of continuity in all dimensions - and not only the time dimension?

(I don't know anything about continuum mechanics other than its name, but it may have something to do with what I'm asking)

Answer 1

The answer is "no" if you take precise definitions of "continuity" and "causality". Then these are different concepts.

Indeed. The "background" for general relativity is a manifold which necessarily have some topological structure -- it locally "looks like" an Euclidean space and you just take standard Euclidean topology to your manifold. You can intuitively understand it with the Einsteinian picture of coordinate system, where you have freely falling observer with a clock at every point. That's what I understand under "continuity". But the structure of a manifold is not enough.

The causal structure -- is some "order" that you establish for the points of your manifold. This is the extra structure and it is not the same as the "continuity" in I way one usually understands it.

To put it simply -- you can always continuously move to causally disconnected points (or events) on your manifold.

Answer 2

Yes, in the context of special relativity and its extensions, causality is pretty much equivalent to locality - you seem to use the word "continuity" for "locality".

In non-relativistic physics, causality requires that the cause must occur before its effect, $t_{cause}\leq t_{effect}$. Because this condition has to hold in all reference frames - much like all laws of physics - it follows that in relativity, the cause must belong to the past light cone of the effect. In other words, signals etc. may propagate at most by the speed of light.

This fact also implies that it must be possible to describe the laws of physics in terms of partial differential equations - which contain both derivatives with respect to time as well as those with respect to space. No direct "action at a distance" is possible in relativity. This extrapolation of the properties of the temporal coordinate to the spatial coordinates is a direct consequence of special relativity that introduces a symmetry between space and time, the Lorentz symmetry.

See many other articles about causality on this server:

http://physics.stackexchange.com/search?q=causality

Answer 3

Just to add to Kostya's answer above, it's worth noting a couple of facts:

1. Causality Implies the Lorentz Group;
2. On the structure of causal spaces and A new topology for curved space–time which incorporates the causal, differential, and conformal structures.

The bottom-line is that the Causal Structure differs from the Metric by a conformal transformation. So, effectively (i.e., modulo a conformal transformation), it doesn't matter whether you supply a Causal Structure or a Metric to your manifold — of course, keep in mind that a point at infinity can cause all sorts of trouble (but usually, in Physics, we'll assume things are "well behaved"... ;-).

To directly answer your question, coupling Luboš's and Kostya's answer: yes, causality and continuity (in the sense given by the metric structure) can be thought of as being the same thing modulo a conformal transformation, i.e., as far as physical events are concerned; having said that, note that acausal regions can be continuously connected, you just can't access those regions physically.

Another way to think of this is to 'split' (foliate) your spacetime manifold in three different regions determined by the light-cone, i.e., use your Causal Structure to foliate your manifold into a spacelike region, a timelike region and a lightlike one. Each one of these can be simply-connected (continuous in the metric sense) and understood in terms of the Causal Structure (modulo a conformal transformation).