# Are all events along a Cauchy surface always space-like separated?

I'm new to relativity. Are all events along a Cauchy surface always space-like separated?

I know that a necessary condition of a Cauchy surface is that every inextendible causal curve intersects the surface exactly once. So if two events along a Cauchy surface aren't space-like separated, then there would exist a causal curve between the two, and hence the curve would intersect the surface twice, contradicting the definition of a Cauchy surface. Is this reasoning sound?

Also, what's the difference between a Cauchy surface and a hypersurface of simultaneity? In Minkowski spacetime, is it the case that every Cauchy surface is a hypersurface of simultaneity and vice versa? What about other spacetimes?

• Related: Quantization surface in QFT Commented Dec 15, 2023 at 8:23
• It's not true that a causal curve has to intersect a Cauchy Surface uniquely. Only timelike curves do.
– P-A
Commented Dec 16, 2023 at 17:40

As we familiar with our ordinary differential calculus we define initial condition in order to get proper solutions, Similarly in GR we define cauchy surface which is achronal,spacelike. Reason behind cauchy surface to be spacelike is we define initial data on it such as any physical matter fields which obeys the all energy conditions which also governed by R.H.S of Einstein equation (Energy-momentum Tensor) suppose its not to be a spacelike then cauchy surface or simply initial conditions evolves after time goes. So it may not be applicable to do physics on it.

The definition of Cauchy surface in a globally hyperbolic spacetime $$(M,g)$$ is a set $$S$$ which

• is achronal
• it intersects every inextendible smooth timelike curve.

REMARKS

• (1) A future-directed smooth causal curve $$\gamma: (a,b) \to M$$, with $$a,b \in [-\infty,+\infty]$$ is future, resp. past, inextendible if there is no $$p \in M$$ such that $$\lim_{s\to b} \gamma(s) =p$$, respectively, $$\lim_{s\to a} \gamma(s) =p$$.

• (2) As $$S$$ is achronal it intersects every inextendible smooth timelike curve exactly once. Indeed, achronal means that there are no distinct points on $$S$$ which can be joined by a smooth timelike curve.

• (3) Instead, acausal is a stronger conditiom: it means that there are no distinct points on $$S$$ which can be joined by a smooth causal curve. In other words the points on an acausal $$S$$ are spacelike separated. $$\blacksquare$$

It is possible to prove that for a Cauchy surface $$S$$ as defined above,

(1) $$S$$ is closed,

(2) $$S$$ must also intersect every smooth causal curve (not necessarily once however, see below),

(3) $$S$$ is a $$C^0$$ embedded submanifold of the spacetime.

An example of Cauchy surface is the plane $$x^0=x^1$$ in 4D Minkowski spacetime. It is clear that this is achronal but not acausal. In other words, the points on $$S$$ are not spacelike separated, and this answers your question.

There is a type of Cauchy surfaces which also are acausal, they are smooth spacelike Cauchy surfaces.

An important result by Bernal and Sanchez establishes the existence of these types of Cauchy surfaces when a generic Cauchy surface exists.

Presumably you mean this type of Cauchy surfaces when you speak of hypersurface of simultaneity. Tha is beacuse it is possible to prove that given such a Cauchy surface $$S$$, there is a (surjective) smooth function $$t: M \to \mathbb{R}$$ whose differential is everywhere timelike and past directed, such that a level set of $$t$$ is exactly $$S$$. That is a so-called temporal function and, using it, the spacetime turns out to be diffeomorphic to $$\mathbb{R} \times S$$ with metric accordingly decomposed $$-dt^2 + h(t)$$.

If $$S$$ is a generic Cauchy surface several weaker theorems are however valid (see the references below).

Some modern important references on these issues are listed below.

Antonio N. Bernal, Miguel Sanchez On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem Commun. Math. Phys. 243, 461–470 (2003)

Antonio N. Bernal, Miguel Sanchez Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes Commun. Math. Phys. 257, 43–50 (2005)

Antonio N. Bernal, Miguel Sanchez Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions Letters in Mathematical Physics (2006) 77:183–197