Can a Null Surface Satisfy the Requirements for a Cauchy Surface?

[PeterDonis]

TL;DR Summary

The definition of a Cauchy surface requires that it be achronal. A null surface meets this criterion. However, it often seems to be assumed that Cauchy surfaces are spacelike. Can a Cauchy surface be null?

The definition of a Cauchy surface, as given in, for example, Wald Section 8.3, is "a closed achronal set $\Sigma$ for which $D(\Sigma) = M$", i.e., every past and future causal curve (timelike or null) through any point in the entire spacetime intersects $\Sigma$.

The definition of "achronal", as given in Section 8.1 of the same reference, is "$I^+(S) \cap S = \emptyset$", i.e., no two points in $S$ can be connected by any timelike curve. Note that this definition only excludes timelike curves, not null curves. That seems to imply that a null surface (more precisely, a 3-surface with one null and two spacelike basis vectors in its tangent space) can be achronal, and hence, if it meets the other requirements above, could be a Cauchy surface.

An example of such a surface would be the 3-surface $t = x$ in Minkowski spacetime. This is a null surface, and seems to meet the other requirements for a Cauchy surface.

Is this correct?

 

[physicsworks]

Although different authors define it differently, I think what you are saying (i.e. Wald's definition and its consequences) is consisent with a definition given by Geroch in his article Domain of Dependence (let me know if you don't have access to it).

 

[robphy]

https://www.google.com/search?q="null+cauchy+surface"

Used in a sentence...

Commun. Math. Phys. 87, 395-415 (1982)
The Unpredictability of Quantum Gravity
S. W. Hawking
https://projecteuclid.org/journals/...bility-of-quantum-gravity/cmp/1103922050.full

p.402
(3.4)
where the integral is taken over a space-like or null Cauchy surface for the past asymptotic region.

Possibly useful:
Class. Quantum Grav. 29 (2012)
The many ways of the characteristic Cauchy problem
Piotr T. Chruściel, Tim-Torben Paetz
https://doi.org/10.1088/0264-9381/29/14/145006
https://arxiv.org/abs/1203.4534

 

[PeterDonis]

An example of such a surface would be the 3-surface $t = x$ in Minkowski spacetime. This is a null surface, and seems to meet the other requirements for a Cauchy surface.

Actually, on thinking this over, it doesn't, at least not if we require every causal curve through any point in the entire spacetime to intersect the surface. The Rindler hyperbolas $x^2 - t^2 = 1/a^2$ for $a > 0$ are timelike curves but do not intersect the surface $t = x$. Only if we restrict the requirement to timelike geodesics will $t = x$ meet the requirements for a Cauchy surface, since every timelike geodesic through any point in Minkowski spacetime will intersect the surface $t = x$. But not every null geodesic will (the null geodesic curve $t = x + k$ for any $k \neq 0$ won't). But I don't think the definition of a Cauchy surface is meant to be so restricted.

So now I have another question: if a Cauchy surface can be null, can anyone give an actual example of a null Cauchy surface?

 

[Ibix]

$t=\sin(x)$ is null at $x=n\pi$, and it satisfies your concerns about Rindler observers and null geodesics.

 

[PeterDonis]

$t=\sin(x)$ is null at $x=n\pi$, and it satisfies your concerns about Rindler observers and null geodesics.

Yes, this would be an example, although I would call it an edge case since the points at which it is null are a set of measure zero (a countable number of isolated points).

 

[PeterDonis]

I would call it an edge case since the points at which it is null are a set of measure zero (a countable number of isolated points).

This sharpens the question I was asking at the end of post #4 to: can anyone give an example of a Cauchy surface that is null everywhere? Or is it possible to prove that a Cauchy surface cannot be null everywhere, but only at at most a set of points of measure zero?

 

[Ibix]

In a (1+1)d Minkowski space "wrapped" into a cylinder with the axial direction being timelike a pair of counter-orbiting light rays form an everywhere null surface that nothing can avoid crossing. You can round off the junctions with short spacelike sections if smoothness is required. I don't like the way a pair of light rays lie in the surface (that seems to violate the separation of "initial data" and "extrapolation"), but it fits the bill for a Cauchy surface as I understand it.

You may be able to find similar surfaces in less artificial spacetimes, e.g. closed FLRW with non-zero $\Lambda$.

 

[PeterDonis]

In a (1+1)d Minkowski space "wrapped" into a cylinder with the axial direction being timelike a pair of counter-orbiting light rays form an everywhere null surface that nothing can avoid crossing.

Such a surface is not achronal; a timelike curve going straight up the cylinder intersects multiple points on the surface. So it's not a Cauchy surface.

 

[Ibix]

Such a surface is not achronal; a timelike curve going straight up the cylinder intersects multiple points on the surface. So it's not a Cauchy surface.

No it wouldn't - each ray completes a half orbit, and you get a single loop around the surface. Think of a Minkowski diagram on a piece of paper. Draw what you can of the future light cone of an event that lies on the centerline of the page - so two 45⁰ lines that terminate at the edges of the diagram. Wrap that into a cylinder. The line is single valued everywhere, with the arguable exception of the meeting points, which you can round off.

 

[PeterDonis]

each ray completes a half orbit, and you get a single loop around the surface.

Ah, I see. This has another "edge case" feature, though: it's not differentiable at the two "join" points. Wald's definition of "achronal" only requires the surface to be $C^0$, so I don't think non-differentiability disqualifies a surface, but it still seems like an "edge case" to me.

So now I need to sharpen my question again. Is there an example of a Cauchy surface that is null everywhere and differentiable everywhere? (The more differentiable, the better, i.e., smooth, $C^\infty$, would be best, but $C^n$ with a larger $n > 0$ is better than smaller $n$, with $C^1$, differentiable, being the minimum.)

 

[Ibix]

Differentiability is why I suggested rounding off the corners, which I think can be done in such a way that the surface is $C^\infty$ everywhere except at the joins between the "rounding" and the null paths, where it may be made $C^n$ for arbitrary finite $n$. But the "rounding" is spacelike so it obviously fails your null everywhere criterion, although I think the non-null region can be arbitrarily small.

 

[PeterDonis]

Differentiability is why I suggested rounding off the corners

Ah, sorry, I missed that.

But the "rounding" is spacelike so it obviously fails your null everywhere criterion, although I think the non-null region can be arbitrarily small.

Yes, by making the path curvature of the rounded corner arbitrarily large.

Now that I think of it, even in the "unrounded" version, where there are sharp corners at the joins, the tangent vector is undefined at the join points, so it is not possible to characterize the surface as null at those points. I think that in itself might be sufficient to violate Wald's definition.

 

[PeterDonis]

But the "rounding" is spacelike so it obviously fails your null everywhere criterion, although I think the non-null region can be arbitrarily small.

Yes, by making the path curvature of the rounded corner arbitrarily large.

Note, though, that the spacelike region still has finite arc length; it is not a set of measure zero.