Confusion about Cauchy surfaces

[Dale]

TL;DR

Cauchy surface rules

I have some questions about Cauchy surfaces (not Cauchy horizons).

I know that a Cauchy surface cannot be timelike, and can be spacelike. What about null? Can a Cauchy surface be lightlike?

Also, do Cauchy surfaces need to “cover” the entire manifold in some sense?

The definitions that I have read get pretty confusing very quickly.

For example, in an inertial frame in flat spacetime (units where $c=1$), is the surface $x=t$ a Cauchy surface? If not, why not?

It is lightlike, so it may fail simply because of that. It does, in some sense, cover the whole spacetime. But there are timelike worldlines that never intersect it. But then again, all timelike geodesics do intersect it.

 

[fresh_42]

Not sure whether this helps and I don't want to keep experts from answering. So this is just a little remark:
https://ncatlab.org/nlab/show/Cauchy+surface
I like nLab because of its normally precise definitions and links.

 

[martinbn]

Usually the Cauchy surfaces are used fir the Cauchy problem ( the initial value problem). Null surfaces are chaeacreristic and not suitable for it. There is a characteristic value problem of course. At the end it is a conventikn and I think the standard assumption is that a Cauchy surface needs to be spacelike.

 

[.Scott]

As I understand it (and that's very limited), asking whether a Cauchy surface can be timelike is like asking if a monotonic function can have a derivative of zero at some points. It kind of depends on what you're going to do with it. So, I'm going along with @martinbn who just posted as I was typing.

I think a Cauchy surface can cover the whole manifold "in some sense". But, depending on what you are using it for, you may not need to be concerned about precisely defining your surface across the entire manifold.

As far as x=t (light-like everywhere) is concerned, whether it's a Cauchy surface or not, I would think it would be pretty useless.

I notice that the link provided by @fresh_42 requires any travel line "intersects Σ precisely in one point" . If travel lines include photons, that would exclude any surface that included any light-like patches.

 

[robphy]

Possibly useful:
https://www.physicsforums.com/threa...he-requirements-for-a-cauchy-surface.1045444/

 

[PeterDonis]

in an inertial frame in flat spacetime (units where $c=1$), is the surface $x=t$ a Cauchy surface? If not, why not?

No, because there are inextendible causal curves that do not intersect it: for example, all of the hyperbolas $x^2 - t^2 = s^2$, where $s^2 > 0$.

all timelike geodesics do intersect it.

The definition in Wald, section 8.3, requires all inextendible causal curves to intersect the surface, not just geodesics.

Note that Wald's definition only requires a Cauchy surface to be achronal, meaning no two points on the surface can be joined by a timelike curve. A null surface is achronal, so there is nothing in Wald's definition that rules out the possibility of a Cauchy surface that is everywhere null. However, no such surface can be a Cauchy surface in flat Minkowski spacetime, for the reason I gave above. I can't think of an example of one in a curved spacetime offhand, but that doesn't mean there isn't one.

 

[Dale]

Not sure whether this helps and I don't want to keep experts from answering. So this is just a little remark:
https://ncatlab.org/nlab/show/Cauchy+surface
I like nLab because of its normally precise definitions and links.

Thanks, that is an unusually clear definition. That definition clearly means that $t=x$ is not a Cauchy surface.

 

[Dale]

Possibly useful:
https://www.physicsforums.com/threa...he-requirements-for-a-cauchy-surface.1045444/

Well, that certainly does make me feel better

 

[Dale]

Wald's definition only requires a Cauchy surface to be achronal, meaning no two points on the surface can be joined by a timelike curve. A null surface is achronal, so there is nothing in Wald's definition that rules out the possibility of a Cauchy surface that is everywhere null.

Yes. That was precisely what was confusing me.