- #1

PeterDonis

Mentor

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- 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?

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?