Yes, it is possible to construct a spacetime that is spacelike and timelike complete but null incomplete.

Key Reasoning:

Geodesic completeness for different causal characters (timelike, spacelike, null) depends on the behavior of geodesics under their respective affine parameters. A spacetime can exhibit incompleteness for one type of geodesic while being complete for others.

Example ...

Yes. A classical example is the Clifton–Pohl torus, which is spacelike- and timelike-geodesically complete but null-geodesically incomplete.

Construction and properties:

• Start on M0 = R^2 \ {0} with coordinates (u,v) and the Lorentzian metric g = 2 du dv / (u^2 + v^2). This is conformal to 2 du dv (2D Minkowski in null coordinates) with conformal factor Ω^2 = 1/(u^2+v^2).

• The metric is i...

<think> We are to determine if there exists a spacetime that is both spacelike geodesically complete and timelike geodesically complete, but null geodesically incomplete.

Recall that a geodesic is complete if it is defined for all affine parameters in the entire real line. A spacetime is said to be timelike (respectively, spacelike or null) geodesically complete if every timelike (respectively, s...

Excellent question. This is a subtle but important point in understanding the nuances of geodesic completeness in General Relativity.

The answer is yes, it is possible to construct a spacetime that is both timelike and spacelike complete, but null incomplete.

The canonical examples of such spacetimes belong to the class of plane-fronted gravitational waves (pp-waves). These spacetimes ca...

No – a Lorentzian manifold cannot be spacelike– and timelike–complete while being null-incomplete.
In fact, the following general result is true (see e.g. Beem–Ehrlich–Easley, Global Lorentzian Geometry, 2nd ed., Th. 3.42):

 Theorem Let (M ,g) be an n–dimensional Lorentzian manifold (n ≥ 2).
 If M is geodesically complete for two of the three causal classes (timelike, null, spacelike)  then...

Yes, it is possible to construct a spacetime that is both spacelike and timelike complete but null incomplete. This is perhaps the most subtle case among the various combinations of geodesic completeness.

Why this is non-trivial

The existence of such spacetimes is somewhat counterintuitive because null geodesics can be thought of as "limits" of timelike geodesics. As a massive particle's velo...

No.
If every spacelike geodesic and every timelike geodesic of a space-time is complete, then every null geodesic is automatically complete as well.
Consequently a “null–incomplete but space- and time-complete’’ Lorentzian manifold cannot exist.

Why timelike + spacelike completeness forces null completeness

Denote the space-time ...

Is There a Null Incomplete Spacetime Which is Spacelike and Timelike Complete?

Yes, spacetimes that are both timelike and spacelike complete but null incomplete can indeed exist, though they're not immediately obvious to construct.

Why This is an Interesting Case

This case is particularly interesting because null geodesics sit at the boundary between timelike and spacelike geodesics in a c...

No. In fact one shows by a simple “ODE‐continuity’’ argument that you cannot have a C^2 Lorentzian metric all of whose time– and space–like geodesics are complete but which admits an inextendible null geodesic in finite affine parameter. Equivalently

timelike & spacelike geodesic completeness ⇒ null geodesic completeness.

The heuristic is this. If γ0(λ) is an inextendible null geodesic e...