By Chretien

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4 The received definition of singularities Any non-maximal spacetime is geodesically incomplete. But such a spacetime is arguably not an acceptable model for actual spacetime. So suppose that we focus on maximal spacetimes. If there are any singularities in such a spacetime they are essential or irremovable. The task now is to find a criterion that will signal when a spacetime is singular. Will geodesic incompleteness suffice? An appeal to the analogy of the better understood case of a Riemannian space would seem to encourage a positive answer.

P. A' on y(v); but it can be shown that y(v) will have finite A' length if and only if it has finite 1 length. ) for a half-curve is a well-defined notion. p. is defined for all C1 curves, not just for geodesies. l. l. if, for example, the curve experiences unbounded acceleration. l. incompleteness of a timelike curve has been developed by Sussmann (1988). Consider a 36 BANGS, CRUNCHES, WHIMPERS, AND SHRIEKS timelike half curve y(r), T 6 [0, TJ), Tt < +00, where T is proper time along the curve.

Today we know that this hope is not realized. Thus we must face the question raised in the second sentence of the quotation: Are such solutions to be "excluded on physical grounds"? 1, spacetime singularities and acausal spacetime structures pose challenges for GTR not unlike the challenge the measurement problem poses for QM. Are the nastier sorts of singularities and acausalities to be excluded on physical grounds? If so, can the exclusion Introducing Space time Singularities and Acausalities 23 be enforced on some non-ad hoc basis within GTR or some natural extension of the theory, or must the theory be modified in some substantial way in order to achieve the exclusion?

### Astrophysics and general relativity by Chretien

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