"Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently.
Disagreement about sign conventions is a frequent source of confusion, frustration, misunderstandings, and even outright errors in scientific work.
A similar convention is used in higher-dimensional relativistic theories; that is, (+,−,−,−,...) or (−,+,+,+,...).
A choice of signature is associated with a variety of names, physics discipline, and notable graduate-level textbooks: The Ricci tensor is defined as the contraction of the Riemann tensor.
Due to the symmetries of the Riemann tensor, these two definitions differ by a minus sign.
It is often considered good form to state explicitly which sign convention is to be used at the beginning of each book or article.