The metaverse has gained popularity recently in many areas including social media, healthcare, education and manufacturing. This work explores the use of the metaverse concept for cultural heritage applications. The motivation is to develop a systematic approach for the construction of a cultural heritage metaverse and to offer, potentially, more effective solutions for tourism guidance, site maintenance, heritage object conservation, etc. We propose a framework for this cultural heritage metaverse with an emphasis on fundamental elements and on characterization of the mapping between the physical and virtual cultural heritage worlds. Efforts are made to analyze the dimensional structures of the cultural heritage metaverse. Specifically, five different dimensions, linearity, planarity, space, time and context, are discussed to better understand this metaverse. The proposed framework and methodology are novel and can be applied to the digitalization of cultural heritage via its metaverse development. This is followed by a detailed case study to illustrate the tangible procedure, constructing a cultural heritage metaverse with a complex and dynamic nature which can be used for different applications, including heritage conservation.