Abstract
This chapter develops the basics of differentiable manifolds and semi-Riemannian geometry for the applications in general relativity. It will introduce finitistic substitutes for basic topological notions. We will see that after basic topological notions are available, the basic notions of semi-Riemannian geometry, i.e., vector, tensor, covariant derivative, parallel transportation, geodesic and Riemann curvature, are all essentially finitistic already. Theorems on the existence of spacetime singularities are good examples for analyzing the applicability of infinite and continuous mathematical models to finite physical things. The last section of this chapter will analyze one of Hawking’s singularity theorems, whose common classical proof is non-constructive.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Field, H. 1980. Science without numbers. Princeton: Princeton University Press.
Naber, G. 1988. Spacetime and singularities: An introduction. Cambridge: Cambridge University Press.
O’Neill, B. 1983. Semi-Riemannian geometry: With applications to relativity. New York: Academic.
Wald, R. 1984. General relativity. Chicago: The University of Chicago Press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Ye, F. (2011). Semi-Riemannian Geometry. In: Strict Finitism and the Logic of Mathematical Applications. Synthese Library, vol 355. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1347-5_8
Download citation
DOI: https://doi.org/10.1007/978-94-007-1347-5_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-1346-8
Online ISBN: 978-94-007-1347-5
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)