Abstract
Generalizing from flat to arbitrary spacetime, we develop the basic formalism of differential geometry, which is the mathematical foundation of the general theory of relativity. Vectors, tensors and the important concept of the covariant derivative are introduced. The latter allows us to compare vectors and tensors at different points. The generalization of symmetry transformations with Killing fields is also introduced.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In fact this choice is not unique. There are at least two frequently used local cartesian coordinate systems: Riemann normal coordinates and Fermi normal coordinates.
- 2.
Note that in flat spacetime we have the relations \(\frac{\partial }{\partial x^\mu } =(\partial _t, \partial _x,\partial _y,\partial _z)=\partial _\mu \equiv _{,\mu }\) and \(\frac{\partial }{\partial x_\mu } =(\partial _t, -\partial _x,-\partial _y,-\partial _z)=\partial ^\mu \equiv ^{,\mu }\)
- 3.
Killing, Wilhelm Karl Joseph, German mathematician, *Burbach 10.5.1847, †Münster 11.2.1923.
- 4.
Lie, Sophus, Norwegian mathematician, *Nordfjordeit 17.12.1842, †Kristiania 18.2.1899.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hentschke, R., Hölbling, C. (2020). Introduction to Multidimensional Calculus. In: A Short Course in General Relativity and Cosmology. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-46384-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-46384-7_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-46383-0
Online ISBN: 978-3-030-46384-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)