Abstract
The differential geometry of the last chapter covered most of what is needed for many applications. However, it lacked the essential ingredient of a metric. Almost all spaces (manifolds) encountered in physics have a natural metric which is either known from the beginning, or is derived from some of its physical properties (general theory of relativity). In this chapter, we look at spaces whose connections are tied to their metrics.
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Notes
- 1.
Although we are restricting the discussion to O(n), it also applies to the more general group O(n−ν,ν).
- 2.
Note that \(d{\pmb{\epsilon}}^{\theta}=0\) does not imply ω 12=0.
- 3.
The classic and comprehensive book Gravitation, by Misner, Thorne, and Wheeler, has a thorough discussion of Newtonian gravity in the language of geometry in Chap. 13 and is highly recommended.
- 4.
It was Maxwell’s discovery of the inconsistency of the pre-Maxwellian equations of electromagnetism with charge conservation that prompted him to change not only the fourth equation (to make the entire set of equations consistent with the charge conservation), but also the course of human history.
- 5.
In GTR, it is customary to use the convention that Greek indices run from 0 to 3, i.e., they include both space and time, while Latin indices encompass only the space components.
- 6.
The reader may be surprised to see the two words “space” and “time” juxtaposed with no hyphen; but this is common practice in relativity.
- 7.
Recall that the 4-momentum of special relativity is \(p^{\mu}=m\dot{x}^{\mu}\).
- 8.
In the Taylor expansion of any potential V(r) about the equilibrium position r 0 of a particle (of unit mass), it is the second derivative term that resembles Hooke’s potential, \({{\frac{1}{2}} }kx^{2}\) with \(k=(d^{2}V/dr^{2})_{r_{0}}\).
References
Flanders, H.: Differential Forms with Applications to Physical Sciences. Dover, New York (1989)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Wiley, New York (1963)
Misner, C., Thorne, K., Wheeler, J.: Gravitation. Freeman, New York (1973)
Wald, R.: General Relativity. University of Chicago Press, Chicago (1984)
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Hassani, S. (2013). Riemannian Geometry. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_37
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DOI: https://doi.org/10.1007/978-3-319-01195-0_37
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