Abstract
In the first three sections of Chapter 8 we had a discussion on static spherically symmetric metrics and the vacuum Schwarzschild solution, which focused mainly on finding the solution. In view of the essentialness of the Schwarzschild solution, this chapter will further discuss several intimately related problems: Sect. 9.1 discusses the timelike and null geodesics in Schwarzschild spacetime; Sect. 9.2 introduces three experimental tests of general relativity posed by Einstein using the vacuum Schwarzschild solution in his early years, namely the gravitational redshift, the precession of the perihelion of Mercury and the bending of starlight in the Sun’s gravitational field; Sect. 9.3 discusses the spacetime geometric structure and physical states in the interior of a spherically symmetric star, as well as the evolution of a spherically symmetric star; Sect. 9.4 analyzes the theory of the extension of the Schwarzschild spacetime in detail.
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Notes
- 1.
- 2.
When doing the quantitative calculation, it is better to go back to the International System of Units (SI), i.e., to fill in the physical constants G and c. From Appendix A one can see that M/r is actually \((GM/c^2)/r\). The mass of the Sun M corresponds to \(GM/c^2\cong 1.5\) km, while the distance between the perihelion of Mercury and the Sun is about \(5\times 10^7\) km, and hence \((GM/c^2)/r\ll 1\).
- 3.
For example, an analysis based on the very-long-baseline interferometry (VLBI) database gives an result which is \(0.99983\pm 0.00045\) times the predicted value of general relativity, where the standard error is reduced to \(4.5\times 10^{-4}\), see Shapiro et al. (2004).
- 4.
Generally speaking, the pressure p is not only a function of the density \(\rho \), but also depends on the specific entropy (i.e., the average entropy per nucleus) and the chemical components of the star. Only when the specific entropy and chemical components are the same everywhere inside the star can p be solely a function of \(\rho \), and the equation of state be expressed as \(f(p,\rho )=0\). The specific entropy of a normal star (including the Sun) is not everywhere the same. However, the specific entropy inside a white dwarf or neutron star, which will be discussed later, can be considered as vanishing everywhere. The discussion in the main text is valid for the study of these “abnormal celestial bodies”.
- 5.
A more precise statement is: since it releases a large amount of high-energy neutrinos, a few seconds after the formation of the neutron star it has \(E_{\text {F}}\gg k_{\text{ B }}T\).
- 6.
- 7.
Also note that a singularity may not be a point, since in the 4-dimensional language \(r=0\) (or \(r=2M\)) represents a hypersurface instead of a point.
- 8.
The precise mathematical definition is: a spacetime \((M,g_{ab})\) is said to be inextendible if there does not exist a spacetime \((M', g_{ab}')\) such that there exists an isometry between the proper subsets of \((M,g_{ab})\) and \((M^\prime , g_{ab}^\prime )\).
- 9.
It only differs from the Minkowski metric up to a diffeomorphism, and thus they are equivalent (they have the same geometry, see Sect. 8.10.2).
- 10.
Later we will see that the light waves received by the exterior observers will have stronger and stronger redshift. Thus, only if we assume (theoretically) that the observer is sensitive to light of any wavelength and intensity can they observe this phenomenon.
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Liang, C., Zhou, B. (2023). Schwarzschild Spacetimes. In: Differential Geometry and General Relativity. Graduate Texts in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-99-0022-0_9
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