Abstract
The Schwarzschild metric is a relevant example of an exact solution of the Einstein equations with important physical applications. It is the only spherically symmetric vacuum solution of the Einstein equations and usually it can well approximate the gravitational field of slowly-rotating astrophysical objects like stars and planets.
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 these calculations we ignore the dimensional difference between t and the space coordinates and we do not write the speed of light c to simplify the equations. This is equivalent to employing units in which \(c=1\), which is a convention widely used among the gravity and particle physics communities.
- 2.
As in Sect. 1.8, we use the notation \(L_z\) because this is also the axial component of the angular momentum (since \(\theta = \pi /2\)) and we do not want to call it L because it may generate confusion with the Lagrangian.
- 3.
Note that for a massive particle we choose \(\lambda = \tau \) the particle proper time. In such a case, for a static particle at infinity we have \(\dot{t} = 1\) and \(E=c^2\) (because we are assuming \(m=1\), otherwise we would have \(E=mc^2\)); that is, the particle energy is just the rest mass. For a static particle at smaller radii, \(E < c^2\) because the (Newtonian) gravitational potential energy is negative. Note also that \(p_t = - E/c\) is conserved while the temporal component of the 4-momentum, \(p^t\), is not a constant of motion. The same is true for \(p_\phi \) and \(p^\phi \): only \(p_\phi \) is conserved.
- 4.
Note that the metric is ill-defined even at \(r = 0\), which is a true spacetime singularity and cannot be removed by a coordinate transformation. The Kretschmann scalar diverges at \(r = 0\).
- 5.
The symbol \(\mathscr {I}\) is usually pronounced “scri”.
References
C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W. H. Freeman and Company, San Francisco, 1973)
P.K. Townsend, gr-qc/9707012
Author information
Authors and Affiliations
Corresponding author
Problems
Problems
8.1
Let us consider a massive particle orbiting a geodesic circular orbit in the Schwarzschild spacetime. Calculate the relation between the particle proper time and the coordinate time t of the Schwarzschild metric.
8.2
Let us consider the Penrose diagram for the Minkowski spacetime, Fig. 8.2. We have a massive particle that emits an electromagnetic pulse at \(t=0\). Show the trajectories of the massive particle and of the electromagnetic pulse in the Penrose diagram.
8.3
Let us consider the Penrose diagram for the maximal extension of the Schwarzschild spacetime, Fig. 8.3. Show the future light-cone of an event in region I, of an event inside the black hole, and of an event inside the white hole.
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Bambi, C. (2018). Schwarzschild Spacetime. In: Introduction to General Relativity. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-13-1090-4_8
Download citation
DOI: https://doi.org/10.1007/978-981-13-1090-4_8
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-1089-8
Online ISBN: 978-981-13-1090-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)