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# 6: Vacuum Solutions

In this chapter we investigate general relativity in regions of space that have no matter to act as sources of the gravitational field. We will not, however, limit ourselves to calculating spacetimes in cases in which the entire universe has no matter. For example, we will be able to calculate general-relativistic effects in the region surrounding the earth, including a full calculation of the geodetic effect, which was estimated in Section 5.5 only to within an order of magnitude. We can have sources, but we just won’t describe the metric in the regions where the sources exist, e.g., inside the earth. The advantage of accepting this limitation is that in regions of empty space, we don’t have to worry about the details of the stress-energy tensor or how it relates to curvature. As should be plausible based on the physical motivation given in Section 5.1, the field equations in a vacuum are simply $$R_{ab} = 0$$.

• 6.1: Event Horizons
One seemingly trivial way to generate solutions to the field equations in vacuum is simply to start with a flat Lorentzian spacetime and do a change of coordinates. This might seem pointless, since it would simply give a new description (and probably a less convenient and descriptive one) of the same old, boring, flat spacetime. It turns out, however, that some very interesting things can happen when we do this.
• 6.2: The Schwarzschild Metric (Part 1)
We now set ourselves the goal of finding the metric describing the static spacetime outside a spherically symmetric, nonrotating, body of mass m. This problem was first solved by Karl Schwarzschild in 1915. One byproduct of finding this metric will be the ability to calculate the geodetic effect exactly, but it will have more far reaching consequences, including the existence of black holes.
• 6.2: The Schwarzschild Metric (Part 2)
We now calculate the geodetic effect on Gravity Probe B, including all the niggling factors of 3 and π. To make the physics clear, we approach the actual calculation through a series of warmups.
• 6.3: Black Holes (Part 1)
A provocative feature of the Schwarzschild metric is that it has elements that blow up at r=0 and at r=2m. If this is a description of the sun, for example, then these singularities are of no physical significance, since we only solved the Einstein field equation for the vacuum region outside the sun, whereas r=2m would lie about 3 km from the sun’s center. Furthermore, it is possible that one or both of these singularities is nothing more than a spot where our coordinate system misbehaves.
• 6.3: Black Holes (Part 2)
Singularities can also occur without any blow-up in the curvature. An example of this is a conical singularity.
• 6.4: Degenerate Solutions
So it appears that the signature of spacetime is something that is not knowable a priori, and must be determined by experiment. When a thing is supposed to be experimentally observable, general relativity tells us that it had better be coordinate-independent. Is this so? A proposition from linear algebra called Sylvester’s law of inertia encourages us to believe that it is.
• 6.E: Vacuum Solutions (Exercises)

Thumbnail: This artist's concept illustrates a supermassive black hole with millions to billions times the mass of our sun. Supermassive black holes are enormously dense objects buried at the hearts of galaxies. Image used with permission (Public Domain; NASA/JPL-Caltech).