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7: Symmetries

This chapter is not required in order to understand the later material.

• 7.1: Killing Vectors
A Killing vector field, or simply a Killing vector, after Wilhelm Killing origiantes from the argument that a metric may be invariant when every point in spacetime is systematically shifted by some infinitesimal amount. When all the points in a space are displaced as specified by the Killing vector, they flow without expansion or compression.  Although the term “Killing vector” is singular, it refers to the entire field of vectors, each of which differs in general from the others.
• 7.2: Spherical Symmetry
More work is required to link the existence of Killing vectors to the existence of a specific symmetry such as spherical symmetry. When we talk about spherical symmetry in the context of Newtonian gravity or Maxwell’s equations, we may say, “The fields only depend on r ,” implicitly assuming that there is an r coordinate that has a definite meaning for a given choice of origin. But coordinates in relativity are not guaranteed to have any particular physical interpretation.
• 7.3: Penrose Diagrams and Causality
A Penrose diagram, also known as a Penrose-Carter diagram or causal diagram, can be used to visualize spacetime with a symmetry, so the relevant properties the whole thing by considering a lower-dimensional part of it. For example,  for a spacetime that is spherically symmetric, then we can reduce the four-dimensional to a two-dimensional one, with each point representing a two-sphere.
• 7.4: Static and Stationary Spacetimes (Part 1)
When we set out to describe a generic spacetime, the Alice in Wonderland quality of the experience is partly because coordinate invariance allows our time and distance scales to be arbitrarily rescaled, but also partly because the landscape can change from one moment to the next. The situation is drastically simplified when the spacetime has a timelike Killing vector. Such a spacetime is said to be stationary.
• 7.5: Static and Stationary Spacetimes (Part 2)
Birkhoff’s theorem is similar to a set of theorems called no-hair theorems describing black holes. The most general no-hair theorem states that a black hole is completely characterized by its mass, charge, and angular momentum. Other than these three numbers, nobody on the outside can recover any information that was possessed by the matter and energy that were sucked into the black hole.
• 7.6: The Uniform Gravitational Field Revisited
There is no global solution to the Einstein field equations that uniquely and satisfactorily embodies all of our Newtonian ideas about a uniform field.
• 7.E: Symmetries (Exercises)

Thumbnail: Penrose diagram for flat spacetime.