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In addition, we have the following relations: C0101 = C2323 = C0123 = 2 2
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(9.21)
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Lastly, we develop some notation to represent the Ricci tensor by a set of scalars. This requires four real scalars and three complex ones. These are
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1 = Rabl a l b , 2
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1 = Rabl a m b , 2
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1 = Rab m a m b 2 (9.22)
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1 = Rab l a n b + m a m b , 4 1 = Rab n a n b 2 = 1 R 24
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1 = Rab n a m b , 2
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While we could go about nding Rab and the components of the Weyl tensor and then using them to calculate the scalars, that would miss the point. We want to avoid using those other quantities and calculate everything from the null vectors directly. It won t come as much of a surprise that there is a way to do it. First we calculate the spin coef cients directly from the null tetrad using (9.15). The next step relies on a monstrous looking set of equations called the Newman-Penrose identities. These threatening equations use the directional derivatives (9.13) applied to the spin coef cients to obtain the Weyl and Ricci scalars.
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Null Tetrads and the Petrov
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There are 18 identities in all, so we won t state them all here. Instead we will list 2 that will be useful for a calculation in the next example. This will be enough to communicate the avor of the method. The reader who is interested in understanding the method in detail can consult Grif ths (1991) or Chandrasekhar (1992). Two equations that will be of use to us in calculating the Weyl scalars and Ricci scalars are = ( + ) (3 ) + 3 + + = 2 + + ( + ) + ( 3 ) +
22 4
(9.23) (9.24)
Before we work an example, let s take a moment to discuss some physical interpretation of these de nitions.
Physical Interpretation and the Petrov Classi cation
In vacuum, the curvature tensor and the Weyl tensor coincide. Therefore, in many cases we need to study only the Weyl tensor. The Petrov classi cation, which describes the algebraic symmetries of the Weyl tensor, can be very useful in light of these considerations. To understand the meaning of the classi cations, think in terms of matrices and eigenvectors. The eigenvectors of a matrix can be degenerate and occur in multiplicities. The same thing happens here and the Weyl tensor has a set of eigenbivectors that can occur in multiplicities. An eigenbivector satis es 1 ab cd C cd V = V ab 2 where is a scalar. For mathematical reasons, which are beyond the scope of our present investigation, the Weyl tensor can have at most four distinct eigenbivectors. Physically, the Petrov classi cation is a way to characterize a spacetime by the number of principal null directions it admits. The multiplicities of the eigenbivectors correspond to the number of principal null directions. If an eigenbivector is unique, we will call it simple. We will refer to the other eigenbivectors (and therefore the null directions) by the number of times they are repeated. If
Null Tetrads and the Petrov
we say that there is a triple null direction, this means that three null directions coincide. There are six basic types by which a spacetime can be classi ed in the Petrov scheme which we now summarize: Type I. All four principal null directions are distinct (there are four simple principal null directions). This is also known as an algebraically general spacetime. The remaining types are known as algebraically special. Type II. There are two simple null directions and one double null direction.
Type III. There is a single distinct null direction and a triple null direction. This type corresponds to longitudinal gravity waves with shear due to tidal effects. Type D. There are two double principal null directions.
The Petrov type D is associated with the gravitational eld of a star or black hole (Schwarzschild or Kerr vacuum). The two principal null directions correspond to ingoing and outgoing congruences of light rays. Type N. There is a single principal null direction of multiplicity 4. This corresponds to transverse gravity waves. Type O. The Weyl tensor vanishes and the spacetime is conformally at.
We can learn about the principal null directions and the number of times they are repeated by examining A . There are three situations of note: 1. A principal null direction is repeated two times: The nonzero components of the Weyl tensor are 2 , 3 , and 4 . 2. A principal null direction is repeated three times (Type III): The nonzero components of the Weyl tensor are 3 and 4 . 3. A principal null direction is repeated four times (Type N): The nonzero component of the Weyl tensor is 4 . Vanishing components of the Weyl tensor also tell us about the null vectors l a and n a . For example, if 0 = 0 then l a is parallel with the principal null directions, while if 4 = 0 then n a is parallel with the principal null directions. If 0 = 1 = 0 then l a is aligned with repeated principal null directions.
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