Notation and conventions
Unless otherwise and explicitly stated, we use geometrized units where
![G = c = 1](article0x.gif)
, so that energy and
time have units of length. Geometric objects are denoted with boldface symbols, whereas their components
are not. We also adopt the
![(− + + + ...)](article1x.gif)
convention for the metric. For reference, the following is a list of
symbols that are used often throughout the text.
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Total number of spacetime dimensions (we always consider one timelike |
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and spacelike dimensions). |
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Curvature radius of (A)dS spacetime, related to the (negative) positive |
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cosmological constant in the Einstein equations ( ) |
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through . |
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BH mass. |
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BH rotation parameter. |
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Radius of the BH’s event horizon in the chosen coordinates. |
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Fourier transform variable. The time dependence of any field is . |
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For stable spacetimes, . |
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Spin of the field. |
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Integer angular number, related to the eigenvalue |
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of scalar spherical harmonics in dimensions. |
![a,b, ...,h](article18x.gif) |
Index range referred to as “early lower case Latin indices” |
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(likewise for upper case indices). |
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Index range referred to as “late lower case Latin indices” |
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(likewise for upper case indices). |
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Spacetime metric; greek indices run from 0 to . |
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, Christoffel symbol associated with the |
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spacetime metric . |
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, Riemann curvature tensor of the |
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-dimensional spacetime. |
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-dimensional covariant derivative associated with . |
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Induced metric, also known as first fundamental form, on |
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-dimensional spatial hypersurface; latin indices run from 1 to . |
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Extrinsic curvature, also known as second fundamental form, on |
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-dimensional spatial hypersurface. |
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-dimensional Christoffel symbol associated with . |
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-dimensional Riemann curvature tensor of the spatial hypersurface. |
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-dimensional covariant derivative associated with . |
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-dimensional sphere. |