Black Holes: The Membrane Paradigm by Kip S. Thorne

Author:Kip S. Thorne [Thorne, Kip S.]
Language: eng
Format: epub
ISBN: 9780300037708
Amazon: 0300037708
Goodreads: 1173972
Published: 2022-07-05T19:15:06+00:00

186

Gravitational Interaction of a Black Hole with Nearby Matter restriction corresponds to aH � 3 x Io· 6 or °‘H2 � 1o- : 1 • For further discussion see the passage following Eq. (6.77c), below.

Our placing the stretched horizon at constant lapse function (rather than, e.g., at constant radial coordinate r) has the advantage that those FIDOs who are initially on the stretched horizon remain always on it -

aside from fractional errors of order att2 which we ignore. See PT, Sec.

III B 1 , for a proof. As we develop the membrane forma1ism for perturbed holes, we shall frequently make use of the FIOOs who live on the stretched horizon. For example, the properties of the stretched horizon’s membrane will be defined in terms of physica1 measurements that these FIDOs make.

B. Gravitational Perturbations outside the Stretched Horizon We tum attention now to a quantitative description of the exterior of a perturbed black hole. Since we shall use tidal gravity to carry the information about the perturbations, before quantifying those perturbations it will be necessary to study in detail the tidal fields of an unperturbed Kerr hole. This we shall do in subsection I . Then in subsection 2

we shall identify a gauge- and slicing-invariant piece of the tidal fields of a perturbed hole - a piece that vanishes in the unperturbed case - and use it, along with the materia1 energy density, energy flux, and stress, to characterize the perturbations.

1 . The tidal fields of an unperturbed Kerr black hole Each FIDO in the absolute space around an unperturbed Kerr hole has an orthonormal set of spatial basis vectors � on which he can project spatial vectors and tensors; and if he chooses to adopt a local-Lorentz, 4-dimensional spacetime view of physics, he can augment these � with a

.. time basis vector” eo (which is equal to his 4-velocity). For example, if the background hole is nonrotating (Schwarzschild), the most natural choice for his basis vectors is [Eqs. (2.3)) e- -

r

[•-2Mr I’·· .!.. e· - _!_ .!.. e =

…!._ .

ar ‘

9

r ao ‘

• rsin6 a”’ ‘

I

.!_ .

(l-2M

(6.6)

/r)‘h at ‘

and if it is rotating (Kerr), the most natural choice is lEq. (3.9)1

� a

a

a

_

e- - — —

e· - - -

e

-

-=

-

·

r

p

ar ‘

9

p 86 ‘

• ”” l!J4’ ‘

”’…!._’

e· - _!_[..!. +

(6. 7)

0

Ol

iJt

iJ4’ ‘

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