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6
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85037192189
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At high relativistic order it is also necessary to satisfy a strictly “in-going” boundary condition at the horizon. See
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At high relativistic order it is also necessary to satisfy a strictly “in-going” boundary condition at the horizon. See 2.
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11
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85037178513
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Throughout this paper we use the—not a very mathematically rigorous—concept of a “test-charge.” By test-charge we mean a particle of essentially infinitesimal extent (we describe it as a Dirac (Formula presented)-function). The particle has bare mass (Formula presented) which is small compared to the mass M of the black hole [i.e., (Formula presented)]; therefore we neglect perturbations in the background geometry. Also, the magnitude of the charge is small enough that energy associated with the field does perturb the background geometry. For a scalar charge we will always use the symbol “(Formula presented),” and for an electric charge “(Formula presented).”
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Throughout this paper we use the—not a very mathematically rigorous—concept of a “test-charge.” By test-charge we mean a particle of essentially infinitesimal extent (we describe it as a Dirac (Formula presented)-function). The particle has bare mass (Formula presented) which is small compared to the mass M of the black hole [i.e., (Formula presented)]; therefore we neglect perturbations in the background geometry. Also, the magnitude of the charge is small enough that energy associated with the field does perturb the background geometry. For a scalar charge we will always use the symbol “(Formula presented),” and for an electric charge “(Formula presented).”
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12
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85037225388
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It is clear from Eq. (B5) that the horizon (Formula presented) is a surface of constant “potential;” therefore, in this sense (only), the horizon behaves like a conductor. Computing the lines of force gives a different picture
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It is clear from Eq. (B5) that the horizon (Formula presented) is a surface of constant “potential;” therefore, in this sense (only), the horizon behaves like a conductor. Computing the lines of force gives a different picture.
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14
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85037182192
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We use three different types of coordinates to describe the Schwarzschild geometry: isotropic coordinate, Schwarzschild coordinates and harmonic coordinates. The radial coordinates will be denoted r, (Formula presented) and (Formula presented) respectively. They are related to each other by (Formula presented). The horizon of the black hole is located at (Formula presented) (Formula presented), and (Formula presented). The coordinates with an over-bar refer to coordinates in the freely-falling frame. The spacetime event where the force measurement is made is (Formula presented). We adopt units in which (Formula presented)
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We use three different types of coordinates to describe the Schwarzschild geometry: isotropic coordinate, Schwarzschild coordinates and harmonic coordinates. The radial coordinates will be denoted r, (Formula presented) and (Formula presented) respectively. They are related to each other by (Formula presented). The horizon of the black hole is located at (Formula presented) (Formula presented), and (Formula presented). The coordinates with an over-bar refer to coordinates in the freely-falling frame. The spacetime event where the force measurement is made is (Formula presented). We adopt units in which (Formula presented).
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20
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85037206423
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The simplest example of this is the field of an isolated electric charge e in flat spacetime where the field is (Formula presented). (Here d is the distance from the charge.) However, if the origin of the coordinates is located somewhere other than the charge we can also expand (Formula presented). (See
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The simplest example of this is the field of an isolated electric charge e in flat spacetime where the field is (Formula presented). (Here d is the distance from the charge.) However, if the origin of the coordinates is located somewhere other than the charge we can also expand (Formula presented). (See 21, p. 102.) Clearly, a simple expression like (Formula presented) is more convenient than the infinite sum when we need to know the behavior near the singularity.
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25
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85037197895
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J. Hadamard, Lectures on Cauchy’s Problem (Yale University Press, New Haven, CT, 1923). This reference is referred to as Hadamard
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J. Hadamard, Lectures on Cauchy’s Problem (Yale University Press, New Haven, CT, 1923). This reference is referred to as Hadamard.
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29
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85037239913
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Our convention for the Dirac-(Formula presented) function is (Formula presented) and (Formula presented). However, we pedantically write the scalar quantities (Formula presented) and (Formula presented)
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Our convention for the Dirac-(Formula presented) function is (Formula presented) and (Formula presented). However, we pedantically write the scalar quantities (Formula presented) and (Formula presented).
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30
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85037201066
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An extremely simple example of the Hadamard elementary solution to keep in mind: for Poisson’s equation in flat space (Formula presented), the Hadamard elementary solution is just the standard Green’s function (Formula presented). Notice (Formula presented) is a solution to the homogeneous equation everywhere except at the source point where it is singular. The Hadamard elementary solution we are constructing will have this property
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An extremely simple example of the Hadamard elementary solution to keep in mind: for Poisson’s equation in flat space (Formula presented), the Hadamard elementary solution is just the standard Green’s function (Formula presented). Notice (Formula presented) is a solution to the homogeneous equation everywhere except at the source point where it is singular. The Hadamard elementary solution we are constructing will have this property.
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32
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85037215556
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general, if a particular solution does not satisfy the boundary conditions, you patch it up by adding a solution to the homogeneous equation to fix the boundary values. However, in the present case it is a good thing that the particular solution we found satisfies the boundary conditions, because, in Schwarzschild spacetime, there are no static solutions to the homogeneous scalar field equation
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In general, if a particular solution does not satisfy the boundary conditions, you patch it up by adding a solution to the homogeneous equation to fix the boundary values. However, in the present case it is a good thing that the particular solution we found satisfies the boundary conditions, because, in Schwarzschild spacetime, there are no static solutions to the homogeneous scalar field equation 33.
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39
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84914783674
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Cambridge University Press, London, S.W. Hawking, W. Israel
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T. Damour, in 300 Years of Gravitation, edited by S.W. Hawking and W. Israel (Cambridge University Press, London, 1987).
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(1987)
300 Years of Gravitation
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Damour, T.1
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44
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0003498504
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Academic Press, New York
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I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980).
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(1980)
Table of Integrals, Series, and Products
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Gradshteyn, I.S.1
Ryzhik, I.M.2
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