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Volumn 61, Issue 8, 2000, Pages

Self-force on a static scalar test charge outside a Schwarzschild black hole

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EID: 17044436949     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.61.084014     Document Type: Article
Times cited : (72)

References (45)
  • 6
    • 85037192189 scopus 로고    scopus 로고
    • At high relativistic order it is also necessary to satisfy a strictly “in-going” boundary condition at the horizon. See
    • At high relativistic order it is also necessary to satisfy a strictly “in-going” boundary condition at the horizon. See 2.
  • 11
    • 85037178513 scopus 로고    scopus 로고
    • 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).”
    • 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).”
  • 12
    • 85037225388 scopus 로고    scopus 로고
    • 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
    • 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.
  • 13
  • 14
    • 85037182192 scopus 로고    scopus 로고
    • 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)
    • 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).
  • 20
    • 85037206423 scopus 로고    scopus 로고
    • 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
    • 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.
  • 25
    • 85037197895 scopus 로고    scopus 로고
    • J. Hadamard, Lectures on Cauchy’s Problem (Yale University Press, New Haven, CT, 1923). This reference is referred to as Hadamard
    • J. Hadamard, Lectures on Cauchy’s Problem (Yale University Press, New Haven, CT, 1923). This reference is referred to as Hadamard.
  • 29
    • 85037239913 scopus 로고    scopus 로고
    • 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)
    • 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).
  • 30
    • 85037201066 scopus 로고    scopus 로고
    • 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
    • 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.
  • 32
    • 85037215556 scopus 로고    scopus 로고
    • 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
    • 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.
  • 39
    • 84914783674 scopus 로고
    • Cambridge University Press, London, S.W. Hawking, W. Israel
    • T. Damour, in 300 Years of Gravitation, edited by S.W. Hawking and W. Israel (Cambridge University Press, London, 1987).
    • (1987) 300 Years of Gravitation
    • Damour, T.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.