Coordinate effect: Vaidya solutions without integrating the field equations
- PDF / 322,981 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 82 Downloads / 186 Views
Coordinate effect: Vaidya solutions without integrating the field equations E. G. Mychelkin1 · M. A. Makukov1 Received: 3 September 2020 / Accepted: 5 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We extend Vaidya’s algorithm for the description of a central mass losing or gaining energy due to electromagnetic-type radiation (‘null dust’) to the case of arbitrary radial corpuscular radiation. We also demonstrate the remarkable possibility of purely algebraic deduction of the Vaidya solution without integrating the field equations, and interpret this possibility as an artifact of curvature coordinates. Since Vaidya’s approach by itself cannot lead to certain dependence of mass on spacetime coordinates, the search for a corresponding mass-function represents an independent issue. In this regard, as a perspective, we discuss an outlook on the problem of variable masses as a whole. Keywords Vaidya metric · Integrability conditions · Corpuscular radiation · Curvature coordinates · Isotropic coordinates
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Symmetries in the Vaidya problem . . . . . . . . . . . . . . . . 3 Vaidya’s solution as an algebraic consequence of the main ansatz 4 Corpuscular radial emission . . . . . . . . . . . . . . . . . . . . 5 Cosmological term in curvature and isotropic coordinates . . . . 6 Relation of Vaidya algorithm to isotropic and null coordinates . . 7 First integral and mass-function . . . . . . . . . . . . . . . . . . 8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Outlook on the problem of variable mass in general relativity . . 10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
M. A. Makukov [email protected] E. G. Mychelkin [email protected]
1
Fesenkov Astrophysical Institute, Almaty, Kazakhstan 0123456789().: V,-vol
123
113
Page 2 of 14
E. G. Mychelkin et al.
1 Introduction In general relativity (GR), the curvature coordinates play a special role. Indeed, for example, in the case of spherical symmetry and given the interval ds 2 = eν(r ) dt 2 − eλ(r ) dr 2 − r 2 dΩ 2 ,
(1)
the angular part of the metric (the unit sphere) remains flat, greatly simplifying the calculations. Moreover, in case of mutual reciprocity of the first two metric coefficients, g00 g11 = −1, as is the case for the Schwarzschild solution, 2m 2m −1 2 dr − r 2 dΩ 2 , ds 2 = 1 − dt 2 − 1 − r r
(2)
the Einstein equations become linear [1]. As opposed to the Schwarzschild
Data Loading...
