2004/09/12 | S黑洞密度[Schwarzschild Black Hole--Constant Density]
类别(Ω〖物理〗) | 评论(1) | 阅读(131) | 发表于 14:09
As in the case of the Schwarzschild solution for empty space with an isotropic static gravitational field, the case of a static gravitating spherical body of constant density can also be found exactly, as was first done by Schwarzschild (1916).Let M be the total mass, R the radius, the density, the mass inside radius r,P the pressure,andthe gravitational potential, then
(1)

(2)

(3)

(4)

For a spherically symmetric metric,
(5)

(6)

With constant,
(7)

and (6) becomes
(8)

Integrating,
(9)

Expanding the denominator of the left side,
(10)

Using the integral
(11)

(for q < 0) with
(12)

(13)

so the indefinite integral of the left side is

(14)

and the left side of (9) becomes


(15)

Use the integral
(16)

with
(17)

(18)

so solve the right side of (9),


(19)

Equating (15) and (19) gives
(20)

(21)

Solving for P,
(22)

Using
(23)

and solving
(24)

For spherical symmetry, the metric coefficients are then
(25)

(26)

(Weinberg 1972, p. 331).
According to (24),when
(27)

(28)

(29)

which is unphysical, so it must be the case that
(30)

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