The Gravitational Field of a Body with Rotational Symmetry in Einstein's Theory of Gravitation

[摘要]

Einstein's set of field equations in vaccuo

Gμυ = 0

is reduced to such a form that simple problems like the sphere (Schwarzschild's solution), the infinite plane and the infinite cylinder can be solved. The fundamental quadratic differential forms for the latter two cases are respectively

ds² = - [(1+4πσz)^-1dz² = - [(1+4πσz)^-1dz + dρ² + ρ²dφ²] + 1+4πσz)dt²,

ds² = - c²₄ρ^-2[(1+4mlogρ)^-1dρ² + ρ²dφ²] - dz² + (1+4mlogρ)dt²,

where σ is the surface density of matter on the plane, z=0; m the linear density of matter on the cylinder, ρ=const.; (ρ,z,φ) the cylindrical coordinates; c₄ an indeterminate constant and the velocity of light is unity. Setting g₄₄ = the Newtonian potential + const., we can get the solution of the general gravitational problem for a body whose mass is distributed symmetrically about an axis provided we can solve

2δ/δψ[(1-2Mψ)δn/δψ] + δ²/δθ²e²ⁿ = 0 (M = mass of the body).

The gravitational field of an oblate spheroidal homoeoid is characterized by

ds² = - ψ^-4(1-2Mψ)^-1dψ² - ψ^-2dE² - ψ^-2cos²Edψ² + (1-2Mψ)dt²,

where ψ = k^-1cot^-1(sinhη), M = mass of the homoeoid whose equation is c²ρ²a²z² = a²c², k² = a²-c² and E,η are related to the cylindrical coordinates (ρ,z,φ) by ρ+iz = kcos(E+iη). Analogous expressions for a prolate spheroidal homoeoid are obtainable. The oblateness of the homoeoid causes a slight increase in the advance of the perihelion of a planet's orbit derived from Schwarzschild's solution.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[效力级别] theory of gravitation

[关键词] Physics;theory of gravitation;gravitational field of a body