Zhou, Peiyuan (1928) The gravitational field of a body with rotational symmetry in Einstein's theory of gravitation. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-06282004-092506
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.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Subject Keywords:||Physics; theory of gravitation; gravitational field of a body|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||1 January 1928|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||29 Jun 2004|
|Last Modified:||16 Mar 2016 19:07|
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