Citation
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:etd06282004092506
Abstract
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
ds^{2} =  [(1+4πσz)^{1}dz^{2} =  [(1+4πσz)^{1}dz + dρ^{2} + ρ^{2}dφ^{2}] + 1+4πσz)dt^{2},
ds^{2} =  c^{2}_{4}ρ^{2}[(1+4mlogρ)^{1}dρ^{2} + ρ^{2}dφ^{2}]  dz^{2} + (1+4mlogρ)dt^{2},
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; c_{4} an indeterminate constant and the velocity of light is unity. Setting g_{44} = 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[(12Mψ)δn/δψ] + δ^{2}/δθ^{2}e^{2n} = 0 (M = mass of the body).
The gravitational field of an oblate spheroidal homoeoid is characterized by
ds^{2} =  ψ^{4}(12Mψ)^{1}dψ^{2}  ψ^{2}dE^{2}  ψ^{2}cos^{2}Edψ^{2} + (12Mψ)dt^{2},
where ψ = k^{1}cot^{1}(sinhη), M = mass of the homoeoid whose equation is c^{2}ρ^{2}a^{2}z^{2} = a^{2}c^{2}, k^{2} = a^{2}c^{2} 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 
Major Option:  Physics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 January 1928 
Record Number:  CaltechETD:etd06282004092506 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd06282004092506 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2747 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  29 Jun 2004 
Last Modified:  16 Mar 2016 19:07 
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