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


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. doi:10.7907/GH13-VA35.


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 + ρ22] + 1+4πσz)dt2,

ds2 = - c24ρ-2[(1+4mlogρ)-12 + ρ22] - 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)-12 - 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
Major Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Bell, Eric Temple
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1928
Additional Information:Author's name on title page is spelled P'ei Yüan Chou.
Record Number:CaltechETD:etd-06282004-092506
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2747
Deposited By: Imported from ETD-db
Deposited On:29 Jun 2004
Last Modified:03 Oct 2019 23:20

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