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

## 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. doi:10.7907/GH13-VA35. https://resolver.caltech.edu/CaltechETD:etd-06282004-092506

## 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

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.)) Physics; theory of gravitation; gravitational field of a body California Institute of Technology Physics, Mathematics and Astronomy Physics Public (worldwide access) Bell, Eric Temple Unknown, Unknown 1 January 1928 Author's name on title page is spelled P'ei Yüan Chou. CaltechETD:etd-06282004-092506 https://resolver.caltech.edu/CaltechETD:etd-06282004-092506 10.7907/GH13-VA35 No commercial reproduction, distribution, display or performance rights in this work are provided. 2747 CaltechTHESIS Imported from ETD-db 29 Jun 2004 03 Oct 2019 23:20

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