Einstein's Equations, Lagrangians for General Relativity, and ADM Formalism

Author

Timothy CorbettFollow

Date of Award

Spring 4-29-2022

Document Type

Honors Project

Degree Name

Bachelor of Science

Department

Mathematics

Department Chair or Program Director

Esunge, Julius

First Advisor

Chiang, Yuan-Jen

Second Advisor

Esunge, Julius

Third Advisor

Sumner, Suzanne

Major or Concentration

Mathematics

Abstract

Einstein’s equations describe the relation of spacetime curvature and present matter. We will consider the case of a Lorentzian manifold diffeomorphic to ℝ × Σ, where time t ∈ ℝ and the three-dimensional manifold Σ represents space. In a homogeneous and isotropic universe, the three-dimensional manifold Σ of the Lorentzian manifold has Riemann curvature tensor ⁽³⁾R = KI, where K is a constant and I is the 3 × 3 identity matrix. Space is flat when K = 0, spherical when K is positive, and hyperbolic when K is negative. In this paper, we will show models agreeing with each. We will derive the Schwarzschild metric, find Einstein’s equations from the Lagrangian, and analyze them using Arnowitt-Deser-Misner formalism.

Recommended Citation

Corbett, Timothy, "Einstein's Equations, Lagrangians for General Relativity, and ADM Formalism" (2022). Student Research Submissions. 454.
https://scholar.umw.edu/student_research/454

Rights

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