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Exploring Non-Relativistic Gravity: Analytical Insights from Kocharyan's Framework

This weekly report delves into Yu-Chiao Chang's analysis of A. A. Kocharyan's paper, "Is Non-Relativistic Gravity Possible?" The work focuses on the mathematical structures involved, such as the supremespace of symmetric tensors and positive definite metrics. By contrasting Kocharyan's dynamic system approach with Miao Li's conditions, the report highlights potential avenues for advancing non-relativistic gravity theories, particularly within the ADM formalism. This paper presents a more rigorous perspective than prior models, urging further investigation into its implications for theoretical physics.

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Exploring Non-Relativistic Gravity: Analytical Insights from Kocharyan's Framework

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  1. Weekly Report 100208 Yu-Chiao Chang

  2. Non-relativistic gravity • A.A. Kocharyan’s paper: “Is non-relativistic gravity possible? ” • Superspace: • Let M be an oriented without boundary manifold. Let be the space of all symmetric two tensors on M and let be the manifold of positive definite Riemannian metrics on M.

  3. The tangent bundle of is . • Let be the space of all symmetric two contravariant tensor densities on M. The cotangent bundle of is • We have a natural pairing between given by:

  4. The DeWitt metric on is given by • The metric G has an inverse metric given by

  5. Hamiltonian formulation • We investigate a dynamical system on given by an invariant action: • Note thatthis Kocharyan considers N is time dependent only, different from Miao Li’s condition, this may result in different conclusion!

  6. The canonical momenta conjugate to are • and the Hamiltonian is

  7. Hamiltonian equations have the following form:

  8. Following the setup, we will consider the constraint structure and comparing with Miao Li’s argument! • Motivation of studying this paper is to find possible ways to rewrite Cook’s theory in ADM formalism, at least some hints. • This paper takes a more mathematical point of view to the non-relativistic gravity, higher than Horava’s theory, so if we want to develop non-relativistic gravity, I think we should pay some attention to this one.

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