External time canonical formalism for gravity in terms of embedding theory

TL;DR

This paper explores a gravity formulation where spacetime is a surface in flat space, using an 'external' time coordinate, and develops a canonical formalism that aligns with general relativity, aiding potential quantization.

Contribution

It introduces a gauge fixing approach for external time in embedding gravity and constructs a canonical formalism with a first-class constraint algebra.

Findings

The gauge fixing preserves all equations of motion.

The canonical formalism is equivalent to general relativity.

The approach facilitates quantization and embedding theory studies.

Abstract

The formulation of gravity theory is considered where space-time is a 4-dimensional surface in flat ten-dimensional space. The possibility of using the "external" time (the time of ambient space) in this approach is investigated. The transition to the "external" time is realized with the help of partial gauge fixing, the coordinate condition which equates the timelike coordinate of the surface and time of the ambient space. It is shown that by using such a gauge condition in the action, the loss of any equations of motion does not take place, although it can happen in the general case. A version of the canonical formalism of the theory is studied in which certain additional constraints are imposed, providing the equivalence of the approach under consideration and general relativity. The corresponding first-class constraint algebra is obtained. It is proved that using the gauge directly in the action leads to the same result as gauge fixing in the constraint algebra, despite the artificial introduction of some of the constraints into the theory. Application of the "external" time may be useful for attempts to quantize the theory as well as for studies in embedding theory where 4-dimensional surfaces fill the whole ambient space and coordinates on the surfaces are not introduced.