[Paper Review] Rigid Quasilocal Frames

Paul L. McGrath|arXiv (Cornell University)|Feb 6, 2014

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TL;DR

This paper introduces rigid quasilocal frames (RQFs) as a geometric framework in general relativity to define energy, momentum, and angular momentum quasilocal quantities without relying on spacetime symmetries. By enforcing zero expansion and shear on a timelike worldtube boundary, RQFs naturally incorporate six degrees of freedom and enable exact, geometric conservation laws that reveal gravitational energy and momentum transfer via the equivalence principle, with key results including a relativistic analogue of the Poynting vector and Archimedes' law.

ABSTRACT

In this thesis, I examine in detail the properties of rigid quasilocal frames (RQF), which have been proposed as a geometrically natural way to define spatially extended reference frames in general relativity. I also explore their usefulness, in particular, as a tool for constructing completely general conservation laws that do not rely on the presence of spacetime symmetries and include both matter and gravitational contributions without the need for any ad hoc structures such as pseudotensors. In doing so, I show how the RQF approach affords a deeper understanding of the nature of gravitational fluxes via the equivalence principle. Finally, I apply the RQF formalism to explore Ehrenfest's rotating disk paradox, a generalization of Archimedes' law to curved spacetime, tidal interactions for Earth's and Jupiter's moons, and more.

Motivation & Objective

• To develop a geometrically natural framework for defining extended systems in dynamical spacetimes using rigid quasilocal frames (RQFs).
• To define quasilocal energy, momentum, and angular momentum without relying on global spacetime symmetries or pseudotensors.
• To derive exact, geometric conservation laws for energy and momentum fluxes that reveal the physical mechanisms of gravitational energy transfer.
• To resolve long-standing paradoxes such as Ehrenfest's rotating disk by showing time-dependent rotation introduces non-locality in time.
• To demonstrate that the full curvature of spacetime, including gravitational contributions, is necessary to fully describe electromagnetic and gravitational energy fluxes.

Proposed method

• Define an RQF as a two-parameter family of timelike worldlines forming a worldtube boundary with topology ℝ×S², satisfying zero expansion and shear for rigidity.
• Use conformal Killing vector (CKV) fields—three boosts and three rotations—on the two-sphere boundary to represent the six motional degrees of freedom.
• Construct quasilocal energy, momentum, and angular momentum using the Brown-York matter plus gravity boundary stress-energy-momentum tensor.
• Derive conservation laws by integrating fluxes of the boundary SEM tensor across the RQF, ensuring only physically relevant fluxes (e.g., due to acceleration) appear.
• Apply the formalism to the small-sphere limit and time-dependent rotations to analyze non-locality and curvature effects.
• Use the linear momentum conservation law to derive a fully general relativistic analogue of Archimedes' law for buoyant forces in curved spacetime.

Experimental results

Research questions

• RQ1 How can energy, momentum, and angular momentum be defined quasilocally in general relativity without assuming spacetime symmetries?
• RQ2 What is the physical mechanism behind gravitational energy and momentum transfer, and how can it be captured geometrically via the equivalence principle?
• RQ3 Why has Ehrenfest’s rigid rotating disk paradox remained unresolved, and can RQFs reveal the underlying non-locality in time?
• RQ4 How do electromagnetic and gravitational energy fluxes contribute equally to energy changes in accelerating systems, such as Bell’s spaceships?
• RQ5 Can a fully general relativistic analogue of Archimedes’ law be derived that includes both matter and gravitational field contributions?

Key findings

• The RQF formalism provides a complete set of six time-dependent degrees of freedom corresponding to Lorentz transformations on the two-sphere boundary, ensuring geometric rigidity via zero expansion and shear.
• Quasilocal energy and momentum conservation laws are derived using the Brown-York boundary stress-energy-momentum tensor, which includes both matter and gravitational contributions.
• The rate of change of energy is proportional to the product of acceleration and momentum, with an exact gravitational analogue of the electromagnetic Poynting vector explaining this transfer.
• In the case of Bell’s spaceship in an electromagnetic field, half the change in electromagnetic energy comes from the standard Poynting flux and half from the gravitational Poynting vector, showing that flat spacetime electromagnetism is incomplete.
• A new, exact general relativistic analogue of Archimedes’ law is derived, showing that the weight of matter and gravitational fields in a region is supported by stresses (buoyant forces) on the boundary.
• The formalism reveals that tidal heating and tidal torque between two bodies arise not from forces on a bulge, but from acceleration relative to mass, providing a deeper, equivalence-principle-based explanation.