Global Conservation of Charge vs. Local Conservation of Energy and Momentum in General Relativity

In the framework of General Relativity (GR), the concept of global conservation laws is more nuanced than in classical physics, particularly due to the curvature of spacetime and the lack of a universal time. This article delves into why charge is a globally conserved quantity while energy and momentum may not be, exploring the underlying physics through the principles of global symmetry and Noether’s theorem.

Charge Conservation and Global Symmetry

Charge conservation arises from the global U(1) symmetry associated with electromagnetism. Noether’s theorem, a fundamental principle in theoretical physics, asserts that for every continuous symmetry of the action of a physical system, there is a corresponding conserved quantity. In the case of electromagnetism, this symmetry is tied to the invariance of the system under phase transformations of the wave function.

Invariant Nature of Charge

Electric charge is a scalar quantity that remains invariant under general coordinate transformations in GR. This invariance under transformations is crucial because it ensures that the total charge remains constant regardless of how spacetime is curved or how observers move.

Global vs. Local Conservation

Charge conservation can be formulated globally, meaning that if you sum the charge over a closed surface, it will remain constant in time. This is a result of the continuity equation derived from Maxwell's equations, which hold in curved spacetime.

Energy and Momentum Conservation

Local Conservation

While energy and momentum are conserved locally—i.e., within small regions of spacetime—they do not have a global conservation law in GR due to the dynamic nature of spacetime. The curvature of spacetime affects how energy and momentum are defined and conserved.

Lack of Global Time

In GR, there is no single time coordinate that applies globally due to the presence of gravitational fields and the curvature of spacetime. This makes defining a global energy conservation law problematic. Energy can be defined in specific cases, such as asymptotically flat spacetimes, but not universally across all spacetimes.

Energy-Momentum Tensor

In GR, energy and momentum are described by the stress-energy tensor, which couples to the curvature of spacetime. The conservation law of the stress-energy tensor, (mu T^{mu u} 0), indicates local conservation but integrating this over a curved region does not yield a straightforward global conservation law.

Summary

The special status of charge in terms of global conservation in GR is indeed tied to its invariance and the symmetries of electromagnetism, contrasting with the more complicated treatment of energy and momentum.