Indeterminacy of Charge Density in Electromagnetic Fields and Energy Density in Gravitational Fields

Einstein had an important hint that allowed him to theoretically construct general relativity using only 'pen and paper (fountain pen).' That hint was the 'equivalence principle.' It is the idea that for a freely falling observer, gravity does not exist, and for an accelerating observer, gravity is created. This equivalence principle was what truly pierced the essence of gravity.

Due to the equivalence principle, the energy distribution of a gravitational field does not behave as a tensor component under general coordinate transformations. By changing coordinates, the energy density of the gravitational field can always be made zero at any arbitrary spacetime point. Furthermore, the amount of gravitational field energy contained within a closed finite volume also changes depending on the coordinates. In other words, the energy of a gravitational field strongly depends on the observer and the coordinate system the observer prepares. This suggests that not only gravity but the universe itself may be a holographic entity based on quantum information.

The holographic principle states that this space, which appears three-dimensional, is ultimately just information to an observer, and that this information exists on a two-dimensional boundary at spatial infinity or on the event horizon of a black hole.

By the way, something similar is happening with electromagnetic fields. While the source of a gravitational field is energy density, the source of an electromagnetic field is charge density. And in quantum mechanics, because of quantum superposition states, we know that charge density is also not determined before observation.

Although the value of the total charge in the entire space is determined, how much of that charge is accumulated in which spatial region is not determined before measurement by an observer. This is not just because the observer does not know; the value of the charge density itself does not exist in reality. The idea of quantum mechanics is that measurement creates the value of the charge density. This is a consequence of the violation of Bell's inequality in quantum entangled states.

The non-existence of charge distribution in quantum electrodynamics, which is a gauge theory, and the non-existence of energy density in general relativity, while different in form, are highly likely to be deeply connected. Such a perspective can be obtained from the theory of compactification of higher-dimensional spacetime, which is also used in superstring theory. The idea of this compactification originated with Kaluza-Klein (KK) theory. In this view, electromagnetic fields are themselves merely components of gravitational fields in curved higher-dimensional spacetime. KK theory is touched upon in the article below.

Therefore, it would not be strange for gravitational fields and electromagnetic fields, and their sources, energy density and charge density, to be related to each other. However, in KK theory, charge corresponds to momentum rather than energy. In relativity, energy and momentum form a relativistic vector, so they are connected to each other. Therefore, there must be a connection between the fact that charge density does not exist before measurement and the fact that the energy density of a gravitational field depends on the observer.

However, quantum electrodynamics is a quantum theory, while general relativity is a classical theory. In fact, effects of quantum gravity that survive in the classical regime are known to exist in general relativity. For example, in the AdS/CFT correspondence theory, in the theory of three-dimensional anti-de Sitter classical spacetime, a central charge 'c' is created that is inversely proportional to the absolute value of the cosmological constant. This value of 'c', which is a quantum mechanical concept, can be said to be one of the survivors of quantum gravity. Similarly, it is an interesting possibility that the observer dependence of the energy density of a gravitational field shares the same root as the observer dependence of charge density in quantum gravity theory. In this way, general relativity, which is a theory of spacetime, and electromagnetism, which is a theory of matter, are closely connected to each other.