In Newtonian mechanics, energy is defined up to an additive constant. You can set the zero of potential energy wherever you like, and there’s no concept of an absolute “rest energy” tied to mass. If the velocity of a particle is zero, its kinetic energy is zero. What matters is the relative energy levels between two states, not the absolute energy of any given state. In contrast, in special relativity this rest energy is well defined by the residual “rest energy” of an particle in special relativity.
In quantum mechanics and quantum field theory, only relative energies are again physically significant. The equations of these theories, such as the Schrödinger equation, only depend on differences in energy levels, and an arbitrary additive constant to the energy magnitudes has no observable consequences. The energy levels are often set relative to a chosen zero-point energy, which are arbitrarily defined, and particles masses and energies are measured relative to this vacuum value, without knowing the value of the vacuum energy itself. This is due to the fact that the solution of adding a constant energy shift in the Hamiltonian differs from the original solution only by an overall phase, which is unobservable. In fact, only energy differences affect measurable quantities, like probabilities, forces, and dynamics. The estimated energy density of the vacuum from just quantum electrodynamics is infinite (if one does not assume a cutoff at a particular energy scale such as the Planck energy), which is not a problem for the theory since this infinite energy can be renormalized to zero. We covered this topic in our previous post “The Planck Energy: Vacuum Catastrophe, Higg’s hierarchy problem, and G”.
Although vacuum energy is unobservable in quantum mechanics, differences in vacuum energies are physically significant and result in measurable effects. According to quantum mechanics, empty space is filled with virtual particles with quantized energies and discrete wavelengths. If boundary conditions are imposed on a particular region of spacetime by two conducting but uncharged parallel plates separated a few nanometers, fewer wavelengths are allowed inside with respect to outside, creating a negative pressure effect inside and attractive forces between the plates, pushing them together. This effect, called the Casimir effect, results in a force that has been measured and matches the prediction of quantum mechanics. Although a different interpretation for the origin of the force in terms of Van der Vaal forces is sometimes debated, the interesting result is that a negative energy density arises between the plates in the quantum field theory interpretation when we scale the vacuum energy density to be zero at infinity, so that it is relative to the space around it. Note that the negative energy arises in particular geometric configurations, such as two parallel plates, but it is positive for others, such as for two half spherical plates, which show repulsive forces.
Despite energy being relative in special and General Relativity in terms of observer’s frames, in General Relativity, only absolute values of energy density enter Einstein’s field equations, and spacetime curvature depends on the specific energy density distribution. Spacetime curvature is sourced by the energy-momentum tensor, making the absolute value of energy density physically meaningful, even though an observer moving relative to the source might measure a different energy. In other words, gravity is sensitive to absolute energy, not to relative energy. This is due to the fact that the metric is a dynamical field.
Now the problem arises when we consider the negative energy density of the Casimir effect and gravity, because the gravitational effect of a negative energy density is paradoxical, or at least weird. We covered this topic in our previous post “Does Antimatter Fall Down? and Negative Mass”, but we will explain it again in short. There are two interpretations for the negative energy (or negative mass) gravitational interaction. If gravity is mediated by a spin-2 field, as expected in quantum field theory, like charges attract while alike charges repel (with the charge in gravity being mass). But this is not the interaction one derives from negative masses in Newtonian gravity and in General Relativity. Bondi showed that while two positive masses attract, two negative masses repel, and a positive and a negative mass chase each other up to the speed of light in a so called “runaway motion” paradox, which can lead to infinite energy sources and perpetual motion machines.
One could think that the sum of the positive mass of the plates plus the negative energy of the Casimir effect should always turn out to be positive or zero, since reducing the positive mass of the plates would make them less strong as boundary conditions and the negative energy from the Casimir effect would be smaller. But the sign of the energy from the Casimir effect depends on the geometry of the plates as explained earlier, and this reasoning is not as simple as it seems.
So which view is correct? Would the plates antigravitate if they were light enough? Is energy truly relative or absolute? We will address this problem under different assumptions.
If we assume vacuum energy to be zero at infinity, and General Relativity is correct and absolute values of energy gravitate, under the spin-2 interpretation of the negative mass interaction, the negative energy density of the Casimir effect should antigravitate, and if it wasn’t for the positive mass of the plates, they would rise up away from the Earth.
If under the same assumptions, Bondi’s interpretation of the negative mass interaction holds, then the negative energy density of the Casimir effect should accelerate towards the Earth and fall as usual, but the insignificant force exerted by the negative energy should negligibly push the Earth away from the plates. But remember this interaction is paradoxical, so it is probably not the case.
If the vacuum energy is not zero at infinity but small and positive, as the cosmological constant of dark energy suggests, the observable Casimir forces are the same, since they only depend on the relative energies, and it will still create a region of lower energy density between the parallel plates than outside them. One could think that with a sufficiently large positive energy vacuum outside, negative energy densities might not even arise inside. But this is not the case, since the Casimir force and negative energy estimation are proportional to an inverse power of the distance between the plates, meaning that it diverges negatively, and it can be arbitrarily negatively large with arbitrarily small distances. And with negative energy densities, we are back to the previous assumptions. Even if one could eliminate the negative energy through a huge positive energy of the vacuum at infinity, gravity would be sensitive to this huge vacuum energy and we would see an enormous effect in the expansion of the universe, which is not the case.
But what if General Relativity is not entirely correct, and only relative energies gravitate? What if a shift in all energies resulted in no gravitational observational difference? This would solve the problem, since we could always get rid of the negative energy density by rescaling it in our theory of gravity, and we could forget about its problematic interaction the same way we do in quantum mechanics. And it makes sense, because the measurement of the mass of a macroscopic body, such as the Sun or a planet, is always tied to the gravitational constant in the standard gravitational parameter. It is not directly measurable in an absolute sense, but is instead always inferred through its gravitational interaction with other masses. Thus, we cannot measure the mass of a large body without referring to an external interaction with another body, and this suggests that even macroscopic masses are not purely intrinsic but relational in a fundamental sense.
Making gravity “unsensitive” to absolute values of energy would also have implications in theories of quantum gravity. Quoting Feynman: “The gravitational field produced by the energy in the electromagnetic field in a vacuum-where there’s no light, just quiet, nothing-should be normous, so enormous, it would be obvious. The fact is, it’s zero! Or so small that it’s completely in disagreement with what we’d expect from the field theory. This problem is sometimes called the cosmological constant problem. It suggests that we’re missing something in our formulation of the theory of gravity. It’s even possible that the cause of the trouble-the infinities-arises from the gravity interacting with its own energy in a vacuum. And we started off wrong because we already know there’s something wrong with the idea that gravity should interact with the energy of a vacuum. So, I think the first thing we should understand is how to formulate gravity so that itdoesn’t interact with the energy in a vacuum. Or maybe we need to formulate the field theories so there isn’t any energy in a vacuum in the first place.”.
In Newtonian gravity and mechanics, a rescaling in all masses or lengths has an effect on observables such as velocities, showing that the dynamics of a system depends on the absolute values of the masses and lengths. This is due to the gravitational constant being a fixed parameter. General Relativity, in which not only mass but also energy is the source of gravity, is not scale-invariant under masses or lengths either.
This can be fixed within the Newtonian framework through Sciama’s relationship for the gravitational constant, in which Newtonian gravity becomes symmetric (or scale-invariant) under multiplying all masses and/or all lengths by the same factor. Additionally, MOND is only space-time scale invariant in the deep MOND low acceleration regime (because in the Newtonian regime it reduces to the Newtonian framework), and it is why it predicts flat galaxy rotation curves with constant tangential velocities, in agreement with the observed Tully-Fisher law. But with the addition of Sciama’s relationship for the gravitational constant to MOND, the so called Machian MOND formulation, MOND acquires also mass and space-time scale invariance at all regimes of application. Thus, Machian MOND exhibits a larger symmetry group of full-scale invariance in mass, space, and time.
Does the property in Machian MOND of full-scale invariance in mass, space, and time, point towards a more fundamental understanding of gravity? Is Machian MOND an approximation in the weak field limit of a more complete theory of gravity?
References:
Manuel Uruena, MOND as a Transformation Between Non-inertial Reference Frames Via Sciama’s Interpretation of Mach’s Principle, International Journal of Theoretical Physics, Volume 63, article number 271, 2024