The Origins of Mass: Between Inertia and Gravitation

What is mass?

Mass is one of the most fundamental properties of matter. It determines a body's resistance to acceleration (inertial mass) and the gravitational force it exerts or undergoes (gravitational mass). Long considered intrinsic, mass is now understood as an emergent phenomenon within the framework of the Standard Model of particle physics.

Difference between inertial mass and gravitational mass

Inertial mass quantifies an object's resistance to any change in its state of motion. It appears in Newton's second law: \( \vec{F} = m_i \cdot \vec{a} \). The greater the inertial mass, the more force is needed to produce the same acceleration.

Gravitational mass determines the intensity of the gravitational force an object undergoes or exerts. In Newton's law, it appears as: \( F = G \cdot \dfrac{m_g M}{r^2} \), with \( m_g \) being the gravitational mass.

Fundamental fact: in all experiments, these two masses are numerically equivalent. This experimental observation, that a heavy or light object falls with the same acceleration in a vacuum, forms the basis of Einstein's equivalence principle. It is central to general relativity.

However, this equivalence is a postulate and not a theoretical necessity. Any difference, however slight, between \( m_i \) and \( m_g \) could reveal new physics. Missions like MICROSCOPE have tested this principle to 1 part in \( 10^{15} \), with no detected deviation to date.

• References:
• • Galileo G., Dialogue Concerning the Two Chief World Systems, 1632.
• • Einstein A., The Foundation of the General Theory of Relativity, 1916.
• • Touboul P. et al., MICROSCOPE Mission: First Results of a Space Test of the Equivalence Principle, Phys. Rev. Lett. 119, 231101 (2017).
• • Will C.M., The Confrontation between General Relativity and Experiment, Living Rev. Relativity 17, 4 (2014).

The vacuum is not empty: The Higgs field

In the Standard Model, the vacuum is not empty: it is filled with a scalar field called the Higgs field. During the spontaneous breaking of electroweak symmetry, this field acquires a non-zero value throughout space. Particles interact with this field, and this interaction gives them a mass proportional to their coupling with the field. The Higgs boson, discovered in 2012, is the quantum excitation of this field.

Why protons and neutrons are massive

Quarks (the constituent elements of protons and neutrons) have a very small mass from the Higgs field, but most of the nucleon's mass (more than 98%) comes from the binding energy of quarks via the strong interaction (quantum chromodynamics). This phenomenon is a striking example of the mass-energy equivalence: \(E=mc^2\).

• References:
• • Higgs P., Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13, 508 (1964).
• • Aad G. et al., Observation of a New Particle in the Search for the Standard Model Higgs Boson, Phys. Lett. B 716, 1–29 (2012).
• • Peskin M.E. & Schroeder D.V., An Introduction to Quantum Field Theory, Addison-Wesley (1995).
• • Particle Data Group, pdg.lbl.gov (new window).

Mass in special relativity and general relativity

Special relativity (\(E=mc^2\)) establishes that mass and energy are equivalent.

General relativity (\( R_{\mu\nu} - \frac{1}{2} R\, g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \)) describes mass (and energy) as the source of spacetime curvature. Any form of energy, including that of a field or the vacuum, curves space and produces a gravitational effect.

The mystery persists

If the Higgs field today explains the origin of the mass of elementary particles via their interaction with the quantum vacuum, the profound nature of this field, its stability over time, and especially the precise value of its vacuum energy remain theoretical enigmas. Indeed, the vacuum energy predicted by relativistic quantum mechanics, and thus its equivalent gravitational mass, should generate a colossal curvature of spacetime, incompatible with the moderate expansion observed in the universe. This inconsistency constitutes the famous cosmological constant problem, related to the observed value of \( \Lambda \) in Einstein's equations:

This discrepancy of more than 120 orders of magnitude between the expected theoretical value and the measured value of the cosmological constant is undoubtedly the largest gap ever encountered between theory and observation in fundamental physics. It raises profound questions about the relationship between inertial mass, gravitational mass, and the very structure of the quantum vacuum.

Cosmological constant: a discrepancy of 120 orders of magnitude

The cosmological constant \( \Lambda \), initially introduced by Einstein in his equations of general relativity, corresponds to a vacuum energy density that influences the expansion of the Universe. It is related to an effective energy density by:

Cosmological observations, particularly those from the cosmic microwave background, indicate that this density is extremely low. However, quantum field theory, which considers the contributions of all vacuum fluctuations up to a high-energy cutoff (often the Planck scale), predicts an immense vacuum density.

The discrepancy between the theoretical prediction \( \rho_{\text{vide}}^{\mathrm{th}} \) and the measured value \( \rho_{\Lambda}^{\mathrm{obs}} \) is of the order of:

This discrepancy of more than 120 orders of magnitude is unprecedented in the history of theoretical physics. It highlights a fundamental disagreement between general relativity (gravitation) and quantum mechanics (fields). The cosmological constant problem is one of the greatest mysteries of fundamental physics.

In the natural units used in particle physics, this density is expressed in GeV\(^4\):

• \( \rho_{\text{vide}}^{\mathrm{th}} \sim 10^{72} \ \text{GeV}^4 \)
• \( \rho_{\Lambda}^{\mathrm{obs}} \sim 10^{-47} \ \text{GeV}^4 \)

This profound disagreement indicates that something essential is missing in our understanding of the quantum vacuum or in the mechanism of gravitation on a large scale. This paradox is at the heart of research on a unified theory of quantum gravity.

• References:
• • Weinberg S., The Cosmological Constant Problem, Rev. Mod. Phys. 61, 1 (1989).
• • Carroll S.M., The Cosmological Constant, Living Rev. Relativity 4, 1 (2001).
• • Planck Collaboration, Cosmological parameters, A&A 641, A6 (2020).