In less technical language, what exactly is a "gravitational form factor"?

Question

The term "gravitational form factor" is a term I don't recall ever seeing before the year 2018 (about three decades after I started reading physics papers).

I have read several recent papers about them, most recently, this one, claiming to have experimentally determined a gravitational form factor for a pion, and also this one predicting gravitational form factors for spin-0 hadrons, and this one, touching upon energy momentum form factors. This paper from 2016 (which I looked up today) is a bit more clear, but still leaves me less than 100% sure that I have the concept down.

I had thought I knew what a form factor meant outside of the gravitational context in particle physics (see also this previous answer at Physics.SE), but after reading those papers, I'm not sure I really understood that either. After reading these papers, I'm still fuzzy on just what a gravitational form factor is, or means, or in what kind of application you would use it.

Could someone explain the concept in less technical language than is used in these papers what a "gravitational form factor" is, what it means, and how it would be used?

Answer 1

A qualitative answer by an experimentalist:

This article gives the description in words of why form factors are used:

How BIG ARE the elementary particles? How is their charge distributed? These questions are tackled with form factors, which are measures of the charge and magnetic‐moment distributions in the particles. Scattering of electrons on nucleons, and recent measurements made with electron–positron colliding beams, give form factors for the proton, neutron and pion.

Elementary particle experiments are scattering experiments that try to define the interactions of tiny composite entities obeying quantum mechanical and special relativity algebras. These are studied in the energy momentum space, and the measurements can, using Fourier transforms, give information about space, and thus give a measure of the size of the composite particles.

In this answer about form factors , the format and use of form factors to decide on the size of composite objects is described.

In more mathematical terms, the cross section for this scattering is given by the Rosenbluth formula $$\sigma =\sigma _{0}\left[ W_{2}+2W_{1}\tan ^{2}(\frac{ \theta }{2})\right]$$ where $\sigma_0$ is the classical cross section (Rutherford for spinless particles, Mott for spin-1/2 particles) and $W_1$ and $W_2$ are the form factors. A particle is called point-like if the form factors don't depend on the momentum transfer $Q^2$. Otherwise, the size of the particle is related to the Fourier transform of the form factors.

It seems that this gravitational form factors business is extending the notion by trying to see the effect of gravitational interactions on the form factors, and because of general relativity,this means space distortions are adapted to the form factor tools.

You ask:

what a "gravitational form factor" is, what it means,

In analogy it should give the form of the mass distribution of a complex, quantum mechanical system, as probed by the gravitational interaction.

and how it would be used?

in validating a general relativity model , and giving a new tool for studying hadron components.

The strong force , due to its high couplings cannot be used to probe quarks within the hadron, the way one uses charge distributions to probe the behavior of hadrons within a nucleus. Due to the high complexity of a hadron, see an illustration for a proton here, the electromagnetic form factor is highly complicated. The gravitational interaction gives a tool to measure the form of a hadron in analogy.

Answer 2

Gravitational form factors characterize the energy, spin and stress distributions within a subatomic particle, such as the proton, the pion, the deuteron (atomic nucleus of deuterium), and even the electron (which acquires a "structure" due to the dressing of the photons and other sea electron/positrons).

Form factors are the generalization of the multipole distributions in nonrelativistic quantum mechanics (NRQM). You can think of them as the Fourier transform of the one-body densities (OBD), even though there are caveats. We of course have different types of densities, e.g. charge density, magnetic charge density, axial charge density, energy density, momentum flow density, spin density, pressure, and shear. Each density is defined from a specific operator. For example, the charge density is defined from $\rho = J^0(x)$ operator. Similarly the magnetic density (actually magnetization density) is defined from the current density $\bf J$. Combined, they form the covariant 4-current: $J^\mu = \bar\psi \gamma^\mu \psi$. And the corresponding hadron matrix elements $\langle p', s'|J^\mu|p, s\rangle$ can be covariantly decomposed into Lorentz structures multiplied by electromagnetic form factors, similar to that of the multipole expansion in NRQM.

In this context, the energy, momentum flow and stress densities are defined from the stress-energy tensor $T^{\mu\nu}$. The corresponding form factors are conventionally called the gravitational form factors. Part of the rational is that the stress-energy tensor couples to the gravitational interaction in general relativity. In principle, the gravitational form factors can be probed by the gravitational force, similar to the probing of the electromagnetic form factors by the electromagnetic force mediated by the photon.

One can define further densities, such as the axial charge density and pseudoscalar density from the axial-vector current $J^\mu_A = \bar\psi \gamma^\mu \gamma_5 \psi$. The corresponding form factors are called the axial form factors. Similarly, these form factors can be probed by the coupling to the weak interaction mediated by W/Z.

In reality, the measuring of the gravitational form factors are far more challenging than other form factors. The reason is that gravity on the level of the proton is far too feeble. The present measurements are based on an idea that, roughly speaking, the coupling of two spin-1 photons looks like a spin-2 graviton. Therefore, one can measure the gravitational form factors in a two-photon process, or its crossing, Compton scattering. That's what all those articles discuss. Of course, since we are speaking of approximate equivalence, the actually measurements as well as the subsequent extraction of these observables are quite involved.