# What is the experimental evidence for creation of a gravitational field by electrons or other leptons?

What experimental evidence do we have that leptons (electrons, muons, tau leptons, neutrinos) create—rather than merely respond to—a gravitational field?

General relativity (GR) predicts that all forms of mass-energy gravitate, that is, generate a curvature of spacetime proportional to their associated energy-momentum tensor. GR has yet to be empirically contradicted, so we have strong theoretical expectations here. My question is about the status of experimental evidence for this prediction.

Massless photons empirically feel gravitation, seen in gravitational lensing. I am aware of the concepts of active and passive gravitational mass. Whether experimental evidence exists for creation of a gravitational field by photons has been addressed (Do photons bend spacetime or not?); in summary, as I understand it, we do not yet know.

What about leptons? The charged leptons (electrons etc.) would seem to be the most promising candidates given that they have far greater mass than neutrinos and photons.

A related question (Active gravitational mass of the electron) was asked nearly a decade ago, and does not address other leptons or recent advances.

First of all we should keep in mind that in order to quantify possible deviations from established theory of general relativity we must work in a wider frameworks that potentially allow e.g. deviations from momentum conservation and make distinction between passive and active gravitational and inertial masses.

For a general information about such frameworks and experimental tests of general relativity see this review:

Specifically about gravitational field of leptons, one possible test would be the measurement of (non)equivalence of passive and active gravitational masses for bodies with different composition. Since the number of electrons per unit mass is different for different elements, if electrons do not produce gravity (do not contribute to active gravitational mass) but still contribute to passive mass then the ratio of passive to active gravitational masses would be different for different elements. This could be verified experimentally. So far, direct laboratory measurements do not have enough precision, but with Lunar Laser Ranging the precision could be greatly improved:

• Bartlett, D. F., & Van Buren, D. (1986). Equivalence of active and passive gravitational mass using the moon. Physical review letters, 57(1), 21, doi:10.1103/PhysRevLett.57.21.

Lunar crust and mantle have different elemental abundancies (mainly in iron and aluminium) and also have about $$10\,\text{km}$$ offset between their respective centers of mass. If the active and passive gravitational mass for iron and aluminium were different, there would be a momentum non-conserving self-force. Laser ranging places the following limit on the difference of ratios for those two mass ratios: $$\left|\frac{(m_A/m_P)_\text{Al}−(m_A/m_P)_\text{Fe}}{(m_A/m_P)_\text{Fe}}\right|< 4×10^{−12}.$$

Since electrons contribute $$2.64×10^{-4}$$ to the (inertial) mass of aluminium and $$2.55×10^{-4}$$ to the mass of iron, this experimental bound is sufficient to conclude that electron contribution to the active mass has the same coefficient as nuclei within achieved precision.

• Very helpful, thank you. Pursuing this last result: has any distinction been drawn between the active gravitational mass of bound versus free electrons? It seems that in principle these may not be equivalent, understanding of course the very distinction between active and passive grav. masses remains hypothetical and subject to the bounds you discuss. Jan 15 at 20:45
• active gravitational mass of bound versus free electrons? Since the binding of electrons in atom is electromagnetic in origin, the possible difference in AM could be if EM field energy also contributes differently to active mass. But EM field already contributes a lot to inertial mass of all baryons and charged leptons (and different amounts for different species) so this possible effect would (probably) have been noticed even for lower-precision Cavendish experiments. So even existing tests would conclude that active mass of leptons (either free or bound) and EM field coincide with passive. Jan 16 at 8:44

The theory which identifies inertial and gravitational masses, General Relativity, has a great deal of experimental support. Mass is just a curvature of spacetime, with both gravity and inertia arising from this fact. If you were to describe a particle which responds to curvature but produces none itself, then it would not have inertial rest mass either (e.g. a photon). Possessing inertia without creating gravity contradicts the fundamental equations of General Relativity and is therefore regarded as impossible. The many experimental confirmations of General Relativity support this position.

Electrons

Electrons have inertial mass. For example a helium nucleus has a different inertial mass from a helium ion. The (inertial) mass of the electron is significant in various engineering fields, such as semiconductors.

More directly, by weighing ions, one can gain a measure of the weight of the electrons which have been stripped off them. This is a ticklish experiment, due to the far stronger electrostatic force arising from ionisation, but weights can sometimes be derived from other observed parameters, such as the amount of laser energy needed to levitate the ion (though I cannot recall any specific experiment made with this in mind).

Neutrinos

Neutrinos are leptons which oscillate between three recognised variants; the electron, muon and tau neutrinos.

This oscillation was first proposed on a theoretical basis to rectify some predictions which did not agree with observation. Crucially, the predicted oscillations depend on whether neutrinos have rest mass or not. This was, at that time, an open question.

Subsequent experiments have confirmed that neutrinos oscillate according to a theoretical model which gives them rest mass. Models which offer no rest mass however fail to offer any oscillations which match observation.

The rest masses of the individual variants or flavors are not known with any great precision, but they are all greater than zero.

General relativity in its classical formulation does not use the language or mathematics of fields, having substituted the curvature induced by the stress energy tensor. So the title :

What is the experimental evidence for creation of a gravitational field by electrons or other leptons?

is not involved in general relativity concepts as the content of the question does.

In addition, as electrons, leptons in general are quantum mechanical entities, and gravity has not been definitively quantized,I think the answer should be withing classical Newtonian gravitation, where classical fields can be defined and are dependent on the mass of the objects. After all the success of general relativity is partly because the classical theory of gravity emerges naturally from the higher mass and energy theory.

By the fact that the mass of all these particles has been measured experimentally,it is evident that leptons also will create gravitational field around them. For electrons there exists the experimental verification of the effects of earth tides on the Cern LEP electron positron beams .

There is no mathematical reason that the classical theory cannot be extended into the general relativity frame the way it is done for all other masses.

• This question asks about experimental foundation for GR, namely the equivalence of various forms of energy in its gravitational action. So discussing it from the framework of GR that presupposes this equivalence from the start is wrong. Jan 15 at 7:36
• @A.V.S. in the content, not the title. As there are no gravitational fields in GR and energy is a local concept, the question is not well formed. I am arguing that there is enough experimental evidence that leptons have mass, and also that Newtonian gravity emerges from the theory of GR, so as far as leptons go this should be enough. Jan 15 at 9:13
• “ General relativity in its classical formulation does not use the language or mathematics of fields” The metric and curvature are fields and GR is a field theory. Jan 15 at 9:29
• @mmeent they are fields in the mathematical sense but not gravitational fields.the term as used in physics, imo Jan 15 at 9:45
• They are also fields in the sense normally used by physicists. The issue you seemingly want to address is that GR is not a linear field theory, making it impossible to isolate the contribution of an individual source. Jan 15 at 9:59

There is no distinction between active and passive mass in mainstream physics. In Newton physics masses interact by $$E= -G m_1 m_2 /r$$. Which of the two masses should be the active and which the passive? Therefore the answer is that measuring the electronic contribution to weight is enough to establish that electrons have a gravitational potential.