Let's consider an horizontal ring of radius $R$ and mass $M$, with a point mass $m$ hanging from a spring, suspended at a distance of $\frac{R}{\sqrt{2}}$ from the center of a ring.

This distance maximize gravitational field of the ring (I'll won't show that here) which is here $\frac{2}{\sqrt{27}}\frac{GM}{R^2}$. If the ring rotate with frequency $\nu$ its kinetic energy $\frac{1}{2}I\omega^2$ can be written $K = 2 \pi^2 M R^2 \nu^2$.

enter image description here

If we suppose that this energy shows itself with a (not only inertial, but also gravitational) mass $\frac{K}{c^2}$, the suspended weight is increased because of the increasing $\Delta g$ of the field, due to this kinetic mass: $$ \Delta g = \frac{4\pi^2}{\sqrt{27}} \frac{G}{c^2} M \nu^2 $$ Trying some substitution we see that this is (not surprisingly) a very small variation of the gravitational field: in SI units we have $\Delta g \approx 10^{-26} M \nu^2 $: seemingly any balance has an hope to measure the corresponding increase of weight $\Delta p = m \Delta g$. But consider that the increase of the weight causes a tiny fall $\Delta h = \frac{\Delta p}{k}$ where $k$ is the constant of the spring. Now, $f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}$ (I call $f$ is the proper frequency of the upper system), so $$ \Delta h = \frac{\Delta g}{4 \pi^2 f^2} $$ This is a very small fall (if the ring weight a ton, spin at 100 Hz, and the proper frequency of the upper system is 1 mHz, we have only $\Delta g \approx 10^{-19}$ m/s$^2$ and $\Delta h \approx$ 1 fm), but let's consider an interferometer like the one in the figure (it is fixed to the laboratory, but bottom mirror is anchored to the suspended mass):

enter image description here

Interferometers are extraordinary tools, but so small wavelength are surely not compatible with the vibrations of the very massive fast spinning ring (don't forget that the bottom mirror is fixed to the suspended mass, that has to be near the ring).

You can think to a thermodynamic experiment too (if a tank of 10 tons of water increase its temperature of 100 K, this will increase above gravitational field of about $10^{-18}\,\,\frac{\textrm{m}}{\textrm{s}^2}$: almost the same order of magnitude of the ring experiment described above): this time we have no vibrations, but we should to know very well thermal effects on the spring (inevitably near the tank) when the heated system change its temperature...

If you are thinking about the possibility to do something like that for measuring the hypothetical gravitational mass of magnetic field, let's play with orders of magnitude: consider a solenoid of radius 1 m, length 10 m and field 10 T. If $\rho = \frac{B^2}{2 \mu_0 c^2}= \frac{\epsilon_0 B^2}{2} \approx 5 \cdot 10^{-10} $ kg/m$^3$ we have that the hypothetical mass of the magnetic field of this solenoid is about 0,05 mg. So the increase of gravitational field on the top of the solenoid has order of magnitude $\frac{7\cdot 10^{-11}\cdot 5 \cdot 10^{-8}}{5^2}\approx 10^{-19}$ m/s$^2$. We see that approximately in this kind of experiments we find always the same (enormously small) order of magnitude.

You can think that if interferometers that work with $\lambda \approx $1 fm or smaller exist (they exist?) experiments like these could be doable. But anyway probably uncontrollable spurious effects would be much greater than the one we would measure. Like casual variations of the earth field (earth is not a rigid body) or simply a flying fly 100 m over the apparatus! Maybe this problems can be solved comparing with the motion of a second suspended mass on the side of the system, but there is another problem: we couldn't really stop a such suspended mass in a way that there are not oscillation of the order of fm (if only because of thermal reasons), so we couldn't stop the fringe shifting.

Without this condition we couldn't start the experiment (this is not completely true, we could study the variation of the harmonic oscillation in the two different conditions: when the "equilibrium" is reached, the motions before and after the $g$ increasing will look identical, but one will notice a greater fringe shifting in one direction during the variation of the gravitational field).

I wonder if, despite all the seemingly insurmountable difficulties, similar experiments (or different, but with the purpose to check the gravitational properties of different kind of energy) were ever tried.

  • $\begingroup$ Hi Fausto, I have edited paragraphs into your text, otherwise it is , to me anyway, difficult to read. I also added two extra tags and shortened the title. If my edit is approved, will you please check and amend it as you see fit. +1 $\endgroup$ – user163104 Aug 6 '17 at 23:02
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    $\begingroup$ Thank you for editing and tagging, I only re-edit the title. $\endgroup$ – Fausto Vezzaro Aug 9 '17 at 19:30
  • $\begingroup$ See Kreuzer, Phys. Rev. 169 (1968) 1007 and Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21. This is the sort of test of GR that is usually stated in terms of the PPN test theory. A good review article is Will, arxiv.org/abs/1403.7377 . $\endgroup$ – Ben Crowell Aug 9 '17 at 20:37

Gravitational radiation measured at LIGO used Kms sized interferometers and measured oscillations of a fraction of the nucleus size. Can't remember exactly, maybe 1/100th or so. A nucleus is 1-15 fm.

I have not checked your math to insure that fm is the right order of magnitude for the different cases, but if it is it might be possible. Now, in LIGO they detected oscillations in the acoustic range, and separated them from the environment. It took a few years to design it and build it. But it might not adapt easily to your conditions. Here's why:

-the detections used spatial diversity (a pair of interferometers thousands of Kms apart). The diversity was used to eliminate noise false alarms. Makes a big difference, but am not sure the equivalent SNR improvement.

-they detected oscillatory behavior at a range of expected frequencies. You need to detect a change in a pseudo-static measurement. You'll have to think about that, see if there a design way to do it.

See the LIGO site at http://www.ligo.org There's papers and descriptions of the design and calculations.

  • $\begingroup$ Thank you. I remember it was small, didn't remember the 1/10000. So it'd be that fraction of a fm. I think that accuracy is more than amazing. On the human hair, well at least we have our standards :) $\endgroup$ – Bob Bee Aug 7 '17 at 18:13
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    $\begingroup$ Well, it's either fade away or grey out. The alternatives are not healthy ones. $\endgroup$ – Bob Bee Aug 8 '17 at 0:39
  • $\begingroup$ Thank you for the answer. Don't waste your time in checking my math: calculus are easy and correct. Probably are questionable the choice of experimental data inserted in the formulas (examples: is it possible to have a ring of a ton spinning a 100 Hz? And what about a mass-spring system with proper frequency of 1 mHz? Are such experimental hypothesis realistic?). In addiction, in the real world of experiments surely one would have to handle problems that in these schematisations doesn't come out. $\endgroup$ – Fausto Vezzaro Aug 9 '17 at 19:27
  • $\begingroup$ As for LIGO, that measurements are shocking, but the fact that such measurements are doable in an experiment, doesn't imply automatically that they are doable in a different one (as you pointed out). Time ago I tried to study the effects of Mercury perihelion precession exploiting the hypothesis that in the (approximately) inertial system of the sun the "kinetic mass" of the planet is inertial and gravitational: you will find 14"/century, very far from the real value. I wonder if in terrestrial experiments for some reason things can go in a different way. $\endgroup$ – Fausto Vezzaro Aug 9 '17 at 19:35
  • $\begingroup$ Possibly the best test for the gravitational effects of various kinds of energy will be cosmological & astronomical observations. For example, if two pulsars are orbiting each other in highly elliptical orbits, their gravitational potential energy and their kinetic energy will be changing continuously; and the resulting effects on their orbits (timing, precession, and decay due to gravitational radiation) will show up over time. Perhaps observation of a binary system of magnetars could confirm that magnetic field energy has an effective gravitational mass. $\endgroup$ – S. McGrew Feb 26 '19 at 3:32

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