What is the smallest acceleration ever measured?

Question

I'm having an argument (discussion) with a referee. I claim that the effects of expansion can be seen in local dynamics. That the orbit of Saturn is affected infinitesimally by the expansion of space and if we had an instrument sensitive enough, we could measure it.

He claims that there are dozens if not thousands of experiments that prove that geometry of spacetime is locally flat and if your instrument isn't sensitive enough to measure any curvature, then you should consider it flat.

I have trouble understanding this. We know for a fact that all of our laboratories on Earth are in curved spacetime. How can any experiment on Earth ever measure flat spacetime?

I know we have pendulums that can measure the attraction between two masses, but even those masses in a perfectly balanced pendulum will feel the tidal forces. So what is the smallest acceleration we can measure in a lab experiment?

Answer 1

The expansion of space doesn't affect the orbit of Saturn. And I don't think the smallest acceleration that can be measured is the right approach.

In Newtonian mechanics, the orbit of Mercury would be a perfect ellipse if nothing disturbed it. But there are other planets. The main disturbance is Jupiter. but all the planets contribute. The net result is that the orientation of the ellipse is predicted to rotate by $532.3035$ arc sec/century. One of the long standing mysteries of physics around 1900 was that the measured rate was $574.10$ are sec/century. When Einstein developed General Relativity, he was able to explain the missing $43$ arc sec/century as an effect of the curvature of spacetime. It was a triumph. See Tests of general relativity

$43$ arc sec/century is small, but measurable. Saturn is harder. Equation 1029 in this site makes it easy to calculate for Saturn. Here are a couple links of orbital parameters and periods. I get $0.014$ arc sec/century.

Another page from the same site shows how to calculate orbital perturbations from planets. For Saturn, the predicted precession is $18.36$ arc sec/year. The observed value is $19.50$. So it appears that the effect of curvature is at the edge of what can be measured. But it is swamped by uncertainties in bigger effects.

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Curvature of spacetime, aka force of gravity, can be directly measured in the lab. As you said, it isn't hard to set up a torsion balance that can measure the attraction of $1$ kg masses. This was first done before $1800$ in the Cavendish experiment. The Eötvös experiment is a very precise version of the same idea.

The slowing of clocks in gravity and the gravitational redshift of light that travels upward can be measured. GPS would not work without corrections for General Relativity.

Two jets carrying precise clocks flew east and west around the world at high speed. Afterward the clocks disagreed by the amount predicted by General Relativity.

General Relativity matters for this kind of precise measurements. For every day work, Newtonian mechanics is good enough. It isn't a question of if spacetime is perfectly flat. It is if spacetime is flat enough not to notice. Typically when two $1$ kg spheres roll on a flat table, their gravitational attraction isn't noticeable.

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The smallest acceleration I can think of that is relevant to curved spacetime is in LIGO. Gravitational waves are minute disturbances to spacetime. To measure them, LIGO has 4 km long cavities that must not vary in length more than a ten thousandth the diameter of a proton. LIGO is sensitive to waves at about $10$ - $100$ Hz. Given $d = \frac{1}{2}at^2$, $a$ must be less than $10^{-24} m/s^2$

Answer 2

This is a partial answer on the question whether

the orbit of Saturn is affected infinitesimally by the expansion of space and if we had an instrument sensitive enough, we could measure it.

Whether bound states expand with the expansion of the universe:

The answer is: obviously not all as there would be no way to observe/measure the expansion of the universe. In this paper a simple model is developed within present theories to gauge the extent to which bound states can be affected by the expansion of the universe.

The abstract:

As the separation between galaxies increases owing to the expansion of the universe, galaxies themselves and smaller bound structures do not grow. An accurate description of the dynamics of cosmic structures requires the full apparatus of general relativity. In order to gain a fairly satisfactory understanding of what does not expand in an expanding universe, however, it suffices to take the harmonic oscillator as prototype of a bound system. More precisely, we show that a study of the quantum dynamics of a nonrelativistic harmonic oscillator in an expanding universe makes it clear that most bound systems do not take part in the overall cosmic expansion. The analysis is elementary and indicates that whether a bound structure partakes in the expansion partially or not at all is essentially determined by a characteristic time scale associated with it.

italics mine

The conclusion:

Our model is so simple that it lends itself to a full and exact quantum treatment. This, in spite of the model’s crudeness, permits a clear analysis of to what extent a bound system grows influenced by the overall cosmic expansion. It turns out that the degree of expansion of a bound system appears to be fundamentally determined by a characteristic time scale associated with the system.Of course such a time scale is strongly correlated to the size of the bound structure, and our elementary model indicates that even galaxy clusters essentially do not grow in response to the general Hubble flow.

So this would solve the argument of Saturn in the negative.

Answer 3

So what is the smallest acceleration we can product/measure in a lab experiment

I'll approach this question from theoretical perspective. Even at absolute zero temperature molecules does not stop moving - they have zero-point energy which is described by : $$ E_0 = \frac {\hbar \omega_0}{2} $$

,where $\omega_0$ is particle fluctuation angular frequency at $0~K$ temperature. Equating that to molecule kinetic energy : $$ \frac {mv^2}{2} = \frac {\hbar \omega_0}{2} $$

One can extract smallest possible molecule speed, which is : $$ v_{min} = \sqrt { \frac {\hbar\omega_0^~}{m} } $$

,Where $m$ is particle rest mass, of which object at hand is composed

Smallest possible acceleration can be expressed as : $$ a_{min} = \frac {v_{min}}{t_{max}} $$

Substituting minimum speed, due to zero-point energy and acknowledging, that no time span can be greater then universe age, gives :

$$ a_{min} = \tau_{u}^{-1}~\sqrt { \frac {\hbar\omega_0^~}{m} }$$

Where $\tau_{u}$ is universe age.