# How do measurements of $G$ account for variations in the local gravitational field?

One of the notable discrepancies of modern physics is the current disagreement of measurements of the gravitational constant $$G$$, well beyond reported uncertainties (and agreeing only to about 1 part in $$10^5$$). Part of the difficulty is that it's impossible to "screen out" the gravitational field of the environment as one screens out environmental electromagnetic fields with a Faraday cage.

For consider a test mass $$m$$ in an environment which includes

1. a field mass of 1 kg at a distance of 1 cm, producing a gravitational field of $$G \cdot 10^4 kg/m^2$$

2. the Earth, of course, producing a gravitational field of $$\sim 10 m / s^2 \approx G \cdot 10^{11} kg / m^2$$ (although note that typically one is measuring gravitational fields in a horizontal direction, where the component of the Earth's gravitational field is many orders of magnitude smaller)

3. the Sun and moon, producing a gravitational field variation over the course of a day of $$\sim 10 \mu m /s^2 \approx G \cdot 10^5 kg/m^2$$

4. a $$1000 kg$$ car parked $$100 m$$ away, producing a gravitational field of $$G \cdot 10^{-1} kg/m^2$$

5. a $$100 kg$$ human sitting $$10 m$$ away, producing a gravitational field of $$G \cdot 10^0 kg/m^2$$

6. a $$10^6 kg$$ cloud floating $$1 km$$ away, producing a gravitational field of $$G \cdot 10^0 kg/m^2$$

In an article such as this one, the figure on p. 7 indicates that one measures responses to variations in the gravitational field of the field mass occurring at frequencies of $$\sim 3\,$$mHz, i.e. periods of a few minutes. I'm confident that on these timescales we understand the Earth, Sun and Moon well enough to model their gravitational fields to the necessary accuracy and account for (2) and (3).

But I worry about sources like (4),(5), and (6). On the timescale of a few minutes and distance scales of $$10-1000 m$$, most universities (where such experiments typically take place) exhibit significant variation in the distribution of humans, cars, and clouds! And the back-of-the-envelope considerations above suggest that these sources of error have about the right relative order of magnitude ($$\geq$$ 1 part in $$10^5$$) to be significant.

That's why I'm confused that I haven't been able to find any discussion of such sources of error in measurements of $$G$$ (though I'm far from an expert), and leads to my

Questions:

1. Do the details of the experimental setup in measurements of $$G$$ render the above back-of-the-envelope considerations moot?

2. If so, this must have been an important design consideration for these experiments -- similar to the fact (which is touched on in any discussion of $$G$$ measurements) that one intentionally measures horizontal gravitational fields to avoid dealing with most of the Earth's gravitational field. So where can I find some discussion of this point?

3. If not, how are these sources of error accounted for? In the unlikely event that they're important but not accounted for, can they explain the discrepancies in the literature on measurements of $$G$$?

A fair criticism of this question is that I didn't actually, y'know, look at more than a paper or two in trying to find out the answer for myself. So I've taken a look at two papers reporting measurements of $$G$$.

I was really hoping that this would resolve my confusion. Unfortunately, their error budgets do not, as far as I can tell (but I'm not actually a physicist), include any budget for variation in the local gravitational field -- only for variations coming from the field mass itself.

• Just nitpicking, 3mHz corresponds to the period of 1/0.003 ≈ 5.5 minutes, not 1 hour. But that's still a good question! – kkm Jun 27 at 15:57
• @kkm Thanks! I edited accordingly. I suppose this makes the problem worse for cars and people, but perhaps ameliorates it a bit for clouds. It's puzzling, though -- you'd think that if these effects were not taken into account, a single experiment would have trouble getting measurements of $G$ that were even self-consistent -- each group would be seeing fluctuations in $G$ over the course of a day and day-to-day. – tcamps Jun 27 at 17:56
• Researcher in this field have certainly calculated and understood the effects of local masses (people, cars, trains, clouds, mountains, aquifers, etc.) on their measurements. It’s almost a bit presumptuous to suppose they haven’t. I don’t know if these fluctuations limit their measurements or not. I would need to look at the error budgets in recent papers to see if these effects limit current measurements. This is why I post a comment and not an answer. I know researchers think about these things because I’ve heard about them in ligo and gravimeter talks. – jgerber Jun 27 at 18:40
• I think you should rephrase this question. In its current form, it implies that the physicists performing G measurements don't consider those sorts of details. – PM 2Ring Jun 27 at 19:11
• @EmilioPisanty Thanks! I've tried to tone it down a bit more now. – tcamps Jun 27 at 20:23