Now turn our attention from electromagnetism to gravity.
The key property of gravity is that it affects all bodies equivalently, with a force that is proportional to the body's mass: this means that all objects in the same gravitational field will experience the same overall acceleration, regardless of their composition. This is called the principle of equivalence for gravitation.
This means that it is impossible to measure directly an overall gravitational field, since a uniform field will accelerate all parts of the apparatus equivalently. The best you can do is to measure the change in the gravitational field over the length of your measuring apparatus. For instance, when you measure the "gravitational field of the Earth", you are using the Earth as a stable reference, effectively making the Earth a part of your apparatus: you are actually measuring the difference between the local field at a particular location, and the average field over the Earth. An external field, such as the Sun or the Galaxy, accelerates the whole Earth, including your apparatus, and this overall acceleration is not directly detectable. (You can infer it by expanding your "measurement apparatus" to include more distant reference points, e.g. measuring your motion relative to other planets or stars.)
While the overall acceleration from a distant gravitating body is undetectable, the change in its gravity from one location to another can be measured. This is sometimes called the gravity gradient 𝑔′, or the tidal field, because it is responsible for creating ocean tides on the Earth, as shown in the following diagram:
The point on the Earth's surface nearest to the Moon is being pulled towards the Moon slightly more than is the Earth as a whole, while the opposite point on the surface is being pulled slightly less. Thus the Earth as a whole is being stretched along the Earth-Moon axis. Meanwhile, the gravitational field at the North pole is deflected slightly downward relative to the average over the Earth, and mirrorwise for the South pole. Thus points on the plane transverse to the Earth-Moon axis are squeezed inward towards this axis. The overall effect is shown by the curved arrows. Since the Earth's oceans respond to this force more readily than its solid crust, the result is that the oceans bulge out along this axis, producing high tides. Similar but smaller tides are created by the Sun's gravitational field.
Now the overall gravitational field of the Moon dies off as 1/𝑟2, just as with electromagnetism. But what we measure is the change in this field between two points separated by some displacement. For displacements 𝑑 much smaller than the distance 𝑟, the relative change in gravity is proportional to 𝑑/𝑟, leading to a tidal field:
where 𝐺 is Newton's gravitational constant and 𝑀 is the mass of the source of gravity. As with the electric dipole field, subtracting off the overall gravitational field cancels the leading-order 1/𝑟2 term, leaving a field that scales as 𝑑/𝑟3, and a field gradient that scales as 1/𝑟3. (For more discussion of this similarity, click here.)
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