The 1/r³ scaling of the electric dipole field of a pair of opposing electric charges, and the similar scaling of the gravitational tidal field of an individual mass, is not just a coincidence. Rather, it is due to the fact that with both fields there is a symmetry between the source of the field and the charge or masses used to measure it.
In electromagnetism, one rarely has large unbalanced charges. Rather, the field at large distances is usually created by two (or more) opposing charges slightly separated from one another. But the field at that distant location can be directly felt by a small "test charge". By contrast, with gravity you typically do have large isolated masses producing fields at large distances. However, you cannot directly measure this field with an individual "test mass", since that mass will accelerate along with the rest of the measuring apparatus. Instead, you must measure the relative force between two (or more) test masses slightly displaced from one another. In both the electromagnetic and gravity cases, you are measuring the difference between two 1/r² fields with slight displacements.
Quantitatively, suppose you have a pair of charges ±Q separated by a displacement s (with the positive charge at the "head" of the displacement vector), and you measure the field using a charge q at a position r, where r is much larger than s. The force you measure will be:
Now, suppose you have a mass M, and you measure the tidal field with a pair of small masses m with a relative displacement d at a position r, where r is much larger than s. Then the relative force you measure (on the mass at the "head" of the displacement vector) will be:
Note that the magnitude of the force goes as GMmd/r³, multiplied by a dimensionless geometric factor. Dividing by d to get the tidal force (gradient), and by m to get the tidal field, we have our earlier result that g′ goes as GM/r³.
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additional mathematical detail.
Start: Gravitational waves demystified
Analogy: Electromagnetic fields
Electromagnetic field of an accelerated charge
Derivation of the radiative electromagnetic field
Gravitational tidal field
Equivalence between dipole and tidal field
Formulae and details
Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum