Gravitational tidal field

Now let us shift 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: a uniform field will accelerate all parts of the apparatus equivalently, and hence be undetectable. 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 average field over the Earth, and the local field at a particular location. 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 of course infer it by increasing the size of your "apparatus", e.g. by measuring your motion relative to other planets or stars.)

However, the effect of gravity from a remote object, such as the Sun or Moon, can be felt by the manner in which it changes from location to location. This is sometimes called the gravity gradient g′, or the tidal field, because it is responsible for creating ocean tides on the Earth. This is 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 the field at the South pole is slightly upward. 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 change in gravity is linear in d, 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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Sections marked with provide optional additional mathematical detail.

Start: Gravitational waves demystified
Analogy: Electromagnetic fields
Electromagnetic field of an accelerated charge
  Derivation of the radiative electromagnetic field
Electromagnetic waves
Gravitational tidal field
  Equivalence between dipole and tidal field
Gravitaional waves
  Formulae and details
Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum