Physical fields
What is a field? A field is a technical term for a way of describing the way something varies with position in space, and is represented by a set of numbers for each point in space, that specifies the magnitude, direction, and any other attributes of the physical behaviour at that point.
Gravitational fields
Because the force of gravitation has direction as well as magnitude, and because space has three dimensions, the gravitational field in classical physics needs three numbers at every point in space. Luckily, this infinite set of numbers can be be summarized by a simple rule called the inverse-square law. This law says that in the vicinity of a massive object, far from the influence of any other objects, the force on a second object varies in inverse proportion to the square of the distance between the two objects, and is directed toward the attracting object. Thus, halving the distance between the objects makes the force 2 x 2 = 4 times as great, and tripling the distance makes the force 1/3 x 1/3 = 1/9 times as great.
Potential energy
If you pull two gravitating objects apart, you use energy to move them against the force, as you feel when you go up a slope, separating yourself slightly more from the earth. The potential energy of a body near a large gravitating mass can be illustrated to some extent by a graph of the form shown below, which shows the potential energy at points in a plane through the centre of the mass. The graph for three dimensions obviously cannot be drawn. This does not matter, in that calculations can be done without graphs and without the need to visualise. Nevertheless, some scientists have used visual images to help in working out a new theory, even if they discard the images later when the theory has been fully worked out.
Limitations of diagrams
The region around an attracting object is sometimes called a potential well, because of the shape of the diagram. Such a diagram is often used to suggest the way that curvature of space causes gravitational attraction, but this is very misleading, as it is a two-dimensional surface embedded in 3-space, whereas curved space and curved space-time are not, apparently, embedded in anything. True curvature is defined by the observed geometry on or in the surface or space, not on its appearance when embedded. Thus a cylinder is not a curved surface in this sense, because the geometry on it is Euclidean. What really counts is the geometry observed by people dwelling in the space, who of course cannot stand outside it and view it. Returning to the classical theory, a perfectly smooth and frictionless model of this shape would apparently illustrate the potential well of Newtonian gravitation. A small ball rolling in it without friction could execute circular orbits, given a suitable initial velocity and position, but for elliptical orbits it would not work, because the ball would have a vertical component of momentum, which has no counterpart in reality. In some cases the ball could leave the steep surface and fall freely in space. Thus the model’s suitability for Newtonian gravitation is as illusory as it is in the Einsteinian case. The diagram is only a diagram, perfectly accurate and realistic as such, but not a physical model. Even a simple diagram showing lines of force diverging from a point is misleading if taken literally, because it conveys the impression that the force falls away inversely proportionally to the distance, rather than the square of the distance. Caution needs to be exercised when interpreting any illustration of a physical system; it could be said that only the fundamental equations are completely trustworthy. Nevertheless, some people like to see a diagram or an artist’s impression, because it gives them something that they can understand more easily than an equation. Artists’ impressions are often labelled as such, but the limitations of diagrams are not always pointed out. A real model would not be frictionless, and the ball would rapidly fall into the hole. Some charitable organisations have a collecting box with this shape on top, and you can watch your coin spiralling in as it loses energy through friction.
Einstein’s theory of gravitation
The Einstein theory of gravitation does not require forces: it attributes the observed behaviour to non-Euclidean space-time, commonly referred to as curved space-time. It is not possible to create meaningful diagrams of this on a two-dimensional page, and most efforts are more like artists impressions. The equations are far more complicated than those of Newton. Where Newtonian gravitation requires three numbers per point, Einstein gravitation requires ten, and it predicts new effects, such as the Lense-Thirring effect, or frame dragging, and gravitomagnetism. Some scientists express a wish for theories to be simple, elegant or beautiful, but we must distinguish between the basic ideas, the equations that represent those ideas, and the difficulty of calculating the behaviour predicted by those equations.