### MACH'S PRINCIPLE

Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology, Routledge Inc., New York. Reprinted with the author's permission. To order this book click here: http://www.routledge-ny.com/books.cfm?isbn=0415923549

The principle that the masses of objects are somehow determined by the gravitational effect of all the other matter in the Universe. More precisely, the inertial mass m of an object (as defined by Newton's second Law of motion, F = ma, as the `reluctance' of the object to be accelerated) is asserted to be not a property intrinsic to the object, but a consequence of the net effect of all other objects. A corollary of this principle is that the concept of mass is entirely meaningless in an empty universe.

Mach's principle is a very deep physical idea of great historical and philosophical importance, but the essence of it goes back much further than Ernst Mach (1838-1916) himself. In 1686, Isaac Newton discussed a similar idea in the Principia. Newton was concerned with what happens when bodies undergo rotation. He knew that a rotating body underwent acceleration towards the centre of rotation, and he was interested in what happened, for example, when a bucket full of water was spun around a vertical axis. What we see if we do this experiment (as Newton himself did) is that the surface of the water, which is flat when the bucket is not rotating, becomes curved when it begins to rotate. This curvature is caused by the centrifugal forces experienced in the rotating frame of the bucket pushing the water outwards from the centre, making the surface of the water concave. This shape remains if we suddenly stop rotating the bucket, which shows that relative motion between the bucket and the water has nothing to do with this effect. In some sense, the acceleration is absolute.

Newton had no problem with this because his laws of motion were formulated in terms of absolute time and space, but it is at odds with the principle of relativity. What should count is the relative motion of the water. But what is it relative to? One of the first suggestions was made by Bishop Berkeley (1685-1753). He had been impressed by Galileo's principle of relativity (see special relativity). He claimed, as later did Mach, that the acceleration was relative, but relative to the fixed stars (or, as we would now put it, to the large-scale distribution of matter in the Universe). Because masses are measurable only in terms of forces and accelerations, Berkeley was essentially arguing that the inertia of the bucket of water is determined by cosmological considerations. The surface of the bucket would look the same if the bucket were at rest but the entire Universe were rotating around it.

Albert Einstein was profoundly influenced by Mach's version of this argument, and he sought to incorporate it explicitly in his theory of gravitation, general relativity; but he was not successful. Many gravitation theorists have sought to remedy this failing in alternative theories of gravity. For example, in the Brans-Dicke theory of gravity there is an additional scalar field over and above the usual matter terms in Einstein's theory. The role of this field is to ensure that the strength of gravity described by the Newtonian gravitational constant G is coupled to the expansion of the Universe; G therefore changes with time in this theory. This is an essentially Machian idea because the effect of changing G can be seen, in some senses, as changing the inertial masses of particles as the Universe expands.