## Machs Principle

In formulating his general theory of relativity, Einstein relies on an idea that he attributes to Ernst Mach, who lived in Germany two centuries after Newton. The fundamental equations of general relativity assert that the curvature of space and time can be determined from a set of functions (the energy-momentum tensor) describing the distribution of matter and energy. This is not unrelated to Mach's statement that "mass everywhere determines inertia" statement, but it seems that Einstein molded and reformulated Mach's idea, ultimately presenting a clearer and truer version of it.

Mach states such ideas in the following sections of his book the Science of Mechanics, where he also objected to Newton's idea that there is an absolute space.

If, in a material spatial system, there are masses with different velocities, which can enter into mutual relations with one another; these masses present to us forces. We can only decide how great these forces are when we know the velocities to which those masses are to be brought. Resting masses too are forces if all the masses do not rest. . . . All masses and all velocities, and consequently all forces, are relative. There is no decision about relative and absolute which we can possibly meet, to which we are forced, or from which we can obtain any intellectual or other advantage.

This is a clear and lucid statement of the relativity of motion. Mach also proposed to define the notion of mass in terms of acceleration and Newton's third law (action/reaction). The Web site "From Stargazers to Starships" provides a good summary of Mach's views on this:

When two compact objects act on each other, they accelerate in opposite directions, and the ratio of their accelerations is always the same.

In a sense, all three of Newton's laws follow from the preceding statement. Adopting the notion of mass definable along this line, given two bodies A and B, Mach points out that only the ratio of the masses of A and B can be defined using concepts of inertia and acceleration. One then defines the mass of 1 liter of water to be 1 kilogram and this, together with Mach's principle, allows to determine all other masses. We will return to these ideas in Chapter 6.