Polynomial hierarchy

In theoretical physics, particularly in discussions of gravitation theories, a Mach principle is any of a class of principles which are more specific statements of Mach's principle.

The broad notion is that "mass there influences inertia here". Any statement which—though possibly far more specific than this—follows in this spirit may be classified as a "Mach principle". The truth of these statements depends on the particular statement. (The truth also depends on the theory of gravity, though Einstein's general relativity is the most frequently discussed theory.)

Examples

Hermann Bondi and Joseph Samuel have listed eleven distinct statements which can be called Mach principles, labelled by Mach0 through Mach10. Though their list is not necessarily exhaustive, it does give a flavor for the variety possible.

• Mach0: The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames.
• Mach1: Newton’s gravitational constant G is a dynamical field.
• Mach2: An isolated body in otherwise empty space has no inertia.
• Mach3: Local inertial frames are affected by the cosmic motion and distribution of matter.
• Mach4: The universe is spatially closed.
• Mach5: The total energy, angular and linear momentum of the universe are zero.
• Mach6: Inertial mass is affected by the global distribution of matter.
• Mach7: If you take away all matter, there is no more space.
• Mach8: $\mathrm{\Omega }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}4\pi \rho G{T}^2$ is a definite number, of order unity, where $\rho$ is the mean density of matter in the universe, and $T$ is the Hubble time.
• Mach9: The theory contains no absolute elements.
• Mach10: Overall rigid rotations and translations of a system are unobservable.

See also

• Mach's principle
• Democratic principle