This is a very short post about Carl Brans’s paper ‘Mach’s Principle and the Locally Measured Gravitational Constant in General Relativity’. It was published right after the Brans-Dicke theory in 1961, where he discusses Mach’s principle, and it is available in the references. The introduction starts talking about Mach and the ratio between the inertial mass of a body and the active gravitational mass.
Carl Brans states: let us see what this ratio might be in a static universe consisting only of a mass shell of radius big R and inertial mass big M (these will be the mass and size of the universe), together with a relatively small body of inertial mass small m and a test particle. We have a picture to understand this thought experiment from Brans.
We expect that the acceleration of the test particle is independent from the mass of the test particle itself, from the equivalence principle. Then he continues stating that the acceleration certainly depends on the mass small m, and because of Machian arguments about the inertial properties of the test body being determined by the distribution of matter in the universe and not intrinsic of the body, it can conceivably also depend on big M and big R, the mass distribution of the universe. We explained Mach’s principle in our previous post “Mach’s principle explained”. He puts this into mathematical form with the simple equation shown, and the as we all known, he relates the function F containing the mass and radius of the shell (or the universe) to the gravitational constant, so that the equation here is literally the usual gravitational acceleration equation. This is the justification for Brans-Dicke, in which the gravitational constant depends on the potential of the universe, and the scalar field of Brans-Dicke, etc. We covered Brans-Dicke in our previous post “Einstein’s true reason for the Cosmological Constant”. The speed of light here is missing, because it is set to one.
Don’t forget about the equation he comes up with, but before showing you why it is important, let me explain why General Relativity is not Machian. An analogous case of Newton’s shell theorem, by which for a spherically symmetric shell, no net gravitational force is exerted by the shell on any body inside, happens in General Relativity by Birkhoff’s theorem. Note that in Birkhoff’s theorem, the mass of the body inside the shell must be null (i.e., a test particle), because otherwise, it would distort the space-time around it and the spherical distribution of mass would cease to be spherical, breaking the spherical symmetry (unless the mass is located exactly in the center of the shell). This means that the spacetime inside the shell is the usual flat Minkowski spacetime, in which there is no gravity or any modification to inertia. The dynamics inside the shell, contrary to Mach’s ideas as correctly pointed out by Carl Brans, do not depend on the mass of the shell. Moreover, in MOND, this shell theorem is no longer valid, as every point mass within an spherical hollow mass distribution experiences a very small force directed toward the center of the mass distribution (the same happens in General Relativity if the body inside the shell has non-zero mass). As a remainder, the shell is a simplification for the boundary conditions of the distribution of mass of the universe.
But now consider 1983’s MOND from Mordehai Milgrom and the equation that Brans worked out. We explained MOND in our previous post “Does MOND work? and evidence for modified inertia”. Actually, MOND also has an equation for the acceleration of a test particle at a distance small r from the mass small m, in which Milgrom introduces to the usual Newtonian acceleration, an interpolating function based on true acceleration and an acceleration constant. But the acceleration constant matches the gravitational acceleration induced by the universe, and the true acceleration can be substituted by the Newtonian field intensity, so that Milgrom’s acceleration equation is the usual Newtonian acceleration with a function F depending on big M, big R, small m and small.
Look at the similarities between Carl Bran’s Machian acceleration from 1961 and Milgrom’s MOND. They both imply that local dynamics are governed by the mass distribution at large. But notice that while Bran’s motivation is theoretical, Milgrom’s motivation is empirical, because MOND correctly explains the dynamics of galaxy rotation curves. The important question now is how to implement this Machian effect into a relativistic theory. Brans-Dicke failed because it modified gravity in the high acceleration regime, where we know gravity strictly follows General Relativity. And regarding MOND, TEVES has shown problems and it loses the power of its simplicity by the addition of a scalar and vector field with corresponding new parameters.
In Independent Physics you will find our journey towards a successful modification of our understanding of gravity for more than one year, and many more interesting topics explained.
References:
Carl H. Brans, Mach’s Principle and the Locally Measured Gravitational Constant in General Relativity, 1961