Derivation of Mach's Principle » 22 May 08

Mach’s Principle is a phrase coined by Einstein to denote the idea that inertia, the resistance of mass to changes in motion, is not a fundamental property of that mass alone, but something that depends on its relationship to all other masses in the universe. This research note gives a mathematical reason behind this mystery.

Although the Theory of Relativity was about motion being relative to the observer, inertial resistance to changes in motion is not relative and does not depend on the observer at all, and that is what intrigued Einstein. For example, when mass is forced to move into a circular pathway, it will resist that force and pull outward against it. That is why stirred tea presses outward and up the inner wall of a mug. But the tea will do this regardless of whether you stand still, spin around, or run past the mug.

Such inertial effects must therefore be independent of the observer. The motion leading to such effects must be measured relative to something absolute, and that absolute is the fixed background of stars in the sky. When something spins, it spins relative to the stars. When something accelerates, it accelerates relative to the stars. Somehow masses far away affect how mass behaves right here.

That is a big problem because how can local and distant masses possibly interact over such vast distances, and how would this interaction create inertia? Apparently, science hasn’t officially solved this problem.

But with the right postulate it can be easily solved, as will be demonstrated below. I will give a layman’s summary of the rest of this research note and then provide the mathematical derivation of Mach’s Principle, of inertial resistance and centrifugal force as a function of the mass of the universe.


Of all fields in existence, gravity has the longest reach. The field connecting local to distant masses is simply gravity, or more precisely, the gravitational potential.

By “gravity” we usually mean the gravitational force field that pulls masses together and makes things fall to the ground. But this force field only arises from gradients in the gravitational potential. If the potential varies over distance, then a force field exists that tugs on masses caught in the field. Potential is more fundamental than the force field, it is the underlying component of gravity.

Every mass has a gravitational field, but whereas the force fields from all masses in the universe cancel each other out, the gravitational potentials do not. So the combined potential fields from all masses in the universe creates an ambient potential throughout the universe. Therefore all masses are immersed in the gravitational potential of all other masses. The interaction between a mass and this ambient field is what leads to inertial effects.

Moving with constant speed and direction does nothing but change the locally experienced value of ambient gravitational potential. Each velocity comes with its own value of potential. This has the effect of dilating time and contracting length relative to slower moving or stationary observers, as predicted by Special Relativity.

Accelerating through this field creates a compression of the field in front of the mass and expansion in the back. The accelerating mass then exists within a field gradient, meaning a gravitational potential that is no longer uniform. This creates a gravitational force field pointing opposite the direction of motion. That causes the mass to resist acceleration, which is the basic inertial property of mass.

As for circular movement and centrifugal force, note that each distance from the center of curvature has a different velocity. Consider a spinning disk: the edge is moving faster than points closer to the center. Since with each velocity comes a different gravitational potential, a gradient in the gravitational potential exists between center and edge of the disk. Therefore, circular motion creates a local gravitational force field pulling outward and away from the center. And that is centrifugal force, another byproduct of inertial resistance to changes in motion.

All these inertial phenomena ultimately depend on masses in the rest of the universe, as stated in Mach’s Principle, because it is the combined gravitational potential of these that lead to resistance to changes of motion by individual masses.

Ambient Gravitational Potential

First step is to calculate an approximation of the ambient gravitational potential field of the universe. The basic equation of gravitational potential:

`varphi = -(G M)/r`

`G` is the gravitational constant

`M` is mass

`r` is distance from center of mass

This must be integrated over all mass in the universe. For simplicity we will assume a uniform spherical distribution of mass around a central point. Then all we need is the radius of the universe and its average mass density and integrate. Because it is mass surrounding a point, rather than a point some distance from the center of mass, the potential is positive instead of negative:

`varphi = G rho int_0^R r d r int_0^pi sin phi d phi int_0^(2pi) d theta = 2 pi R^2 G rho`

`rho` is average density of universe

`R` is radius of universe

Values for radius and density of the universe depends on whether it is the visible, observable, or total universe being considered, whether the curvature of space is included, and other variables. Ranges given:

`rho = 4.5xx10^-26` to `18xx10^-26 (kg)/(m^3)`

`R = 1.30xx10^26` to `4.34xx10^26 m`

The ambient gravitational potential has the following range of possible values:

`varphi_a = 3.16xx10^16` to `142xx10^16 m^2/s^2`

A better way to find the ambient potential is to use an equation from General Relativity for how a local gravitational potential affects the local rate of time. Since in flat spacetime there is no local potential and the rate of time is undisturbed, then conversely a local potential that is equal and opposite to the ambient potential will stop time. So we set the following equation to infinity and solve for the local potential, then the opposite is the ambient potential.

`t = t_0/(sqrt(1+(2varphi_l)/c^2))=oo`

`varphi_l = -1/2 c^2`

`varphi_a = 1/2 c^2 = 4.46xx10^16 m^2/s^2`

The result falls within the range given above and is more accurate since it depends only on the speed of light, which is well known. Since the ambient potential, which comes from the gravitational potential fields of masses in the universe, is solely a function of the speed of light, it may therefore be inferred that in a less massive universe with a lesser ambient potential, the speed of light is also lower.

For the rest of this note, the ambient potential will be taken as:

`varphi_a = 1/2 c^2`

Gravitational and Electromagnetic Potentials

Showing how the ambient potential leads to local inertial effects requires a postulate of mine that links electromagnetism to gravity. A postulate is an idea that cannot be derived or proven from previous ideas, but gains validity from the consistent success in applying it. The postulate is as follows:

`varphi = beta nabla cdot vec A`

This equation states that the gravitational potential is proportional to the divergence of the magnetic vector potential. That is the missing link between EM and gravity. The beta is a constant of proportionality to be empirically determined.

The vector potential `vec A` is a gradient in the scalar superpotential `chi`, and the latter is a scalar field of pure flux forming the substrate of spacetime. See my other science research notes for a thorough explanation of the vector potential and scalar superpotential.

The following relations hold true between these:

`vec A = nabla chi`

`nabla cdot vec A = nabla^2 chi`

`varphi = beta nabla^2 chi`

The gravitational force field `vec g` is the negative gradient of the gravitational potential `varphi`, so:

`vec(g) = -nabla varphi`

`vec(g) = -beta nabla (nabla cdot vec A)`

Wave Equations

Wave equations exist for the scalar and vector potentials. These are just your typical wave equations relating how spatial variations of a wave relate to temporal variations:

`1/beta varphi = nabla cdot vec A = nabla^2 chi=1/c^2 (del^2 chi)/(del t^2)`

`-1/beta vec(g) =nabla (nabla cdot vec A)=1/c^2 (del^2 vec A)/(del t^2)`
(for the case of curl-free vector potential)

These wave equations are important because they link motion through the ambient gravitational potential with the alteration of potential for that mass.

Uniform Velocity through Ambient Potential

In the first case under consideration, we have a mass moving with constant speed and direction through the ambient gravitational potential of the universe. This field fundamentally consists of scalar superpotential varying over space, and may be written out mathematically as a function of position `x`:

`nabla cdot vec A = (d^2 chi)/(d x^2) = (varphi_a)/(beta)`

`chi(x) = 1/2 (varphi_a)/(beta) x^2 + C_1 x + C_2`

`chi(x) = 1/2 varphi_a/beta x^2` for `C_1, C_2 = 0`

Motion through space causes the superpotential to vary over time for the traveling mass. It is like mile markers showing different values at different distances, and thus the observed marker showing different values at different times on a road trip. To find this rate of change, we differentiate the above equation twice with respect to time:

`(d chi)/(dt)=1/2*(varphi_a)/(beta) (2*x (dx)/(dt))`

`(d^2 chi)/(dt^2)=(varphi_a)/(beta)*(x*(d^2 x)/(dt^2) + (dx/dt)^2)`

`(d^2 chi)/(dt^2)=(varphi_a)/(beta)*(x*a) + (varphi_a)/(beta)*v^2`

Since the velocity is steady, there is no acceleration `a` and the first term on the right is zero. Then we are left with:

`(d^2 chi)/(dt^2)=(varphi_a)/(beta) *v^2`

A mass moving with constant velocity therefore experiences a scalar superpotential that changes as the square of time elapsed. We can substitute this into the wave equation:

`(d^2 chi) /(d x^2) = 1/c^2 (d^2 chi_a) /( d t^2)`

`(d^2 chi) /(d x^2) = 1/c^2 (varphi_a)/(beta) *v^2`

The left side of this equation represents a newly generated gravitational potential. Because this new potential is in the frame of reference of the moving mass itself, a minus sign must be affixed to switch back to the stationary reference frame where the ambient potential resides so that both potentials can be properly summed:

`varphi_l = 1/beta (d^2 chi_l) /(d x^2) = -1/beta (d^2 chi) /(d x^2)`

`varphi_l = -(v^2/c^2) varphi_a`

What an interesting result! The local gravitational potential `varphi_l` is a function of velocity. It is simply the ambient potential times the squared ratio between velocity and speed of light. For the moving mass, the total potential at any point would be the sum of local and ambient values:

`varphi_T = varphi_l + varphi_a`

`varphi_T = varphi_a(1-v^2/c^2)`

At zero velocity, the total potential just equals the ambient. For two moving masses, if both have the same velocity then there will be zero difference in `varphi_l` between them and each will appear to the other as being situated in the same ambient potential. This is in accordance with Special Relativity where all that counts is the relative velocity between two observers.

The new potential may be written more simply if we substitute the actual value of ambient potential into the equation:

`varphi_l =- v^2/c^2 varphi_a = -v^2/c^2 (1/2 c^2) = -1/2 v^2`

Except for the minus sign, which is a matter of convention and frame chosen, this is the kinetic energy equation without the mass variable. Gravitational potential generated by velocity is a type of “kinetic potential.” Kinetic energy isn’t normally thought about in terms of gravitational potential, but that is what it appears to be.

Time dilation and scale contraction of Special Relativity then come down to the ratio between local and ambient gravitational potentials:

`t = t_0/(sqrt(1-v^2/c^2)) = t_0/(sqrt(1+varphi_l/varphi_a))`

`l = l_0 sqrt(1-v^2/c^2) = l_0 sqrt(1+varphi_l/varphi_a)`

Additionally, the most famous physics equation may be rewritten in terms of the ambient potential:

`E = mc^2`

`E = 2 m varphi_a`

This suggests that the intrinsic energy of matter is essentially its gravitational potential energy relative to the rest of the universe. Picture a rubber sheet with a small part pinched from below, pulled and held downward. This illustrates energy as matter, as stable potential rather than kinetic energy. When the pulled portion is released, the stored potential energy flies out in all directions, which illustrates the annihilation of matter and its conversion back into kinetic / electromagnetic energy.

Linear Acceleration and Inertia

For mass accelerating in a straight line, each moment in time and position in space comes with its own velocity, and thus its own gravitational potential. So there will be a different `varphi_l` for different values of `x`. This comprises a gradient, which in turn generates a gravitational force field.

We can take the “kinetic potential” equation and rewrite the velocity variable in terms of acceleration and position:

`varphi_l = -1/2 v^2`

`varphi_l = -1/2 (sqrt(2 x a))^2`

`varphi_l = -x a`

Then to get the gravitational field experienced by a moving mass due to its acceleration, we change signs (multiply by -1) to switch reference frames back to the moving mass and take the gradient or spatial derivative of this local gravitational potential:

`vec(g) = -nabla*(-1)*varphi_l = -d/dx (x a)`

`vec(g) = -a`

As you can see, the induced gravitational field is equal and opposite the acceleration. This means an accelerating mass will experience a backward pull proportional to the rate of acceleration, which is identically the property of inertia. The force of this pull is equal to the gravitational force field times the mass, and so the force needed to accelerate an object is:

`F = ma`

Centrifugal Force

In the case of rotation or mass moving around a circular path, each point along the radius of curvature has a different tangential velocity and thus a different local gravitational potential.

Tangential velocity is a function of angular velocity `omega` and radius `r`, and these can be plugged into the kinetic potential equation and differentiated with respect to radial position to get the gravitational field produced by circular motion:

`v = omega r`

`varphi_l = -1/2 omega^2 r^2`

`vec (g) = -nabla*(-1)* varphi_l = 1/2 d/(dr) (omega^2 r^2)`

`vec (g) = omega^2 r = (v^2/r^2)r = v^2/r`

`F = (mv^2)/r`

This indicates that the force needed to keep a mass moving along a curved path (instead of flying outward back into a straight path) is a function of its mass, tangential velocity, and radius. This is the standard physics equation for centripetal / centrifugal force, except I interpret it as a gravitational force acting on the mass due to a gradient of potentials existing along the radius of curvature.


With the postulate that the gravitational potential is the divergence in the vector potential, that all masses in the universe create an ambient potential, and the wave equation for the scalar superpotential, in the end I have derived the Equivalence Principle, Mach’s Principle, and Newton’s First and Second Laws.

Further Reading

See my latest paper: A Brief Introduction to Scalar Physics.