# Mach's principle and a reference frame for acceleration

Although there are different statements of Mach's principle, one statement could be that acceleration is meaningless unless it can be defined relative to something. The distance stars provide a fixed reference frame against which we can define inertial frames and measure acceleration and distant matter in the universe might even somehow causes inertia.

Mach's principle seems to have fallen out of fashion in modern physics, most answers to this question Is Mach's Principle Wrong? doubt Mach's principle.

So as an alternative can a body itself provide a reference frame for acceleration?

When a body is accelerated it always gets deformed. One part of it is pushed or pulled and that influences the other parts that then also get accelerated.

A simple example is three masses attached by equal springs in an equilateral triangle shape. In an inertial frame the shape is an equilateral triangle and the potential energy stored by the springs is a minimum (diagram A)

If we take one mass and pull it, one mass is accelerated first and the triangle is deformed into an isosceles triangle, with greater stored potential energy. The other two masses are then pulled along and follow the first. The system knows it's been accelerated, but not because of any knowledge of the distant stars, but because it has been deformed, (diagram B).

If the masses are dropped in a gravitational field, (diagram C) the masses are in the equilateral triangle shape, even though they are accelerating relative to the distant stars. This seems fine with General Relativity and the equivalence principle that say the 'falling frame' is the same as an inertial frame. So again in the inertial frame there is no deformation.

So can the body itself be the reference when defining inertial frames?

In Newton's bucket experiment, the liquid in the bucket takes on the curved shape when rotating relative to the distant stars - true (independent of whether the bucket rotates too, or is suddenly grabbed and stopped). However it could be said that the liquid 'knows' it's accelerating as it's been deformed when going from the stationary state to the rotating state. The acceleration varies with different radii, for cicular motion $$a=-\omega^2r$$ and the liquid is deformed in the sense that parts of it further from the middle experience higher acceleration.

Thus since the acceleration isn't constant throughout the body, as in a gravitational field, the body knows, not by reference to the distant stars, but due to deformation and it's internal potential energy being increased that an acceleration is occurring.

Have there been any attempts to explain the inertia of bodies along these lines?

Could it be that the resistance to motion we experience as inertia, is due to the forces needed to cause cause compressions and deformations - and mass (inertia) doesn't exist at all?

It could be argued that it doesn't matter that an extended object is deformed during an acceleration as we know that the individual components atoms, protons, electrons etc...have mass of their own.

But the argument could be repeated at a smaller scale. Could the inertia of say a proton be due to forces needed to deform it as it accelerates, the same for the electron. It would just be necessary to deny the existence of truly point particles.

• Here's a link that might interest you, if it hasn't already: arxiv.org/abs/1910.10819 . (Arxiv has it as pop-sci, but the same author has many hep papers found on Arxiv by his name.) Dec 17, 2021 at 4:05
• @ Thankyou Edouard, will have a look Dec 17, 2021 at 9:33
• You can't observe acceleration relatively to infinitely far away objects. At best you can observe rotation. Mach's principle (whatever it might be, I have never seen a really strong definition of it), is simply useless. Jun 18 at 6:49

There's no need to invoke deformation to explain acceleration in relativity. If a body is measured to have acceleration in one inertial frame, then all inertial frames will agree that it is accelerating. For extended bodies acceleration can be directly measured with an accelerometer (this is called "proper acceleration").

This was a difficulty in Newtonian physics because gravity appeared to cause acceleration which would not appear on an accelerometer. In general relativity this is no longer an issue -- gravity is treated as a fictitious force similar to the apparent "centrifugal force" experienced in a merry-go-round. We understand that the apple falling to the Earth is not undergoing (proper) acceleration and remains inertial. It only appears to be accelerating to observers on the surface of the Earth because they are following a non-inertial path, being accelerated by the forces keeping the Earth from collapsing.

• Yes, this is mentioned in the few sentences under the diagram, it adds to the argument that the distant stars are not involved as the cause of inertia. May 15 at 7:36
• @JohnHunter There was never a reason to believe that (effectively infinitely) distant objects make any difference to local physics. That has never been observed. Jun 18 at 6:48

I'm not a physicist or a mathematician, but I spent a lot of time trying to understand this problem.

The conclusion I have conceived is as follows:

Each material particle has its own individual gravitational field. This gravitational field is infinitely extensive.

When the particle changes its position, due to a force applied to it, this new position must also be adopted by the spherical extension of its field, over time. When a force is applied to the particle for a certain time, the extension of its gravitational field "tries" to be brief and correct its distribution, with each new position of the particle that is accelerated, but there is a delay, this correction we already know, no it can be instantaneous.

So didactically, this works as if the force applied to the particle tends to separate it, or to dissociate it from its gravitational field, as it is extensive.

I realized then that the inertia that we feel when we apply a force to an object, results from the deformation (work) that we cause in the distribution of its gravitational field, which must always be spherical.

The greater the force, the greater the deformation in the spherical distribution of this field.

In Newton's Bucket's experiment, note that the gravitational field of each particle of water passes through its walls as if they did not exist, and when, due to a force, a particle in the bucket, moves absolutely in relation to the spherical distribution of its field, the reaction to deformation of the field appears like inértia as I said above.

My argument is that the gravitational field immediately outside the bucket always "knows" absolutely with a little delay, that its particle has changed the position due to the rotation of the system

In my understanding, the absolute frame of reference that Newton claimed to exist, was simply the extension of the gravitational field of each particle (which is correct), which in this experiment behaves in this way, but which does not function as an absolute frame of reference (what is wrong) for the other water particles, because each has its own field.

In the Equivalence Principle, in my view, the reason that justifies the equality of the masses is that the gravitational field of the specimen (object) is the same in the case of the inertial or gravitational mass.

Realize that when we try to push an object, using force, we act as if we want to move it away, or to dissociate it from its gravitational field, which reacts in the form of the inertia that appears. This attribute is called inertial mass.

On the other hand, when on the Earth's surface, we support this same object so that it does not fall to the ground, we must observe that our strength prevents the object from falling, and accompanies all its extensive field that is being attracted to the center of the planet. This impediment causes the same deformation in the field, if the forces applied in both cases are identical. We call this mass, or inertia, or deformation, a gravitational mass.

Inertia is the attribute that appears whenever we try, by applied force, to dissociate the object from its gravitational field.

Note that I dismissed the Mach's Principle for finding it unnecessary.

Sorry about my English.

• Yes, are you trying to explain inertia as due to the masses gravitational interaction with other bodies, or due to a reaction back from the spherical field when we try and accelerate a body? The trouble with the first is time delay, even if information travelled at the speed of light - apparently Feynman and Wheeler used 'retarded waves' to try and get around this. In the question it's hinted that inertia may not exist at all, but the resistance to motion we feel as inertia may be due to the force needed to compress or stretch a body. May 24, 2021 at 8:14
• the question has been edited to include more discussion of the origin of inertia. May 24, 2021 at 12:31
• @John Hunter - In my understanding, a particle's inertia arises only when we apply a force to it, as if we want to separate it from its individual gravitational field. However, this gravitational field that was deformed by force, reacts in a finite time, in an attempt to follow the particle's movement at each new position of it, while the force is maintained. This reaction of the field to the applied force is called inertia, but it is only a deformation in the individual particle gravity field. Therefore, I do not see a consequence as a result of a relationship between particles. May 24, 2021 at 22:14
• OK, interesting answer, someone with more knowledge of General Relativity might comment on whether the gravitational field would do what your suggesting... May 24, 2021 at 22:37
• I think the value of inertia depends on c. In our Universe the value of c is approximately 298 .10 ^ 3 km/s. However, in a hypothetical universe, where the speed c is supposed to be twice our own (for example), in the equation m = E / c ^ 2, the inertia value decreases 4 times. That is, there the objects will be 75% lighter. I am considering that c, can also be the reaction speed of the gravitational field. May 24, 2021 at 23:52