# What is the origin of the inertia?

Is there any explanation why it is harder to move an object with more mass than an object with lesser mass? What kind of force is opposing our force? Is it finalized currently and well known what the origin of inertia is, or not yet 100%? I read some explanations that are linked to general relativity and Mach's principle, but can someone please tell me in the case of current physics, it is well understood?

• just a guess, but Newton's third law? Which I believe is a consequence of conservation of momentum. Commented Jun 21, 2022 at 18:54
• I would say that mass is nothing but how hard it is to move an object. Now why some objects are harder to move than others... that's a good question. Commented Jun 22, 2022 at 2:03
• Commented Jun 22, 2022 at 6:56
• In short: mass, energy and momentum are extensive properties, and force is related to these as F=dp/dt. "Extensive" in this context means properties that scale with size/amount. Two electrons should require twice the force to accelerate the same amount as one electron, etc. As for the origin of mass... that's another question entirely. Commented Jun 22, 2022 at 7:05
• One possible answer is at arxiv.org/abs/1910.10819 . I believe it's had some observational support, from Cai and from Lior Shamir, in evidence suggesting that our local universe, like the others in a version of an inflationary multiverse that does not require hypothetical iinflaton or graviton particles, is rotating. The preprint that I've cited (which is one of many that can be found by its author's name on Cornell University's Arxiv website) depends on an odd mix of General Relativity with Einstein-Cartan Theory, so it might help to know that ECT reduces to GR in vacuum. Commented Jun 22, 2022 at 11:13

Currently, there is no scientific consensus on why inertia exists, or why the geodesic through spacetime is the least path of energy (ultimately leading to: why do we need to expend energy to move an object away).

Although, we know how much force is needed, we don't know why, and so this question may be well suited for either opinions, philosophy, or just to take it as a given property of mass (the resistance to change in motion).

As for "why it is harder to move an object with more mass than an object with lesser mass", I would be cheating you if I say $$F=ma$$ because you could question why that specific equation should apply to nature. I can only say observation until we discover more fundamental truth.

By fundamental truth, I am indicating a possible theory or law that dictates the formation of present physics laws and tells why nature works. If your curiosity is satisfied with that theory, congratulations! But, if you are an extremely curious person, you might ask, "Why this set of principles", and ultimately, one of two scenarios will happen.

1. You will keep asking "why" questions and never reach a final answer (unless you believe the statement by Hawking that asking why or who caused the Big Bang to happen makes no sense because there was no time existing before it) or, 2) You will have to allow something to be true and accept that humans can never get a final answer.
• The observation is the fundamental truth. Commented Jun 21, 2022 at 12:36
• Your last point, reminds me of Richard Feynman - Why? Commented Jun 22, 2022 at 3:50
• @Bruce Wayne YES, because it is from the legend Commented Jun 22, 2022 at 7:22

If it is not an explanation, it is a good argument:

Suppose $$2$$ solids, each with a volume $$V$$ and the same material. They are separated, and each one receives a net force $$F$$. As a consequence they get an acceleration $$a$$. If we look at this system of $$2$$ objects, a force of $$2F$$ is being applied, resulting in an acceleration $$a$$.

If we join that solids, making an unique object with the double of the mass, it is perfectly reasonable to expect that it is necessary a force $$2F$$ to get the same acceleration $$a$$. Otherwise we should conclude that just by taking them nearby, their behavior would change.

What is orgin of inertia

The idea of inertia goes back to Newton who wrote his famous equation $$m \frac{d\vec{v}}{dt}=\vec{F} \tag{1},$$ with $$m$$ standing for the inertial mass. The equation (1) expresses mathematically a physical observation that an object at rest tends to stay at rest and an object in motion tends to stay in motion. Now, let us think about what would happen when $$m$$ goes to zero. First, we re-write equation (1) in terms of vector length $$v$$ and vector direction $$\hat{v}$$. It applies $$\vec{v} \equiv v~\hat{v},~~~\hat{v}~\cdot \hat{v}=1,~~~\frac{d\hat{v}}{dt} \cdot \hat{v}=0.\tag{2}$$ The equation (1) reads then $$m \frac{d{v}}{dt}\hat{v}+m v \frac{d\hat{v}}{dt}=\vec{F},\tag{3}$$ and because of equation(2) can be expressed as $$(\vec{F}\cdot\hat{v}) \hat{v}+m v \frac{d\hat{v}}{dt}=\vec{F}.\tag{4}$$ In the limit of $$m\rightarrow 0$$ we obtain $$(\vec{F}\cdot\hat{v}) \hat{v}=\vec{F}~~~\implies \hat{v}=\hat{F}\tag{5}.$$ With this result we can calculate object's trajectory but without any time specification to its momentarily position. That situation has striking resemblance to the calculation of trajectories in Feynman’s famous paths integral method.

It seems that inertial mass and time are two sides of the same coin.

For example, if we scale mass $$m(\epsilon)=\epsilon~m,~~~0\le\epsilon\le\infty,\tag{6}$$ and correspondingly, the fundamental physical constants as $$c\rightarrow\epsilon^{-1/2}~c,~~~h\rightarrow\epsilon^{1/2}~h,~~~G\rightarrow\epsilon^{-2}~G,~~~k_{B}\rightarrow \epsilon^{0}~ k_{B},~~~\tag{7}$$ then all physical equations (Dirac, Einstein, Maxwell, etc.) stay unchanged but become elliptic if the $$\epsilon$$ goes to zero, i.e. the inertial mass vanishes. All terms with time derivatives in these equation disapear.

By the way, the connection between mass scaling and time (step) is well-known in the finite element analysis.

• can you explain me how to get to equation 4 from 2? Commented Jun 22, 2022 at 7:26
• **4 from 3 and 2 Commented Jun 22, 2022 at 7:26
• @Aveer, you have to multiply (scalar multiplication) both sides of equation (3) with $\hat{v}$ and take into account equation (2). The result should be $m\frac{d v}{dt}=\vec{F}\cdot\hat{v}$.
– JanG
Commented Jun 22, 2022 at 9:07
• @JanG what did you mean that inertial mass and time are two sides of the same coin? Commented Jun 15 at 20:02
• @user36093 That without inertia, time wouldn't exist.
– JanG
Commented Jun 16 at 7:58

Inertia is not directly measurable and its definition reflects that. Inertia is defined as the resistance to motion. So we must be capable of measuring motion to measure inertia. Now speed is defined as the change in distance over a duration divided by that duration. So we need to examine both space & time to understand inertia.

Lets tackle space first. Now Newton in his Principia wrote:

Absolute space, of its own nature, without reference to anything external, always remains homogeneous and immobile.

Newton here, does not discount the possibility of a larger universe within which our universe is embedded and through which it moves. But without evidence, he refers only to our space, this is what he means by "of it's own nature, without reference to anything external".

In contemporary physics both space and spacetime are modelled what are called manifolds. These are locally Euclidean. This is pretty much what Newton means by homogeneity. The universe is "immobile" in the sense that its parts do not move, likewise a manifold. What I mean by this, is not that things within cannot move, they obviously can. But that one part of space cannot move to another part of space. In contemporary language, it's topology is fixed or "immobile".

Liebniz argued against the "absolute space" of Newton and for a relative notion. He said that space was described by all the distances between all particles. So if there was no masses/particles in the universe, then space, in a sense, disappears. Now the term distance, in comtemporary terms, is the metric. The metric is what allows us to measure both distances and angles. So to sharpen Liebniz's argument we should say that without masses the metric loses all meaning and we can no longer measure distances and angles. This does not mean that space itself disappears. What we have done is separated the notions of metric and space. In Newtons & Liebniz's discussion they were identified. Space is the possibility of placing something, somewhere. The metric is the possibility of measuring distances. They are not the same thing. Without a metric, space still can exist, we do not have a void.

Now, after Einstein we can examine this argument since a universe without matter is a cosmological vacuum and this can be modelled by General Relativity. What we find, in the absence of a cosmological force (this is the vacuum pressure which is usually referred to by its coupling, the cosmological constant), is that the universe must have vanishing Ricci curvature. This follows from Einstein's field equations. Moreover, it also means the scalar curvature vanishes. Since the Riemann curvature, by Ricci decomposition, is a sum of the conformal, Ricci and scalar curvature, we can see that the Riemann curvature, in this situation, reduces to exactly the conformal curvature.

Now, the conformal curvature is invariant under conformal changes to the metric. This means locally we can change the metric by some factor without changing the conformal curvature. And this means that distance, and since the metric measures spacetime and not just space, duration becomes undefined notions here. However, the notion of angle remains valid as this is invariant under conformal changes.

Now recall we said inertia is defined as the resistance to motion. So we must be capable of measuring motion to measure inertia. But speed is defined as the change in distance over a duration divided by that duration. But we can neither measure distances or durations so the notion of speed also becomes immeasurable. We can do a little better with rotational speed but ultimately this is not measurable either. This is because rotational speed is the change of angle divided by the duration. Now whilst angles are measurable, duration is not. Hence angular speed is not measurable either. Since we cannot measure linear or rotational speeds then we cannot measure linear and rotational inertia either.

More, Newtons characterisation of space as "homogeneous" remains pretty much valid in this larger context of spacetime. Spacetime is still modelled by manifolds and these are "homogeneous". However, he also characterised them as "immobile". On the whole, the usual spacetimes peoole examine are immobile in that their topology does not change. But General Relativity does admit wormholes and these enact topology change. So his secomd characterisation is true upto these kind of topology changes. But these tend to take exotic forms of matter, so in the vacuum we can discount them.

Thus after Einstein, we can see that Liebniz is correct and Newton was wrong:

Spacetime, of its own nature, and without reference to anything external, and without anything internal (ie masses or particles) is homogeneous and immobile. It is also relational in both Liebniz's and Machs sense, and not absolute.

We also see Mach was correct - without any masses in the universe, the notion of both linear and rotational inertia no longer makes sense.

Note, however, Newton was not entirely incorrect. With his acute physical understanding he spotted something absolute in the character of space through his bucket experiment. It is not the measurement of rotational inertia or momentum but that of angle. But perhaps this suggests a possible modification of GR to align even more closely to Liebniz's notion of distance being immeasurable when there is nothing there to give measure. I mean a theory where even angle also becomes meaningless in empty universe.

Moreover, there is another notion that is absolute in empty spacetime and that is its causality structure. Although without particles in the universe, one is hard put to say what events can occur (actually there are events in empty spacetime in 4 spacetime dimensions or higher - gravitational waves), the possible cause and effect structure is preserved and makes sense. This makes it absolute.

• "resistance to motion" is a little too close to Aristotle, while "resistance to change in velocity" is closer to Newton Commented Jun 23, 2022 at 12:54
• @Henry: Well, Newton read Aristotle - so I don't see it being a problem. Commented Jun 25, 2022 at 12:03