Is there still no known origin of the law of inertia?

The quote by Feynman is stating the obvious, that the law of inertia has no deeper explanation, other than when used, together with other laws, principles and postulates to set up a theoretical physical model for mechanics.

In general laws, principles, postulates are extra axioms used in physics models so that the theoretical model fits observations and predicts future ones.

At the moment there does not exist a Theory Of Everything (TOE)for physics. If one such does emerge in the future, it could be possible that the number of physics laws,postulates, principles will be reduced to that one mathematical theory and its axioms. We have not reached that level, if it is ever reached. In such a TOE it could be that inertia would be theorem and not an axiom, i.e. it would be more in mathematical formulation economical to have it as a theorem. (In a theory the place of axioms and theorems can be interchanged, one chooses the simplest mathematically as axioms)

There is still no answer to the question of why the law of inertia should hold (and you can say the same of every law from which it is supposed to be deduced). But suppose it didn't hold. What would this imply for the motion of an object in, say, empty space?

The law of inertia states that the momentum of the object stays the same forever, as long as no force acts on it. So the velocity vector of the object is constant during its motion. So what will happen if this is not the case?

I'm sure you can imagine that if the momentum of the object is not constant during its motion, the object would behave in a way that we have never seen. So while the "why" of this law is not known, it certainly is backed up by our observation. The world would look very different. Maybe in a parallel Universe, the law of inertia doesn't hold. Try to imagine how and if we could live there. One can say that the law of inertia is an empirical law, a law based on our observations.

Of course, the (ultimate) base of all our Natural laws is based on observation but empirical laws "beg" for a deeper explanation, like the empirical Bohr model for the energy levels in an atom was totally accounted for by quantum mechanics (which was fully developed years later) by means of the Schrödinger equation. There is no doubt though that Bohr played a role in the development of QM. But in the case of the law of inertia, there is no deeper theory (not yet, and I doubt there will ever be) that explains this law.

The law of inertia can be seen as the result of the translation invariance of the laws of physics. Of course it's a matter of taste whether you think translation invariance makes a more intuitively appealing axiom than the law of inertia itself.

If one imagines on the origin of the universe as coming from nothing, one immediately realize that:

1. The net energy in the universe must be zero and the universe should be a free meal, i.e, if some energy is created by stretching up something at a point A, then, somewhere in the universe this thing (or equivalent) is going to be stretched down at a point B. This is the principle of conservation of energy in a nutshell.

2. The total amount of “movement” in the universe must be zero, not only because energy is zero, but because there is nowhere for the universe to go as well, i.e, if something is moving right next to point A, another thing(s) must be moving left near point B to counterbalance this motion. This is the principle of conservation of momentum in another nutshell.

People in the vicinity of point A or B may observe the existence of energy and motion, but it just doesn’t exist at the big scale. Therefore, if a state of motion is observed somewhere, such a state is demanded to remain the same until it is passed or transferred into another object. This explains why the state of motion should persist (half of the inertia principle). What this doesn’t explain, is why the motion appears to resist the change, in other words, why motion is not transferred from object 1 to object 2 at a zero time (the other half).

The fact is that there is another principle that seems to be a part of the universe’s behavior (I don’t know the reason why) but is well established.

3. The transfer of any peace of information (yes, motion is information) cannot be done at a speed higher than the speed of light in vacuum $c$.

Principle 3) demands motion to be passed at a finite amount of time, making an object appear to resist gaining or losing motion. Combining all this together, one realizes that inertia is a consequence of the basic structure of the universe.

The law of inertia can be expressed in terms of spacetime geometry as, "free massive particles follow timelike geodesics". The geodesic equation is an expression of the least action principle. Thus to say, "a free particle obeys the law of inertia", is the same as to say, "a free particle obeys the least action principle".

Another representation of the law of inertia using the least action principle is the Noether theorem stating that a momentum of a free particle is conserved in a space with a continuous translation symmetry.

The origin of the least (or more precisely, stationary) action principle is in quantum mechanics, "if we consider the classical description as a limiting case of the quantum formalism of path integration, in which stationary paths are obtained as a result of interference of amplitudes along all possible paths." (Principle of Least Action)

Thus the origin of the law of inertia is in the wave properties of matter. The stationary path of inertial motion is a result of a constructive interference of quantum waves (similar to the Fermat's Principle in optics).

Well, i think that a motivation for why bodies not subjected to forces obey the inertia principle can be stated: our universe has a space-time that, in this condition, looks homogeneus and isotropic. If anyone tries to write up the equation of motion for a body moving in such space-time, a lot of properties are strictly forbidden and the only solution left is a uniform straight motion. However, note that this answer is not totally complete, because now the focus has been shifted to the reason why the space-time of our universe looks exactly homogeneus and isotropic, in absence of charges or fields (i would add).

Answering this question really requires a more philosophical take. For one, insofar at least as physics is concerned, the only way you can talk of an "origin" of something is to derive it from a more basic set of principles. Yet that assumes that there is such a more basic set of principles, and that may not be the case, and it's reasonable to imagine it is not the case: there can very easily be a finite set of groundwork, complete principles sufficient to generate all phenomena in the Universe.

The reason we can think this is that it is logically possible, and to show that we need only exhibit a consistent but imaginary example. Consider a Universe composed of a single kind of point particle whose laws of motion were literally just Newton's laws of motion, perhaps with some suitable forces. It's not our Universe, but it's a possible one (c.f. Modal logic). If you were to pose this question in such a Universe, it would have no answer in the way physics, as a mode of study, conceives of such a thing, because these would be its most basic laws with no deeper ones underlying them.

Hence, the only way physics would provide an answer to this is if it turns out that in the real Universe, there is a deeper set of laws in which inertia is not a fundamental phenomenon. Since we don't have a set of laws we can confide in as being complete, it is still possible, but then again, it may not be.

And past that, the question effectively amounts to "why is the Universe we live in the way it is?" even when all discoverable explanations have been exhausted, and this likely goes out of the realm of empirical science altogether. Empirical science does have limitations, and here is it (and those that say otherwise, and/or that say that nothing else is "worth" asking about are, in my mind, of a rather limited, if not hubristic, mindset, but that's for another discussion in likely another forum).

OK, this doesn't really answer the question. The answer is: "right now, we don't know. Moreover, it is reasonable to suspect that we may not ever be able to 'know', and we may very well come to a point where that is the most reasonable conclusion given the balance of evidence that will be available at that time."

The main metaphysical point was made in the answer of anna v: in any theory based on mathematical reasoning the role of axioms and theorems (i.e. that which can be deduced from the axioms) can be interchanged. We normally put as axioms the simplest things we can that are sufficient to allow the theorems to be deduced. Determining which are simpler can sometimes be subjective or a matter of taste. Since the law of inertia is already a fairly simple statement, most other statements will be deemed more complicated and therefore less deserving of being called axiomatic in any given formulation of physics.

If we don't want to make the law of inertia axiomatic, then we have a choice of other things which we might argue are simple enough to warrant being used as axioms, from which inertia can be deduced. Here are some.

1. Symmetry of the Lagrangian with respect to translation in time and space (in classical mechanics), leading to conservation of energy and momentum.

2. The claim that the worldline of an object in free fall is a timelike geodesic of spacetime. (Such a worldline can also be described as a line of maximal proper time between any given pair of events on the line.)

3. Like (1.) but now invoking quantum mechanics.

I have a slight preference for (2) over (1) here, but of course since the world is quantum physical, one might argue that no classical reasoning can be adequate. However, it seems to me that the law of inertia could be asserted as axiomatic in any formulation of physics, and thus used to somewhat reduce the set of other things one would have to assert in order to give a complete statement of one's theory, whether present-day quantum theory or any future development.

For me a lot of the answers here went over my head but I think I have an easy explanation for anyone else in my situation -- Relativity.

There is no absolute frame of reference which means that something is only "Moving" relative to something else. Nothing is ever moving in it's own frame of reference unless it is being acted upon by another force. (I'm not sure the concept of motion exists in your own frame of reference, only acceleration. That would make a good follow-up question)

As an example: If you fire a bullet from your spaceship, as soon as the acceleration stops the bullet is just sitting in space in it's own frame of reference and the ship is moving away from it.

If the bullet "Slowed" and you looked at it from the bullet's frame of reference it would look like the ship was accelerating towards the bullet--Why would it do that? Wouldn't it just stay still? If it turned (not in a straight line) it would look as though the ship suddenly accelerated to the side. The only sensible option is for both to sit still until something else accelerates them.

Once you think about it this way and subtract some earthbound constants like friction, gravity and a very commonly accepted point of reference it would be surprising anyone would expect any different behavior.

So in essence, the part of the Law of inertia that says a body in motion remains in motion is the exact same thing as the first part "A body at rest remains at rest". There is no difference between the two except for where the viewer is located.

To answer this question properly requires probing deep - an exercise in deconstructing space and time. This is, in fact, what mathematicians have done over the past century or more, separating out the different elements of the infrastructure into different layers. From the vantage point of a programmer, this frame of mind is quite familiar: one starts with a base type, and builds from it, derived types, each one adding more infrastructure onto the type it is derived from.

At the rock bottom layer is a set of points. That's Layer 0.

They are endowed with a "topology" which gives them sufficient structure to determine such things as "continuity", "contiguity", "connectedness", "interiors", "boundaries", and so forth. Geometry, pursued at this level, is sometimes called "rubber sheet geometry", because at this low level there is no concept of congruence, shape or similarity. The topological structure is Layer 1. In this layer, there is also enough to conceive of the notion of continuous sequences of points, called "paths" or "curves" or "trajectories".

The geometry should be sufficiently structured to support the notion of "rate of change", so we can talk about gradients, velocities and so forth. The additional structure required is called the structure of a "differential manifold". It makes it so that the geometry can be displayed in a series of maps (an "atlas") that consistently mesh together. So, in the vicinity of a point, it looks and acts much like the continuum of an ordinary geometry on which you can do calculus.

That's Layer 2. In this geometry, at Layer 2, there is now not only enough infrastructure to conceive of a notion of paths, but also such notions as its "gradient" or "speed".

Classically, a geometry is purely spatial and involves only spatial relations. However, since the time (at least) of Galileo, who entertained the idea of a symmetry transform that mixed the time coordinate in with the spatial coordinates (that transform now being called a "boost"; i.e. a transform from one frame to another that is moving uniformly with respect to it), the two have been intertwined. This marriage of the two was essentially an elopement that was not fully consummated until around 300 years later, when it also emerged that a "boost" also mixes spatial coordinates in with the time coordinate and alters temporal relations such that what is deemed to be simultaneous, when boosting to another frame, is no longer simultaneous.

The point of the last paragraph is that the notion of geometry necessarily expands to include time, and so would be rightly called a "chrono-geometry"; the object of its study no longer being a "space" but a "space-time".

In a space-time, one not only has the "paths" of purely spatial geometry, but also sequences of points that ascend in time, which are called "trajectories" or "worldlines". The "gradient" of a path becomes the "velocity" of a worldline. So, now we have enough to talk about motion and to do calculus with it. It is at Layer 2, that the basic law of kinematics emerges: Velocity = rate of change of Position with respect to time.

However, at Layer 2, there is not enough infrastructure speak of the "curvature" of a path or the "acceleration" of a worldline! That requires additional infrastructure - and (voilá!) that's the very infrastructure your question pertains to!

It's called a "connection". A connection does two things. For purely spatial geometries, it endows paths with a notion of straightness by determining whether a direction remains "the same" at different points along the path. A path that keeps the same direction is one that is then deemed to be "straight" and is called a "geodesic" (technically, it is only called an "autoparallel", the term "geodesic" only pertains to Layer 4 below). For the curved surface of the Earth, for instance, the geodesics at two nearby points on the equator heading north would be initially parallel but would both proceed along their respective lines of longitude to the north pole, where they converge and meet. The example of such curves on the Earth is the origin of the term "geodesic".

The extra structure of a connection gives you Layer 3.

In the space surrounding the Earth, the spatial geodesics are well-approximated by the paths taken by rays of light, so that a light beam traces out a spatial geodesic.

For trajectories, the connection determines which motions keep the "same velocity" from one time to the next; i.e. which trajectories are non-accelerating or "inertial". Out of all the worldlines in the chronogeometry, the connection determines a subset of them to be inertial. These, too, are called geodesics.

The chief property of the connection is that through each point in each direction runs a unique geodesic.

The question you're asking amounts to the question: "where does the connection come from" or "what determines the connection"? That is, out of all the possible worldlines or paths traversing the space-time, why are some of them straight or inertial and others not?

In Physics, the extra structure is simply assumed to be an extra layer of infrastructure that's just there. In most physical theories, a chrono-geometry with layers 0, 1, 2 and 3 is generally assumed to be there as a precondition to whatever theory is posed.

For dynamic and geometric theories of gravity, the connection is itself subject to dynamical laws that determine how it changes from one point in time to the next. But that it actually be there, in the first place, is assumed at the outset, not explained away. The outstanding example of this is the Einstein Equations. The Einstein equations are formulated on top of a geometry endowed with Layer 3 infrastructure, so the existence of Layer 3 is there as a precondition. The theory assumes that there is a connection, the equations help determine what it is. But the fact that there is that extra infrastructure of a connection is there, is an assumption or precondition. Left unanswered is why there should be any Layer 3 infrastructure at all.

So, when you're asking "what determines which motions are inertial", whether you realize it or not, you're really also asking about where all these other parts of what comprise Layer 3 come from. That includes the question: what determines which curves are "straight"?

Finally, at Layer 4, one has the "metric". For spatial geometry, this provides the additional infrastructure of angles, congruence, path length (and area and volume), orthogonality and a semblance of the Pythagorean relation. For chrono-geometries it provides the infrastructure needed for the concept of duration, time measure, and of a "space-time orthogonality" relation.

(A spatial direction in a chrono-geometry is orthogonal to a temporal direction if a path oriented in that direction is seen as being "simultaneous" from the vantage point of someone on a trajectory oriented in the time-like direction. So space-time orthogonality gives us a local version of "simultaneity".)

To reach Layer 4 from Layer 2, it is enough to just add in a metric. A metric determines a connection and notions of geodesic and inertiality by the "least distance" and "greatest time" principles.

For paths in a purely spatial geometry with a metric, the geodesics are the paths that provide the shortest connection between its nearby points. I say "nearby" with the example of the Earth in mind. The Greenwich mean line is roughly a geodesic, that wraps around the other side of the world as the 180 degree line. Any two points on it can be traversed between in two ways: one directly, and the other by going the opposite way around the world. Only the direct way is the "shortest" way.

For chrono-geometries, the corresponding notion is that of the "inertial" worldline. These are the worldlines that connect two points on its path in the "greatest" amount of time. Thus, for instance, an inertial motion between the Earth and the moon would register more clock time than a motion that quickly accelerates to a high speed upon leaving the Earth and quickly decelerates back to a stop, upon reaching the moon. The amount of time dilation for the worldline is directly related to how much from inertial the worldline deviated.

When the structure of a metric is added directly onto Layer 2 to reach Layer 4, skipping the intermediate stage at Layer 3, the connection derived from the metric is called a "Levi-Civita Connection" and gives you the required infrastructure for Layer 3. Such a geometry is called a Riemannian manifold - if it is a purely spatial geometry. If it is a chrono-geometry, it's called Lorentzian, which is a subclass of what are called the "pseudo-Riemannian" manifolds.

Pseudo-Riemannian manifolds are a larger class of chrono-geometries that permit two or more time-like dimensions, while Lorentzian manifolds have only one, and Riemannian manifolds have none, but only spatial dimensions.

The space-like or time-like nature of a dimension is determined by the metric itself. The metric gives you the approximate semblance of a Cartesian coordinate grid surrounding each point (the prime example, of course, being a segment of the Earth's surface when it is mapped on a flat sheet with a coordinate grid) ... but with the proviso that the Pythagorean relation goes like $α(Δx² + Δy² + Δz²) - β(Δt²)$ for suitable coefficients $α$ and $β$ (e.g. $α = 1$, $β = c²$). The time-like dimensions carry signs that are opposite from the space-like dimensions in the Pythagorean relation.

The distinction between space-like and time-like directions is metrical: the concept exists only at Layer 4. There isn't enough infrastructure at Layer 3 to tell space-like apart from time-like.

A good down-to-Earth example of this, by the way (literally), is this: what happens if you were to take the flight distances between 4 cities (like New York, Chicago, Miami and Houston) and treat the flight paths as straight lines? That is, what if you were to pretend to be a Flat Earther and pretend that all flight paths were not just geodesics but outright straight lines? Could you fit the distances onto a tetrahedron in a Euclidean geometry? The answer turns out to be no! If you actually look up the distances and run through the calculations, you'll find that they require a 2+1 dimensional geometry to fit into.

(If you expand the exercise to include two more cities, like Los Angeles and Seattle, and treat the 15 flight paths all as straight lines, you may very well find that the 15 distances require a 3+2 dimensional geometry to fit into and that they won't fit in either a 5+0 dimensional space, nor even a 4+1 dimensional space!)

So, another answer to your question is to skip Layer 3 and go straight to Layer 4. The question now becomes: "why is there any metric at all?"

It is possible to introduce both the metric and connection independently onto Layer 2. Then one can distinguish between the "native" connection and the Levi-Civita connection given to you by the metric. The difference between the two connections is then called the "contorsion". If we require that metrical relations be preserved along the geodesics (and inertial worldlines) then the connection is called "metrical" and the geometry, itself, is called Riemann-Cartan.

The distinction between "autoparallel" and "geodesic" is here. If the connection is Levi-Civita, then the autoparallel curves are the same as the geodesics. Otherwise, if both a connection and metric are present - as independent objects - then geodesics and autoparallel curves generally do not coincide. It will be the autoparallel curves that are "inertial" and "straight", while the geodesics will be the shortest or longest-duration curves, but not generally inertial or straight. "Autoparallel" is the more general term, since it applies to both Level 3 and Level 4, while "geodesic" only has meaning at Level 4.

There is also an intermediate layer, Layer 3½, which includes only the conformal part of the metric. This is the metric up to change in scale (and sign). This is enough infrastructure to see the distinction between spatial and temporal dimensions, which Layer 3 can't see, enough to recognize geometric similarity and angles (but not congruence or length), and also enough to distinguish which directions are space-like, past-pointing or future-pointing, but not enough to allow you to define a unique Levi-Civita connection, nor an unambiguous notion of "geodesic", but only "auto-parallel".

In addition, it is also possible to only go part way with Layer 4, by allowing the metric to be of rank less than 4. The geometry for non-relativistic theory is a Newton-Cartan chrono-geometry, its metric being only rank 1: $Δt^2$ ... which corresponds to $α = 0$, $β = 1$. There is no notion of spatial geometry, per se, native to this structure. Instead, one has to resort to the inverse metric, which is given by the structure $$frac{1}{α} left(left(frac{∂}{∂x}right)^2 + left(frac{∂}{∂y}right)^2 + left(frac{∂}{∂z}right)^2right) - frac{1}{β} left(frac{∂}{∂t}right)^2$$ rescale it (by multiplying by $αβ$) to $$β left(left(frac{∂}{∂x}right)^2 + left(frac{∂}{∂y}right)^2 + left(frac{∂}{∂z}right)^2right) - α left(frac{∂}{∂t}right)^2$$ and then apply $(α,β) = (0,1)$ to get the Poisson Operator: $$left(frac{∂}{∂x}right)^2 + left(frac{∂}{∂y}right)^2 + left(frac{∂}{∂z}right)^2.$$

The metric and its inverse have to be treated as independent, though related, objects. The Newton-Cartan space is the chrono-geometry which has these two structures added onto it. This is a generalization of the Riemann-Cartan space-times of Layer 4. Unlike the Levi-Civita connection, a connection in a Newton-Cartan space-time is not uniquely determined by the metric.

This can be ameliorated by embedding the geometry into a 5D geometry equipped with an extra 1D invariant $$Δx² + Δy² + Δz² + 2β Δt Δu + αβ Δu² = 0$$ with the Proper Time invariant given by: $$Δs ≡ Δt + α Δu.$$

For each value of $(α,β)$ the 4D chronogeometry embeds into the 5D where the metrical relation/constraint, upon substituting for $Δu$, would reduce to the relation $$β Δs² = β Δt² - α (Δx² + Δy² + Δz²)$$ suitable for use when $β ≠ 0$. So, $s$ plays the role of historical time and is invariant. For $α = 0$ and $β ≠ 0$, it gives you a 5D geometry suitable for Newtonian physics - called the Bargmann Geometry.

The curved version of this metric is used in 5D cosmology, for $αβ ≠ 0$ (which means both Euclidean 4D and Lorentzian 3+1D), because it can be equivalently described as the metric obtained by substituting the proper time $s$ for $u$ as: $$Δx² + Δy² + Δz² + frac{β}{α} (Δs² - Δt²) = Re left(Δx² + Δy² + Δz² - frac{β}{α} left(Δ(t + is)right)^2right)$$ a metric with complex time $t + is$. It changes between locally Euclidean and locally Minkowski signature when the sign of $αβ$ changes.

Why am I personally interested, also, in the question of the origin of "inertia"? Well, you've seen the recent reports and videos released from the Pentagon about those strange vehicles that zip about with extremely fast and sharp stop and go actions, leaving behind no sound, turbulence or wake, making a complete mockery of the law of inertia, almost as if they're trying to showboat and flaunt it. Whatever is driving those vehicles, it is as if they found a way to actually shield the effect of inertia - not just in the vehicle itself, but in the surrounding space.

(The reports go much further with this, indicating that the craft were able to start at tens of miles up in the air, zip suddenly down to a few feet over the ocean and come to a dead stop - all in a fraction of a second - without any sound, sonic boom, wake or friction burn.)

Never mind whether it is for real or not. Just the mere idea sparks curiosity and raises the question: whether and how it is possible to accomplish that within the known geometric framework just laid out. Within that framework, it's easy to describe: the vehicles are messing around with the infrastructure provided at Layer 3, altering the connection in such a way as to make the fast stop-and-go motion the one that is inertial at each point along its path, instead of the stationary or slow-moving motions ordinary objects would take in that same setting.

Do we need to go outside known physics for this, and are we seeing a demonstration of that as-yet-unknown physics being flaunted before us?

So, the question you're asking now gets put at the top of the heap: is there something more that determines the connection than just the Einstein law of gravity? Something that can actually be engineered in the way that allows one to shield from the effects of inertia at rapid acceleration and deceleration both within the vehicle and in the area surrounding it?