# The Concept of Inertia

1. Jan 20, 2012

### Mechanic

I’ve struggled with the concept of inertia as derived from Newton’s first law of motion for the following reason:

IF, as postulated by Einstein, gravitational acceleration is not caused by the application of a force to an object but is caused by the curvature of space-time…

AND IF space-time is curved at all locations in the universe to some extent, however small…

THEN all free objects at all locations everywhere will accelerate without the application of any force.

…which directly contradicts the definition of inertia as the property of objects to resist acceleration unless acted upon by an external force. In fact the opposite of this definition of inertia true. Let’s call the acceleration at all locations in the universe due to space-time curvature the “ambient” acceleration for emphasis. Then we can say that all free objects will accelerate without any force acting on them meaning that inertia is not that property of an object that causes it to resist acceleration, rather inertia is that property of an object that causes it to resist deviating from the ambient acceleration. A (real = measure-able) force is required to cause an object to deviate from the ambient acceleration.

This definition of inertia makes sense to me and has the added benefit of by-passing much of the confusion surrounding attempts to define inertial and non-inertial reference frames.

2. Jan 21, 2012

### Simon Bridge

Your confusion comes from awkward descriptions of space-time and what it means to be "curved". The curvature does not cause the acceleration, there is no force. In GR the force of gravity is a pseudo-force like the centrifugal force or any of the mysetrious effects in non-inertial reference frames.

So the curvature of space-time does not cause an acceleration. What it does is change the appearance of the path when viewed in 3D. Light, for eg, has no mass and yet it's path is bent around massive objects. Light always travels the quickest distance between two points, which is a curve.

The law of inertial works for forces and gravity is not a force.

3. Jan 21, 2012

### Mechanic

There is no confusion. Saying that gravitational acceleration is caused by the curvature of space-time is a commonly used short-hand and is immaterial. My point is – and which you appear to agree with – gravitational acceleration is not caused by a force, thus objects throughout the universe are accelerating with no force acting upon them and a force is required for them to deviate from that acceleration - which counters the common description of inertia.
And your point that gravity is not a force is incorrect – or unclear. Step on a weight scale and watch the dial move due to the force of gravity. Seen in this way gravity causes an obvious, observable force. If your point was that acceleration due to gravity is not caused by a force then of course I agree.
Again, saying that inertia is the property of matter that causes it to resist acceleration unless acted upon by a force is incorrect because all free objects in the universe are accelerating with no force acting upon them. Inertia is the property of matter that causes it to resist deviating from the ambient acceleration unless acted upon by a force.

4. Jan 21, 2012

### Mechanic

It would be understandable if the next point to be explored was that Newton’s laws of motion are only valid in inertial reference frames – which is entirely correct. But the last comment in my original post alludes to the problems with definitions inertial and non-inertial reference frames and examples of them, not the least of which is that the laws of motion are only valid in inertial reference frames, but strictly speaking there are no inertial reference frames anywhere in the universe, so those laws are never valid. Many of the examples of the different types of reference frames – even in widely used physics texts today – are wrong and/or misleading.

5. Jan 21, 2012

### Ken G

The distinction you are making is not wrong, but it is nonstandard use of terminology. The key point is, the role of gravity is to establish the inertial paths, and inertia resists deviations from the inertial paths. What you want to call acceleration is largely an issue of convention, so depends on a coordinate choice, but the standard choice is to define acceleration by that which is measured by an accelerometer that was built in free fall (or more likely, were compensated for the gravity when built). As such, gravity does not cause acceleration-- an object in free fall, under any gravity, registers no acceleration on an accelerometer. But it does follow the inertial path.

6. Jan 21, 2012

### Mechanic

Thanks for the reply. We are in mutual agreement on much of this so please don’t take this as nit-picking but…Yes, terminology here is tricky because in fact the most standard definition of acceleration – and the one I am using - is the change in velocity as a function of time, regardless of whether an accelerometer is present or not. Your description of ”proper acceleration” as acceleration that is measurable by an accelerometer is acknowledged. And you correctly home in on the criticality of defining the coordinate system relative to which acceleration is measured when the term is used as I use it – the change in velocity vs time. I say, for example, that two remote free objects in space (all other objects are so far away their gravitational effects are negligible) will accelerate relative to each other without any external forces acting on them,and that defines the necessary coordinate system – the coordinate system at rest relative to one of the objects. After all, the mathematician can conjure up any number of coordinate systems, but what does that tell us about the underlying physics? The physics comes from the physical interactions between the two objects so let’s measure those interactions relative to those objects. And I still maintain that a better definition of inertia is - Inertia is the property of matter that causes it to resist deviating from the ambient acceleration unless acted upon by a force. And, based on your last post you may even agree with me when my definition of acceleration is used????

7. Jan 21, 2012

### Ken G

That is the standard definition of "coordinate acceleration", which really doesn't have much of the physical meaning of acceleration. Proper acceleration is the only kind of physically meaningful acceleration that should be discussed in relativistic contexts. For example, you mentioned the person on a scale, and that person does indeed have proper acceleration, but not because of gravity, it's because of the scale. That is also what the scale measures-- the scale does not measure gravity. That's clear from the fact that if the scale is on an elevator in free fall, it registers nothing, even though the gravity is just the same.
There is no such thing as a coordinate system that is at rest, there is only a coordinate system in which the object is not moving. Give me any two objects, doing anything, and I can find a coordinate system where neither object is instantaneously moving, yet both objects have arbitrary coordinate acceleration. It's not physical. If one wants to say that inertia resists acceleration, then one must be using a physically meaningful version of acceleration, and that's proper acceleration, not coordinate acceleration.
And that's the point-- the underlying physics must refer to coordinate-independent invariants like proper acceleration, not coordinate dependent numbers like coordinate acceleration. In invariant language, inertia does measure difficulty in getting acceleration-- real acceleration.
Interactions happen at the objects themselves, so only coordinate systems local to the objects can refer to such physical concepts. But global coordinate systems, like what you are talking about, connect the two objects across all that space and time in between, and that connection gives us considerable freedom in choosing our coordinates. That is very much one of the keys of GR-- if you choose global coordinates that connect two objects by a chain of observers that have no mutual motion, those will not be inertial observers in the presence of the tidal effects of gravity, and noninertial observers are not the ones to ask about how inertia behaves. Even in the absence of gravity there's no concept of ambient acceleration that is physically meaningful-- if I use curvilinear coordinates (like polar coordinates), I am using a chain of observers who have smoothly varing orientations, and even an object moving in a straight line at constant speed will exhibit coordinate acceleration in those coordinates (the radial and azimuthal velocities will vary with time but it cancels with how the observer orientations are varying to give zero proper acceleration even in Galilean relativity).
That definition of acceleration is not usable. Instead, what is well defined is the concept of an invariant-- and the need to base the laws of physics on things that all observers will objectively agree on (i.e., not ambient acceleration or any other kind of purely coordinate acceleration).

8. Jan 21, 2012

### Simon Bridge

... and since it is a short-hand and not a literal or physical acceleration, there is no reason to expect it to follow the kinds of rules you point out it does not in fact follow. When you replace the short-hand with what it is a shorthand for, your objection vanishes.

Go do it, and you will see.

For the specifics - thanks Ken G.

9. Jan 22, 2012

### Mechanic

Thanks for the comment. I think I agree with you but am not sure. And in any case, the point my argument relies on is simply the claim that gravitational acceleration is not caused by a force – which everyone agrees with. Whether I describe what the cause of that acceleration is or not to the satisfaction of all does not matter – as long as I am correct in identifying what is not the cause.

So, I’m still comfortable with my view of inertia as described in this thread. I’ll proceed with the impact to the second law of motion in a separate thread. Thanks again.