# Problems with Inertial Reference Frames

1. Feb 4, 2012

### Mechanic

The initial presentation of Newton’s Laws of Motion (NLM) to students often proceeds as follow: 1. The 3 laws are presented, 2. The caveat that the laws are only valid in Inertial Reference Frames (IRFs) is (sheepishly) mentioned, 3. An attempt is made to define an IRF, and 4. Some examples of IRFs and Non-Inertial Reference Frames (NIRFs) are given. After struggling with some of the commonly given examples of IRFs/NIRFs I believe that they are often flawed – even in well respected textbooks. There are 3 such example categories in particular:
1. All inertial frames are in a state of constant, rectilinear motion with respect to one another.
2. An inertial frame of reference is a frame of reference that is not accelerating.
3. We can usually treat reference frames on the surface of the Earth as inertial frames. (Since the Coriolis effect is generally small enough to be ignored.)”

Before criticizing these three examples, let me point out perhaps the most glaring problem with IRFs, which is that if we are to be rigorously precise we must acknowledge that IRFs are only conceptual approximations of reality valid only in infinitesimally small volumes. They simply do not physically exist anywhere in nature, and thus NLM are, strictly speaking, never valid. Nevertheless, assuming for the moment that they do serve some purpose consider the first example above. (I find lengthy posts to sometimes be cumbersome, so I’ll try breaking this into multiple consecutive rapidly submitted sub-posts and see if that helps. See the separate sub-post for the first example.)

(But first…A handy conceptual test, the “Let Go Test” (LGT), used to determine whether an IRF is present or not can be conducted as follows: Imagine some object initially held at rest relative to the reference frame in question. Release the object. If no external forces are exerted on the object and the object does not change position relative to the reference frame then an IRF is present, otherwise a Non-Inertial Reference Frames (NIRFs) is present. This is simply based on Newton’s first and second laws of motion (Please allow the lack of rigor here in the name of brevity – I think you get what I mean).)

(And finally, a very important note: As established elsewhere, gravitational acceleration is not caused by a force. For purposes of this posting, it is not necessary to identify what the cause of gravitational acceleration is - it is only necessary to stipulate that gravitation acceleration is not caused by a force. In this post the assertion that gravitational acceleration is not caused by any force will be rigorously adhered to.)

2. Feb 4, 2012

### Mechanic

Problems with Inertial Reference Frames, cont.
1. As to the first example: “All inertial frames are in a state of constant, rectilinear motion with respect to one another “
This is essentially claiming that IRFs do not accelerate relative to one another – which is wrong. Consider a reference frame, RFA, in free fall above the Earth. (Assuming that IRFs exist…) Conducting a LGT in RFA establishes that it is an IRF. Now, consider another reference frame, RFB, which is also in free fall and is directly above RFA. RFB is also an IRF for the same reasons RFA is an IRF. However since the acceleration of gravity will vary as a function of height above the Earth, RFA and RFB are accelerating at different rates relative to the Earth, and thus RFA and RFB are accelerating relative to each other. More generally, since all free fall accelerations vary at all locations, all reference frames in free fall are accelerating relative to each other. So, not only is the first example above wrong, the exact opposite of it is true. I recommend that it no longer be used.

3. Feb 4, 2012

### Mechanic

Problems with Inertial Reference Frames, cont.
2. As to the second example: “An inertial frame of reference is a frame of reference that is not accelerating. “
(Again, assuming that IRFs exist, which they do not…) Reference frames in free fall are IRFs yet they can clearly be said to be accelerating relative to the Earth. Some may say that reference frames in free fall are not experiencing “proper acceleration”, which is true, yet they are clearly experiencing “relative” or “coordinate” acceleration, which is a type of acceleration and which thus renders the second example inaccurate. Conversely, a reference frame fixed to the surface of the Earth is not an IRF, yet it can be said to have a constant velocity relative to the Earth. In fact, examples can be constructed in which either IRFs or NIRFs are present regardless of whether they are (relatively) accelerating or not, illustrating that the presence or lack of presence of IRFs is completely independent of whether the reference frame is accelerating or not. I recommend that this example no longer be presented.

4. Feb 4, 2012

### Mechanic

Problems with Inertial Reference Frames, cont.
3. As to the third example:
“We can usually treat reference frames on the surface of the Earth as inertial frames. (Since the Coriolis effect is generally small enough to be ignored.)”
As conducting a LGT demonstrates, reference frames fixed on the surface of the Earth are clearly not IRFs – and not just because of the Earth’s rotation. Even if the Earth was not rotating (there would be no Coriolis effect) the released object would accelerate relative to the Earth in the absence of any external forces acting upon it. In any case, even if the Coriolis effect is relatively small the fact that objects will gravitationally accelerate forces the recognition that reference frames on the surface of the Earth are clearly not IRFs. I recommend this example no longer be used.

5. Feb 6, 2012

### Philip Wood

No insightful post from me, I'm afraid. Just reporting that I've long been uneasy with what is commonly written about inertial frames, and appreciate your critique.

6. Feb 6, 2012

### D H

Staff Emeritus
Mechanic's critique was invalid.

He mixed and matched concepts from Newtonian mechanics and general relativity. Inertial frames are quite different in the two. They do not mix and match. Inertial frames in Newtonian mechanics have infinite extent. Inertial frames in general relativity are local. A free-fall frame is inertial in general relativity, but not in Newtonian mechanics. Gravitation is a real force in Newtonian mechanics, but not in general relativity.

That it is still valid to teach Newtonian mechanics is in a sense a falsification of Popper's concept of falsification, at least a naive version of Popper's concept. There is a huge gap between false everywhere and universally true. Quantum mechanics, special relativity, and general relativity certain do show that Newtonian mechanics is not universally true. That does not mean that it is everywhere false.

We don't have that universal truth (yet). Certainly not quantum mechanics or general relativity. Physicists are still in search of a way to unify the strong force with the electroweak force, and gravitation with those other forces. That general relativity admits singularities is viewed by most as indicative of some hidden flaw in the theory. It is still quite valid to teach quantum mechanics and general relativity even with these suspected flaws.

The same goes for Newtonian mechanics. Newtonian mechanics is more than accurate enough to describe what goes on in the smallish velocity / largish distance macroscopic world we typically confront on a day to day basis.

7. Feb 6, 2012

### Philip Wood

Where does he mix and match?

8. Feb 6, 2012

### JDoolin

I'm not entirely sure what you mean here, but I strongly disagree that there is any such thing as a "free-falling" inertial reference frame.

At least from the standpoint of Newton's Third Law:

$$a \equiv \frac{\partial^2 x}{\partial t^2} = \frac{\sum F}{m}$$

I have seen the argument given that since the force of gravity cannot be "felt" by a free-falling observer, that the quantity on the left must be zero. I think it would be really wise to analyze that idea very carefully.

The claim seems to be that the definition of distance is arbitrary. And then because it is arbitrary, we can choose that distance so that it equates to zero. And then you use that zero in a rather rigid (non-arbitrary) fashion, claiming that because you have arbitrarily chosen the distance to be zero, that it is a non-accelerating reference frame.

9. Feb 6, 2012

### D H

Staff Emeritus
A free-fall frame in the vicinity of a massive body is not inertial in Newtonian mechanics because gravitation is a real force in Newtonian mechanics. A free-fall frame, whether or not in the vicinity of a massive body, is a (locally) inertial frame in general relativity. He is mixing and matching by using the general relativistic concept of an inertial frame in a Newtonian concept. He does this in each of his posts in this thread.

10. Feb 6, 2012

### Mechanic

Inertial reference frames (IRFs) are defined as reference frames in which Newton’s Laws of Motion (NLM) are valid. Please cite a reference documenting that there is some other (just as precise) definition of a different type of IRF. Thank you.

11. Feb 6, 2012

### JDoolin

To my knowledge, inertial means "not accelerating"

Free-falling means "accelerating under the force of gravity."

What is the assumption of General Relativity? Is the object in free-fall near a massive body accelerating, or not?

12. Feb 6, 2012

### Mechanic

There are two types of acceleration: 1. “Relative” (also known as “Coordinate”) acceleration and 2. “Proper” acceleration. Relative acceleration may be the more familiar type and is simply the rate at which the velocity of the object changes over time and requires precise definition of the coordinate system in which the motion is measured. Proper acceleration is acceleration that is measurable by an accelerometer. An actual force is always present and measurable with proper acceleration – no so with relative acceleration. The object in free fall above the massive body is undergoing relative acceleration.

13. Feb 6, 2012

### JDoolin

So, by that logic, I am properly accelerating, by standing on the floor, because the g-forces would be measurable by an accelerometer, but if I throw a baseball, and it climbs up into the air, reaches it's peak, and then falls back to the ground, (until it hits) it is not properly accelerating.

Does D H agree with this?

Does anybody except me think that sounds a little misleading?

Last edited: Feb 6, 2012
14. Feb 6, 2012

### JDoolin

Now, I don't claim to know what the "standard" meaning of "inertial frame" is. But what I mean is "not accelerating" It would take a long time for me to explain what I mean by inertial reference frame, but I can try giving examples of what I think are inertial frames, and what are not. Keeping in mind that these are probably controversial examples.

Inertial Reference Frames:
Minkowski, (coordinates are stationary with objects moving through)
Schwarzschild, (coordinates are stationary with objects moving through.)
Milne (Coordinates are stationary with objects moving through)

Noninertial Reference Frames:
FLRW, (coordinates attached to bodies flying away from each other from the big-bang.)
Painleve, (coordinates attached to bodies falling into a black-hole)
Rindler (coordinates attached to a rigid body under constant acceleration)

So what I think is a reasonable definition of an inertial reference frame is that the coordinates themselves are not accelerating. I don't care what is happening to the objects. But I would like to hear the standard General Relativity definition.

15. Feb 6, 2012

### Staff: Mentor

Sounds to me like a pretty reasonable summary of the general relativistic view of what's going on.... Thereby demonstrating that if you poke around at the foundations of classical physics, you'll eventually move beyond classical physics.

I wouldn't say it's misleading, but it is pretty seriously unhelpful if you don't also have a decent understanding of the classical Newtonian view of the same situation. There's more insight to be had from comparing the two views than there ever will be from arguing which one is "right".

16. Feb 6, 2012

### D H

Staff Emeritus

Inertial frames are different in Newtonian mechanics and general relativity. General relativity is a locally realistic theory. Two circumstances are very much alike if there are no local experiments that distinguish one from the other. This is at the very heart of the equivalence principle, which in turn is at the very heart of general relativity.

Newtonian mechanics is a global theory. It implicitly postulates a universal inertial frame, aka God's frame, with respect to which all other inertial frames are neither accelerating nor rotating. There certainly are problems with this point of view. Nonetheless, it works in an amazing number of applications.

Have you heard of Einstein's elevator car thought experiment? If you haven't, google that phrase. I'll give a short synopsis.

Imagine you are in an elevator car (modern terminology: a spaceship) with no windows. Suppose you feel your normal weight on your feet when you stand, or on your rear when you sit. With no windows, how can you distinguish whether
(1a) the car is sitting still on the surface of a planet or
(1b) somewhere out in deep space accelerating at 1g?

Or suppose you find yourself floating about the car, apparently in a zero g environment. With no windows, how can you distinguish whether
(2a) the car is in orbit about some planet or
(2b) is just coasting along somewhere out in deep space?
Once again, the answer is that you can't.

In both situations, there is no local experiment that lets you distinguish between the two alternatives.

Now consider a reference frame centered on this elevator car. Is this an inertial frame or a non-inertial frame? General relativity and Newtonian mechanics agree on cases (1b) and (2b), but disagree on cases (1a) and (2a). GR says that (1a) is not an inertial frame; Newtonian mechanics says that it is. GR says that 2a is an inertial frame; Newtonian mechanics says it is not.

17. Feb 6, 2012

### JDoolin

Equivalence Principle. Hmmmm..

A general point I'd like to make, which is perhaps completely at odds with the equivalence principle, and perhaps the entire philosophy of General Relativity:

When two ideas are subtly different, our prerogative is not to claim that they are equivalent, but in fact, it behooves us to go to extra lengths to distinguish the two ideas, so that people don't confuse them.

Well, I can definitely say I am familiar with the question, at least.

From October 2010:
https://www.physicsforums.com/showpost.php?p=2953679&postcount=87

July, 2011

http://www.spoonfedrelativity.com/pages/Accelerating-Elevator.php

I'm aware that your argument is familiar and entrenched among General Relativity Experts, but it utterly fails to convince me.

Imagine you are trying to explain to me the difference between red and green, and I say there is no difference between the two, because when I close my eyes, I can see neither one. How does this differ, essentially, from your claim that acceleration and gravity are the same, because when we don't look outside, we can't tell the difference?

Any rational person can tell the difference between standing on the ground, and shooting off in a rocket. And, if he had any doubt which situation he were in, he would look outside to check.

Also important, but more technical
The acceleration due to gravity drops off as $\frac{G M}{r^2}$. This radius is generally pretty small.
while the acceleration of a born rigid platform drops off as $g=c^2/r$. And this radius is enormous.
Admittedly, it's not much of a difference in the height of a typical elevator, but with sensitive enough equipment, you might detect it.

Last edited: Feb 6, 2012
18. Feb 7, 2012

### D H

Staff Emeritus
Exactly. (Aside: I don't see how that is at odds with general relativity.)

What the OP has done is just the opposite. He went to extra lengths to conflate ideas from Newtonian mechanics and general relativity. The two frameworks are markedly different with regard to the nature of space, time, gravitation, and inertial frames. It is simply invalid to take those general relativistic concepts into a Newtonian framework. Inertial frames have infinite extent in Newtonian mechanics but are local (and of limited use) in general relativity. Gravitational acceleration results from the real gravitational force in Newtonian mechanics but is a pseudo force that results from using a non-inertial frame in general relativity. The two theories do not mix and match.

Suppose some future Einstein comes up with the theory of everything that melds general relativity and quantum mechanics, subtly modifying both along the way. With this theory, gravitation becomes some kind a quantum interaction. Would this falsify the GR POV that gravitation is a pseudo force and vindicate the Newtonian POV that gravitation is a real force?

The answer is a resounding NO! While general relativity does show that the Newtonian POV is not universally true, it also vindicates the Newtonian POV in the context of smallish velocities, smallish masses, and largish distances. It has to. By Einstein's time, hundreds of years of experiments had shown that, except for a few puzzling exceptions, Newtonian mechanics was "true". Any new theory must necessarily encompass the existing body of knowledge. Has as general relativity had to encompass Newtonian mechanics in those areas where Newtonian mechanics was well-tested, this new theory of everything must encompass general relativity in those areas where general relativity is well-tested.

What the OP fails to see is that there is a huge gap between universally false (e.g., phlogiston theory) and universally true (we have no such theory, yet). Newtonian mechanics lies somewhere within this gap. Presumably, so do quantum mechanics and general relativity. That the two are subtly at odds with another is prima facie evidence that neither is the universal truth.

19. Feb 7, 2012

### JDoolin

Mechanic, you seem to have two different ideas here.

• The NLM (which I'm not familiar with, by the way)
• the local-inertial-free-fall-frame of General Relativity.

Can you make the "most glaring problem with IRF's" argument without reference to the NLM, or is it fundamental to your argument? (In Newtonian Mechanics, it IS possible to go faster than the speed of light, so if your argument relies on Newtonian Mechanics being valid anywhere, ....)

Last edited: Feb 7, 2012
20. Feb 7, 2012

### D H

Staff Emeritus
I believe that Mechanic meant NLM to mean Newton's laws of motion.

It is of course an acronym of his own making.