Problems with Inertial Reference Frames

[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.)

 

[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.

 

[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.

 

[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.

 

[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.

 

[D H]

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.

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.

 

[Philip Wood]

Where does he mix and match?

 

[JDoolin]

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.

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.

 

[D H]

Where does he mix and match?

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.

As established elsewhere, gravitational acceleration is not caused by a force.

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.

Reference frames in free fall are IRFs yet they can clearly be said to be accelerating relative to the Earth.

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.

 

[Mechanic]

He mixed and matched concepts from Newtonian mechanics and general relativity. Inertial frames are quite different in the two.

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.

 

[JDoolin]

A free-fall frame, whether or not in the vicinity of a massive body, is a (locally) inertial frame in general relativity.

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?

 

[Mechanic]

Is the object in free-fall near a massive body accelerating, or not?

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.

 

[JDoolin]

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.

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?

 

[JDoolin]

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.

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.

 

[Nugatory]

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.

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.

Does anybody except me think that sounds a little misleading?

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".

 

[D H]

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?

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?
Answer: You can't.

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 let's 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.

 

[JDoolin]

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.

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.

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?
Answer: You can't.

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 let's 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.

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
https://www.physicsforums.com/showthread.php?t=510985&page=2http://www.spoonfedrelativity.com/pages/Accelerating-Elevator.phpI'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.

 

[D H]

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.

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.

 

[JDoolin]

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.

Inertial frames in general relativity are local. A free-fall frame is inertial in general relativity,

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, ...)

 

[D H]

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

It is of course an acronym of his own making.

 

[Ken G]

To me, the problem here that D_H is addressing is that any statements we make about inertial frames must first choose a theory that we are working in. This is a crucial issue that people often forget-- they treat physics terms as if they were statements about reality, like they were trying to define what an inertial frame "really is." That's not the case-- theories are what give our language meaning, so for physics terms to have the correct meaning, the theory giving them meaning must first be identified. So if we define an inertial frame as one where the postulates of some theory apply (that is how it is defined in Newtonian physics, using Newton's three laws), then the definition relies on the theory and is only as good as the theory.

Hence, Mechanic's criticism of the definition of an inertial frame is valid, but it is no more valid than the same criticism of Newton's three laws-- there is no pedagogical problem there, it's just that Newton's laws aren't perfect. Hence, D_H is right that there is no problem in need of fixing there-- Newton's laws aren't exactly right, so the definition of an inertial frame that gets used with Newton's laws isn't exactly right either (nor is the idea that constant rectilinear motion maps from inertial frame to inertial frame, that just isn't right but it is the meaning that attaches to Newton's laws). The Newtonian approach to inertial frames is useful in many contexts, and so is that definition of an inertial frame, but neither work in general-- they break down when speeds approach c, and then gravity destroys the Newtonian concept of global inertial frames, forcing us to adopt a purely local concept of what inertial motion is.

 

[JDoolin]

(1a) the car is sitting still on the surface of a planet or
(1b) somewhere out in deep space accelerating at 1g?
Answer: You can't.

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 let's 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.

Mechanic said, "IRFs are only conceptual approximations of reality valid only in infinitesimally small volumes." We could argue all day about what Mechanic meant by this, but it isn't really his responsibility to explain what YOU mean by an IRF. The point is, if we apply his argument to what you are describing as inertial reference frames, it appears to be a valid concern.

Question: How does GR justify saying that 2a is an inertial frame?
Answer: By defining an inertial frame to be something that is only valid locally.
Question: How locally?
Answer: infinitesimally small volumes
Question: What does that mean?
Answer: Valid only along a single worldline in space-time; i.e. a single point in space.

Do you agree with those answers?

 

[JDoolin]

theories are what give our language meaning, so for physics terms to have the correct meaning, the theory giving them meaning must first be identified.

I'm in a room. I want to "look outside."

I would like a theory that let's me "look outside."

If a theory of physics is based on the premise that I cannot look outside, then the problem is not with my language. The problem is with the theory.

 

[AlephZero]

I'm in a room. I want to "look outside."

I would like a theory that let's me "look outside."

If a theory of physics is based on the premise that I cannot look outside, then the problem is not with my language. The problem is with the theory.

Let's change your argument a little bit, to see what's wrong with it:

I'm in a spaceship. I want to travel faster than c.

I would like a thery that let's me travel faster than c.

If a theory of physics is based on the premise that I cannot travel faster than c, then the problem is not with my language. The problem is with the theory.

Nope. Not if you want your theory to match the observable universe, anyway.

 

[JDoolin]

Let's change your argument a little bit, to see what's wrong with it:
Nope. Not if you want your theory to match the observable universe, anyway.

Okay, except for AlephZero who has apparently never looked outside before, is there anyone else who can find something wrong with my argument?

 

[Ken G]

I'm in a room. I want to "look outside."

I would like a theory that let's me "look outside."

If a theory of physics is based on the premise that I cannot look outside, then the problem is not with my language. The problem is with the theory.

No theory is based on the premise of not being able to look outside, but the device of not looking outside is merely a way to describe what a particular theory is saying. It almost sounds like you want your theories to be all things to all people, but physics just doesn't work that way. What you are saying is right-- in GR, inertial frames are local, and there does not appear to be any possible exact theory in which inertial frames are global. But in Newtonian physics, they are global, so when Newtonian physics is being described, this language of inertial frames being used. This is not an error in the definition of inertial frames, it is just a difference between two theories of motion. One can criticize the Newtonian approach because it breaks down, but one should really expect that of any theory.

 

[D H]

I am violating my rule #1, don't post while you're upset. But I do need an outlet. That stupid company from Redmond WA just cost me two hours of work. Ouch! I just stepped in a huge stinking pile of Microstuff.

So I'll just take a deep breath, relax, and write about physics.

Mechanic said, "IRFs are only conceptual approximations of reality valid only in infinitesimally small volumes." We could argue all day about what Mechanic meant by this, but it isn't really his responsibility to explain what YOU mean by an IRF. The point is, if we apply his argument to what you are describing as inertial reference frames, it appears to be a valid concern.

No, it isn't, at least not the way he wrote it. He took the general relativity concept of inertial frames over to the Newtonian mechanics world. That is simply invalid. Just because both theories use the word "inertial frame" does not mean that the things called "inertial frames" in general relativity can be used as "inertial frames" in Newtonian mechanics. They are very different things. All that mixing and matching does is to lead to confusion. It does not lead to insight.

Mechanics thinks, and apparently so do you, that inertial frames are something real. They aren't. You can't touch one, you can't take a picture of one. They are mathematical abstractions that we use in our attempts to describe physical reality. They are a map, not the territory. Saying they are real is confusing the map for the territory.If you want to compare two scientific theories, you first need to find some experiments for which each theories will predict some outcome. Then you need to determine the outcomes as predicted by each theory. In determining the outcome predicted by one of those theories you need to work within the framework of that theory. Then you need to switch gears and determine the outcome as predicted by the opposing theory. You cannot mix and match. You need to stay faithful to the theory at hand, then switch to the alternate theory and once again make the computations per the framework of that theory.

In some cases such as Lorentz ether theory versus special relativity you will find that the predicted outcomes are always identical. Choosing between such theories is largely a matter of aesthetics. That is not the case here. Where Newtonian mechanics and general relativity do disagree, it is always general relativity that comes out on top. We have yet to find a situation where general relativity yields an incorrect prediction. General relativity, at least for now, appears to be universally true.

So why do physics instructors still teach Newtonian mechanics? The answer lies in the fact that the two theories do not always disagree, at least not in the context of experimental error. Where they do agree, it is Newtonian mechanics that wins hands down with regard to ease of computation. Those places where they do agree are exactly those situations that we humans normally run up against in our everyday world. This is why it is still valid to teach Newtonian mechanics.

Question: How does GR justify saying that 2a is an inertial frame?

Another place where Mechanic was wrong was in saying that Newtonian mechanics is only valid in inertial frames of reference. People from d'Alembert on have been using pseudo forces to model physics from the context of Newtonian mechanics and from the perspective of a non-inertial frame. One key distinguishing factor between inertial and non-inertial frames is that the laws of physics take on their simplest forms when expressed with respect to an inertial frame of reference. Those pseudo forces just vanish in an inertial frame.

A key distinguishing feature between real forces and pseudo forces is that a pseudo force is always proportional to the mass of the object on which the pseudo force is acting. Now look at gravitation. Gravitational force is always proportional to the mass of the object. Gravitational acceleration can be transformed by choosing the right coordinate system. When you do that, the laws of physics (at least locally) take on their simplest form. In many regards gravitation looks exactly like a pseudo force. In general relativity, it is a pseudo force. A free-fall frame is inertial in general relativity because (at least locally) that is where the laws of physics take on their simplest form.

 

[JDoolin]

Mechanics thinks, and apparently so do you, that inertial frames are something real. They aren't. You can't touch one, you can't take a picture of one. They are mathematical abstractions that we use in our attempts to describe physical reality. They are a map, not the territory. Saying they are real is confusing the map for the territory.

I think what you mean is that inertial frames do not exist within the context of some common interpretations of General Relativity. And I think I agree with you there, because within the context of General Relativity inertial frames are defined as "freely falling." And since those two words "inertial" and "falling" are essentially opposite, of course, no such thing exists.

However, within a broader context, an inertial reference frame is simply a coordinate system where the coordinates are not moving around over time and the origin is not accelerating. If you are trying to say that no such coordinate system exists, then that's where we disagree.

 

[JDoolin]

A free-fall frame is inertial in general relativity because (at least locally) that is where the laws of physics take on their simplest form.

Okay, I will concur with that. When you zoom in close enough on any phenomenon, for a short enough period of time, all of the motion will appear to almost unaccelerated, and almost in straight lines.

But there are a couple of problems with this. (1) observers cannot choose their scale arbitrarily. I can't decide to become small enough and slow enough to perceive the motion of a baseball as a straight line. (2) Observer's don't see every phenomenon in this zoomed in state. They see the phenomena from the distance, where objects go into orbit, or take roughly parabolic paths. (3) No matter how close you zoom in, the word "almost" is still there. "almost" unaccelerated. "almost" straight lines. You cannot zoom in far enough to make the lines perfectly straight.

The one possible exception to this is if I attach the camera to the object, and point the camera at the object, so nothing else is visible, then in that one case, the object will appear to be unaccelerated. So there is a zoom-level where the motion actually does appear unaccelerated. So, this is essentially where the laws of physics take on their simplest form... When you consider an object which looks only at itself.

 

[JDoolin]

It almost sounds like you want your theories to be all things to all people, but physics just doesn't work that way. What you are saying is right-- in GR, inertial frames are local, and there does not appear to be any possible exact theory in which inertial frames are global.

General Relativity Experts believe that there is no possible exact theory in which inertial frames are global;

To me, that statement seems logically equivalent to saying there is no possible exact theory which predicts what you can see when you "look out the window."

This is my problem; that whenever I do a Lorentz Transformation, General Relativity Experts will claim that it is not possible. When I point out that I just did it, so it must be possible, they will continue to claim there is no possible exact theory in which inertial frames are global.

So I don't need GR to be all things to all people. I just need GR to stick to the incredibly tiny domain where it is valid, which is a zero-volume space.

 

[D H]

This is my problem; that whenever I do a Lorentz Transformation, General Relativity Experts will claim that it is not possible. When I point out that I just did it, so it must be possible, they will continue to claim there is no possible exact theory in which inertial frames are global.

So I don't need GR to be all things to all people. I just need GR to stick to the incredibly tiny domain where it is valid, which is a zero-volume space.

Just because you have problems with general relativity does not mean it is wrong. Just because you want the universe to be simple and to be absolutely realistic does not mean that it is.

The fact is that general relativity is the best model of scientific reality we have. It is still a locally realistic theory. The windmill you should be tilting at is quantum mechanics, which is not even locally realistic. Bell's theorem.

You'd still be tilting at windmills, however.

 

[Ken G]

So I don't need GR to be all things to all people. I just need GR to stick to the incredibly tiny domain where it is valid, which is a zero-volume space.

I would say your problem here is conflating the usefulness of GR with the usefulness of the concept of global inertial frames. They aren't at all the same-- one works, the other doesn't. It's not GR's problem that the global inertial frame idea falls apart, indeed this is a feature of GR not a bug, and seems to be a crucial feature to match reality, "looking outside." I am in complete agreement with what D_H is saying also.

 

[JDoolin]

Just because you have problems with general relativity does not mean it is wrong. Just because you want the universe to be simple and to be absolutely realistic does not mean that it is.

The fact is that general relativity is the best model of scientific reality we have. It is still a locally realistic theory. The windmill you should be tilting at is quantum mechanics, which is not even locally realistic. Bell's theorem.

You'd still be tilting at windmills, however.

I'm not the one who said that physics has to be simple. That was you who suggested that we need to zoom-in to a level where the physics is simple. Remember, I want to zoom out to where physics has some complexity.

I am not saying I want the universe to be simple. I am saying that a theory should be compatible with the theories from which it is supposedly derived. If General Relativity is derived from Special Relativity, it should have at least some circumstance where the use of a Lorentz Transformation is permitted.

I'm not asking for something to be "realistic" because I don't know what you mean by realistic. What I would probably suggest instead is a rubric to evaluate a theory. Rather than just say "General Relativity is the best model of scientific reality we have" we need to actually break down General Relativity into the hundreds or thousands of ideas that it encompasses.

Because you have all these things in General Relativity

• Coordinate free General Relativity
• Schwarzschild Coordinates
• Rindler Coordinates
• free-falling-inertial-reference-frames
• Gravitational Lensing
• Parallel Transport
• Painleve Coordinates
• FLRW Coordinates
• Dark matter and dark energy
• de Sitter Universe
• Einstein Field Equations
• Gravitational Time Dilation

I don't have any desire to charge in and try to destroy the whole of General Relativity. There is a lot of good stuff in there. But those parts of General Relativity that are based on this concept that "Global Inertial Reference Frames Don't Exist" need to be surgically removed. Because the correct statement is that "Global Inertial Reference Frames are Observer Dependent."

Having said that, let's go back to this statement:

Just because you want the universe to be simple and to be absolutely realistic does not mean that it is.

Now, isn't it the assumption of General Relativity that there IS a mathematical model to describe the universe? My argument is not that the universe should be simple, but that the universe should be self-consistent, and shared. If an event happens in your universe at some place at some time, that same event must happen in my universe at some place, at some time.

Where GR goes wrong is that you say that if an event happens in your universe at some specific place at some time, it does NOT necessarily happen at any specific place and time in my universe. This is the point I disagree with.

I think if specific places and times can be established for events according to ONE observer, then specific places and times can be established for events according to ANY observer.

 

[Ken G]

If General Relativity is derived from Special Relativity, it should have at least some circumstance where the use of a Lorentz Transformation is permitted.

It does-- over any scales where the changes in gravitational potential are suitably much less than c². But note the key point here-- the length scale is limited, so again we find that the concept of an inertial frame has a necessarily local quality, or at least not completely global (as in cosmology, for example).

Rather than just say "General Relativity is the best model of scientific reality we have" we need to actually break down General Relativity into the hundreds or thousands of ideas that it encompasses.

It doesn't encompass that many ideas, the theory is based on a small set of postulates and ideas. That's why it is such a good theory (along with its spectacular accuracy). But this also means that applying the theory requires idealizations-- and as with any theory, this is a feature not a bug. Note also that a long list of different types of coordinates has very little to do with general relativity, any more than Cartesian vs. polar coordinates has much to do with Newton's laws. The whole point of GR is to be a theory whose "ideas" can be expressed in completely coordinate-free form. That is what makes it a fully objective theory (which, by the way, Newton's laws are not, in particular the first law).

I don't have any desire to charge in and try to destroy the whole of General Relativity. There is a lot of good stuff in there. But those parts of General Relativity that are based on this concept that "Global Inertial Reference Frames Don't Exist" need to be surgically removed. Because the correct statement is that "Global Inertial Reference Frames are Observer Dependent."

No, that is not the correct statement, and for two reasons:
1) there is usually no global inertial frame associated with any observer, because of the presence of horizons. For example, an accelerating observer has a Rindler horizon, and a spinning observer has a horizon at the distance where the angular speed is c. For the Earth, for example, that is roughly the distance of Jupiter, so the "global inertial frame" of an Earth-bound observer cannot even encompass the entire solar system.
2) worse, it is against the genius of GR to encorporate fundamentally observer-dependent concepts into the structure of the theory. To understand GR (and SR for that matter), it is necessary to recognize the importance of the difference between what is an objectively supportable statement about the nature of some situation, which must be expressed in invariant form, versus what is just a matter of coordinates, which is like a word that sounds different in English and Italian. In English, we have the word "love", in Italian, "amore". The words sound totally different, so in your approach to the concept of love, we would have the statement that love is language dependent because amore sounds completely different. However, the whole point of the concept of love is that it ought to be there no matter what language you use, or even if you have invented language at all. When the same cannot be said about the concept of a global inertial frame, it exposes the fact that such a concept is not a physically real object that should appear in any theory of physics. Rather, it is simply a matter of coordinates, which is important in the practice of getting useful numbers, but has no place in any theory of physics. Indeed, that is pretty much the breakthrough realization that underpins all of relativity.

Where GR goes wrong is that you say that if an event happens in your universe at some place at some time, it does NOT happen at any specific place and time in my universe.

GR says no such thing, nor does this claim have anything to do with the concept of a global inertial frame. You are confusing "happening at a place and time" with "being able to be given coordinates that exist in some particular global system." Those are just not the same thing.

 

[JDoolin]

1) there is usually no global inertial frame associated with any observer, because of the presence of horizons. For example, an accelerating observer has a Rindler horizon, and a spinning observer has a horizon at the distance where the angular speed is c. For the Earth, for example, that is roughly the distance of Jupiter, so the "global inertial frame" of an Earth-bound observer cannot even encompass the entire solar system.

Does General Relativity categorize the reference frame of an accelerating observer as an "inertial reference frame?"

Does General Relativity categorize the reference frame of a rotating observer as an "inertial reference frame?"

Does General Relativity categorize the reference frame of a free-falling observer as an "inertial reference frame?"

Does General Relativity have a word for the reference frame of an observer who is NOT accelerating, NOT rotating, and NOT free-falling?

(By the way, these aren't rhetorical questions. I'd really like a simple yes or no answer.)