How Do The Definition Of Inertial In Relativity Relate To General Mechanics

[hprog]

Hi all, I am struggling with the following problem, and so far I have not found a satisfying solution.

The problem is that the definitions of "Inertial" and "Non-Inertial" in relativity appear to me as being at opposites with the ones in Classical Physics (and of course I am probably missing something but my question is what?).
The definition of an inertial frame in which the principle of relativity takes place is defined in classical relativity textbooks (see "Spacetime Physics" by Taylor and Wheeler 1966 and many others) to be an object falling in gravity (such as a rocket, airplane or ship) in which according to the principle of equivalence it should be considered to be an inertial frame.
(Why can we not use directly a true inertial frame without any gravity at all? The answer on that appears to be that such ideal frame is not actually possible [or at least common] because there is gravity all around and any two object in existence will exert gravity on each other.)
On the other hand for the most part in relativity a gravitational frame is considered to be non-inertial (even if the frame is not in orbit or rotation) just because the gravitational force itself, again this is based on the principle of equivalence.

So far for relativity, let us see what general mechanics says on this matter.
In general mechanics any object falling in gravity have gravity as a force acting on it and therefore the object will accelerate towards the source of the gravity (provided that there is no air resistance) and as such the object must be considered non-inertial!
If the gravitational force is actually in orbit or rotation (such as the Earth or other planets and even stars) this is yet another reason for it being non-inertial.
Such objects are non-inertial themselves and can of course not be used as an "inertial frame of reference", since besides the requirement to be inertial it has another requirement and that is that all other inertial frames will appear to the frame as obeying the inertial law of motion, but in this case different objects falling in gravity will see each other as accelerating, and objects on Earth will see similar objects on the moon in rotation.
On the other hand for an object at rest on earth, although there is gravity acting upon the object but there is also an equal and opposite force (the third law of motion) that the Earth is causing on the objects, which balances out the gravitational force and as such the net force equals zero, and the objects remains at rest relative to the earth, thus the objects are inertial (in case of a gravitational field not in orbit).

So how do we relate the two definitions?

Here are I will list my attempts to solve this.

First I attempted to answer that the inertial definition of relativity is only for an object that meets two requirements, 1) that it is in terminal velocity (since the air resistance is balancing out the gravitational force) 2) that the gravitational field is not in orbit or rotation.
However I am not currently satisfied with such an answer for the following reasons:
1) In such case the problem with the objects at rest in the gravitational field remains.
2) There is no way to distinguish with an internal experiment between an object in terminal velocity and an object that is not.
3) More over since objects in terminal velocity and objects that are not in terminal velocity are obeying the same physics laws (and similar can be said that object that the gravitational field is orbiting are also obeying the same physics laws), and as such it will follow that inertial and non-inertial frames are equivalent, something that does not comply with the principle of equivalence.

On the other hand I attempted to answer that the definition for non-inertial for a gravitational frame is only when the gravitational field is in orbit or rotation.
However I am not currently satisfied with such an answer for the following reasons:
1) In such case the problem with the objects falling in the gravitational field remains, as they cannot be considered as inertial.
2) Since objects in a gravitational field in orbit and objects in a gravitational field that is not in orbit are obeying the same physics laws, and as such it will follow that inertial and non-inertial frames are equivalent, something that does not comply with the principle of equivalence.But recently I started to think that the principle of equivalence does not necessarily have to be interpreted as saying that equivalent behavior must also be considered to have the same inertial and non-inertial status, and we might able to say that although gravity and acceleration behave equal and have similar laws of physics they are still not the same when considering the inertial status.
In fact there are ways to distinguish between gravity and acceleration such as tidal gravity, and it might be that the same is with the inertial status.
However if this is the case then we can no longer use the principle of equivalence to say that an object falling in gravity is considered inertial since we just interpreted the principle of equivalence of not saying this at all.
If so then the question remains what is the inertial frame of relativity? and why do the classical relativity textbooks consider object falling in gravity to be inertial?

We may again try to use terminal velocity in a gravity field that is not in orbit, but if so why not use objects at rest in the gravity field?
But the real problem is that since there is gravity everywhere there is no gravity field that is not in orbit and if we are to claim that we can find one then why not go back to the simple definition of relativity in which objects are just moving in uniform motion without any gravitational field involved at all? Help is needed, and satisfying answers are appreciated.
Thanks in advance.

 

[Dale]

So how do we relate the two definitions?

You are correct, the definitions are fundamentally different. The definitions disagree whenever there is a gravitational force in the Newtonian analysis. So you do not relate them to each other. You simply make sure that you understand the differences between the two different definitions.

First I attempted to answer that the inertial definition of relativity is only for an object that meets two requirements, 1) that it is in terminal velocity (since the air resistance is balancing out the gravitational force)

No, such an object is NOT inertial in relativity. The air resistance is an unbalanced real force. An onboard accelerometer will not read 0.

The easiest way to determine if a frame is inertial in relativity is to determine what an accelerometer at rest in that frame would read. If it would read 0 then the frame is inertial, otherwise it is non-inertial.

 

[bcrowell]

The basic issue here is that frames of reference are global in Newtonian mechanics, local in GR.

The Newtonian distinction between inertial and noninertial frames is inherently global. E.g., people aboard the space station can't tell by measurements carried out inside the station that the frame of reference tied to its walls is one that Newton would consider as being noninertial. Only by observing the average state of motion of the universe on large scales can you determine what constitutes an inertial frame according to Newton. E.g., the frame of the "fixed stars" would be considered by Newton to be an excellent approximation to an inertial frame.

In the context of GR, due to cosmological expansion there is clearly no way to carry out a determination of the average state of motion of the universe on large scales. Therefore GR can only have a locally defined distinction between inertial and noninertial frames. Since astronauts aboard the ISS can't tell by local measurements that the station is falling, GR has to treat the frame tied to the walls of the station as inertial.

 

[hprog]

You are correct, the definitions are fundamentally different. The definitions disagree whenever there is a gravitational force in the Newtonian analysis. So you do not relate them to each other. You simply make sure that you understand the differences between the two different definitions.

So can you explain the differences and give me the formal definitions for inertial and non-inertial in relativity vs general mechanics, or at least provide me a reference.

What I have found ("Six Ideas That Shaped Physics Unit R" by Thomas Moore) that the definition of an inertial frame in relativity is that any isolated object will be observed as obeying the inertial law.
However this will not be with objects falling in gravity since different objects falling in gravity will see each other as accelerating and objects falling on Earth will see objects falling on the moon as orbiting and rotating.
But anyway the definition that Moore describes complies with classical physics, or maybe I am wrong?

Can you answer me this? or if you think he is wrong then can you give me the right definition and a reference?
Thanks in advance.

 

[bcrowell]

What I have found ("Six Ideas That Shaped Physics Unit R" by Thomas Moore) that the definition of an inertial frame in relativity is that any isolated object will be observed as obeying the inertial law.
However this will not be with objects falling in gravity since different objects falling in gravity will see each other as accelerating and objects falling on Earth will see objects falling on the moon as orbiting and rotating.[...]

Your counterexamples are nonlocal. Frames of reference in relativity are local. See #3.

 

[hprog]

Your counterexamples are nonlocal. Frames of reference in relativity are local. See #3.

The definition from Moore is for relativity directly.
Actually Einstein writes similar in his book on Relativity (Relativity the Special and General Theory, 1925) "If a mass m is moving uniformly in a straight line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system K', provided that the latter is executing a uniform translatory motion with respect to K".

But so far I do not understand how the difference between local and global inertial frames are solving my problem (not the ones from the initial post and not the one of post #4).
My question is that although a falling objects are simulating inertial frames they still have gravity acting as a force on them, and as such they are not inertial at all.
And similar is with objects resting on earth.
And the fact that we are looking for a local frame does not make it inertial/non-inertial any more than a flight simulation is to be considered flying.

(One of my attempts to solve this problem was that according to GR there might be no longer a need to classify gravity as a force.
However if so then the problem that I presented witht objects resting in gravity might also be considered inertial is now becoming even greater, since gravity is not a force at all.)

And as for the problem of post #4 that other objects falling are not observed to be inertial, can also not be solved by claiming local since after all the rule is violated.
Are you supposed now to say that the Earth might be claimed to be at rest? after all the proof of the orbiting Earth is based on stellar parallax and stellar aberration, clearly non-local.
If you are to say this then the Michelson-Morley experiment no longer proves relativity, as the Earth might be at rest.

But post #4 also deals with local objects as well since two objects falling on Earth next to each other might see each other as accelerating and in this case they are clearly local to each other.
And if you claim that this is non-local then how are we supposed to find two reference frames in uniform motion to each other?
(Your only solution would be with two objects moving uniformly within the same falling rocket, however you do not have now "frames" in uniform motion since the frame is the rocket)

I would appreciated if anyone would provide me with a reference to the exact definitions of inertial that are solving the above problems.

Thanks in advance.

 

[I_am_learning]

I personally find it difficult to accept the following fact.
Someone isn't in inertial frame beacuse at that frame the accelerometer doesn't read 0 which inturn occurs because some force is being applied only to its outer-body/frame and not through out the whole inner body as gravity would have done.

 

[DrGreg]

But post #4 also deals with local objects as well since two objects falling on Earth next to each other might see each other as accelerating and in this case they are clearly local to each other.

That isn't true. If you are falling freely under gravity and you observe another falling object near to you, it will move at constant velocity relative to you. This is true only locally as an approximation; objects further away may well be accelerating relative to you. It is in this sense we refer to a "locally inertial" frame. In general relativity there are no frames that are truly inertial everywhere, only frames which locally approximate an inertial frame in a small region.

 

[Dale]

I would appreciated if anyone would provide me with a reference to the exact definitions of inertial that are solving the above problems.

Here are a few

http://www.zweigmedia.com/diff_geom/Sec9.html
http://people.sissa.it/~rezzolla/lnotes/virgo/node3.html
http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.6
http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.7

 

[hprog]

That isn't true. If you are falling freely under gravity and you observe another falling object near to you, it will move at constant velocity relative to you. This is true only locally as an approximation; objects further away may well be accelerating relative to you. It is in this sense we refer to a "locally inertial" frame. In general relativity there are no frames that are truly inertial everywhere, only frames which locally approximate an inertial frame in a small region.

Your argument would be true if all objects were in constant acceleration, however because of air resistance it is not the case.
Consider two objects falling in gravity, one has already reached the terminal velocity while the other is still accelerating, it is clear that the second object will not be observed as being in uniform motion.
But even if the objects are farther away in which case you consider them to be in non-local and thus non-inertial say if we have two such objects far apart but both are falling in gravity and both are in terminal velocity with the same velocity, so in such case they will see each other as being in uniform motion, then why according to you they consider each other as non-inertial just because they are non-local to each other?

 

[hprog]

Here are a few

http://www.zweigmedia.com/diff_geom/Sec9.html
http://people.sissa.it/~rezzolla/lnotes/virgo/node3.html
http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.6
http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.7

Thanks, I will take a look in this links.
Actually I am very curious to know how objects at rest on Earth are considered to be in non-inertial state with the net force still being zero.

 

[Dale]

Your argument would be true if all objects were in constant acceleration, however because of air resistance it is not the case.

DrGreg is correct. If there is air resistance then the object is not in free fall, by definition.

 

[DrGreg]

Your argument would be true if all objects were in constant acceleration, however because of air resistance it is not the case.

DrGreg is correct. If there is air resistance then the object is not in free fall, by definition.

Yes. When I said "free-fall" I should have said "free-fall in vacuum" and with no other external forces either. Anything experiencing air resistance or any other form of friction is not moving inertially.

 

[GrayGhost]

Actually I am very curious to know how objects at rest on Earth are considered to be in non-inertial state with the net force still being zero.

Hello hpfrog,

Nice to make your aquaintance. I've asked the very same questions, and many just like them, many years ago. I still like to ponder them. I hope this helps you ...

Let's say the Earth did not rotate, and that it travels in (a virtual) uniform translatory motion far from gravity source. The reason you are non-inertial when you stand on said earth, is because "you feel your own weight". As you pointed out, the molecular forces of the Earth you stand on precisely matches the force that demands your downward-freefall. The 2 opposing forces act on your body and compress the molecules of your body (top-to-bottom), hence you feel weight.

In free space far from gravity source, your ship burns its thrusters ... the ship moves fwd and your body doesn't (at first). Then your body collides with the chair-back you sit in, which is fixed to the ship. The molecules of your body are compressed (front-to-back) and you feel the G-force (which again is a weight), because your body is then forced to move with the ship. Anytime your body feels weight, it is resisting a change in its current state of motion, and IOWs experiences an inertial force. Only when your body "feels no inertial force", are you inertial. Floating in deep space, freefalling to Earth (before hitting atmosphere), and orbiting the earth, are examples of inertial motion under the context of general relativity.

In all these cases, the astronaut feels no weight. The "equivalence principle" says that standing stationary in a gravity field is equivalent to accelerating via rocket burn in flat spacetime ... and these 2 cases are examples of being non-inertial.

GrayGhost

 

[Dale]

Actually I am very curious to know how objects at rest on Earth are considered to be in non-inertial state with the net force still being zero.

The net force is not zero. By attaching an accelerometer you can easily determine that there is a net force pointing upwards.

 

[hprog]

OK, thanks all of you for the replies, and I think I already got the idea.
The point here is that the definitions of inertial and non-inertial in relativity have nothing to do with general mechanics any more than sharing vocabulary.
While in Newtonian physics an object is inertial when the net force is zero, and as such objects at rest on Earth are inertial while objects falling are not, instead in relativity inertial is when an objects does not feel a force acting on it although a force might actually be acting on it without the object feeling it, and as such objects falling in gravity are inertial but objects resting on Earth are not.
This is similar to what I proposed in my initial post

But recently I started to think that the principle of equivalence does not necessarily have to be interpreted as saying that equivalent behavior must also be considered to have the same inertial and non-inertial status, and we might able to say that although gravity and acceleration behave equal and have similar laws of physics they are still not the same when considering the inertial status.
In fact there are ways to distinguish between gravity and acceleration such as tidal gravity, and it might be that the same is with the inertial status.
However if this is the case then we can no longer use the principle of equivalence to say that an object falling in gravity is considered inertial since we just interpreted the principle of equivalence of not saying this at all.
If so then the question remains what is the inertial frame of relativity? and why do the classical relativity textbooks consider object falling in gravity to be inertial?

However I just did not realized in my initial post that this is what relativity is considering inertial, and as such everything works out just fine.
Thanks all of you for your help.