# Non inertial or inertial reference frame?

## Homework Statement

Specify and explain whether the following is an inertial or non-inertial observer: An observer is placed on a rock between Andromeda and the Milky Way.

N/a

## The Attempt at a Solution

So here is my understanding, the observer would be situated within an inertial reference frame as it is not specified whether there is an acceleration relative to either of the galaxies, my curiosity is that since the Andromeda galaxy is traveling towards the Milky Way does this mean that the observer is in fact accelerating relative to Andromeda? Or is the observer still stationary relative to Andromeda? Bit confused, I'd be more than grateful for any clarification.

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gneill
Mentor
The observer is in free-fall. Suppose he was enclosed in a sealed lab and could conduct any experiments he wished entirely within that lab (no reference to anything outside the lab). Can you think of any experiment he might perform that would determine whether or not his lab frame of reference is inertial or not?

Erm maybe place a ball somewhere in the lab and see whether it moves or not?

gneill
Mentor
Erm maybe place a ball somewhere in the lab and see whether it moves or not?
And what would happen?

Well if the lab was in an inertial frame then the ball would stay put, if it was non inertial then the ball would move in the opposite direction to the net force on the lab no?

gneill
Mentor
Well if the lab was in an inertial frame then the ball would stay put, if it was non inertial then the ball would move in the opposite direction to the net force on the lab no?
True. But what do you say will happen to the ball in this particular case? Will the ball stay put or not?

True. But what do you say will happen to the ball in this particular case? Will the ball stay put or not?
Sorry, i think i've misunderstood the 'case'. Not sure how you can specify what happens to the ball if the frame of reference is unknown? I guess it just stays put because its frame is not relative to anything? Its a similar case as a particle in space with no other particles in existence right? the particle does not move because there is no relativity.

gneill
Mentor
Sorry, i think i've misunderstood the 'case'. Not sure how you can specify what happens to the ball if the frame of reference is unknown? I guess it just stays put because its frame is not relative to anything?
The idea is to use what you know about the situation to deduce whether or not the frame of reference can be considered to be an inertial one. For the ball to move with respect to the lab there would have to be some motion that ball partakes in that is different from the motion that the lab itself pursues. In this case you know that the lab is in free-fall in whatever small and locally uniform gravitational field that exists this far away from any large bodies.

How do bodies of different mass 'fall' in a uniform gravitational field?

The idea is to use what you know about the situation to deduce whether or not the frame of reference can be considered to be an inertial one. For the ball to move with respect to the lab there would have to be some motion that ball partakes in that is different from the motion that the lab itself pursues. In this case you know that the lab is in free-fall in whatever small and locally uniform gravitational field that exists this far away from any large bodies.

How do bodies of different mass 'fall' in a uniform gravitational field?
They would accelerate in the field relative to whatever mass is creating the field, so they would be within a non-inertial reference frame? I'm still not sure of the answer to my initial question, an observer is stationed on a rock between 2 galaxies. This is all the information i'm given. I imagine the galaxies are both having a gravitational effect on the rock? So does it accelerate towards one of the galaxies?

gneill
Mentor
They would accelerate in the field relative to whatever mass is creating the field, so they would be within a non-inertial reference frame? I'm still not sure of the answer to my initial question, an observer is stationed on a rock between 2 galaxies. This is all the information i'm given. I imagine the galaxies are both having a gravitational effect on the rock? So does it accelerate towards one of the galaxies?
While it may be true that the laboratory is accelerating with respect to the galaxies as it falls in the net gravitational field, the laboratory is hermetically sealed and the observer cannot perform any observations of anything outside the lab (no windows, no doors, no cameras mounted outside, no contact with outside).

Inside the lab, will the ball move with respect to the lab? Do things fall at different rates and/or directions in a uniform gravitational field?

Oh right, yeh think i get what you were getting at now, yeh the ball WILL move with respect to the lab in the field. Still not sure how i relate this to my question though, so do i treat the rock as the lab and the observer as the ball... The observer moves relative to the rock? But then how do i know if hes inertial or not?

gneill
Mentor
Oh right, yeh think i get what you were getting at now, yeh the ball WILL move with respect to the lab in the field. Still not sure how i relate this to my question though, so do i treat the rock as the lab and the observer as the ball... The observer moves relative to the rock? But then how do i know if hes inertial or not?
Why would the ball move with respect to the lab? The lab is not being held stationary in space is it? I thought it was sitting on a rock that's in free-fall; just 'floating' in space.

Why would the ball move with respect to the lab? The lab is not being held stationary in space is it? I thought it was sitting on a rock that's in free-fall; just 'floating' in space.
Ok, i think the lab analogy has confused me a bit, so basically the lab is stationary in space, not accelerating relative to anything therefore the ball in the lab cant be moving with respect to the lab since the ball is just sitting there as there are no net forces on the ball? So the frame of reference of the ball is inertial?

Erm I just thought I'd clarify what the definition of an inertial reference frame is, since the OP has used terms/reasoning which implies that they haven't got a strong grasp on it. From the Wikipedia article:
"An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed."

I italicised the part I felt was important (i.e. things can accelerate according to an inertial frame, but only if a force is acting on it). Also, think of the 'frame of reference' as being a pair of eyes; they have to see the particle (that is not acted on by a force) move in a straight line.

gneill
Mentor
Ok, i think the lab analogy has confused me a bit, so basically the lab is stationary in space, not accelerating relative to anything therefore the ball in the lab cant be moving with respect to the lab since the ball is just sitting there as there are no net forces on the ball? So the frame of reference of the ball is inertial?
Okay, the first thing to grasp is that there is no such thing as 'stationary in space'. This is not because nothing can 'hold still', but because there is no absolute frame of reference that we can call 'space'. That is, there is no preferred frame of reference to which we can assign the designation "absolutely no motion", or "absolute rest". All frames are relative.

In your proposed experiment the ball will NOT move with respect to the lab. Both the lab and the ball will move IDENTICALLY in any uniform background gravitational field. Remember Galileo's dropping of objects of different mass? They took the same time to fall, right?

The point is, the present experiment will not detect that your 'guy on a rock' comprises a non-inertial frame of reference. And in fact, there is NO experiment that an observer in such a situation can do that would distinguish his frame of reference from an inertial frame of reference whether he uses masses, sound, light, chemistry, or anything else. So if the frame of reference is indistinguishable from an inertial frame, what must you conclude?

For future reference, any free-falling body in space follows a path that is called an inertial trajectory I don't think that a frame being accelerated by gravity constitutes an inertial reference frame, and I also think that this means that there can not possibly be an inertial frame in our universe, at least not according to classical physics (i.e. gravitational force acts instantaneously regardless of distance).

Just because the ball remains at rest relative to the lab frame, it doesn't mean that the lab is inertial; the ball is only at rest relative to the lab because it has a force (from gravity) acting on it such that it maintains at rest, but if you were to remove that gravitational force (and hence have it undergo true inertial movement), it will no longer be stationary according to the lab frame anymore (because the lab is being accelerated by the gravity).

gneill
Mentor
I don't think that a frame being accelerated by gravity constitutes an inertial reference frame, and I also think that this means that there can not possibly be an inertial frame in our universe, at least not according to classical physics (i.e. gravitational force acts instantaneously regardless of distance).

Just because the ball remains at rest relative to the lab frame, it doesn't mean that the lab is inertial; the ball is only at rest relative to the lab because it has a force (from gravity) acting on it such that it maintains at rest, but if you were to remove that gravitational force (and hence have it undergo true inertial movement), it will no longer be stationary according to the lab frame anymore (because the lab is being accelerated by the gravity).
If you remove the gravitational force then the ball and lab will still be experiencing the same relative motions. The gravitational force, whether zero value or nonzero value, as long as it's the same, encompasses all of the lab including the lab itself. You cannot switch off gravity in only one part of the lab (if you can, you'll be headed to Stockholm shortly!).

There is NO experiment that you can do within the lab that can detect the lab's motion. All laws of physics behave there as though they take place in an inertial frame of reference.

A body in free-fall comprises an inertial frame of reference.

The point isn't about what we can and can't do though; the ball has a force acting on it, so it no longer has to have a straight line trajectory according to an inertial reference frame, in fact I think it can not possibly have a straight line trajectory (it only has one in a non-inertial reference frame with the same acceleration).

Think of the lab as a spaceship at some hypothetical place with zero gravity, and the ball as a much smaller spaceship inside the lab. You're saying that if they both fire their rockets to have the same acceleration (i.e. they are both accelerated by the same gravitational force), hence their motions would be indistinguishable, that they both constitute inertial reference frames.

Now let's say we attach a straight, easily tilted, rod to both of the spaceships, and have one of the spaceships turn off their rocket. Both spaceships will be accelerating according to each other, but only one of them will have a rod that's not tilted: this is the one with the rockets turned off, and the one that is considered the inertial reference frame.

EDIT: I have to go to sleep now, and when I wake up I'll be busy, so it will be many hours before I can continue this discussion.

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gneill
Mentor
The point isn't about what we can and can't do though;
No, in physics that is precisely the point. There is NO EXPERIMENT that the person within the lab can do that could determine that he is not in an inertial frame of reference. None. Nada. Zip. (We are. of course, assuming that the lab is not rotating!) The lab and its contents are moving inertially.
the ball has a force acting on it, so it no longer has to have a straight line trajectory according to an inertial reference frame, in fact I think it can not possibly have a straight line trajectory (it only has one in a non-inertial reference frame with the same acceleration).
The ball does not have a force acting on it that the observer can detect. Only some external observer could determine that a force is acting on the ball.

It only has to have a straight line trajectory according to the observer in the lab. Actually, to be thorough it should hold for objects that he sets in motion along three mutually orthogonal paths -- all three should move in straight lines.
Think of the lab as a spaceship at some hypothetical place with zero gravity, and the ball as a much smaller spaceship inside the lab. You're saying that if they both fire their rockets to have the same acceleration (i.e. they are both accelerated by the same gravitational force), hence their motions would be indistinguishable, that they both constitute inertial reference frames.
Nope, because an observer in the spaceship, even if he was unaware of the rocket on his own ship firing, if he released an unpowered ball in his cabin would see the ball accelerate. In this case he can certainly tell that he is in a non-inertial frame of reference. Projectiles in his ship would follow curved trajectories (just like here on Earth! This is the equivalence principle at work -- the equivalence between constant acceleration and being held stationary in a uniform gravitational field).
Now let's say we attach a straight, easily tilted, rod to both of the spaceships, and have one of the spaceships turn off their rocket. Both spaceships will be accelerating according to each other, but only one of them will have a rod that's not tilted: this is the one with the rockets turned off, and the one that is considered the inertial reference frame.
Yes, because the unpowered spacecraft and its contents are now moving inertially. It is following a spacetime geodesic path which for gravitationally flat regions of space is as good as we can get to an inertial frame of reference.

It was General Relativity with its geometric view of gravitation that introduced inertially moving frames of reference as inertial frames. Before that it was Newton's absolute space and then Special Relativity's flat space without gravity wherein inertial frames were simply unaccelerated frames in uniform motion. Things are thus a bit more complicated with GR, where non-local inertial frames don't have to be in uniform motion with respect to each other.

For the OP, in order to keep things at an introductory level consistent with Newtonian Mechanics it should be enough to say that a frame of reference associated with a body floating in space far from other gravitating masses such that the effects of gravity are negligible comprises an inertial frame of reference.
EDIT: I have to go to sleep now, and when I wake up I'll be busy, so it will be many hours before I can continue this discussion.

Nope, because an observer in the spaceship, even if he was unaware of the rocket on his own ship firing, if he released an unpowered ball in his cabin would see the ball accelerate. In this case he can certainly tell that he is in a non-inertial frame of reference. Projectiles in his ship would follow curved trajectories (just like here on Earth! This is the equivalence principle at work -- the equivalence between constant acceleration and being held stationary in a uniform gravitational field).
You dismissed my unpowered ball test in order to argue that a gravitationally accelerated frame was inertial, but then you used the same test to argue that the rocket accelerated frame wasn't. The equivalence principle is exactly why I changed the gravitationally accelerated problem to a rocket accelerated one, because you were unwilling to accept 'turning off gravity' (which is what I was referring to when I said "the point isn't about what you can or can't do", I didn't mean to say that the (in)ability to absolutely test for inertial motion was irrelevant).

If the rocket accelerated spaceship is considered an inertial frame, the reason why the "unpowered ball" follows a curved trajectory would be because there is a force applied to it according to that frame, i.e. it disagrees with the frame that considered the ball to be unpowered. This will be exactly the same for the gravitationally accelerated lab/ball; a ball that has no force acting on it according to one frame (say, the surface of the Earth) would have a force acting on it according to the lab/ball frame (say, that is free falling in Earth's gravity) and hence the "unpowered ball" has a curved trajectory in the free falling lab/ball frame.

Essentially you've inadvertently introduced a concept of 'relative forces', and I don't necessarily disagree with it actually. It introduces different 'classes' of inertial references frames: a frame would belong to a certain class if it has the same acceleration as another frame (i.e. they see each other as having no acceleration); all frames will consider all frames within their class to be inertial and they will all agree on a force, but will disagree that frames in other classes are inertial (and hence also disagree on the forces according to other classes); and of course, there would be infinitely many classes of inertial frames (one class corresponding to 'each' acceleration). This would mean, as far as I'm understanding (and it could be all wrong), that every single frame in the universe can be considered inertial, they just can't all be considered inertial all at once.

Since you've brought up general relativity, I just thought I'd mention that everything I've said in this thread was with the assumption of classical mechanics (the definition I gave for an inertial frame was also stated on the Wikipedia page to be a classical mechanical definition), and more specifically, I assumed there was nothing special about a gravitational force as opposed to another force. I've heard many weird stories about general relativity, one which says that gravity is not a force, and another which says that straight lines (in the conventional sense, whatever that means) are no longer straight lines. Basically everything I've said makes no sense then, so I guess that means I agree with you by default in that regard.

gneill
Mentor
You dismissed my unpowered ball test in order to argue that a gravitationally accelerated frame was inertial, but then you used the same test to argue that the rocket accelerated frame wasn't. The equivalence principle is exactly why I changed the gravitationally accelerated problem to a rocket accelerated one, because you were unwilling to accept 'turning off gravity' (which is what I was referring to when I said "the point isn't about what you can or can't do", I didn't mean to say that the (in)ability to absolutely test for inertial motion was irrelevant).

If the rocket accelerated spaceship is considered an inertial frame, the reason why the "unpowered ball" follows a curved trajectory would be because there is a force applied to it according to that frame, i.e. it disagrees with the frame that considered the ball to be unpowered. This will be exactly the same for the gravitationally accelerated lab/ball; a ball that has no force acting on it according to one frame (say, the surface of the Earth) would have a force acting on it according to the lab/ball frame (say, that is free falling in Earth's gravity) and hence the "unpowered ball" has a curved trajectory in the free falling lab/ball frame.
Okay, well the problem with applying rockets to the test masses is that it generates two categories of test object, powered and unpowered, which the observer can distinguish. The existence of the distinction would be evidence of non-inertiality. When no such distinction can be made because the effect is intrinsic to everything (even photons!) then the observer cannot tell that he is not simply floating at rest in space.

A rocket-powered lab is thus a non-inertial frame, as would be a lab that's held in position in space against a pervading gravitational field (in the same way that we're prevented from falling to the center of the Earth by the Earth's surface).
Essentially you've inadvertently introduced a concept of 'relative forces', and I don't necessarily disagree with it actually. It introduces different 'classes' of inertial references frames: a frame would belong to a certain class if it has the same acceleration as another frame (i.e. they see each other as having no acceleration); all frames will consider all frames within their class to be inertial and they will all agree on a force, but will disagree that frames in other classes are inertial (and hence also disagree on the forces according to other classes); and of course, there would be infinitely many classes of inertial frames (one class corresponding to 'each' acceleration). This would mean, as far as I'm understanding (and it could be all wrong), that every single frame in the universe can be considered inertial, they just can't all be considered inertial all at once.
Due to the geometric nature of space in General Relativity, the concept of inertial frames of reference all being related to one another by constant velocity differences has to be abandoned. It's only (approximately) true for local frames of reference where they are said to be co-moving with the local region of space. Fortunately we're still left with the important property that in inertial frames the laws of physics behave the same and have identical, simplest form -- there is no way to distinguish any inertial frame from another to give it a "best inertial frame" award Since you've brought up general relativity, I just thought I'd mention that everything I've said in this thread was with the assumption of classical mechanics (the definition I gave for an inertial frame was also stated on the Wikipedia page to be a classical mechanical definition), and more specifically, I assumed there was nothing special about a gravitational force as opposed to another force. I've heard many weird stories about general relativity, one which says that gravity is not a force, and another which says that straight lines (in the conventional sense, whatever that means) are no longer straight lines. Basically everything I've said makes no sense then, so I guess that means I agree with you by default in that regard.
Yeah, I think we've strayed markedly from what the OP needed as an answer for what is likely a Newtonian Physics framework. Hopefully this digression hasn't totally confused him...