Understanding Inertial Frames: How to Determine if You're in One

[madness]

Is there really such a thing as an inertial frame? How would we know for sure if we were in one? Looking at space, things are moving and accelerating all over the place, it would seem impossible to tell whether a frame is really inertial or not.
Imagine you are in space and everything around you is accelerating past you. How do you know if you are in a non-inertial frame or in an inertial frame where everything around you is experiencing a force?

 

[Gear300]

Is there really such a thing as an inertial frame? How would we know for sure if we were in one? Looking at space, things are moving and accelerating all over the place, it would seem impossible to tell whether a frame is really inertial or not.
Imagine you are in space and everything around you is accelerating past you. How do you know if you are in a non-inertial frame or in an inertial frame where everything around you is experiencing a force?

One of the principles of relativity is that there is no absolute reference frame; all frames are taken relative to another.

 

[madness]

Doesn't sound right to me. In GR the laws of physics are the same in all frames, but inertial frames are special as they correspond to (locally) flat space.

 

[ConradDJ]

Doesn't sound right to me. In GR the laws of physics are the same in all frames, but inertial frames are special as they correspond to (locally) flat space.

Yes, inertial frames are special. But apparently there are none, in the real world, even though it approximates "flat space" very closely, locally. An inertial RF could exist only where there's no gravitational interaction, right?

 

[D H]

Yes, inertial frames are special. But apparently there are none, in the real world, even though it approximates "flat space" very closely, locally. An inertial RF could exist only where there's no gravitational interaction, right?

No.

In Newtonian mechanics inertial reference frames exist by assumption. That you can't find one doesn't mean they don't exist. It just means you can't find one. What one can find is something that is very, very close to inertial. For example, a solar system barycenter frame is dang close to inertial across the breadth of the solar system. The perturbative gravitational accelerations due to the nearby stars and due to the galaxy as a whole are extremely small. Ignoring general relativity is a much more significant source of error than is ignoring the fact that there are gravitational sources outside the solar system.

In general relativity, departure from inertial is measurable. Accelerometers measure translational acceleration and gyroscopes measure rotation with respect to a local inertial frame.

 

[madness]

D_H - are you saying that you simply choose an inertial frame to be one not accelerating with respect to some local matter distribution, or is there some global inertial frame for the whole universe? Since F=ma, how can forces be detected other than by the (relative) acceleration, which doesn't tell you which frame is inertial?
Thinking in terms of Newtonian physics, it seems strange that there should be some set of inertial frames, independent of the matter distribution and motion in the universe. I think Mach's principle and Newton's bucket might be relevant here somehow...

You say that accelerometers and gyroscopes can measure departure with respect to a local inertial frame. Doesn't this require the specification of an inertial frame to begin with? Couldn't this be an arbitrary choice?

 

[belliott4488]

madness -

You probably already know this, but the basic test of whether or not you're in an inertial frame is to see if Newton's laws hold. If you have something at rest in your frame, will it remain at rest or will begin to accelerate in some direction?

Such a test should be sufficient to answer your original question, i.e. are you inertial and the mass around is accelerating, or vice versa?

 

[D H]

D_H - are you saying that you simply choose an inertial frame to be one not accelerating with respect to some local matter distribution, or is there some global inertial frame for the whole universe?

In Newtonian mechanics there is some global inertial frame for the whole universe. All inertial frames are related to this global inertial frame by a constant rotation, constant velocity affine transformation. Just because you can't find such a frame doesn't mean it doesn't exist.

Thinking in terms of Newtonian physics, it seems strange that there should be some set of inertial frames, independent of the matter distribution and motion in the universe.

That is all a bit moot as the real universe is not Newtonian.

You say that accelerometers and gyroscopes can measure departure with respect to a local inertial frame. Doesn't this require the specification of an inertial frame to begin with? Couldn't this be an arbitrary choice?

No. It only requires an accelerometer and a gyroscope. You don't have to tell the accelerometer or gyro what an inertial frame is. How accelerometers and gyroscopes work is a different matter. Mach's Principle, maybe. We don't really know (yet). It's axiomatic for now.

 

[madness]

You say in Newtonian mechanics there is a global inertial frame regardless of the matter distribution - is this your point of view or a definitive fact? Isn't Mach's view of the Newton's bucket argument proof he wouldn't agree?

 

[Cleonis]

Is there really such a thing as an inertial frame? How would we know for sure if we were in one? Looking at space, things are moving and accelerating all over the place, it would seem impossible to tell whether a frame is really inertial or not.
Imagine you are in space and everything around you is accelerating past you. How do you know if you are in a non-inertial frame or in an inertial frame where everything around you is experiencing a force?

As belliot4488 has pointed out, the criterium that singles out an inertial frame from a non-inertial frame is whether the laws of motion hold good.

Take the example of the planet Mars, that is orbiting the Sun is an eccentric orbit. (Of course all planetary orbits are somewhat eccentric, in the case of Mars the eccentricity is somewhat more pronounced.)
Suppose you would map the motion of Mars in a coordinate system that is not stationary with respect to the fixed stars, but, say, co-rotating with the position of Saturn relative to the Sun
Mapped in that coordinate system the orbit of Mars would show discrepancies from the what the law of universal gravitation describes. The orbit of Mars is described by the inverse square law of gravitation exactly _if and only if_ the orbit is mapped in an inertial coordinate system.

The higher the accuracy of your astronomical data, the closer you can pinpoint the solar system's inertial frame.
In Kepler's laws the position of the Sun is thought of as the origin of the coordinate system. Newton showed that in fact Jupiter is so heavy that the Sun and Jupiter are orbiting a common center of mass. (With the common center of mass of Jupiter and the Sun just outside the Sun.)
If you can pinpoint the location of the solar system's inertial frame then the inertial frame must exist in some form.

As you expand your view there is a hierarchy of nested inertial frames. If you zoom out to encompass the entire galaxy then the galaxy's inertial frame is non-moving with respect to the galaxy's common center of mass.

In distinguishing inertial frames from non-inertial ones the following criterium is very unpractical: 'If an object remains in uniform motion (straight line, no acceleration) then the frame that is co-moving with that object is an inertial frame.' The problem, of course, is that to get that motion in a straight line in the first place you'd need a patch of space that is free from gravitational field - but gravitation is everywhere!
The practical criterium is to see whether the observed motion is all exactly according to the laws of motion. That is the case if and only if the motion is mapped in an inertial coordinate system.


Cleonis

 

[madness]

Hmm, yes I knew all this before I posted the thread but was still confused, now I'm not sure why I was confused in the first place. The only question remaining for me is whether (in Newtonian mechanics) the set of inertial frames are defined by the matter distribution of the universe or independent of it. This is essentially the problem of Newton's bucket so I suppose I won't get an answer.

 

[D H]

Inertial frames are axiomatic in Newtonian mechanics. In other words, they have an independent existence apart from the universe. That in Newtonian mechanics the universe obeys Newton's laws of motion in an inertial frame gives you a mechanism for finding an inertial frame. All you have to do is find a reference frame in which all of the objects in the universe obey Newton's laws of motion.

 

[Cleonis]

Hmm, yes I knew all this before I posted the thread but was still confused, now I'm not sure why I was confused in the first place. The only question remaining for me is whether (in Newtonian mechanics) the set of inertial frames are defined by the matter distribution of the universe or independent of it. This is essentially the problem of Newton's bucket so I suppose I won't get an answer.

I think that in the context of Newtonian mechanics the question whether the set of inertial frames is defined by the matter distribution of the universe is beyond scope. Newtonian mechanics simply assumes the existence of inertia.

GR is much more probing as to the origin of inertia (given that in GR the description of inertia and the description of gravitation are unified). Even GR may not go deep enough to answer the question, but GR does touch it, I think.

Cleonis

 

[Naty1]

Is there really such a thing as an inertial frame?

sure. You're in an inertial frame when you feel no forces acting on you.

How would we know for sure if we were in one? Looking at space, things are moving and accelerating all over the place, it would seem impossible to tell whether a frame is really inertial or not.

Much more difficult subject to measurment accuracy...but ...you can tell via DH's comment on measurement:

It only requires an accelerometer and a gyroscope. You don't have to tell the accelerometer or gyro what an inertial frame is.

 

[madness]

Force is determined by the relative acceleration of things around you. If everything around me was constrained to move like I was in a rotating frame, I don't think I could distinguish this.

 

[belliott4488]

Force is determined by the relative acceleration of things around you. If everything around me was constrained to move like I was in a rotating frame, I don't think I could distinguish this.

I'm not following you. What if you just toss a ball in the air and watch its trajectory? In a truly rotating or otherwise non-inertial reference frame, it will follow some weird trajectory that you could not explain without invoking some unknown external forces that act on it. If you're in a truly inertial frame, the ball will follow the simple trajectory predicted by Newton's laws. This is all irrespective of what any of the other objects around you are doing.

Or, are you wondering about distinguishing between an external gravitational field and an accelerating frame with no gravity? Principle of Equivalence? It doesn't sound like you are, but otherwise I'm not getting your question.

 

[Cleonis]

Force is determined by the relative acceleration of things around you. If everything around me was constrained to move like I was in a rotating frame, I don't think I could distinguish this.

As the context here is inertial frames of reference you probably mention 'I don't think I could distinguish this' for a reason.

We have of course that the members of the equivalence class of inertial coordinate systems are indistinguishable. According to our current understanding this indistinguishability is a matter of principle. We assert in thought experiments that even if we would have perfect instruments, and unlimited resources for investigation, we will still not be able to distinguish among the members of the equivalence class of inertial frames.

You seem to describe another type of thought experiment. A thought experiment in which 'everything around you is constrained to move like you are in a rotating frame', with you being unaware of these constraints. That is, the thought experiment you hint at is one in which you are not aware of some circumstance (the constraints), but it is accessible information, you can readily investigate and find out.

I agree you wouldn't be able to distinguish - but only because you're not investigating. Under those circumstances your inability to distinguish has no bearing on physics principles.

Cleonis

 

[madness]

Yes I agree, I was responding to Naty1's comment "You're in an inertial frame when you feel no forces acting on you", which I find dissatisfying. My point is, you can only detect force by the relative acceleration of things around you, so checking to see if you are experiencing an forces won't help.
However, I do understand that by checking the motion of test particles relative to yourself you should be able to detect whether you are in an inertial frame. As I said earlier, I just managed to get myself confused when I made this thread.

 

[D H]

An object has coordinates in all reference frames. An object is not "in" a frame. An object can be used as the basis for defining a frame of reference. So, change Naty1's response to "You can serve as the basis for an inertial frame if you feel no forces acting on you."

This is close to correct in the general relativistic sense of an inertial frame. There is still the problem of rotation. Suppose you are in a spacesuit and are floating out in space well above the Earth's atmosphere. A frame with origin at your center of mass is a locally inertial frame, so long as it is not rotating with respect to the fixed stars. You can't feel the effects of a even a moderate rotation rate. Our best guess as to what comprises a non-rotating reference frame is rotating wrt inertial in milliarcseconds per decade (zero, as far as we can tell). There is no way to tell this kind of rate by "feel".

Naty1's modified statement is absolutely incorrect in the Newtonian sense. Just because you can't feel any forces acting on you does not mean you are motionless wrt some inertial frame. You cannot "feel" gravity, by any means. You can only infer it indirectly. This is part of the reason why G is only known to four decimal places of accuracy.

 

[madness]

How do you know if you are feeling a force other than by the relative acceleration wrt objects around you? Ok an extended body may have stress/strains but in terms of point particles (which extended bodies are made of) how can you tell?

 

[D H]

Get in an elevator and punch up a floor. If it's a glass elevator you can see the relative acceleration of the objects around you. You don't need to see the outside world to know that you are accelerating. You can feel the acceleration in your gut and in your ears. (You've got a bunch of accelerometers in your head called the otolith organs in your inner ear.)

 

[madness]

Ok so what is there to distinguish a force from an (relative) acceleration? I think force is a redundant concept - it is not needed in Lagrangian or Hamiltonian formulations. What is fundamental is the accelerations of objects.

 

[D H]

You missed the point. An object does not need to look outside itself to determine if it is accelerating with respect to a free-falling object or rotating with respect to the fixed stars. An accelerometer will accomplish the former, and a ring laser gyro (or anything else sensitive to the Sagnac effect) will accomplish the latter. A non-rotating frame that is co-moving with a free-falling object is a local inertial frame in general relativity.

 

[madness]

Can you explain to me how (absolute) acceleration is detectable? Imagine a single point particle in an otherwise empty universe - it seems crazy to me that you could tell whether it is accelerating. Adding more particles should only allow relative motion to be measured. I know I'm probably wrong but I find this quite confusing.

 

[D H]

Acceleration and is absolute, even in general relativity. Why this is the case? We don't know that (yet). For now, it is a given. In other words, it is axiomatic -- it is embodied in the equivalence principle. While mathematicians will accept any axiom so long as it is useful, physicists do not. They test them, test them, and test them again. This axiom has been tested. In fact, the equivalence principle is one of the, if not the, most precisely tested of all concepts in physics.

 

[Cleonis]

Can you explain to me how (absolute) acceleration is detectable? Imagine a single point particle in an otherwise empty universe - it seems crazy to me that you could tell whether it is accelerating. Adding more particles should only allow relative motion to be measured. I know I'm probably wrong but I find this quite confusing.

According to our current understanding what is measurable is acceleration with respect to the local inertial frame.

The barest example I can think of is a universe with two electrically charged particles in it. Due to the electromagnetic interaction each of the particles will accelerate with respect to the local inertial frame. That is the prediction.
Assume in this example that gravitation is negligable compared to the electromagnetic interaction, so that the universe can be regarded as geometrically flat spacetime.

Of course, to confirm with a _local_ measurement the predicted acceleration (with respect to the local inertial frame), there would have to be a functional accelerometer device available.

In a universe that can be regarded as geometrically flat spacetime: if you detect you are accelerating with respect to the inertial frame, then what you are detecting is that you are accelerating with respect to the equivalence class of inertial frames of reference. In that sense it's not absolute acceleration what you detect, the acceleration detection does not negate the principle of relativity of inertial motion.

Finally, in order to have this kind of thought experiment at all I'm assuming througout that to have a universe with matter in it there must be spacetime with the property that objects in it have inertia, and that the inertial properties of spacetime are perfectly uniform. Without attributing that property to spacetime there is no thought experiment of this kind.

Cleonis

 

[madness]

Ok I've had a few beers now so my reasoning may be impaired hehe. I understand that acceleration is considered absolute - Newton's laws (if I can avoid GR here) only hold within certain frames. However, I don't see how to tell if you are in an inertial frame other than checking whether Newton's laws hold for things around you. I understand that some extended body could act as an accelerometer - it is a collection of points each evolving under (if in an inertial frame) Newton's laws. But I don't see how a point particle could be equipped with an accelerometer.

 

[D H]

But I don't see how a point particle could be equipped with an accelerometer.

So what? You are not a point mass.

 

[madness]

Right but the point holds that the only way to tell if you are in an inertial frame is to check if everything is evolving under Newton's laws. I consider this an important distinction. There is no way to tell if you are accelerating without checking the relative motion of other objects. You might say "so what?" but it makes a difference to my understanding of the situation.

 

[D H]

Note well: The test given by an accelerometer and gyro is if the object to which the sensors are attached can serve as the basis for an inertial frame in general relativity.

When it comes to Newtonian mechanics, you essentially have to be omniscient to find a true inertial frame. Of course, if you are all seeing, you can see the inertial frames without having to see that all the objects that occupy the (nonexistent) Newtonian universe obey Newton's laws of motion.

Mere mortals must resort to creating toy universes in which the inertial frames are known a priori or must resort to finding frames that are approximately inertial. The former are what high school teachers do full-time and freshman physics professors do part-time. The latter is what physicists and astronomers have been doing for centuries as their real job (when they aren't torturing freshmen with problems of the first kind).