How does an isolated observer know if they're accelerating?

[Shirish]

I'm reading a book on principles of relativity, and am going through the definitions of reference frames and free particles. From what I've understood from there and other answers on SE, a non-inertial reference frame is one in which a free particle is measured to be undergoing acceleration. This requires an observer to identify a free particle in the first place.

If I'm an isolated observer, there can be two scenarios:

1. I see a particle at rest w.r.t. me. How do I tell if that's a free particle without communicating with someone I know is in an inertial frame (and hence can confirm whether or not the particle is free)?
2. I see a particle that's accelerating: how do I know whether I'm in an accelerating frame, or if I'm in an inertial frame but that particle is accelerating?

As someone who started with the subject, it's very confusing for me. I'm not sure if in either scenario it'd be possible for me to confirm whether the particle is free or not.

 

[PeroK]

I'm reading a book on principles of relativity, and am going through the definitions of reference frames and free particles. From what I've understood from there and other answers on SE, a non-inertial reference frame is one in which a free particle is measured to be undergoing acceleration. This requires an observer to identify a free particle in the first place.

If I'm an isolated observer, there can be two scenarios:

1. I see a particle at rest w.r.t. me. How do I tell if that's a free particle without communicating with someone I know is in an inertial frame (and hence can confirm whether or not the particle is free)?
2. I see a particle that's accelerating: how do I know whether I'm in an accelerating frame, or if I'm in an inertial frame but that particle is accelerating?

As someone who started with the subject, it's very confusing for me. I'm not sure if in either scenario it'd be possible for me to confirm whether the particle is free or not.

If you are "really" accelerating (the term is having non-zero "proper" acceleration), then you are subject to a force. This force can, at least theoretically, be measured.

For 1) note that no one is "in" an inertial reference frame. A better way to put this is to say that your rest frame is inertial.

For 2) You can check your rest frame by measuring the net force on you or on a local object that is at rest next to you.

If you want to check Newton's first law, then you would in fact have to measure the force on any particle in the experiment. If you have no measuring equipment, then you can't do much.

 

[Shirish]

If you are "really" accelerating (the term is having non-zero "proper" acceleration), then you are subject to a force. This force can, at least theoretically, be measured.

For 1) note that no one is "in" an inertial reference frame. A better way to put this is to say that your rest frame is inertial.

For 2) You can check your rest frame by measuring the net force on you or on a local object that is at rest next to you.

If you want to check Newton's first law, then you would in fact have to measure the force on any particle in the experiment. If you have no measuring equipment, then you can't do much.

Thanks! But I'm still not clear. In scenario 1) how do I check if I'm accelerating? How do I know if the particle at rest w.r.t. me is a free particle or not?

Also, in scenario 2) how does measuring the net force on myself tell me whether or not my frame is accelerating?

 

[Ibix]

In scenario 1) how do I check if I'm accelerating?

Get a bathroom scale and weigh yourself. If you weigh non-zero then you are moving non-inertially with proper acceleration equal to your weight divided by your mass. If you weigh zero (and, indeed, can't even stand on the scales in any meaningful sense) you are moving inertially. Then you can make careful observations of your unknown particle and measure its velocity with respect to some chosen inertial frame, which will tell you its state.

Also, in scenario 2) how does measuring the net force on myself tell me whether or not my frame is accelerating?

A frame is a matter of choice. Measuring the force on yourself tells you whether or not you are undergoing proper acceleration. You may then choose to use a non-inertial frame in which you are at rest or an inertial frame in which you can only be instantaneously at rest.

 

[Shirish]

Get a bathroom scale and weigh yourself. If you weigh non-zero then you are moving non-inertially with proper acceleration equal to your weight divided by your mass. If you weigh zero (and, indeed, can't even stand on the scales in any meaningful sense) you are moving inertially. Then you can make careful observations of your unknown particle and measure its velocity with respect to some chosen inertial frame, which will tell you its state.

A frame is a matter of choice. Measuring the force on yourself tells you whether or not you are undergoing proper acceleration. You may then choose to use a non-inertial frame in which you are at rest or an inertial frame in which you can only be instantaneously at rest.

So for scenario 1), do you mean that if I measure a force on myself whose source I cannot trace, then I can conclude I'm in an accelerating frame? But then I could be stubborn and say I'm not the one who's accelerating, the weighing scale is accelerating and whatever force I've measured is because of that. How do I make the distinction? Either my rest frame is non-inertial or the weighing scale's rest frame is.

 

[PeroK]

Thanks! But I'm still not clear. In scenario 1) how do I check if I'm accelerating? How do I know if the particle at rest w.r.t. is a free particle or not?

Also, in scenario 2) how does measuring the net force on myself tell me whether or not my frame is accelerating?

Let's start at the beginning. We assume flat spacetime (SR).

One way to think of a reference frame is a (3D) grid of "observers" at regular intervals. Each observer has a fixed coordinate (set of numbers to describe their position) and a clock. We assume that we can achieve two things (either theoretically or practically):

1) We know that the observers are all at rest with respect to each other.

2) We have synchronised all the clocks (in this reference frame).

Each observer records any local event (like a particle going past them or an explosion local to them). In this way, we get a space coordinate and a time coordinate for any event (in our reference frame). After an experiment, we can collate all the data and put together a description of what happened, where and when (in our reference frame).

Note that in this case the reference frame can, in fact, be described as the "rest frame" for each of the observers.

Further, let's assume that each observer can measure whether there is a net force on them or not. This would require an "accelerometer". And, let's assume that that can all confirm that there is no force on them. This means that each observer is moving inertially. And, in our reference at least, they are all at rest.

If we have this set-up, then we know we have an inertial reference frame. In practice, of course, you have to make do with a lot less than this! But, theoretically, we could set the whole thing up like this.

Now a particle enters our experiment. We can track its motion by a sequence of locally recorded events. And all the local observers can feed their data back to a central point. This tells us the trajectory of the particle. We can tell by subsequently studying the data whether the particle moved in a straight line at constant speed or not.

Newton's first law says that if the particle has no force acting on it, then it moves in a straight line at constant speed, or remains at rest.

First, if we believe this law, then we could use this as a way to measure the force on any particle.

If, however, we wanted to check the first law, we would need some way to measure the force on the particle (and other particles) and check that the particles which moved in straight lines at constant speed has no force on them and the ones that did not move in a straight line had forces on them.

That is the theoretical basis of the first law in any case.

 

[Ibix]

So for scenario 1), do you mean that if I measure a force on myself whose source I cannot trace, then I can conclude I'm in an accelerating frame?

No. If you measure a net force on yourself, then you are undergoing proper acceleration. The end. Whether or not you can trace the source of the force (whatever you mean by that) is irrelevant. You are either undergoing proper acceleration or you aren't.

You are not "in" any frame. You may choose any frame you wish to describe your motion. But if you want to use your rest frame (reasonable enough) then it must be a non-inertial frame if you are moving non-inertially.

But then I could be stubborn and say I'm not the one who's accelerating, the weighing scale is accelerating

Is the scale at rest with respect to you? If so, you are either both accelerating or both not. If it is moving with respect to you, you aren't using it right

 

[PeroK]

So for scenario 1), do you mean that if I measure a force on myself whose source I cannot trace, then I can conclude I'm in an accelerating frame? But then I could be stubborn and say I'm not the one who's accelerating, the weighing scale is accelerating and whatever force I've measured is because of that. How do I make the distinction? Either my rest frame is non-inertial or the weighing scale's rest frame is.

Let me answer this. First, it is an assumption of Newton's laws that we can measure and know when a particle is subject to a force. This applies to ourselves. Without this, you can cannot get started with Newtonian mechanics. Newton's laws make no sense if you cannot measure forces.

You also have Newton's third law. If you exert a force on a scale, then the scale exerts the same force on you.

A subtle point: things are either moving inertially or moving non-inertially (accelerating). This defines whether their rest frame is inertial or not. Things are not in a frame or not in a frame. All things exist in all frames.

Your path through spacetime, for example, can be described using any frame, inertial or non-inertial. You are in all of these frames. This is independent of your state of motion.

 

[Shirish]

No. If you measure a net force on yourself, then you are undergoing proper acceleration. The end. Whether or not you can trace the source of the force (whatever you mean by that) is irrelevant. You are either undergoing proper acceleration or you aren't.

You are not "in" any frame. You may choose any frame you wish to describe your motion. But if you want to use your rest frame (reasonable enough) then it must be a non-inertial frame if you are moving non-inertially.

Is the scale at rest with respect to you? If so, you are either both accelerating or both not. If it is moving with respect to you, you aren't using it right

Thanks for the clarification! I'll say in advance - whatever arguments I'm putting up, that's just to clear my understanding and not to challenge your understanding or argue with you for argument's sake. You know better than me for sure.

My apologies since my wording wasn't clear. What I meant was, let's say I'm standing on the weighing scale in empty space and there's a nonzero reading on the scale. Also suppose I'm really really massive, the scale has a comparatively negligible mass but a large positive charge, and I have zero charge. Based on the nonzero reading, what do I conclude:

1. I'm not the one accelerating because there's no force on me at all (maybe there's no massive object to exert gravitational force on me). BUT the weighing scale might be accelerating since there's some negatively charged thing in the distance pulling on the weighing scale, and I'm just in the scale's way causing the nonzero reading.

2. There's no negatively charged thing and hence no force on the scale. Instead, I'm the one accelerating due to a possibly massive object that's pulling me towards it, the scale is just in my way and hence the nonzero reading.

3. Even if there's no way to distinguish between the above two cases in empty space, it doesn't matter. What DOES matter is that either me or the scale has a force acting on it, both I and the scale have the same rest frame and therefore that rest frame is non-inertial.

@PeroK , @Ibix : Did I get it right or anything I've got wrong/missed?

 

[PeroK]

Thanks for the clarification! I'll say in advance - whatever arguments I'm putting up, that's just to clear my understanding and not to challenge your understanding or argue with you for argument's sake. You know better than me for sure.

My apologies since my wording wasn't clear. What I meant was, let's say I'm standing on the weighing scale in empty space and there's a nonzero reading on the scale. Also suppose I'm really really massive, the scale has a comparatively negligible mass but a large positive charge, and I have zero charge. Based on the nonzero reading, what do I conclude:

1. I'm not the one accelerating because there's no force on me at all (maybe there's no massive object to exert gravitational force on me). BUT the weighing scale might be accelerating since there's some negatively charged thing in the distance pulling on the weighing scale, and I'm just in the scale's way causing the nonzero reading.

2. There's no negatively charged thing and hence no force on the scale. Instead, I'm the one accelerating due to a possibly massive object that's pulling me towards it, the scale is just in my way and hence the nonzero reading.

3. Even if there's no way to distinguish between the above two cases in empty space, it doesn't matter. What DOES matter is that either me or the scale has a force acting on it, both I and the scale have the same rest frame and therefore that rest frame is non-inertial.

@PeroK , @Ibix : Did I get it right or anything I've got wrong/missed?

If you and the scale are local to each other and at rest relative to each other, then either you are both moving inertially or you both have the same proper acceleration.

You can see this by imagining your joint motion in an inertial reference frame (IRF). Acceleration is defined by:
$$\vec a = \frac{d^2\vec r}{dt^2}$$

So, you can see that in any IRF:
$$\vec a_{you} = \vec a_{scale}$$

And you can conclude from that that you have the same proper acceleration, zero or non-zero.

All the analysis of possible external forces is superfluous.

 

[Shirish]

If you and the scale are local to each other and at rest relative to each other, then either you are both moving inertially or you both have the same proper acceleration.

You can see this by imagining your joint motion in an inertial reference frame (IRF). Acceleration is defined by:

→a=d2→rdt2a→=d2r→dt2​

So, you can see that in any IRF:

→ayou=→ascalea→you=a→scale​

And you can conclude from that that you have the same proper acceleration, zero or non-zero.

All the analysis of possible external forces is superfluous.

So is it possible for the scale to register a nonzero reading even if both I and the scale have the same proper acceleration? Reason I'm asking this is because if I'm isolated in space with just a weighing scale (which is at rest w.r.t. me), and if the scale can't show a nonzero reading if both the scale and I have the same proper acceleration, then there's no way for me to distinguish whether we're both inertial or both accelerating (hence non-inertial).

Thanks a lot for your help so far. It's just that I'm unclear on how to make the distinction.

 

[PeroK]

So is it possible for the scale to register a nonzero reading even if both I and the scale have the same proper acceleration? Reason I'm asking this is because if I'm isolated in space with just a weighing scale (which is at rest w.r.t. me), and if the scale can't show a nonzero reading if both the scale and I have the same proper acceleration, then there's no way for me to distinguish whether we're both inertial or both accelerating (hence non-inertial).

Thanks a lot for your help so far. It's just that I'm unclear on how to make the distinction.

One conceptual problem you have here is that Special Relativity is called "special" not because it's a brilliant theory (athough it is!) but because it deals with flat spacetime - it does not include a theory of gravity.

The concept of external forces acting at a distance is not a great concept to introduce at this stage.

To answer your question, imagine the following scenario. You and the scale are independently attached to a rig that your friend is pulling. You and the scale accelerate at the same rate because both are being pulled by your friend. And, there need be no force between you and the scale. But, you can measure your acceleration by using a different device that is not directly attached to the rig.

Conceptually, therefore, you need an accelerometer that is somehow not subject to the same Force/Mass ratio as you are. If all objects are subject to the same Force/Mass ratio, then you can't tell the difference between that and moving inertially! Gravity is one example of that. In which case, the extenal force is effectively ficticious - not a force at all! - and you have a part of the equivalence principle:

You cannot tell the difference between free-fall in a gravitational field and inertial motion in gravity-free space.

You've got yourself into slightly deep waters here! The practical point is that if there is a real external force on you, there must be a way to avoid the same external force acting equally on all your possible measuring devices.

PS note that since an EM force is proportional to charge and acceleration is proportional to mass, you can detect real, external EM forces.

 

[PeroK]

If I'm isolated in space with just a weighing scale, the only way I can think of to establish whether I'm in an accelerating frame, is to see if I'm experiencing a pseudo-force (or just a force). Since by definition, proper acceleration is the acceleration measured w.r.t. an inertial frame and in this thought experiment (since I'm isolated) I have no way of identifying an inertial frame, there seems to be no way to measure proper acceleration either.

Now with these limitations, how can I ascertain whether I'm experiencing a force / pseudo-force? From what you said, using an accelerometer - fair enough, but I don't understand what properties should the weighing scale have to qualify as an accelerometer.

It's clear from your answer that since gravitational force is proportional to gravitational mass, which is equal to inertial mass, therefore in a uniform gravitational field both my acceleration and the scale's acceleration would be equal, and hence no way of knowing if I'm inertial or accelerating.

But if we were in a uniform electric field (and let's say both I and the scale were charged), and the ratio of our charges was equal to the ratio of our inertial masses, am I correct in saying that even in this scenario I wouldn't be able to tell if I was inertial or accelerating?

There's theory, which assumes that measurements can be made. As far as the theory is concerned, you don't necessarily have to specify how something is done.

Practically, experimental physics is about outwitting nature, as it were. You try with different combinations of charge and mass.

We know it's not the case that all objects have a charge proportional to their mass. There are two opposite charges, which behave oppositely. Charge can be varied independent of mass.

 

[PeroK]

So the summary is that if the scale is at rest w.r.t. me, then there is no way for me to tell if my rest frame is accelerating or not, correct?

Secondly, even if there IS relative acceleration between me and the scale, I still cannot reliably say whether it's my rest frame that is inertial and the scale's non-inertial, or whether the scale's rest frame is inertial and mine non-inertial. I would still need more non-identical scales to verify whether my frame is inertial. [this point I'm not so sure about]

Also, thanks a ton for your patience and help. You've been single-handedly more helpful so far than the entire Physics Stack Exchange website.

You have to be careful not to disappear down the rabbit-hole of the philosophy of physics. You're not limited to a scale. Experimental physicists use all sorts of clever devices and ingenious methods to make measurements. If you give them a bathroom scale and say "that's it", that's not the game!

Inertial frames don't really exist in any case. They are an idealisation. But, it's the actual physics you can build with the theory that is important. I would get on with SR. Worrying about IRF's is not what SR is about. There's a beautiful theory to be discovered.

For example, Newton used his three laws of motion and his law of gravitation to justify why planets move in elipses round the Sun. You can go back and question the basis of his laws, and root around for practical, theoretical or philosophical flaws in them. But, that huge empirical fact that the theory built on those foundations agrees with experiment is really the acid test for physics.

 

[Nugatory]

So is it possible for the scale to register a nonzero reading even if both I and the scale have the same proper acceleration?

Yes. That’s even the most common situation - it’s what’s going on when you step onto a scale to weigh yourself. You and the scale both have the same proper acceleration, 9.8 meters/sec^2 upwards and the scale is reading something non-zero.

 

[Shirish]

Yes. That’s even the most common situation - it’s what’s going on when you step onto a scale to weigh yourself. You and the scale both have the same proper acceleration, 9.8 meters/sec^2 upwards and the scale is reading something non-zero.

Thanks! That did occur to me, but I asked that question in the context of the thought experiment - i.e. when I'm isolated in space with just the scale. My apologies, I should've been clearer. Wanted to keep the situation as simple as possible with just me (the observer) and the measuring apparatus.

In that case, how do we measure the proper acceleration without being able to identify any inertial reference frame, and let's say we don't have to measure the acceleration since I'm at rest relative to the scale (so both have the same proper acceleration), then how can it be that the scale has a nonzero reading?

 

[Dale]

how do I check if I'm accelerating?

With an accelerometer

then there's no way for me to distinguish whether we're both inertial or both accelerating (hence non-inertial).

You can tell with an accelerometer

how do we measure the proper acceleration without being able to identify any inertial reference frame

With an accelerometer

 

[Shirish]

With an accelerometer

You can tell with an accelerometer

With an accelerometer

I appreciate the witty answer, even though it does not help me much. Thanks regardless :)

 

[PeroK]

I appreciate the witty answer, even though it does not help me much. Thanks regardless :)

You are essentially trying to deal with the case where the laws of physics (for your hypothetical universe) conspire to prevent you doing measurements - in this case a measurement of proper acceleration.

If the universe was thus constructed, then you might have a problem. But, that is not the universe we live in. In our universe you don't have to worry about the practical impossibility of experimental physics.

 

[Shirish]

You are essentially trying to deal with the case where the laws of physics (for your hypothetical universe) conspire to prevent you doing measurements - in this case a measurement of proper acceleration.

If the universe was thus constructed, then you might have a problem. But, that is not the universe we live in. In our universe you don't have to worry about the practical impossibility of experimental physics.

Fair enough. I realize the situation I was trying to force in the question was ill-posed. I will read up more about proper acceleration as I go along, and also try to look for the paper where the concept of accelerometer was first introduced.

 

[PeroK]

Fair enough. I realize the situation I was trying to force in the question was ill-posed. I will read up more about proper acceleration as I go along, and also try to look for the paper where the concept of accelerometer was first introduced.

I'd move on. Sometimes things that you worry about to begin with seem fairly unimportant if you press on with your learning. You can always come back to this if you need to.

 

[hutchphd]

Can one not use two clocks, separated but stationary with respect to the observer as an accelerometer? Any change in the synchrony of the clocks will indicate an acceleration.. (?)

 

[Dale]

I appreciate the witty answer, even though it does not help me much. Thanks regardless :)

Why doesn’t it help you? I am teaching you that there is a very clear experimental way to identify inertial frames and inertial objects. You seem to think it is ambiguous, but it is not. You simply use a very common measurement device.

 

[PeterDonis]

I'm at rest relative to the scale (so both have the same proper acceleration), then how can it be that the scale has a nonzero reading?

You're at rest relative to your bathroom scale when you stand on it, and it shows a nonzero reading. Why is this a problem?

let's say I'm standing on the weighing scale in empty space and there's a nonzero reading on the scale.

This won't happen "in empty space". If you are just floating out in empty space, the scale will read zero. In fact you will have to strap it to your feet if you want to keep "standing" on it (the term "standing" is really a misnomer in this situation since there is no "up" or "down" direction in this situation--all directions are equivalent).

You could have the scale read nonzero if you were inside a rocket with its engine firing and you put the scale on the "floor" formed by the tail end of the rocket and stood on it. In that case, just as if you were standing on Earth, you would be at rest relative to the scale.

 

[Shirish]

Why doesn’t it help you? I am teaching you that there is a very clear experimental way to identify inertial frames and inertial objects. You seem to think it is ambiguous, but it is not. You simply use a very common measurement device.

In that case, I apologize for my remark. Basically I was thinking of it purely as a thought experiment exercise, since I was reading the ED of moving bodies paper and that also seems to consider thought experiments to clarify concepts.

I had read about the accelerometer a bit before you posted your answer: the reason I was a bit confused was that I thought you'd still have to calibrate the accelerometer in a frame you know for sure is inertial. Which led to the question how to identify an inertial frame so that one can calibrate the accelerometer in it: one could just test out Newton's first law in various ways to establish that. I think I have a better picture than when I started.

 

[cianfa72]

You're at rest relative to your bathroom scale when you stand on it, and it shows a nonzero reading.

As far as I can understand, the bathroom scale measures the proper acceleration of the 'human' standing on it by the 'real' force 'transiting' through it: actually when you stand on the scale it 'transmits' the 'real' force driving you away from the inertial motion (away from its 'natural' spacetime geodesic path)

 

[PeterDonis]

the bathroom scale measures the proper acceleration of the 'human' standing on it by the 'real' force 'transiting' through it

No. The bathroom scale measures the strain induced in a spring by the spring being compressed when you stand on it. When you first stand on the scale, there is a short transient period where the spring compresses; then everything is motionless (relative to you) and the scale gives you its reading. Since everything is motionless (relative to you) when the scale gives its reading, there is no force "transiting" through anything.

 

[PeterDonis]

when you stand on the scale it 'transmits' the 'real' force driving you away from the inertial motion

If you mean that whatever force is pushing you away from inertial motion is exerted on you through the scale when you stand on the scale, of course that is true. But that does not require the scale to move relative to you.

 

[A.T.]

... I thought you'd still have to calibrate the accelerometer in a frame you know for sure is inertial. ...

That is true in Newtonian physics. In General Relativity inertial frames where redefined to always match zero accelerometer reading.

 

[cianfa72]

If you mean that whatever force is pushing you away from inertial motion is exerted on you through the scale when you stand on the scale, of course that is true. But that does not require the scale to move relative to you.

Sure, in that sense my wording was 'transmitting a force' when you stand on it. Basically two forces are involved: the first one to push the scale itself away from inertial motion and the other to push you away from inertial motion when you stand on it.

 

[timmdeeg]

Can one not use two clocks, separated but stationary with respect to the observer as an accelerometer? Any change in the synchrony of the clocks will indicate an acceleration.. (?)

Which observer do you mean? One who accelerates stationary with the clock or an inertial observer?

 

[Shirish]

That is true in Newtonian physics. In General Relativity inertial frames where refined to always match zero accelerometer reading.

Could you elaborate? If we calibrate the accelerometer in an inertial frame, and if inertial frames are frames where the accelerometer reading is always zero, won't that be a circular argument? At some point we'll have to make an assumption / axiom and let everything follow from that - but I'm not sure what's an axiom and what's a consequence. So could you explain how the case is different in general relativity from special relativity/Newtonian mechanics in this context?

 

[hutchphd]

I'm afraid I'm getting in the middle...thought the main question was answered.
I mean the guy has a clock in each of spread-out arms. My tenet is that this serves as an accelerometer along the direction of his arms. ... the clocks will immediately change synchrony with any acceleration. No other frames required, just the guy and his clocks , simple question as sanity check for me..

 

[Nugatory]

If we calibrate the accelerometer in an inertial frame...

An ideal accelerometer is a box containing a weight; the weight is attached by six stretched strings to the six sides of the box. Proper acceleration is measured by the tension in the springs, which in turn is calculated from the stretched length of the springs and Hooke's Law.

This accelerometer needs no calibration; the proper acceleration is zero if and only if the tension of each spring is equal to the tension of the opposite spring.

 

[PAllen]

I'm afraid I'm getting in the middle...thought the main question was answered.
I mean the guy has a clock in each of spread-out arms. My tenet is that this serves as an accelerometer along the direction of his arms. ... the clocks will immediately change synchrony with any acceleration. No other frames required, just the guy and his clocks , simple question as sanity check for me..

It depends on your orientation relative to your acceleration. If your proper acceleration is along your feet to head axis, and your hands are outstretched, there will be no change in the comparison of the clocks. If you hold one clock down towards your feet, and one above your head, their synchrony will change.