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

[PeterDonis]

if inertial frames are frames where the accelerometer reading is always zero

They aren't. Inertial frames are frames in which the reference clocks and rulers we use to measure times and distances are in free fall. But that doesn't mean every single object in the universe needs to be in free fall. You can still apply forces to objects, such as accelerometers, and measure their effects. So you can still calibrate an accelerometer in an inertial frame.

 

[PeroK]

@Shirish now you've got dragged into trying to understand GR before you can even start to learn SR!

 

[Shirish]

@Shirish now you've got dragged into trying to understand GR before you can even start to learn SR!

You're right! Baby steps - my bad

 

[Dale]

They aren't. Inertial frames are frames in which the reference clocks and rulers we use to measure times and distances are in free fall.

I would say that there are several equivalent tests for whether a frame is inertial or not.

 

[Dale]

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?

In principle an accelerometer does not need calibration because they are based on defined measurable local effects. Of course, in practice many accelerometers need to be calibrated to get the most accurate reading, but that is not an “in principle” problem.

In a gravitational field it becomes fairly easy to calibrate using an accurate ruler and clock and a free falling object. That is an in principle circular approach (how do you know the object is in free fall), but not a practical problem (we know how to build one pretty well).

 

[pervect]

I should probably note that many of the concerns in this thread disappear with the idea of arbitrary coordinate systems. The underyling philosphy is that coordinates are just labels. It takes a bit of extra work to formulate physics in a way that allows this to happen, Lagrangian mechanics is an example of how to do this. You'd start with lumped Lagrangian mechanics, then go to Lagrangian field theories. Goldstein covers this in the graduate text "Classical Mechanics".

Another thing that's worth pointing out is that the offered defintions of proper acceleration as inpired by GR do not include gravity as a force. Therefore, for instance, if you are riding on one of the "zero-G" aircraft flights, the so-called "Vomit comet", and your stand on a scale, it reads zero, and by the defintions being discussed you are not accelerating, as your proper accleration is zero. Newtonian gravity does things a bit differently, but we already know it's not as good as GR as far as experiment goes, though it's often close enough and it's considerably easier to calculate with.

 

[A.T.]

...if inertial frames are frames where the accelerometer reading is always zero, won't that be a circular argument? ...

It's not an argument, but a definition. In inertial frames of GR accelerometers at rest read zero.

 

[cianfa72]

They aren't. Inertial frames are frames in which the reference clocks and rulers we use to measure times and distances are in free fall.

In GR context, does that imply that free falling reference clocks and rulers we use (to measure times and distances) might be not at rest with each other ?

 

[Dale]

In GR context, does that imply that free falling reference clocks and rulers we use (to measure times and distances) might be not at rest with each other ?

In curved spacetime there are no global inertial frames, only local ones. In local inertial frames the reference clocks are at rest wrt each other to first order.

 

[Nugatory]

In GR context, does that imply that free falling reference clocks and rulers we use (to measure times and distances) might be not at rest with each other ?

Yes. Consider two free-falling parachutists on opposite sides of the earth. Both are in free fall, but also moving towards one another so not at rest relative to one another.

 

[hot pies]

The easiest way to determine if your accelerating is to make a simple pendulum. IF the pendulum makes some angle with the vertical then your accelerating. In other words your frame of reference is one which is non inertial ( provided there is some deflection.)

 

[PeterDonis]

IF the pendulum makes some angle with the vertical then your accelerating.

This is not correct.

If there is a particular "vertical" direction in your vicinity, that will in general be the direction of your acceleration, and a hanging pendulum will align itself along this direction.

 

[hot pies]

if your located on the Earth, moving in a horizontal direction and accelerating along this direction then the pendulum will make some angle since the horizontal component of the tension force ( string of the pendulum) will be unbalanced

 

[Ibix]

if your located on the Earth, moving in a horizontal direction and accelerating along this direction then the pendulum will make some angle since the horizontal component of the tension force ( string of the pendulum) will be unbalanced

That's not an isolated observer, then. That's someone who can see a planet out of their window. And with this scheme you are measuring coordinate acceleration, not proper acceleration, because you are comparing an instrument reading (the direction of the pendulum) to a standard (what you are calling vertical) which you cannot justify without reference to an outside body (the planet). Furthermore, your scheme is blind to the case that your proper acceleration is parallel to the direction you are calling vertical.

 

[Dale]

This is not correct.

If there is a particular "vertical" direction in your vicinity, that will in general be the direction of your acceleration, and a hanging pendulum will align itself along this direction.

It is not too far off though. If there is a “vertical” at all then the pendulum is non inertial. So the device @hot pies described can indeed be used as a simple accelerometer.

 

[Ibix]

It is not too far off though. If there is a “vertical” at all then the pendulum is non inertial. So the device @hot pies described can indeed be used as a simple accelerometer.

Not in the mode of operation he's proposing, though, at least not for an isolated observer. I agree you could use a pendulum by simply displacing it from whatever position it's in and seeing if it starts to return - a minor modification of the standard way of measuring $g$ with a pendulum.

 

[jbriggs444]

Not in the mode of operation he's proposing, though, at least not for an isolated observer. I agree you could use a pendulum by simply displacing it from whatever position it's in and seeing if it starts to return - a minor modification of the standard way of measuring $g$ with a pendulum.

If this is a pendulum on a fixed axis (one degree of freedom) one needs to find three suitably independent directions. If the pendulum stays put in all three directions, there is no proper acceleration.

If this is a pendulum on a ball joint or similar (two degrees of freedom) then one need only find two distinct directions that are not at 180 degrees to each other. If the pendulum stays put in both directions, there is no proper acceleration.

One observes that the pendulum, when it does not stay put, follows an oscillating pattern. [If not, the proper acceleration is not constant and will be tough to quantify]. If so, the midpoint of the oscillation is determined by the direction of the proper acceleration. In the fixed axis case one needs readings from two axes.

With the direction of acceleration determined, one can set up small oscillations and measure the magnitude in the obvious manner.

 

[PeterDonis]

if your located on the Earth, moving in a horizontal direction and accelerating along this direction then the pendulum will make some angle

Some angle with what? If you are accelerating horizontally, the direction you think is "vertical" will change. The pendulum will point in the direction you now think is "vertical".

If you have some external reference other than your own acceleration, you can tell the difference between that external reference's "vertical" and the "vertical" defined by your acceleration. But this thread is about an isolated observer, who has no such external reference.

 

[Shirish]

I have another small clarification related to inertial / non-inertial frames - just want to be doubly sure about it.

Imagine a person and a weighing scale accelerating through empty space towards each other, with the person being pulled by a downward force F (being exerted by some distant, unknown source S) and the scale being pulled upward by a force of the same magnitude F (being exerted by a different distant and unknown source S ′).



Later they collide and the person ends up standing on the scale, but they stop moving since both are being pulled against each other with the same force F.



In this case, do we say that the person's rest frame (and by extension the scale's rest frame since it's also at rest w.r.t. the person) is inertial or non-inertial?

[My thoughts so far: before hitting the scale, if the person used an accelerometer, he/she would detect an acceleration and conclude his/her rest frame was non-inertial. But after hitting the scale, he/she would no longer be accelerating due to the distant source S, since the force is balanced by the scale pushing against the person. Hence, there will be no reading on the accelerometer and the person would conclude that his/her rest frame is inertial. Am I correct in saying this?]

[Tried asking this on Physics SE: https://physics.stackexchange.com/q...-being-pulled-in-opposite-direction-by-the-s/ but like I said, would be nice to have additional confirmation, plus I don't know if 'space frame' is conceptually correct to talk about]

 

[Dale]

My thoughts so far: before hitting the scale, if the person used an accelerometer, he/she would detect an acceleration and conclude his/her rest frame was non-inertial. But after hitting the scale, he/she would no longer be accelerating due to the distant source S, since the force is balanced by the scale pushing against the person. Hence, there will be no reading on the accelerometer and the person would conclude that his/her rest frame is inertial. Am I correct in saying this?

Yes, that is pretty reasonable. Of course, you do have to be careful like this. In principle a reference frame usually is considered to cover all of spacetime, so you would have to consider the entire worldline (and therefore say it is non inertial).

But if you are careful at the borders it is possible to break your spacetime up into patches and identify specific patches as being inertial or non inertial. There is nothing wrong with that in principle, but in practice many people who do that are not careful about the boundaries and can produce errors that way.

 

[Shirish]

Yes, that is pretty reasonable. Of course, you do have to be careful like this. In principle a reference frame usually is considered to cover all of spacetime, so you would have to consider the entire worldline (and therefore say it is non inertial).

But if you are careful at the borders it is possible to break your spacetime up into patches and identify specific patches as being inertial or non inertial. There is nothing wrong with that in principle, but in practice many people who do that are not careful about the boundaries and can produce errors that way.

But one of my takeaways from this thread was that the way to determine whether one's rest frame is inertial / non-inertial is by using an accelerometer. So let's say that person after hitting the scale sees no reading on the accelerometer, even in the thought experiment there's no way for him to determine everything globally or the entire worldline, right?

 

[hutchphd]

He could carry a recording accelerometer and look at the record.
He could carry a spatially separated set of 4 "synchronized" clocks rigidly attached to his person and look for asynchrony.
What does "everything globally" mean...he will likely not be omniscient! It is not clear what you mean.

 

[Shirish]

He could carry a recording accelerometer and look at the record.
He could carry a spatially separated set of 4 "synchronized" clocks rigidly attached to his person and look for asynchrony.
What does "everything globally" mean...he will likely not be omniscient! It is not clear what you mean.

Sorry for the vague wording, by that I mean he can only make local measurements. So he'll have to conclude something about his reference frame, that covers all of spacetime, using only local measurements. I'm not sure how

 

[hutchphd]

I will defer to @Dale (or anyone else who actually knows!) but given a recording 3D accelerometer I don't know why he cannot always trace a path through spacetime from that data and end points

 

[Nugatory]

I mean he can only make local measurements. So he'll have to conclude something about his reference frame, that covers all of spacetime, using only local measurements.

That can’t be done; he can only verify that he is at rest in a local inertial frame.

 

[Dale]

So let's say that person after hitting the scale sees no reading on the accelerometer, even in the thought experiment there's no way for him to determine everything globally or the entire worldline, right?

With only local information you will not be able to say much globally. I was not assuming only local information in my response to you above.

 

[Dale]

I will defer to @Dale (or anyone else who actually knows!) but given a recording 3D accelerometer I don't know why he cannot always trace a path through spacetime from that data and end points

If you have an accurate map in advance then you can do this. It is called inertial navigation. I think it is used by cruise missiles and other military applications.

 

[PeterDonis]

the way to determine whether one's rest frame is inertial / non-inertial at a given instant of time by your clock is by using an accelerometer to see if you have nonzero proper acceleration at that instant of time.

See the additions in bold that I made above. Others have emphasized that an accelerometer is a local measurement; that means local in time as well as in space.

 

[Shubh Goel]

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?

I don't think we know if we are accelerating or not becuase there is no right answer. For one frame of reference you are, for one you are not.

 

[Ibix]

I don't think we know if we are accelerating or not becuase there is no right answer. For one frame of reference you are, for one you are not.

This is true if you are talking about coordinate acceleration. Most of this thread is focussed on proper acceleration, which is an invariant.

 

[Shubh Goel]

This is true if you are talking about coordinate acceleration. Most of this thread is focussed on proper acceleration, which is an invariant.

Okay.

 

[Shirish]

In principle a reference frame usually is considered to cover all of spacetime, so you would have to consider the entire worldline (and therefore say it is non inertial).

Could you elaborate just a bit more on this? More specifically how it follows from considering the entire worldline that the reference frame is non-inertial? Did you mean that if we consider the entirety of that person's path before and after hitting the scale, we'd conclude that that person's rest frame was non-inertial? (It's clear to me from what @PeterDonis said that local spacetime measurements will tell us that the frame is inertial)

Sorry if I abused some of the terminology and got it wrong

 

[Ibix]

Could you elaborate just a bit more on this? More specifically how it follows from considering the entire worldline that the reference frame is non-inertial?

@Dale is making the point that a frame covers spacetime, not just space at the moment you are at now. When you are moving inertially there is an inertial rest frame (covering all of spacetime) in which you are at rest. But you were not always at rest in this frame, so it isn't your rest frame (usually it's called your momentarily comoving inertial frame - MCIF). Your personal rest frame has to be one in which you were always at rest, which can't be inertial if you weren't always inertial. You need to consider your entire history (or at least the bit relevant to your experiment) when defining your rest frame - in other words, your entire worldline.

Health warning: as Dale also notes, there are some subtleties in constructing a non-inertial frame. It isn't quite as simple as slicing up some inertial frames and bolting them together.

 

[Shirish]

@Dale is making the point that a frame covers spacetime, not just space at the moment you are at now. When you are moving inertially there is an inertial rest frame (covering all of spacetime) in which you are at rest. But you were not always at rest in this frame, so it isn't your rest frame (usually it's called your momentarily comoving inertial frame - MCIF). Your personal rest frame has to be one in which you were always at rest, which can't be inertial if you weren't always inertial. You need to consider your entire history (or at least the bit relevant to your experiment) when defining your rest frame - in other words, your entire worldline.

Health warning: as Dale also notes, there are some subtleties in constructing a non-inertial frame. It isn't quite as simple as slicing up some inertial frames and bolting them together.

Ah I see! So if I understood you correctly, even if I do look at my accelerometer reading and see nothing, that only allows me to say that my momentarily comoving reference frame is inertial. If I see a reading, I can only say that my MCRF is non-inertial. If I need to make any conclusions about my rest frame (at least as far as the experiment is concerned), then I'd need to "stitch together" MCRF's at each point of time (spacetime?) throughout the experiment - i.e. my rest frame is like a composition of the various MCRF's.

Also, I can define a free particle as one whose rest frame (defined / measured in the above sense) is inertial.

Hope I got most of it right. Thank you for the very clear wording! I've realized it's not good to rely too much on intuition since I've been thinking of RF's in terms of space and time being separate.

 

[Ibix]

Ah I see! So if I understood you correctly, even if I do look at my accelerometer reading and see nothing, that only allows me to say that my momentarily comoving reference frame is inertial. If I see a reading, I can only say that my MCRF is non-inertial. If I need to make any conclusions about my rest frame (at least as far as the experiment is concerned), then I'd need to "stitch together" MCRF's at each point of time (spacetime?) throughout the experiment - i.e. my rest frame is like a composition of the various MCRF's.

I think you have some of the terminology backwards, but re-reading my last post I think I managed to write it slightly ambiguously.

Whatever you are doing (accelerating or not), you have a velocity. You can use that instantaneous velocity to define an inertial reference frame in which, just for a moment, you are at rest. This is the "Momentarily Comoving Inertial Frame" (MCIF). It's inertial by definition and is a perfectly normal inertial rest frame. It just gets a special name because you are instantaneously at rest in (or co-moving with) it. In another instant you may or may not be at rest in this frame.

If you look at your accelerometer and see it reads zero then you know that part of your personal rest frame is the same as an inertial frame - the part during the period when your accelerometer read zero. However, the whole thing is only inertial if your accelerometer always reads zero. Otherwise there have to be parts where it doesn't look like an inertial frame, so the whole thing isn't inertial.

You are correct that you can stitch together a string of parts of MCIFs to create a rest frame for you. However, you have to be very careful - the reason being the slippery "when" that I wrote in "when your accelerometer read zero" above. Different inertial reference frames have different notions of simultaneity, so you'll always find that your parts of spacetime that MCIF #1 calls "while you were at rest in MCIF #1" overlap with parts of spacetime that MCIF #2 calls "while you were at rest in MCIF #2", and there are parts of spacetime that no MCIF calls "while you were at rest in this MCIF". Care is needed when you stitch them together to make sure you don't end up with overlaps and missing areas, and you end up with something of a mess anyway. There are better ways to do it if you ever need to create non-inertial frames.