# When Does an Observer Become an Inertial Observer?

Discussion in 'Physics & Math' started by Mike_Fontenot, Aug 2, 2019.

1. ### arfa branecall me arfValued Senior Member

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So this business of the existence of points in spacetime. Physics works just fine when it treats the earth and moon as "pointlike", it's a matter of scale after all.

A point is a thing Newton says must exist at the centre of any massive object, the physics is based on this idea. But the idea is classical, or, more or less, based on anthropocentric "intervals" of time. As these intervals get smaller, the universe gets less and less classical--but Newton didn't know that.

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3. ### James RJust this guy, you know?Staff Member

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You take issue with my statement that velocity can be either constant or changing?

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5. ### James RJust this guy, you know?Staff Member

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I'm not sure what you mean by an "instantaneous derivative".

All derivatives are defined as limits. I gave the definition of velocity earlier, explicitly as a limit. See also my post #17.

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7. ### exchemistValued Senior Member

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I wonder if he means that such things as velocity and acceleration have instantaneous values - continuing to have finite values at any given instant (and thus pointing out how ridiculous Quarkhead’s assertion in post 8 is, by the way).

8. ### James RJust this guy, you know?Staff Member

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Thanks. I didn't read it that way, but that's certainly a possible reading. I guess we'll find out soon enough.

9. ### arfa branecall me arfValued Senior Member

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Your post isn't directed at me, but:

How does an observer tell their velocity is changing or not changing? This is only possible if they can compare velocities--from point to point.

That's simply because velocity is local; it's so localised that it only pertains to one point at a time; "at a time" actually means "at any instant of time". At any instant of time then, an observer has one and only one velocity; if they also have an acceleration (vector), then strictly speaking they aren't inertial.

However, they don't accelerate in an instant of time because that's physically impossible, therefore they have one and only one velocity at any instant, and so must be "instantaneously inertial".
Which is the point QH is making (me too, and thousands of other physicists) . . .

But what is an instant of time? According to you it corresponds to a limit in one dimension.

Last edited: Aug 7, 2019
10. ### Mike_FontenotRegistered Senior Member

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His accelerometer shows a non-zero reading if and only if he's accelerating at that instant in his life. That assumes that you are talking about a special relativity scenario (where there are no significant masses involved), not a general relativity scenario.

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11. ### exchemistValued Senior Member

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Surely this has to be wrong. Just because a velocity has a single value at a given instant does not entitle one to conclude that there is no acceleration.

Who are these thousands of physicists who think acceleration disappears at a given instant? Can you supply a link?

12. ### arfa branecall me arfValued Senior Member

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I don't know; what I stated was "they don't accelerate in an instant of time".

The concept of being inertial "instantaneously" isn't wrong, sorry.

13. ### exchemistValued Senior Member

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You need to back up that assertion. I do not believe thousands of physicists think this, as you claim.

The notion of being “instantaneously inertial” seems absurd. If a body is accelerating, that acceleration has a defined, non-zero value at any given instant along its path. And an observer moving with the object will measure a corresponding force at any given instant. You cannot wish away the value of the acceleration, just by considering only a single instant.

The idea that you can pretend it is not accelerating at a chosen instant would apply equally to velocity. It is like looking at one frame of a film with a galloping horse and claiming that at that instant the horse is not moving. It’s ridiculous.

14. ### arfa branecall me arfValued Senior Member

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Yea. Zeno's paradoxes are similarly "ridiculous", although they weren't resolved satisfactorily for, well, quite a while.

If velocity is an instantaneous derivative then an observer in (relative) motion is inertial at any instant, even if the motion includes acceleration.

It doesn't make acceleration disappear, because there are no instants of time. It really is that simple; although calculus says what it says and limits exist.

15. ### exchemistValued Senior Member

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Now you are making it worse.

James has already asked you what you mean by an “instantaneous derivative”. Did he ever get a reply ? I’ve never heard of this term and suspect you have just made it up. Can you give an example of a derivative that is “instantaneous” and one that is not, and point out the difference?

Furthermore to say there are no instants of time is obvious nonsense. t=0 and t=1 sec are two instants, one second apart. An instant is just a point on a time axis, just as a location in space is a point on spatial axes.

16. ### phytiRegistered Senior Member

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To calculate velocity requires two observations of a moving object,
which satisfies the definition v=dx/dt, spatial interval/time interval.
In the case of a curve, with a fixed point p and a variable point q,
the slope of line pq changes as q moves toward p.
The fictitious limit of the slope, when supposedly p and q are coincident,
is by definition the tangent and velocity at p.
But the definition of a vector, with magnitude and direction,
is contradicted since both intervals = 0.

17. ### QuarkHeadRemedial Math StudentValued Senior Member

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This lack of restraint on your part is impolite and uncharacteristic. My "ridiculous" assertion was that, given sufficiently small regions of space and of time, acceleration reduces to linear velocity.

My source was Einstein (Annalen dur Physik 38 (1912) where he states "....the laws of Euclidean geometry ...do not hold in a uniformly rotating system in which because of Lorentz contraction the ratio of circumference to diameter should be different from $\pi$". (translation by A. Pais)

Tell me under what circumstances can Lorentz contraction apply other than to objects in uniform linear velocity relative to each other.

18. ### exchemistValued Senior Member

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Well I’m sorry if my comment seems intemperate, but on the face of it what you suggest does seem ridiculous. As I said in post 30 acceleration does not vanish if sufficiently small intervals are considered. After all the whole basis of differential calculus is that as intervals tend to zero their ratio may continue to have a well-defined value. James made the same point earlier in the thread. Surely you can see that this statement of yours needs at least more qualification and explanation before it can make sense to readers.

I do not think it will do to just argue backwards from the fact that Einstein may have applied Lorentz contraction to rotation. Did Einstein actually say acceleration disappears at infinitesimal intervals? I bet he didn’t.

Can you explain in more detail what your argument is?

19. ### Neddy BateValued Senior Member

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In SR, if there is acceleration, it can be useful to consider the instantaneous velocity for a sufficiently small time interval, and assume it is inertial for that small time. This is only a useful approximation for certain purposes, and it does not imply that acceleration does not really exist, nor does it imply that acceleration is the same thing as inertial. An accelerometer would detect the difference.

To use Quarkhead's example above, the length contraction of a uniformly rotating system can be calculated using its instantaneous velocity, and the result is very much the same as it would be for uniform linear velocity. But an accelertometer co-moving with the rotating system would detect a force that would not be present on a truly inertial accelerometer.

Edit: Sorry exchemist, you and I were posting at the same time, and I essentially said the same thing you had already said.

20. ### QuarkHeadRemedial Math StudentValued Senior Member

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You can bet all you like, but it may be more profitable to THINK. Recall that rotation is acceleration. Recall also that the space-like trajectory of a rotating body is circular (let's assume so, anyway). Further recall that any circle can be regarded as a polygonal with infinitely many (straight sides.

If you cannot see where this is going, I am sorry - it is quite clear to me.

21. ### arfa branecall me arfValued Senior Member

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Don't make me pull my old undergrad calculus book out of storage.

Oh wait, here it is. Here's what it says in the introductory part (to the derivative):

"Having figured out the rocket's instantaneous velocity at the moment when t = 3, let's reanalyse the process . . ."
Before this it refers to instantaneous speed (but we can handle that, right?). Instantaneous because the result of the calculations reduces to a value at a single point of time, otherwise known as an instant of time. Hence instantaneous velocity!

We already know that velocity is a derivative, so must be an instantaneous derivative (if we want everything in the same logical domain, just for laughs).

As for acceleration: Newton says an object accelerates if an external force acts to change the local velocity. What kind of clue is that?

Last edited: Aug 8, 2019
22. ### arfa branecall me arfValued Senior Member

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Obviously.
Then there should be no problem at all with synchronising a pair of clocks and using them to measure this 1 second interval with exactly the same accuracy? Which should be with perfect accuracy? Which is nonsensical, there is no such thing in physics.

Well ok. let's have a pair of atomic clocks so we can assume they're both as accurate as we can get. But of course, being an experiment we need to write down all the sources of potential error. so what are they?

23. ### el esRegistered Senior Member

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Acceleration isn't about speed at one instant of time, but about CHANGE of speed over instants of time.
If periods of acceleration are inertial, then the equivalence of acceleration and gravitation goes out the window and GR has lost it's foundation.
That wouldn't be the end of the world. Some future QGT may consist of discrete actions.