# 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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I could try explaining it all over again; but I've decided to just get on with my life. Sorry if you've seen something on sciforums that you haven't seen before (although as I may have mentioned, it's something that's been in use for quite a long time, probably at least since before you were born). Suck it up bro.

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3. ### DaveC426913Valued Senior Member

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I thought we were talking about standard terms - no explaining necessary. Wouldn't that be what Ex was taking issue with?

Rhetorical question. I'll step back to let the thread resume its course.

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5. ### exchemistValued Senior Member

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Thanks for the intervention Dave. Indeed, you see this just the same as I do - and James too, I think , since he seemed (per post 23) similarly perplexed by Arfa's use of the term "instantaneous derivative".

The only reason for me challenging Arfa on his idiosyncratic terminology was that he seemed to be using it to construct what looked to me like a false idea, to the effect that one can ignore the existence of acceleration if one considers the velocity at a single instant. Whereas it is plain that any acceleration being experienced by a body continues to have a finite value at any individual instant, just as velocity and position do. So one cannot make an inertial frame of reference just by freeze-framing the film of the action.

Neddy has now clarified that although indeed acceleration remains at a given instant, one is in fact allowed to apply Lorentz's contraction formula to accelerating frames of reference.

So we can let the issue of Arfa's terminology lie.

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7. ### exchemistValued Senior Member

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Thanks, that seems fairly clear.

8. ### arfa branecall me arfValued Senior Member

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--https://www.intmath.com/differentiation/4-derivative-instantaneous-rate-change.php
--https://www.themathpage.com/aCalc/instantaneous-velocity.htm

. . .

You're making the mistaken assumption that I meant acceleration can be ignored, when I said no such thing. What I said was that acceleration is physically impossible in an instant of time. Thus, in an instant of time there is no acceleration, but, there can be an acceleration vector. This 'reduces' to a linear velocity as QH states.

But that only makes sense in calculus, which has the limit of a function to play with. In experiments the best you can say is that an accelerating object has a velocity at each instant of time, even if these "instants of time" can't be actually measured.

So your skepticism and disbelief don't really address what I actually said. They might address what you thought I said, but that's a different story.

Last edited: Aug 14, 2019
9. ### exchemistValued Senior Member

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You said, in post 11: " ....an accelerating object also has instantaneous velocities (at each instant!); therefore it seems to follow that an object (or observer) is inertial at every instant."

Which is not so, because - as you now concede - there is "an acceleration vector". If there is an acceleration, the situation is not inertial. Obviously.

But it does not matter, as in the meantime I have got the answer to my query from Neddy.

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10. ### arfa branecall me arfValued Senior Member

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And there you are, continuing with your misunderstanding.
If there is an acceleration, then the velocity at one instant will be different to the velocity at another instant.
But at any instant the velocity can have one value and one direction. Thus, at any instant the object is inertial, and Lorentz contraction makes sense.
This isn't correct, Lorentz contraction doesn't apply to accelerating frames, but to inertial frames (as QH states earlier in this thread).

11. ### arfa branecall me arfValued Senior Member

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I know that my previous post looks like it contradicts itself. But it seems to come down to what is meant by a frame of reference.

I found this in Wikipedia's page on inertial frames:
--https://en.wikipedia.org/wiki/Inertial_frame_of_reference

12. ### exchemistValued Senior Member

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Take it up with Neddy then. I find he makes more sense than you.

13. ### TabbyStarRegistered Member

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Learning here. I have read the entire thread. I am stuck at a basic or fundamental level though. My heads spins;(

Is this issue similar to Zeno and motion? In my head I struggle with understanding motion (velocity) and time. I feel that time exists but motion is almost an illusion...to me I mean.

Example: I kick a stone on the ground. It moves from point A to point B. Distance and time lapse occurred. Yet at any point I can reference a snapshot (single time frame) where the stone is static in that instance. If I divide the time lapse by infinite slices, there always are infinite static frames where no velocity or motion took place (in a single frame). Yet I understand the stone did move and a force had to move it (my foot).

Apologize in advance for my input if it side tracks the discussion briefly. Just trying to learn here.

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

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Let's take a concrete example. Suppose an object moves along the x axis with position given as a function of time as follows:

$x(t) = \frac{1}{2}\alpha t^2$,

where $\alpha$ is a constant.

The instantaneous velocity (i.e. the "velocity at a point in time") is then:

$v(t) = \frac{dx}{dt} = \alpha t$

The instantaneous acceleration (i.e. "the acceleration at a point in time") is:

$a(t) = \frac{dv}{dt} = \alpha$.

Suppose, for example, that the constant $\alpha = 3$ ms$^{-2}$. Consider a particular time, say $t=2$ seconds after we start the clock.

At the given time, the object is at $x=6$ m from the origin of the position coordinate, travelling in the positive x direction at $6$ m/s, and accelerating at $3$ ms$^{-2}$.

An observer riding on this object at this instant in time is experiencing an acceleration of $3$ ms$^{-2}$, so he is clearly not inertial. Not in this instant of time, or in any other instant, while riding on this object.

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

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Yeah, so I tried that, and there are lots of references at the top of the list to things like "derivative as an instantaneous rate of change", but nothing much that actually uses the term "instantaneous derivative". No doubt, further down in the search results, it will be possible to find examples of people who don't know much about physics using the term.

I'm still wondering who these thousands upon thousands of working physicists are who regularly use the term "instantaneous derivative".

I've just been having a conversation with Magical Realist in another thread. No doubt you'll find 5 million hits regarding belief in Bigfoot, too. That doesn't mean the idea that Bigfoot is real is actually worth anything.

Give me a quote from one textbook that uses the term "instantaneous derivative" in a sentence.

I don't think Newton used the term "derivative" at all. Again, a quote from Newton might be useful at this point. You know, where he talked about "instantaneous derivatives" in the Principia, for instance - not that I recall him doing that anywhere in the book, which I've read from cover to cover.

A time rate of change is a derivative with respect to time. The term "instantaneous" as a description of a type of derivative doesn't seem to exist. There is "instantaneous velocity" and "instantaneous acceleration" and the like, but nobody writes "velocity is the instantaneous derivative of position" or anything like that.

Well, I'm told that I've been off living in a land that is separate from thousands of physicists who are all writing textbooks, none of which I've ever come across in my travels. I'm hoping you can bring me up to speed by introducing me to one or two of these physicists and/or textbooks.

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

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TabbyStar:

You are puzzled because when you take a snapshot photograph of something, the thing doesn't move in the photograph? When you took the photo, you only captured a particular moment in time.

Besides, the thing is still in motion at the instant you take that photo. The proof is that when you take the next photo, even a very short time after this one, the thing will have moved in the photo. If it wasn't moving in the previous photo, then it couldn't have got from there to here - at least not unless it sneakily moved in between photos, stopping just before you took the next photo. Which makes it kind of strange that, no matter what you do, you can never catch it in the act of sneakily moving.

I find it puzzling that people are surprised to find that they can't see motion when they have effectively done away with the time element that allows it to happen.

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17. ### arfa branecall me arfValued Senior Member

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So in your part of the universe, instantaneous velocity and instantaneous acceleration exist, but you aren't allowed to call velocity or acceleration a derivative, because that would mean both are instantaneous . . . derivatives?

Or there's a rule someone wrote down that says it's ok to call velocity a derivative, or instantaneous, but not both? Nobody writes "velocity is the instantaneous derivative of position", but it's ok to write "instantaneous velocity is the derivative of position" (or IS IT??).

Who cares anymore?

18. ### billvonValued Senior Member

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No. Both are derivatives, period. Instantaneous or continuous describes the sort of data/process you are working with, not the sort of differential equation (or integral) you are using.
Nope, no rule. It just doesn't make any sense, like calling an equation a high altitude integral because you did it in an airplane.

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

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It's all very well to pretend you can't read what's written, but it gets a bit old after a while.

Clearly, if I was against calling velocity a derivative, I wouldn't (in this very thread, just a few posts up) have referred to velocity as the derivative of position with respect to time.

The term "instantaneous velocity" is used to make a distinction between that and "average velocity" (which is taken over a larger time interval).

To be clear: my contention is that the term "instantaneous derivative" is not used by competent physicists. I'm still waiting on you to provide any evidence that shows I'm wrong about that.

If you cared a bit more, you might get it right eventually.

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20. ### QuarkHeadRemedial Math StudentValued Senior Member

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Correct
.Not sure that "inertial" applies (look up Newton's Law of Inertia a.k.a. his 1st Law of motion). At a stretch you might say "locally inertial", but otherwise you are correct, I believe.

The consequence is profound - since the space metric and the time metric, relative to an accelerating object, differ as you compare one small "locally inertial" region of space and of time to another, and because of the Equivalence Principle, and switching to unified spacetime, we may say that in the presence of a gravitational source the spacetime metric is not the same in two small distinct spacetime regions.

And since by definition the curvature field is the second derivative of the metric field this implies a non-zero curvature field in the presence of a source

21. ### Neddy BateValued Senior Member

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Lorentz contraction applies whenever there is relative velocity. Any relative velocity. It does not have to be uniform linear velocity.

If you (and QH) are saying that special relativity can be applied to accelerating frames, but only because we analyze one instant of time, then most everyone else in this thread finds that reasoning bizarre.

If you (and QH) are saying that special relativity cannot be applied to accelerating frames, and that general relativity is required instead, then that is a common misconception:

http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

Excerpt:
"It's a common misconception that special relativity cannot handle accelerating objects or accelerating reference frames. Sometimes it's claimed that general relativity is required for these situations, the reason being given that special relativity only applies to inertial frames. This is not true. Special relativity treats accelerating frames differently from inertial frames, but can still deal with accelerating frames."

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22. ### exchemistValued Senior Member

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Once again, very illuminating.

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23. ### QuarkHeadRemedial Math StudentValued Senior Member

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And your argument in support of this claim is.........

That is their problem, not ours

Neither of us said any such thing. Take squint at post #77