Inertial and non inertial frames

[TrickyDicky]

Look, everyone has limits to their understanding. Your problem is that you believe that you understand things much better than you actually do.

You could use some of this too, don't you think?

 

[TrickyDicky]

This is almost true by definition. Locally, you can use inertial Cartesian coordinates even in curved spacetime. You can use coordinates that are defined within a local chart, that are only good within a small region. Pretty much the definition of "flat spacetime" is the existence of a global chart, a single chart covering all of spacetime within which one can use inertial Cartesian coordinates.

I have not implied nothing that contradicts this in any of my posts, you are clearly turning things round to fit what you think I've said.
The problem is when you want to include an extended accelerated frame, , it is in that case when you need non-inertial coordinates and you only cover a part of the flat spacetime manifold so you run into horizons. We are not discussing the case of a single particle at a spacetime point here.

 

[stevendaryl]

I have not implied nothing that contradicts this in any of my posts, you are clearly turning things round to fit what you think I've said.

You said several times that one cannot globally describe accelerated motion in flat spacetime using inertial Cartesian coordinates. That's not true.

The problem is when you want to include an extended accelerated frame, it is in that case when you need non-inertial coordinates and you only cover a part of the flat spacetime manifold so you run into horizons. we are not discussing the case of a single particle at a spacetime point here.

That's right. You cannot use inertial Cartesian coordinates to describe a noninertial frame. But you can globally describe accelerated motion using an inertial frame. It's not the frame of the accelerated particle, but that doesn't matter.

 

[stevendaryl]

You could use some of this too, don't you think?

I'm painfully aware of the things that I don't understand. This isn't one of them.

 

[stevendaryl]

Wow, you finally got it .
You're welcome.

Then why did you say "Nope" when I wrote

If you use inertial Cartesian coordinates, you can describe accelerated motion globally in SR, with no horizons.

 

[TrickyDicky]

Then why did you say "Nope" when I wrote
"If you use inertial Cartesian coordinates, you can describe accelerated motion globally in SR, with no horizons."

Because you can't, you can do it locally, it is called 4-acceleration tangent vector at a point. Any way I'm not sure what you mean by describe accelerated motion globally, we are tlking about Minkowski flat spacetime, this manifold is homogeneous and isotropic in the 4-dimensions, not just the spatial ones like is the case in GR, if you introduce a non-inertial frame globally to the manifold you'll have to attach non-inertial coordinates, and Christoffel corrections to keep the manifold homogeneous and isotropic.
This is not necessary if you describe accelerated motion locally at a point, and you can even construct a finite curved worldline of accelerated motion by describing subsequent points, but you cannot extend it indefinitely.

Look, we are not going to agree, let's agree to disagree at least.

 

[stevendaryl]

Because you can't, you can do it locally, it is called 4-acceleration tangent vector at a point.

Oh, my gosh. Yes, you can certainly describe accelerated motion globally using inertial Cartesian coordinates. I think you're confusing two different things:

(1) Noninertial motion.
(2) Noninertial frames.

You can use an inertial frame to (GLOBALLY in flat spacetime) describe noninertial motion.

 

[stevendaryl]

Look, we are not going to agree, let's agree to disagree at least.

No, I'm not going to "agree to disagree". This isn't a matter of opinion. Flat spacetime implies that there is a GLOBAL inertial Cartesian coordinate system that can be used to describe all physics.

 

[TrickyDicky]

Oh, my gosh. Yes, you can certainly describe accelerated motion globally using inertial Cartesian coordinates. I think you're confusing two different things:

(1) Noninertial motion.
(2) Noninertial frames.

You can use an inertial frame to (GLOBALLY in flat spacetime) describe noninertial motion.

You are the one talking about noninertial motion without defining it, I'm talking a bout noninertial frames.
You can use inertial frames to describe noninertial motion if by noninertial motion you mean what I'm calling 4-acceleration tangent vector at at a point, that is the derivative at a point along a curve constructed by different snapshots at subsequent times of the proper time parameter tau of Minkowski spacetime reperesented with inertial coordinates.

 

[TrickyDicky]

Flat spacetime implies that there is a GLOBAL inertial Cartesian coordinate system that can be used to describe all physics.

Yeah, sure, I wonder why Einstein even thought of the necessity to use a curved spacetime to describe all physics.
Are you kidding, now?

 

[TrickyDicky]

In other words the reason we can describe "accelerated motion" locally in flat spacetime with inertial cordinates is the Equivalence principle, this is what is implied when in the wikipedia page it is claimed that the implicit knowledge about GR is used.

 

[PAllen]

Yeah, sure, I wonder why Einstein even thought of the necessity to use a curved spacetime to describe all physics.
Are you kidding, now?

All physics except gravity. That was implicit in 'flat spacetime'. So, yes, you can jump on the failure to make explicit the qualifying clause.

 

[PAllen]

In other words the reason we can describe "accelerated motion" locally in flat spacetime with inertial cordinates is the Equivalence principle, this is what is implied when in the wikipedia page it is claimed that the implicit knowledge about GR is used.

I disagree completely with this view. As long as one is ignoring gravity, you can describe accelerated motion (e.g. accelerated radiating charges colliding with each other and with neutral charges) in one inertial frame, with no implicit use of equivalence principle. Only if you need to bring in gravity does the equivalence principle come into play. The use of one inertial frame for such situations is, in fact, how all such physics is actually computed. Nobody in their right mind uses either accelerated frames, curilinear coordinates, or GR for situations gravity can be ignored.

 

[Muphrid]

Yeah, sure, I wonder why Einstein even thought of the necessity to use a curved spacetime to describe all physics.
Are you kidding, now?

stevendaryl's statement is correct. If the spacetime is flat, there exists a plain old cartesian coordinate system that can describe everything everywhere because all the tangent spaces are the same.

You make sweeping, general statements that on their face are incorrect. You do not clarify them properly until someone else clarifies them for you. You are condescending and arrogant when you should be humble and inquisitive. You think you know way more than you actually do. Take a lesson from this thread, sir. You have much more to learn. People here are only going to be willing to help you do so if you tone down this attitude and be more circumspect, more realistic, about what you actually understand.

 

[stevendaryl]

You are the one talking about noninertial motion without defining it,

I didn't realize there was any question as to what "noninertial motion" means. It means "accelerated motion", or motion such that the following quantity is nonzero:

$\dfrac{D}{d \tau} U^\alpha = \dfrac{d}{d \tau} U^\alpha + \Gamma^\alpha_{\beta \gamma} U^\beta U^\gamma$

I'm talking about noninertial frames.

You're talking about lots of things, but I'm particularly objecting your claim (which I thought you were making, but you're not very clear) that one cannot globally describe accelerated particles using inertial Cartesian coordinates.

You can use inertial frames to describe noninertial motion if by noninertial motion you mean what I'm calling 4-acceleration tangent vector at at a point, that is the derivative at a point along a curve constructed by different snapshots at subsequent times of the proper time parameter tau of Minkowski spacetime reperesented with inertial coordinates.

Is there some other notion of describing motion, other than describing how 4-velocity changes with time?

 

[stevendaryl]

Yeah, sure, I wonder why Einstein even thought of the necessity to use a curved spacetime to describe all physics.
Are you kidding, now?

I said that in FLAT spacetime, you can describe physics using inertial Cartesian coordinates. I didn't say that all spacetimes were flat.

 

[stevendaryl]

In other words the reason we can describe "accelerated motion" locally in flat spacetime with inertial coordinates is the Equivalence principle, this is what is implied when in the wikipedia page it is claimed that the implicit knowledge about GR is used.

Oh my gosh! No, that's not true at all. The equivalence principle allows us to use SR locally to describe motion in a gravitational field. In the absence of gravity, you don't need the equivalence principle to describe accelerated motion.

There are too many confusions in what you're saying to address all of them, but you are misunderstanding the point of the equivalence principle and why it was important to Einstein in the development of General Relativity. If you understand Special Relativity, then you don't need the equivalence principle in order to describe physics in a curvilinear or accelerated coordinate system. You just need calculus: take the description in terms of inertial Cartesian coordinates, and perform a coordinate transformation to get a description in terms of noninertial coordinates.

What you will find if you do that is that when described using noninertial coordinates, there are location-dependent effects, such as: a clock at the rear of an accelerating rocket runs slower than a clock at the front of an accelerating rocket. You don't need the equivalence principle to deduce this effect: it follows from pure Special Relativity, plus calculus. (You do need something called the "clock hypothesis", which is that the rate of a clock doesn't depend on its acceleration but only on its instantaneous velocity: http://en.wikipedia.org/wiki/Clock_hypothesis)

What the equivalence principle allows you to do is to solve problems involving clocks in a gravitational field by transforming to an equivalent problem involving accelerating clocks, which can be solved using SR. The importance is that you reduce a new problem (the behavior of clocks in a gravitational field) to a solved problem (the behavior of accelerating clocks in SR).

 

[D H]

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