Inertia/Non-inertial frame - isotropy

[gionole]

It's known that in inertial frame, space is isotropic. (statement of this where I have taken out of is attached as image)

When we talk about an uniform accelerated train, ground frame is considered as inertial frame(at least in newtonian mechanics). So if ground frame is considered such as inertial, then if we bring an example of dropping a ball, space should be isotropic, but we know that it's not(due to $mgy$). So I end up in a contradiction.

Where am I making a mistake ? is it that I'm wrong that ground frame can never be considered inertial or what exactly ?

 

[Dale]

at least in newtonian mechanics

The statement you quoted is about relativity, not Newtonian mechanics. In GR inertial frames are free fall frames (and they are local). The ground frame is not a free fall frame

 

[gionole]

Q1: So the image I attached means that the author is talking about GR. Is it galillean or general relativity ?

Q2: I'm afraid I didn't grasp "In GR inertial frames are free fall frames (and they are local)."

 

[Dale]

Q1: So the image I attached means that the author is talking about GR. Is it galillean or general relativity ?

Sorry. GR is general relativity, almost always. I should have clarified.

Q2: I'm afraid I didn't grasp "In GR inertial frames are free fall frames (and they are local)."

In GR an object is inertial if an attached accelerometer reads 0.

On the ground, an accelerometer reads an acceleration of $g$ in the upwards direction. So the ground is not inertial.

An accelerometer in free fall reads $0$. So free fall is inertial.

 

[gionole]

Inertial frame is a frame in which acceleration is 0. That's clear.

Ofc, earth accelerates as well and we all do, so since it's not 0, ground frame is not inertial. That's clear too.

1. In free fall, object that's falling down to earth definitely has an acceleration downwards. So what is meant by: "free fall is inertial" ? It definitely has acceleration.

2. I believe, "free fall frame" is the frame the object that's falling down looks at the universe from. (i know it's not a good choice of word). Right ? If so, when it's falling down, it knows that it's accelerating, so how is free fall frame inertial ?

 

[Dale]

1. In free fall, object that's falling down to earth definitely has an acceleration downwards. So what is meant by: "free fall is inertial" ? It definitely has acceleration.

No. In GR it does not have acceleration. An attached accelerometer reads 0. This is called proper acceleration

 

[gionole]

That's what I was curious. How does that happen ? by free fall, don't you mean the object that's falling down to earth under gravity ? if so, then how can it not have acceleration ?

 

[Dale]

That's what I was curious. How does that happen ? by free fall, don't you mean the object that's falling down to earth under gravity ? if so, then how can it not have acceleration ?

I have told you twice already. An attached accelerometer reads 0. That is what determines proper acceleration. Asking a fourth time is not going to change any of the previous 3 answers.

Edit: Remember, gravity is not a force in GR, it is spacetime curvature. Since there is no force there is no acceleration.

The free falling object is not accelerating down, the ground is accelerating up. This is confirmed by the accelerometer readings

 

[A.T.]

if so, then how can it not have acceleration ?

You are confusing frame dependent "coordinate acceleration" (dv/dt), with frame invariant "proper acceleration" (what an accelerometer measures).

 

[gionole]

@Dale I have been reading about accelerometer and now I understand why it reads 0 in free fall.

Now, I'm ready to ask different questions.

1. before I ask actual questions, I wanted to confirm something so we're on both the same page. You mentioned "free fall frame" - here is what I believe "free fall frame" is - Imagine a huge box is falling down under gravity and I'm in a box inside. To me, I'm at rest. I have an accelerometer in my hand, it reads 0, so what I see around me is my "free fall frame". So I am in this box, I look around and proudly say: to me, my frame is inertial. Do you think everything I said is correct ?

2. If we say that ground frame is not inertial(which I agree), then if you're sitting in a constant speed moving car that's moving on ground ofc, to you, is the frame inside car inertial ? if you have acceloremeter in your hand, would it show 0 or not 0 ?

3. I believe in newtonian mechanics though, people treat ground frame and any "body" that's moving with constant speed on ground to be inertial. I think that's statement is correct.

 

[A.T.]

You mentioned "free fall frame" - here is what I believe "free fall frame" is - Imagine a huge box is falling down under gravity and I'm in a box inside. To me, I'm at rest. I have an accelerometer in my hand, it reads 0, so what I see around me is my "free fall frame". So I am in this box, I look around and proudly say: to me, my frame is inertial. Do you think everything I said is correct ?

https://en.wikipedia.org/wiki/Equivalence_principle

2. If we say that ground frame is not inertial(which I agree), then if you're sitting in a constant speed moving car that's moving on ground ofc, to you, is the frame inside car inertial ? if you have acceloremeter in your hand, would it show 0 or not 0 ?

Every smart phone has an accelerometer. So why you don't try it?

3. I believe in newtonian mechanics though, people treat ground frame and any "body" that's moving with constant speed on ground to be inertial. I think that's statement is correct.

Newtonian mechanics models gravity differently than GR and is a good approximation for many situations.

 

[Dale]

1. before I ask actual questions, I wanted to confirm something so we're on both the same page. You mentioned "free fall frame" - here is what I believe "free fall frame" is - Imagine a huge box is falling down under gravity and I'm in a box inside. To me, I'm at rest. I have an accelerometer in my hand, it reads 0, so what I see around me is my "free fall frame". So I am in this box, I look around and proudly say: to me, my frame is inertial. Do you think everything I said is correct ?

Yes. That captures the essence of the equivalence principle.

2. If we say that ground frame is not inertial(which I agree), then if you're sitting in a constant speed moving car that's moving on ground ofc, to you, is the frame inside car inertial ? if you have acceloremeter in your hand, would it show 0 or not 0 ?

You can just try This yourself. It is not zero, but don’t take my word for it. Test it yourself.

3. I believe in newtonian mechanics though, people treat ground frame and any "body" that's moving with constant speed on ground to be inertial. I think that's statement is correct.

Yes. In Newtonian mechanics gravity is a force, so free fall frames are not inertial. This is the heart of the conflict in your OP.

 

[gionole]

You can just try This yourself. It is not zero, but don’t take my word for it. Test it yourself.

Thank you <3

Since it's not 0, then in GR, even constant speed moving car is not inertial frame, while in newtonian mechanics, it's considered inertial. That's clear now and thanks for confirming equivalence principle.

My confusion was that I correctly showed isotropy for the experiment of throwing ball in a constant speed moving train. If you do the experiment such as in any direction while you're in a constant speed moving train, you will realize that space to you will be isotropic.(maybe not 100%, due to train standing on ground) or not only that, if train is accelerating, and observer is on ground, to him, ground frame can be considered inertial.
Then It got me thinking that $mgy$ (fall under gravity) is non isotropic and if ground frame was inertial, how can inertial frame be non-isotropic and hence my confusion.

I think the quote(the image) I attached can even be said not only in GR, but in newtonian(as long as you forget about free fall and effects of earth which is spinning and rotating).

 

[Dale]

My confusion was that I correctly showed isotropy for the experiment of throwing ball in a constant speed moving train. If you do the experiment such as in any direction while you're in a constant speed moving train, you will realize that space to you will be isotropic.

Do you mean homogenous? It is definitely not isotropic. Up is different from left and forward.

 

[gionole]

@Dale

Long time, no see. Thanks for the discussions so far, have been focusing on my current job and didn't have time to go through all this.

I wanted to double check something, so if I understand the below and you agree, I think I finally grasp everything.

I realized one thing though. Landau says that: in inertial frame, space is homogeneous and isotropic and you told me that this statement is only true in GR.

I agree with you about isotropy. if we check the validity of the statement in newtonian mechanics, it fails, because clearly, in newtonian, ground frame is inertial, but it's not isotropic, hence Landau must be meaning general relativity in which I fully agree that space is isotropic.

Now, note that statement "space is homogeneous in inertial frame" holds true in both, GR and newtonian mechanics. In newtonian mechanics, ground frame is inertial and clearly, space is homogeneous. Whether you drop a ball here or 10m from here, it will behave the same.

My final assumption is:

"in inertial frame, space is isotropic" - this only holds true if we talk about in General relativity and it fails in newtonian mechanics.

"in inertial frame, space is homogeneous" - this holds true in both, General relativity and newtonian mechanics as well.

Thoughts about my final assumption ? do you find it true ?

 

[vanhees71]

Of course in Newtonian physics space is homogeneous and isotropic for any observer (not only for inertial observers as in SR and GR), because space for any observer is simply the standard 3D Euclidean affine manifold.

In SR any standard inertial observer (i.e. with the standard foliation of Minkowski space) observes also a 3D Euclidean manifold as "space". It's the hypersurface defined by constant coordinate time in the standard Einsteinian construction of Lorentzian coordinates, where $g_{\mu \nu}=\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$.

In GR a free-falling, non-rotating observer is in a local inertial system and can define Lorentzian coordinates in each point along its (time-like trajectory) by constructing a tetrad field along his trajectory.

 

[Dale]

in inertial frame, space is homogeneous and isotropic and you told me that this statement is only true in GR

I think that if you look back you will see that I said it is only locally true in GR. That is because inertial frames are only local in GR. Inertial frames are global in SR.

But yes, there is an unnecessary historical distinction between inertial frames in relativity (SR and GR) vs Newtonian physics. I say unnecessary because it is also possible to formulate Newtonian gravity as a curvature of spacetime. That is called Newton-Cartan gravity. It is identical to classical Newtonian physics in terms of all physical predictions, but formulated in a way that is conceptually closer to GR. Inertial frames are free falling frames, and locally gravity is a fictitious force, and real gravity is tidal effects which are represented as curved spacetime.

@vanhees71 is also correct with his point that standard Newtonian physics considers gravity as a real force on top of a Euclidean space rather than as part of space or spacetime. So in that sense Newtonian space, considered separate from gravitational fields on top of it, is also homogenous and isotropic.

 

[gionole]

I think that if you look back you will see that I said it is only locally true in GR. That is because inertial frames are only local in GR. Inertial frames are global in SR.

But yes, there is an unnecessary historical distinction between inertial frames in relativity (SR and GR) vs Newtonian physics. I say unnecessary because it is also possible to formulate Newtonian gravity as a curvature of spacetime. That is called Newton-Cartan gravity. It is identical to classical Newtonian physics in terms of all physical predictions, but formulated in a way that is conceptually closer to GR. Inertial frames are free falling frames, and locally gravity is a fictitious force, and real gravity is tidal effects which are represented as curved spacetime.

I think I understand your comments and still think that my assumptions in reply #15 is still correct. My bad that I didn't yet study general relativity, so just to make my brain satisfied with this topic at this point, I think #15 summarizes everything in a correct way. In newtonian mechanics, inertial frames are not really isotropic as the example of ground frame is not isotropic, but newtonian says that ground frame is inertial, while in GR locally, the statement of inertial frame being isotropic holds true as seen in free falling frame.

What I also added in #15 was that whether you choose inertial frame in GR or newtonian, space is homogeneous in both, while this was not true for isotropy.

Thoughts ? later on, I will come back to this thread once I cover general relativity course.

 

[Dale]

I think your #15 is reasonable. But do recognize that reasonable people may still disagree. You have good reasons and justifications for your claims, but other people may use the words with different meanings that lead them to different statements.

 

[gionole]

I think your #15 is reasonable. But do recognize that reasonable people may still disagree. You have good reasons and justifications for your claims, but other people may use the words with different meanings that lead them to different statements.

Amazing. I think that's enough before I cover GR and when I cover, things will make more sense mentioned in this thread. Let me think on it today and hopefully, I have no questions remaining. Will update the thread in a couple of hours. Huge thanks @Dale

 

[gionole]

Thanks so much. I think I cleared everything about this topic. There is one question that's bugging me and I think I came up with my own answer. Just posting this, if you could confirm - no worries if you're not free.

Question:
To check homogeneity of space, it's simple using active transformation method. You don't change frames, but only the starting position of test object and observe it in the same frame.

Why does passive transformation method give a solid way of checking homogeneity ? In this, we don't change the location of the object, but we just observe the same thing from different location/frame. The thing is, in this method, we don't change test object location in absolute sense in space. We just observe it from 2 different locations, while in active way, we actually change its location(i.e conduct 2nd experiment elsewhere).

I am trying to understand why passive transformation exactly detects homogenity or inhomogeneity ? I have some intuition why it works, but just want to be more content with myself.
One explanation I came up with by myself is the following:

When we observe the object from two different frames, we're essentially testing different points in space as we observe the object's motion from two different points in space. Our different frames are essentially differet points of space, right ? So if object's motion looks the same in each frame, then we tested different points(origin points of such frames) in a solid way. It's true that with this method, we don't test the same points as in active method, but it doesn't make difference at all in terms of checking homogeneity/isotropy.

 

[Dale]

I don’t really systematically do either the active or passive methods. I just look at the free particle eqts of motion and see if there are any inhomogenous or anisotropic terms.

 

[gionole]

I don’t really systematically do either the active or passive methods. I just look at the free particle eqts of motion and see if there are any inhomogenous or anisotropic terms.

I was trying to clearly grasp the passive method because it's used in lagrangian and $x$ is substituted by $x'+a$ and then seeing whether $x'(t)$ is different than $x(t)$