Understanding the General Relativity view of gravity on Earth - Comments

[Dale]

The GPS satellites are not nothing but made of something.

This is a complete strawman. Nobody is claiming this.

I'm not familiar with the technical details and how the reference frame(s) are defined, but for sure the set of satellites realize a reference frame.

No part of the GPS system is at rest in the ECI frame. And there is no sense in which the material of the GPS system "is" the ECI frame.

To the contrary, I'm complaining about people that take the coordinates for the world,

This is another strawman. Nobody is doing that here.

but I think it's only a semantical issue, and we shouldn't discuss it further here.

I certainly agree with that.

 

[harrylin]

I agree, that is what matters most. The disagreement is (at this level) a purely semantic one. The semantics are different, so I tried to capture that.

Yes, I simply gave a suggestion for less ambiguous phrasing.

[..] What is the Landau vocabulary that you are talking about?

As I said, Landau uses the term "locally inertial system of reference" (similarly others use "local inertial frame") for non-Galilean reference systems that locally can be used just like Galilean reference systems.

PS: and once more, that skillfully avoids the contradictory definitions that PAllen described in post #43

 

[A.T.]

Once more, Landau uses the term...

Once more, how would you call the following frames according to Landau's conventions?
- A frame at rest to the surface of a non-rotating planet
- A frame free falling towards that planet

 

[PAllen]

Yes, I simply gave a suggestion for less ambiguous phrasing.

As I said, Landau uses the term "locally inertial system of reference" (similarly others use "local inertial frame") for non-Galilean reference systems that locally can be used just like Galilean reference systems.

PS: and once more, that skillfully avoids the contradictory definitions that PAllen described in post #43

How does it avoid it? The Earth lab is an inertial frame per Newton and an accelerated frame per relativity.

 

[Dale]

Yes, I simply gave a suggestion for less ambiguous phrasing.

As I said, Landau uses the term "locally inertial system of reference" (similarly others use "local inertial frame") for non-Galilean reference systems that locally can be used just like Galilean reference systems.

PS: and once more, that skillfully avoids the contradictory definitions that PAllen described in post #43

I actually think that it makes the situation worse, not better. With the purported Landau definition of a local inertial frame you have that in Newtonian mechanics the apple frame and the ground frame are both local inertial frames. Since the two sets of frames accelerate wrt each other locally I think that is more confusing and contradictory than the usual terminology.

 

[PWiz]

I actually think that it makes the situation worse, not better. With the purported Landau definition of a local inertial frame you have that in Newtonian mechanics the apple frame and the ground frame are both local inertial frames. Since the two sets of frames accelerate wrt each other locally I think that is an untenable situation.

So in these kind of situations, GR says that if two or more observers which have 0 proper acceleration being acted upon them notice each other's paths to diverge/converge (i.e. notice coordinate acceleration of the other observer and can read each other's accelerometer's reading as being 0), they can conclude that spacetime around them must have non-zero intrinsic spacetime curvature, right?

 

[Dale]

So in these kind of situations, GR says that if two or more observers which have 0 proper acceleration being acted upon them notice each other's paths to diverge/converge (i.e. notice coordinate acceleration of the other observer and can read each other's accelerometer's reading as being 0), they can conclude that spacetime around them must have non-zero intrinsic spacetime curvature, right?

Yes, exactly.

 

[Ramanujan143]

I don't understand what it means that " 5° N line is constantly turning to the north and the 5° S line is constantly turning to the south".
Could someone explain?

If it is hard to see at first then consider the 89.9 degree latitude line. This is a tight little circle around the pole, so to stay on the latitude line you have to constantly turn towards the pole.

The same thing happens on the 5 degree latitude line, it just is not as tight of a turn.

sorry i still don't get it. I understand that the north latitude line is turning in a circular path, but it's not really turning towards the north pole just towards the axis of the north pole. and the same with the south latitude line, and arent the north and south pole on the same axis? so doesn't that mean both latitude lines are turning in the same direction. or am i thinking this because I'm visualizing this in three dimensions? thanks.

 

[A.T.]

or am i thinking this because I'm visualizing this in three dimensions?.

Yes, you seem to think about the 3D embedding space, which has no physical significance. The axis is not part of the 2D surface which represents curved space-time here. Only that 2D surface matters in this analogy. Try this applet, which shows the space-time geometry along a radial line:

http://www.adamtoons.de/physics/gravitation.swf

 

[Dale]

sorry i still don't get it. I understand that the north latitude line is turning in a circular path, but it's not really turning towards the north pole just towards the axis of the north pole. and the same with the south latitude line, and arent the north and south pole on the same axis? so doesn't that mean both latitude lines are turning in the same direction. or am i thinking this because I'm visualizing this in three dimensions? thanks.

For understanding Riemannian geometry on a sphere you have to consider only the 2D curved surface of the sphere, not the 3D flat space it is embedded in. The axis is not part of the surface, so in the geometry of the surface it is not something you can turn towards.

On a sphere the "straight lines" (aka geodesics) are great circles. All other paths must turn, including latitude lines other than the equator.

 

[harrylin]

I actually think that it makes the situation worse, not better. With the purported Landau definition of a local inertial frame you have that in Newtonian mechanics the apple frame and the ground frame are both local inertial frames. Since the two sets of frames accelerate wrt each other locally I think that is more confusing and contradictory than the usual terminology.

The so-called "ECI" frame is in good approximation a Galilean frame; that is non-ambiguous. And "Local inertial frame" means exactly what some people here confusingly call "inertial frame"; in Newtonian mechanics only the falling apple frame is such a "local inertial frame".

How does it avoid it? The Earth lab is an inertial frame per Newton and an accelerated frame per relativity.

See here above; and also per Newton the Earth lab measures "proper acceleration" if one uses Wikipedia's definition of that term as it's simply what an accelerometer indicates.

 

[A.T.]

And "Local inertial frame" means exactly what some people here confusingly call "inertial frame"; in Newtonian mechanics only the falling apple frame is such a "local inertial frame".

So how would you categorize the following frames according to your interpretation of Newtonian mechanics ?
- A frame at rest to the surface of a non-rotating planet
- A frame free falling towards that planet

 

[harrylin]

So how would you categorize the following frames according to your interpretation of Newtonian mechanics ?
- A frame at rest to the surface of a non-rotating planet
- A frame free falling towards that planet

Once more, Landau managed to introduce definitions that are consistent throughout; it's a mistake to think along the lines of "according to your interpretation of Newtonian mechanics". A reference system that is at rest to the surface of a non-rotating planet is in good approximation a Galilean frame and a system that is falling towards that planet has the modern label "local inertial frame".

 

[A.T.]

Once more, Landau managed to introduce definitions that are consistent throughout;

Then please apply only Landau's definitions the following two frames consistently:
- A frame at rest to the surface of a non-rotating planet
- A frame free falling towards that planet

 

[vanhees71]

The so-called "ECI" frame is in good approximation a Galilean frame; that is non-ambiguous. And "Local inertial frame" means exactly what some people here confusingly call "inertial frame"; in Newtonian mechanics only the falling apple frame is such a "local inertial frame".

See here above; and also per Newton the Earth lab measures "proper acceleration" if one uses Wikipedia's definition of that term as it's simply what an accelerometer indicates.

SCNR: ..., and the international space station is a material realization of such a(n approximate) local inertial reference frame! In a sense it's the most straight-forward realization of such a frame: Just let a body fall freely :-).

 

[PAllen]

Once more, Landau managed to introduce definitions that are consistent throughout; it's a mistake to think along the lines of "according to your interpretation of Newtonian mechanics". A reference system that is at rest to the surface of a non-rotating planet is in good approximation a Galilean frame and a system that is falling towards that planet has the modern label "local inertial frame".

But, per relativity, it is NOT. It is an accelerated frame, period. It can be made part of an extended coordinate system (ECI), but that is NOT a frame, in general relavivity. The frame in which the Earth lab is at rest is pure and simple an accelerated frame in GR. There is no avoiding the contradiction between this and the Newtonian view that the Earth lab materializes and inertial frame, object in it are subject to the force of gravity.

 

[Dale]

The so-called "ECI" frame is in good approximation a Galilean frame; that is non-ambiguous. And "Local inertial frame" means exactly what some people here confusingly call "inertial frame"; in Newtonian mechanics only the falling apple frame is such a "local inertial frame".

According to your description, the ECI frame is also a local inertial frame, considered on the scale of the Earth. The apple frame is a "local inertial frame", the ECI frame is also a "local inertial frame" and yet the two frames accelerate relative to each other. Therein lies the problem.

I believe that the reference you posted earlier gave examples of the center of mass of the Jupiter/moon system and the solar system as examples of local inertial frames. Those frames accelerate relative to each other.

 

[atyy]

The so-called "ECI" frame is in good approximation a Galilean frame; that is non-ambiguous. And "Local inertial frame" means exactly what some people here confusingly call "inertial frame"; in Newtonian mechanics only the falling apple frame is such a "local inertial frame".

See here above; and also per Newton the Earth lab measures "proper acceleration" if one uses Wikipedia's definition of that term as it's simply what an accelerometer indicates.

As far as I can tell, the discussion is purely about terminology. Anyway, I just thought I'd point out a modern discussion (Rovelli) of exactly the passage in Newton you mentioned. Rovelli uses Newtonian "inertial" and Newtonian "noninertial" frames closer to what, say, DaleSpam uses. However, the case of the free falling frame in Newtonian gravity clearly carries over to what one calls a local inertial frame in general relativity, and it applies especially to gravity because of the equivalence principle. So Rovelli does distinguish the concept and attributes it to Newton (among others), quoting the same passage you did. However, he is aware that terminology is tricky, so in the Newtonian context, he uses the terms "in a sufficiently small region" (which could clearly be synonymous with "local") and "free falling reference system".

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf (p42, comments just before Eq 2.116 and also footnote 19)

 

[harrylin]

Then please apply only Landau's definitions the following two frames consistently:
- A frame at rest to the surface of a non-rotating planet
- A frame free falling towards that planet

Already done in #73

 

[harrylin]

But, per relativity, it is NOT.[..] The frame in which the Earth lab is at rest is pure and simple an accelerated frame in GR. [..].

Please back up your claim and cite a reference according to which "in relativity", a reference system that is at rest to the surface of a non-rotating planet is not in good approximation a Galilean frame. As far as I know Galilean frames are uniquely defined, there is no ambiguity like with the term "inertial".

 

[A.T.]

Already done in #73

So per Landau a "local inertial frame" is accelerating relative to a "Galilean frame"?

As far as I know Galilean frames are uniquely defined,

What is the definition of "Galilean frame"?

 

[harrylin]

As far as I can tell, the discussion is purely about terminology. Anyway, I just thought I'd point out a modern discussion (Rovelli) of exactly the passage in Newton you mentioned. Rovelli uses Newtonian "inertial" and Newtonian "noninertial" frames closer to what, say, DaleSpam uses. However, the case of the free falling frame in Newtonian gravity clearly carries over to what one calls a local inertial frame in general relativity, and it applies especially to gravity because of the equivalence principle. So Rovelli does distinguish the concept and attributes it to Newton (among others), quoting the same passage you did. However, he is aware that terminology is tricky, so in the Newtonian context, he uses the terms "in a sufficiently small region" (which could clearly be synonymous with "local") and "free falling reference system".

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf (p42, comments just before Eq 2.116 and also footnote 19)

Thanks for the ref.
Yes it's only a little nitpicking about terminology, how to improve explanations to be totally non-ambiguous by using phrasing that is theory independent. Indeed "free falling reference system" is IMHO even better than "local inertial frame". In that way the term "inertial" can be avoided entirely.

 

[harrylin]

So per Landau a "local inertial frame" is accelerating relative to a "Galilean frame"?
What is the definition of "Galilean frame"?

Yes of course. Galilean reference systems are hypothetical systems that are not influenced by any forces or fields; their (non-local) operational definition is that they move uniformly in straight line relative to each other (and of course motion is defined in 3D).
(Landau: in a galilean reference system, any free motion takes place at a constant speed in magnitude and direction. [..] Thus there is an infinite number of galilean reference systems that are in constant straight line and uniform motion relative to each other.")

 

[vanhees71]

Isn't it very simple? The reference frame defined by rods being at rest with respect to the Earth is not a local inertial frame, because objects fall down due to gravity. Rods fixed on a freely falling non-rotating body define such a local inertial frame.

The reason, why in GR we don't consider the Earth frame not as a local inertial frame is that we don't consider gravity to be a force, while this is the case in Newtonian mechanics, so that in Newtonian mechanics the Earth frame can be considered as an approximate inertial frame (it's not exactly as any of the nice Foucault pendulums in countless science museums and physics departments on the world prove :-)).

 

[harrylin]

Isn't it very simple? The reference frame defined by rods being at rest with respect to the Earth is not a local inertial frame, because objects fall down due to gravity. Rods fixed on a freely falling non-rotating body define such a local inertial frame.

The reason, why in GR we don't consider the Earth frame not as a local inertial frame is that we don't consider gravity to be a force, while this is the case in Newtonian mechanics, so that in Newtonian mechanics the Earth frame can be considered as an approximate inertial frame (it's not exactly as any of the nice Foucault pendulums in countless science museums and physics departments on the world prove :-)).

Yes but it's even simpler: "local inertial frame" means "free falling reference system". The ECI frame does not constitute a "free falling reference system" for objects near the Earth in any theory.

 

[A.T.]

(Landau: in a galilean reference system, any free motion takes place at a constant speed in magnitude and direction. [..] Thus there is an infinite number of galilean reference systems that are in constant straight line and uniform motion relative to each other.")

And what does "free motion" mean here? The employed model of gravity (Newtonian force vs. GR) determines which object is "force free".

 

[Dale]

Indeed "free falling reference system" is IMHO even better than "local inertial frame". In that way the term "inertial" can be avoided entirely.

That is good phrasing.

 

[Dale]

The ECI frame does not constitute a "free falling reference system" for objects near the Earth in any theory.

Yes, it does. The ECI is in free fall about the sun.

Galilean reference systems are hypothetical systems that are not influenced by any forces or fields

As I said, Landau uses the term "locally inertial system of reference" (similarly others use "local inertial frame") for non-Galilean reference systems that locally can be used just like Galilean reference systems.

By these definitions all Galilean frames are also local inertial frames, since a Galilean reference system is clearly a reference system that locally can be used just like a Galilean reference system. So again, these Landau local inertial frames can accelerate relative to each other.

 

[PAllen]

Please back up your claim and cite a reference according to which "in relativity", a reference system that is at rest to the surface of a non-rotating planet is not in good approximation a Galilean frame. As far as I know Galilean frames are uniquely defined, there is no ambiguity like with the term "inertial".

No book on GR written 1970s or later that I have seen even mentions Galilean frames. On the other hand, MTW has a whole section on "Proper Reference Frames" in general relativity, which is is my primary reference on the matter [I can't give a page number at this moment because I am on vacation; also my internet access is limited]. Numerous papers on Fermi-Normal coordinates espouse the same approach. My posts earlier on this, specifically formulas I gave in discussion with Peter Donnis, come from this discussion.

In the framework of "Proper Reference Frames", the ECI frame is the (insert local if you must) inertial frame of a non-spinning observer in the center of the earth. It has exact Minkowski metric at the origin and vanishing connection components at the origin (which is why it is inertial). Of course this reference frame includes the surface of the earth, but it is completely different from a reference frame 'of a lab on the surface'. The latter is defined by using the lab center as the origin, the lab center clock as the standard of time, and ruler measurements from the lab center. The result is completely different frame than the ECI. This lab frame is an accelerating frame, because the:

- mathematically: the connection coefficients do not vanish at the origin
- physically: the origin of the frame (lab center) experiences proper acceleration

In contrast, in Newtonian physics, the lab frame would be identical to the ECI frame [assuming a non-rotating earth] except for translation of origin. They would both be inertial frames.

[Note: ECI stands for "earth centered inertial" frame, and as used with GR, it has a metric varying radially from the center, with connection coefficients becoming non-vanishing away from the center. This gets at why I think local frames are more than just 'at a point' definitions. They are useful to describe physics in a possibly substantial spatial region and over a long period of time. The fundamental limit on their extension is only due to break down of forming a valid coordinate chart. In practice, they often lose utility before running into such fundamental issues (e.g. incorporating the sun in ECI is both complex and useless, but mathematically possible, in principle, in GR. Fermi-Normal coordinates do not yet break down, but they become intractable and useless.]

 

[harrylin]

Yes, it does. The ECI is in free fall about the sun.

Once more: the ECI frame is a free falling reference system of the Earth, but does not correspond to the free falling local reference system of a group of particles near the Earth.

By these definitions all Galilean frames are also local inertial frames, since a Galilean reference system is clearly a reference system that locally can be used just like a Galilean reference system. So again, these Landau local inertial frames can accelerate relative to each other.

No, by definition Galilean frames do not accelerate relative to each other. Free falling reference systems only mimic Galilean frames locally for the physics.