Understanding the General Relativity view of gravity on Earth - Comments

[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.

 

[harrylin]

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

Surely you can answer that question yourself: are systems that are affected by gravitational fields, generally in "constant straight line and uniform motion relative to each other"?

Anyway, for sure the elaborations here were many times more than what textbook authors assume to be sufficient; it won't be useful to comment or clarify more.

 

[Dale]

Free falling reference systems only mimic Galilean frames locally for the physics.

Galilean frames clearly mimic Galilean frames also.

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.

Yes, that is the problem. Two different free falling reference frames do not correspond to each other and are not equivalent even though they both cover some of the same events.

 

[vanhees71]

But it's clear that an observer on the surface of the Earth is not inertial by definition, because he is not freely falling because of the electromagnetic interactions of the material (together with Pauli blocking for that matter) around with the observer. I think, it's very clear that the ECI is not a local inertial reference frame. Why some authors call such a frame Galilean is one of the great mysteries of the textbook writers, which I never understood. Galilei-Newton spacetime is very different from the general-relativistic (Einstein-Hilbert) spacetime. I'd prefer to call the reference frames that are realized by freely falling bodies (note again, that's a very real issue!) local inertial frames.

 

[Dale]

I think, it's very clear that the ECI is not a local inertial reference frame... I'd prefer to call the reference frames that are realized by freely falling bodies (note again, that's a very real issue!) local inertial frames.

On the contrary, it is very clear that the ECI is a local inertial reference frame (after all, the "I" in "ECI" is for "Inertial"). It is freely falling around the sun and therefore clearly qualifies as a "local inertial frame" per your usage and per the Landau usage.

In fact @harrylin is incorrect in claiming that the ECI is a "Galilean reference frame", but I share your distaste for the term. The ECI is a local inertial frame: it is free-falling around the sun, which is in turn free falling around the galaxy, ...

But it's clear that an observer on the surface of the Earth is not inertial by definition, because he is not freely falling because of the electromagnetic interactions of the material (together with Pauli blocking for that matter) around with the observer.

Sure, (neglecting rotation) such an observer is not inertial. However, in Newtonian mechanics they are at rest in an inertial frame, the ECI. They are acted on by two real forces, the contact force and gravity, which cancel each other out. So although the observer itself is not inertial, their rest frame (the ECI) is inertial.

 

[PeterDonis]

it is very clear that the ECI is a local inertial reference frame (after all, the "I" in "ECI" is for "Inertial")

It might be "inertial" in the Newtonian sense, yes, but not in the GR sense. Which, of course, just underscores the ambiguity in terminology that has driven much of this thread.

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 means the ECI is not a local inertial frame in the standard GR sense; such a frame would have vanishing connection coefficients everywhere within its domain. (The fact that the connection coefficients must vanish, to the accuracy of measurement, is what restricts the domain of a local inertial frame to a small patch of spacetime.) What you're describing, in GR terms, are more like Fermi normal coordinates centered on a freely falling worldline; such coordinates are not a local inertial frame because they can cover an entire "world tube" centered on the worldline, not just a small patch centered on a particular event. and the connection coefficients can become non-vanishing off the centered worldline because of spacetime curvature.

Also, as I understand it, the ECI frame, from a GR point of view, takes the metric for Fermi normal coordinates centered on a freely falling worldline, and adds in the Earth's gravitational potential "by hand" in the appropriate metric coefficients. (See, for example, the treatment in section 3 of the Living Reviews article on relativity in the GPS http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html .) This means that, in GR terms, the ECI is not even an inertial frame in a small patch of spacetime; its metric is not Minkowski anywhere, because of the gravitational potential.

So the only sense in which the ECI could be said to be "inertial" is the Newtonian sense in which DaleSpam is using the term here. (The main intent of the "I" in ECI appears to be to signify that it is non-rotating, as opposed to the ECEF frame which rotates with the Earth. In GR terms, once again, this would mean Fermi normal coordinates, not local inertial coordinates--but then we still have the Earth's gravitational potential added in, as above.)

 

[Dale]

It might be "inertial" in the Newtonian sense, yes, but not in the GR sense. Which, of course, just underscores the ambiguity in terminology that has driven much of this thread.

Yes. The GPS system does not use GR in its computations, other than a correction for time dilation. Other than that, it treats spacetime as flat.

 

[PeterDonis]

The GPS system does not use GR in its computations, other than a correction for time dilation.

If you mean that the calculations of the satellite orbits (which are essential to the position data sent to receivers) don't require GR, that's true; GR effects are much too small to matter for anything other than time dilation.

Other than that, it treats spacetime as flat.

That's not quite true; if you look at the metric in the Living Reviews article I referenced, it has a correction term in the spatial part of the metric as well. But that correction term turns out to be small enough that it can be ignored (it's a factor of $c^2$ smaller than the correction to $g_{00}$). So in practical terms, yes, the GPS coordinates are assumed to be Euclidean in the spatial part.

 

[PAllen]

Well, to Peter's comment on my description, I make the following notes:

MTW does not define a proper reference frame ultralocal at all. It defines it geometrically, with the result being Fermi-Normal coordinates extended to allow for rotation of the tetrad relative to Fermi-Walker transport of a starting tetrad. In inertial frame is simply the special case where connection components vanish exactly at the origin world line (the metric is Minkowski exactly at the origin world line in all cases). Thus, ECI would a an inertial frame per this definition.

Note, from the Living Review article you reference, the author seems to agree:

"For the GPS it means that synchronization of the entire system of ground-based and orbiting atomic clocks is performed in the local inertial frame, or ECI coordinate system "

"because in the underlying earth-centered locally inertial (ECI) coordinate system"

Finally, the metric given, if naturally extended to the center (they don't bother with this since subterranean GPS is not a realized product), would have Minkowski metric an vanishing connection at the origin.

 

[PeterDonis]

MTW does not define a proper reference frame ultralocal at all.

But MTW also makes a sharp distinction between a "local inertial frame" (my term, I'd have to go back and look to see exactly what term(s) MTW uses for this), which only covers a small patch of spacetime, and what you are calling a "proper reference frame", which MTW calls Fermi normal coordinates and which can cover a "world tube" around any chosen worldline. The ECI frame is definitely not a local inertial frame by MTW's definition. Whether it qualifies as Fermi normal coordinates is more problematic, because of the extra terms in the metric due to the gravitational potential. See below.

from the Living Review article you reference, the author seems to agree

He agrees on an ambiguous use of terminology, yes. He is using "local inertial frame" in the Newtonian sense (or perhaps the "Fermi normal" sense--but see below), not the GR sense (i.e., the MTW sense I referred to above). Unfortunately this seems to be very common.

the metric given, if naturally extended to the center (they don't bother with this since subterranean GPS is not a realized product), would have Minkowski metric an vanishing connection at the origin.

It would if you rescaled the potential $V$ to be zero at the center of the Earth, yes (the standard ECI frame does not do this; the potential is effectively zero on the geoid, so it would be negative at the center of the Earth).

However, that isn't enough to make ECI coordinates the same as Fermi normal coordinates. At least as I read MTW's discussion of those, they assume that the object following the chosen worldline is a test object, and does not produce any spacetime curvature on its own; the nonzero connection coefficients as you move away from the chosen worldline can only be due to spacetime curvature from other sources (for example, the Sun). The Earth clearly does not meet this requirement.

 

[PAllen]

But MTW also makes a sharp distinction between a "local inertial frame" (my term, I'd have to go back and look to see exactly what term(s) MTW uses for this), which only covers a small patch of spacetime, and what you are calling a "proper reference frame", which MTW calls Fermi normal coordinates and which can cover a "world tube" around any chosen worldline. The ECI frame is definitely not a local inertial frame by MTW's definition. Whether it qualifies as Fermi normal coordinates is more problematic, because of the extra terms in the metric due to the gravitational potential. See below.

I don't recall an MTW definition of local inertial frame separate from the section on Proper Reference Frame (which has inertial frame as a special case). However, I can't check right now because I have no access to my books.

He agrees on an ambiguous use of terminology, yes. He is using "local inertial frame" in the Newtonian sense (or perhaps the "Fermi normal" sense--but see below), not the GR sense (i.e., the MTW sense I referred to above). Unfortunately this seems to be very common.

Of course all we can do is guess what Ashby meant. I don't see his usage as Newtonian.

It would if you rescaled the potential $V$ to be zero at the center of the Earth, yes (the standard ECI frame does not do this; the potential is effectively zero on the geoid, so it would be negative at the center of the Earth).

Yes, I agree on this. Almost as soon as I wrote my prior post I realized you would have to reset the zero point of the potential (which is arbitrary anyway).

However, that isn't enough to make ECI coordinates the same as Fermi normal coordinates. At least as I read MTW's discussion of those, they assume that the object following the chosen worldline is a test object, and does not produce any spacetime curvature on its own; the nonzero connection coefficients as you move away from the chosen worldline can only be due to spacetime curvature from other sources (for example, the Sun). The Earth clearly does not meet this requirement.

My read is different from yours. The frame is based on a world line not a body of any kind (test or otherwise). The center of the Earth is perfectly ok. Nothing in their derivation restricts the world line to being in vacuum (any mix of Weyl and Ricci curvature is accommodated by their construction).

 

[Dale]

Does anyone know if Newton Cartan spacetime is curved, or if it is only space that is curved since the space and time parts are metrically distinct? (If metrically distinct is even a valid term)

I don't know much about Newton Cartan gravity, but it might go a long way to harmonizing terminology like this.

 

[PeterDonis]

I don't recall an MTW definition of local inertial frame separate from the section on Proper Reference Frame

I'll check my copy when I get a chance.

Nothing in their derivation restricts the world line to being in vacuum (any mix of Weyl and Ricci curvature is accommodated by their construction).

I'll take a look, it's been a while since I reviewed that chapter and I may be misremembering.

 

[PAllen]

My read is different from yours. The frame is based on a world line not a body of any kind (test or otherwise). The center of the Earth is perfectly ok. Nothing in their derivation restricts the world line to being in vacuum (any mix of Weyl and Ricci curvature is accommodated by their construction).

Ok, so I guess a precise description of ECI is 'almost' what I wrote many posts ago, which was:

" the (inertial) frame of a non-spinning observer at the center of the earth" [you can think of this as a test body or small lab, though only an origin world line figures in the construction]

The correction is that ECI starts from the above, and rescales such that g00 is 1 at the geoid rather than the center, for more convenient usage at and outside the Earth's surface. To me, it is still functionally an inertial frame of a center of Earth observer, despite this coordinate transform, but this is a judgement.

 

[cianfa72]

A comment on this old insight.

@Dale said

Inertial frame: a coordinate system where inertial objects have no coordinate acceleration.

I believe no coordinate acceleration refers to 2nd derivative of spacelike coordinates w.r.t. timelike coordinate.

In inertial frames, the line formed by an inertial object’s coordinates is a straight line.

This should be just a necessary condition.To be an inertial reference frame (aka inertial coordinate chart/system) we must add the requirement of vanishing geodesic deviation for them (i.e. straight worldlines of inertial objects starting parallel remain parallel).

 

[Dale]

To be an inertial reference frame (aka inertial coordinate chart/system) we must add the requirement of vanishing geodesic deviation for them (i.e. straight worldlines of inertial objects starting parallel remain parallel).

Hmm, is that condition actually required? Is it possible to have a region of curved spacetime with geodesic deviation and a chart where all inertial objects have worldlines that are straight lines in the chart? I can’t think of an example, but certainly admit that there may be one that I haven’t considered.