Understanding the General Relativity View of Gravity on Earth
Often students have difficulty reconciling the General Relativity (GR) view of gravity versus their own experience with gravity on the surface of Earth. This article covers some of the basic concepts and explains how they work with “every day” gravity.
Table of Contents
Important Concepts
- Spacetime: the combination of 3 dimensions of space and 1 dimension of time into a unified 4D “manifold”.
- Coordinate systems: mappings between events in spacetime and 4 numbers called coordinates. Usually, the coordinates involve a 1-time coordinate and 3 spatial coordinates.
- Proper acceleration: the acceleration measured by an ideal accelerometer. This is the acceleration that you physically “feel”.
- Coordinate acceleration: the 2nd derivative of position in some given coordinate system.
- Inertial frame: a coordinate system where inertial objects have no coordinate acceleration. In inertial frames, the line formed by an inertial object’s coordinates is a straight line.
- Curvature: when a surface is not flat, for example, the surface of a sphere. The geometry is different in curved manifolds versus flat ones. For instance, the angles in a triangle may not sum to ##\pi##.
Non-inertial Frames and Fictitious Forces
Physics is traditionally taught using inertial frames, however, there is no restriction that prevents the use of non-inertial frames (e.g. a rotating reference frame). In a non-inertial frame inertial objects undergo coordinate acceleration, and in order to use Newton’s 2nd law in such frames additional “fictitious forces” (e.g. the centrifugal and Coriolis forces) are introduced.
These fictitious forces all share a few properties:
- They are proportional to the mass of the object
- They cannot be detected by an accelerometer
- They disappear in inertial frames
The Force of Gravity: Newton vs. Einstein
In Newtonian mechanics, gravity is considered to be a real force, even though it shares the first two properties of fictitious forces listed. This makes Newtonian gravity a bit of a strange force. You cannot determine if a given reference frame is inertial or not simply by using accelerometers, you have to additionally know the distribution of mass nearby to correct your accelerometer readings for the presence of gravity.
In GR, this is simplified by considering gravity to be a fictitious force just like any other force which is proportional to the mass and cannot be measured by an accelerometer. This means that in GR an apple in free fall to the ground is considered to be inertial, while in Newtonian gravity the apple is non-inertial. Similarly, the rest frame of the free-falling apple is an inertial frame in GR with no fictitious forces, while in Newtonian mechanics it is a non-inertial frame with a fictitious force that is equal and opposite to the force of gravity.
The Apple Falling
As mentioned above, an apple freely falling to the ground is considered to be inertial in GR. If we make a coordinate system where the apple is at rest, it is an inertial coordinate system. In this reference frame, the apple does not accelerate, instead, the ground accelerates upwards at g and slams into the apple. This corresponds to the fact that an accelerometer attached to the apple reads 0, while an accelerometer attached to the ground reads 1 g upwards.
This is a bit odd. What is causing the ground to accelerate?
If you draw a free-body diagram of a small section of the ground, you find that there is a rather large real upward pressure force on the bottom of the section, and this upward pressure is not balanced by a corresponding large downward pressure on the top. Therefore, there is a net upward force on the ground, which is responsible for the upward acceleration. Although this seems like a strange way to think, at first, it is clear that this approach still accounts for whether or not the apple splatters when it hits the ground, and so forth.
Spacetime Geometry
Now consider a “spacetime diagram” of the apple, where time is plotted on one axis and the apple’s vertical position is plotted on the other axis. In such a diagram the apple can be represented by a line (called a “worldline”) which shows its position at each point in time. If we take the frame where the apple is at rest then the apple travels along a straight line which is parallel to the time axis. A second apple that started free-falling a moment earlier or later would have some constant velocity relative to the first apple, so its worldline would also be a straight line, but not parallel to the time axis.
So objects at rest relative to each other have parallel worldlines while objects that are moving relative to each other have non-parallel worldlines. Similarly, objects at rest relative to a reference frame have worldlines that are parallel to the time axis.
Now, consider a point on the ground. This point is initially at rest relative to the apple (and therefore the reference frame), so its worldline starts out parallel to the time axis. However, by the time the ground collides with the apple, it has gained some relative velocity and is no longer parallel. In other words, the ground’s worldline is curved.
So inertial objects (accelerometer reads 0) have straight worldlines while non-inertial objects have worldlines that are curved in a direction and by an amount given by their proper acceleration.
Curved Spacetime and Tidal Gravity
While this is all well and good for a single apple falling in a uniform gravitational field, what happens when we consider tidal gravity (gravity that varies over space) such as two apples falling on opposite sides of the world?
Let’s suppose that there is a hole completely through the Earth and no atmosphere so that we can neglect the ground and the air (let’s also neglect the rotation of the Earth). In this case, the two apples will start on opposite sides of the Earth and eventually collide when they reach the middle. When they start they will be at rest relative to each other, and when they collide they will have a considerable velocity relative to each other.
How can that work with our spacetime view? We already established that being at rest means that their worldlines are parallel, and having relative velocity means that their worldlines are not parallel. So the two apples’ worldlines start parallel and then wind up non-parallel. However, we also established that having 0 proper acceleration (i.e. being inertial and having 0 accelerometer reading) means that the worldline is straight. So both apples’ worldlines are straight. How can we reconcile the straight worldlines with the fact that they go from being parallel to intersecting? The answer is that in the presence of tidal gravity spacetime is curved, not flat.
A straight line in curved space is called a geodesic, for example, a great circle is a geodesic on the surface of a sphere. A longitude line is a great circle, and therefore a geodesic. Consider two nearby longitude lines at the equator they are parallel but at the pole they intersect, despite being everywhere straight. So curved spaces have the necessary geometric properties.
Using this idea of curved spacetime, we can describe how the two apples can have 0 proper acceleration everywhere (straight worldlines) and yet accelerate relative to each other (initially parallel, but later intersecting). However, this has some consequences. The most critical is that there is no longer any such thing as an inertial frame that covers both apples, inertial frames are now “local” meaning that you can only use them when tidal effects are negligible (spacetime curvature is negligible).
The Surface of the Earth
In the previous section we neglected the ground, but now let’s consider the ground. If the ground is accelerating upwards and the direction corresponding to “up” changes around the globe, then it seems that the surface of the Earth should be expanding, with the distance from one point to another continually increasing.
This reasoning is incorrect in curved spacetime. In flat spacetime, it would be correct that the surface of the Earth could not be accelerating (proper acceleration) outwards while retaining a constant radius, but spacetime is curved so it can indeed accelerate (proper acceleration) outwards while retaining a constant radius.
To understand the importance of curvature, consider two latitude lines on a sphere. For simplicity consider the latitude lines 5° N and 5° S. As you follow those lines around the sphere, they maintain a constant distance from each other. However, the 5° N line is constantly turning to the north and the 5° S line is constantly turning to the south. So they are turning away from each other but maintaining a constant distance. This is impossible on a flat surface, but possible on a curved surface.
In the geometry of the spacetime around the Earth (i.e. Schwarzschild spacetime) you can take any two points on the surface of the Earth and find that they are accelerating (covariant derivative = proper acceleration) in different directions, and yet, because the spacetime is curved, the distance between them does not change (e.g. as measured by radar).
Summary and Key Points
- In GR the force of gravity is considered to be a fictitious force, and inertial reference frames are free-fall frames where falling apples have straight worldlines and the ground continuously accelerates upwards at g.
- In order to model tidal gravity (where gravity varies from location to location), we must use curved spacetime.
- This allows for two apples to each have straight (geodesic) worldlines but still, accelerate relative to each other.
- It also allows for the ground to continuously accelerate upwards without the Earth expanding.
- However, it means that you cannot make an inertial frame that covers the whole Earth, and so inertial frames are only local.
Education: PhD in biomedical engineering and MBA
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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 vaccuum (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.
“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 vaccuum (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.
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.
“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. :wink: 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 vaccuum (any mix of Weyl and Ricci curvature is accommodated by their construction).
“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. :wink: 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.
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.
“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.
“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.
“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 [URL=’http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html’]here[/URL].) 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.)
“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 [USER=293502]@harrylin[/USER] 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.
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.
“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.
“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.
“Yes, it does. The ECI is in free fall about the sun.[/quote]
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.
[quote]
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.
“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.]
“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.
” 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.
“(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”.
“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.
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 :-)).
“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.”)
“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”.
[URL]http://www.cpt.univ-mrs.fr/~rovelli/book.pdf[/URL] (p42, comments just before Eq 2.116 and also footnote 19)”
Thanks for the ref. :smile:
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.
“Already done in [URL=’https://www.physicsforums.com/threads/understanding-the-general-relativity-view-of-gravity-on-earth-comments.824068/page-4#post-5188194′]#73[/URL]”
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”?
“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”.
“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 [URL=’https://www.physicsforums.com/threads/understanding-the-general-relativity-view-of-gravity-on-earth-comments.824068/page-4#post-5188194′]#73[/URL]
“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”.
[URL]http://www.cpt.univ-mrs.fr/~rovelli/book.pdf[/URL] (p42, comments just before Eq 2.116 and also footnote 19)
“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.