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.
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“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.
“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 :-).
“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
“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”.
“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
“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.
“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 doesnt 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.
“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:
[URL]http://www.adamtoons.de/physics/gravitation.swf[/URL]
“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.
“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?
“Yes, I simply gave a suggestion for less ambiguous phrasing. :cool:
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 [URL=’https://www.physicsforums.com/threads/understanding-the-general-relativity-view-of-gravity-on-earth-comments.824068/page-3#post-5178260′]#43[/URL]”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.
“Yes, I simply gave a suggestion for less ambiguous phrasing. :cool:
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 [URL=’https://www.physicsforums.com/threads/understanding-the-general-relativity-view-of-gravity-on-earth-comments.824068/page-3#post-5178260′]#43[/URL]”
How does it avoid it? The earth lab is an inertial frame per Newton and an accelerated frame per relativity.
“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
“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. [/quote]
Yes, I simply gave a suggestion for less ambiguous phrasing. :cool:
[quote] [..] 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 [URL=’https://www.physicsforums.com/threads/understanding-the-general-relativity-view-of-gravity-on-earth-comments.824068/page-3#post-5178260′]#43[/URL]
“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.
The GPS satellites are not nothing but made of something. 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.
To the contrary, I’m complaining about people that take the coordinates for the world, but I think it’s only a semantical issue, and we shouldn’t discuss it further here.
“Then explain to me, how to define a frame in practice without realizing it somehow as a material object.”The way GPS does it is a good example. Do you not understand that there is no material object in the GPS system which “is” the ECI frame?
But in any case this misses the heart of the issue. Even when you do attach a reference frame to a material object, the frame and the object are not the same thing. The frame is a mathematical tool for the analysis, the object is a physical material thing.
Do you not understand the difference between a material object and a mathematical tool? Were you not the one who was complaining that people mistake the world for the coordinates? It seems like that comment was introspective, because I don’t see anyone else here showing any indication of that confusion besides yourself.
“Then explain to me, how to define a frame in practice without realizing it somehow as a material object.”
One can define a reference frame, without any physical objects at rest in that frame, simply by stating that the frame moves a velocity v relative to some physical object.
Then explain to me, how to define a frame in practice without realizing it somehow as a material object.
“As theoretical physicists we often forget that the world does not consist of quadruples of numbers (coordinates)”I don’t think that anyone here is making that mistake. However, avoiding this mistake does not require making the alternate mistake of saying that a physical object “is” a related mathematical quantity.
“from which you can evaluate the coordinates in whatever other frame you want”Precisely. This can be done because the frame is not the material object, it is a mathematical quantity.
Sure, but the GPS satellites are a material realization of some reference frame (or many local reference frames), from which you can evaluate the coordinates in whatever other frame you want. Without a material realization of some reference frame you cannot measure anything. As theoretical physicists we often forget that the world does not consist of quadruples of numbers (coordinates) but that you have to realize reference frames in order to map the space-time manifold to subsets of ##mathbb{R}^4##.
“to realize such a frame you need indeed some materialization of it, i.e., a measurement apparatus prepared such that it is at rest relative to the center-of-momentum frame.”No, you don’t. We know quite well the solar system’s center of momentum frame despite not having any measurement apparatus at rest relative to the frame. Similarly, the GPS Earth centered inertial frame is extremely well realized without any at-rest measurement apparatus.
This started from an exceedingly minor nitpick about a well written post, and I don’t want to make any more of it than we already have. Please do not read more into it than was there.
“Sure! But to realize such a frame you need indeed some materialization of it, i.e., a measurement apparatus prepared such that it is at rest relative to the center-of-momentum frame”
Not sure what you mean by “realize a reference frame”. But to define a reference frame, one certainly doesn’t need any physical object to be at rest in that frame.
“Consider an otherwise isolated system of two equal-mass classical charges. What would you consider to be the most natural reference frame? I would consider the inertial center of momentum frame most natural, not one of the non inertial frames attached to the charges.”
Sure! But to realize such a frame you need indeed some materialization of it, i.e., a measurement apparatus prepared such that it is at rest relative to the center-of-momentum frame, which then should be an inertial frame (which of course can be tested either as soon as you have materially realized this frame).
“g00 will take the approximate form 1 + 2a z, which is 1 for z=0”
Ah, right, got it.
“I’d say, a reference frame is always determined by some physical object. How else should it be realized?”
Consider an otherwise isolated system of two equal-mass classical charges. What would you consider to be the most natural reference frame? I would consider the inertial center of momentum frame most natural, not one of the non inertial frames attached to the charges.
“I assume that by “almost Minkowski” you mean “Minkowski except for the extra term in ##g_{00}##”, correct?”
No, g00 will take the approximate form 1 + 2a z, which is 1 for z=0. Note:
– It is a universal feature of FN coordinates that the metric is exactly Minkowski at the origin, for all time (but not so for connection coefficients, unless the origin world line – time axis – is an inertial world line).
– To see this from Rindler coordinates, translate the origin to z = 1/g (the world line whose proper acceleration is g). Get g00 = (1 + gz)[SUP]2[/SUP], consistent with the general first order value for g00 in FN coordinates (In general, for FN coordinates, we have proper acceleration as a function of time, and and varying in direction over time. Then, the first order form of g00 is 1+2 a[SUB]j[/SUB](t)x[SUP]j[/SUP], with summation implied. )
[edit: I guess another aspect of almost Minkowski is that deviation from Minkowski is first order proportional to distance from the origin. The smaller the ‘lab’ the less deviation anywhere from Minkowski metric. Again, the connection coefficients encode the acceleration, which does not vanish locally. This is mathematically what I mean by ‘almost Minkowski except for acceleration’].
“That seems right to me. The only minor detail is that I wouldn’t say that a physical object is a reference frame, but I realize that saying things like “the frame of the lab” or “the frame where the lab is at rest” makes the wording more cumbersome.”
I’d say, a reference frame is always determined by some physical object. How else should it be realized?
“Fermi-Normal coordinates in which the metric is almost Minkowski for the lab (exactly at e.g. lab center), for all time, but there are time independent, nearly constant, connection coefficients.”
I assume that by “almost Minkowski” you mean “Minkowski except for the extra term in ##g_{00}##”, correct?
”
….
Relativistic Terminology:
1) A lab sitting on earth is an accelerated frame. It can be made part of some natural coordinate system that is asymptotically Minkowski at infinity because the spacetime is asymptotically flat. There are several such natural coordinates (e.g. standard exterior Schwarzschild, isotropic exterior Schwarzschild, etc.).
…..
”
I should add that the earth lab coordinates that best capture experience of ‘inertial except for nearly uniform acceleration’ are also Fermi-Normal coordinates in which the metric is almost Minkowski for the lab (exactly at e.g. lab center), for all time, but there are time independent, nearly constant, connection coefficients. These coordinates are NOT useful globally, and are NOT the same as any of the convenient global coordinates. This is important to note in reference to all frames being local in GR. Note also, such coordinates are close to Rindler coordinates in flat spacetime with the origin suitably translated – up to the order of tidal effects.
“That seems right to me. The only minor detail is that I wouldn’t say that a physical object is a reference frame, but I realize that saying things like “the frame of the lab” or “the frame where the lab is at rest” makes the wording more cumbersome.”
Agreed.
That seems right to me. The only minor detail is that I wouldn’t say that a physical object is a reference frame, but I realize that saying things like “the frame of the lab” or “the frame where the lab is at rest” makes the wording more cumbersome.
To clarify terminology, I would like to propose a specific situation and what I think is essentially universal modern terminology. For any that disagree, this should hopefully focus discussion. Consider:
1) A lab sitting sitting on earth under the fictitious assumption that earth is isolated in an empty universe, and is not rotating.
2) A space lab in orbit around earth.
Newtonian terminology:
1) The earth lab is an inertial frame, and as with any true inertial frame in Newtonian physics, it is global in extent – covering the whole universe.
2) The space lab is an accelerating frame, with the special feature that it locally only can be treated as if it were inertial. Unlike a true inertial frame, there is no way to give it global extent while preserving the characteristics of simulating an inertial frame.
Relativistic Terminology:
1) A lab sitting on earth is an accelerated frame. It can be made part of some natural coordinate system that is asymptotically Minkowski at infinity because the spacetime is asymptotically flat. There are several such natural coordinates (e.g. standard exterior Schwarzschild, isotropic exterior Schwarzschild, etc.).
2) The space lab is an inertial frame. Like any frame in GR, it is local. There are no useful global coordinates based on the space lab. The quasi-local coordinates that correspond to this inertial frame are Fermi-Normal coordinates, which will be indistinguishable in properties from Minkowski coordinates over the space station, for all time (not just for some finite period). This latter feature is due to this being a true inertial frame in GR. (The Fermi-Normal coordinates are primarily useful within a reasonable size world tube encompassing the space lab’s history).
“Once more, what most matters for physics is the methods of calculation; and Landau gives a good example of vocabulary that is reasonably theory neutral.”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.
Note, that the quote from Newton is not limited to gravity. For example, a bunch of electrons interacting with each other in the presence of an external uniform E-field would also satisfy the description given by Newton and could use the same simplification mentioned by him. However, neither GR nor Newtonian physics would consider such a frame to be an inertial frame. So I do not think that the description is intended to be a definition of an inertial frame. I think it is intended to be a description of the utility of non-inertial frames.
What is the Landau vocabulary that you are talking about?
“As far as I know, “inertial frame” was not part of the vocabulary at that time, and it is besides the point. A group of free falling bodies towards a planet could according to Newton’s mechanics be used for local calculations as if they are in straight uniform motion, discounting the acceleration from the planet’s gravitation. On that point there is no disagreement between Newton and Einstein.
PS compare with modern usage:
“in a gravitational field the particle moves so that its world point moves along an extremal or, as it is called, a geodesic [..]; however, since in the presence of the gravitational field space-time is not galilean, this line is not a “straight line”, and the real spatial motion of the particle is neither uniform nor rectilinear. [..]by a suitable choice of the coordinate system one can always [turn] an arbitrary point of pace-time [into] a locally-inertial system of reference [which] means the elimination of the gravitational field in the given infinitesimal element of space-time”
– Landau & Lifchitz (Fields)”
I would have to say that parts of this wording are not modern, common usage. Especially e.g. “real spatial motion” is a concept with no plausible definition. Neither can “straight line” be defined in some way other than geodesic to make the statement that a geodesic is not straight. I would call modern books on GR as e.g. Wald, Carroll, Straumann.
“In their “mechanics” book, L&L describe “inertial frames” as…”
According to your interpretation of their definition, which frame is inertial:
– A frame at rest to the surface of a non-rotating planet?
– A frame free falling towards that planet?
– Both?
“If neither that quote nor the Newton quote use the term “inertial frame”, then there can obviously be no contradiction in how they use it. That doesn’t change the fact that the term is being used differently in classical Mechanics and GR.”
Once more, that depends on the definitions. In their “mechanics” book, L&L describe “inertial frames” as “Galilean” reference systems (it’s even the term they use in the older English version) and as you saw, they state in their “fields” book that the real spatial motion of the particle is neither uniform nor rectilinear while you state that it is actually considered inertial. In order to distinguish the concepts, they use “local inertial frames”.
“As far as I know, “inertial frame” was not part of the vocabulary at that time, and it is besides the point. A group of free falling bodies towards a planet could according to Newton’s mechanics be used for local calculations as if they are in straight uniform motion, discounting the acceleration from the planet’s gravitation. On that point there is no disagreement between Newton and Einstein.
PS compare with modern usage:
“in a gravitational field the particle moves so that its world point moves along an extremal or, as it is called, a geodesic [..]; however, since in the presence of the gravitational field space-time is not galilean, this line is not a “straight line”, and the real spatial motion of the particle is neither uniform nor rectilinear. [..]by a suitable choice of the coordinate system one can always [turn] an arbitrary point of pace-time [into] a locally-inertial system of reference [which] means the elimination of the gravitational field in the given infinitesimal element of space-time”
– Landau & Lifchitz (Fields)”
Not sure if L & L cover this later, but you can do much more than that. For an inertial particle (free fall, geodesic), you can introduce inertial coordinates that are spatially local but temporally global. That is, in mathematical terms, the metric remains diag(1,-1,-1,-1) and the connection components vanish, at the spatial origin, for all time. These are Fermi-Normal coordinates.
“they completely avoid the use of “inertial frame””If neither that quote nor the Newton quote use the term “inertial frame”, then there can obviously be no contradiction in how they use it. That doesn’t change the fact that the term is being used differently in classical Mechanics and GR.
“That doesn’t answer the question how what I wrote contradicts the Landau quote. And I’m not trying to avoid anything, but clarify by acknowledging conceptual differences.”
Again you put words in my mouth that I did not say (“not according to” is not synonymous with “contradict”!). In the cited part they completely avoid the use of “inertial frame” which different people may interpret differently; rather they use terms that are understood the same by everyone.
“I’d also avoid the expression “fictitious force”. It’s somewhat misleading. What you do is to write down an equation of motion of a particle in Newtonian physics using a non-inertial frame. From the kinetic part you move everything to the right-hand side, so that the equation reads
$$m ddot{vec{x}}=vec{F}(vec{x},dot{vec{x}}),$$
and then you call the right-hand side “force”, although it’s not precisely a force but contains parts from the kinetic term (unfortunately even giving names like Coriolis and centrifugal force), which go away when writing the equation in an inertial frame.
Everything becomes very easy when using Hamilton’s principle, which is form invariant under arbitrary point transformations (change of generalized coordinates) in the Lagrangian or even under the larger group of canonical transformations in phase space (symplectomorphisms).”
Right – Alonso&Finn neatly avoided that issue by simply writing the equation as coordinate acceleration relative to a rotating frame. No need at all to invoke a “fictitious force”.
“What you wrote appears to introduce just the inconsistency of definitions that we are trying to avoid.”That doesn’t answer the question how what I wrote contradicts the Landau quote. And I’m not trying to avoid anything, but clarify by acknowledging conceptual differences.
I’d also avoid the expression “fictitious force”. It’s somewhat misleading. What you do is to write down an equation of motion of a particle in Newtonian physics using a non-inertial frame. From the kinetic part you move everything to the right-hand side, so that the equation reads
$$m ddot{vec{x}}=vec{F}(vec{x},dot{vec{x}}),$$
and then you call the right-hand side “force”, although it’s not precisely a force but contains parts from the kinetic term (unfortunately even giving names like Coriolis and centrifugal force), which go away when writing the equation in an inertial frame.
Everything becomes very easy when using Hamilton’s principle, which is form invariant under arbitrary point transformations (change of generalized coordinates) in the Lagrangian or even under the larger group of canonical transformations in phase space (symplectomorphisms).
“The difference is if you consider there to be a fictitious force which is locally canceling the real force (Newton) or if you consider there to be no force locally, either real or fictitious (Einstein). The former represents a convenient non inertial frame and the latter represents an inertial frame. [/quote]
Once more, what most matters for physics is the methods of calculation; and Landau gives a good example of vocabulary that is reasonably theory neutral.
[quote]
I will have to read the rest of the reference, but at least the section 2.6 does not seem to support your usage.”
“He even applied this reasoning to the entire solar system, in order to justify treating it as an isolated system: if there were any outside force acting on it, it must have been acting more or less equally and in parallel directions on all parts of the system.
It may be alleged that the sun and planets are impelled by some other force equally and in the direction of parallel lines; but by such a force (by Cor. VI of the Laws of Motion) no change would happen in the situation of the planets to one another, nor any sensible effect follow; but our business is with the causes of sensible effects. Let us, therefore, neglect every such force as imaginary and precarious, and of no use in the phenomena of the heavens….(1729, volume 2 p. 558)”
“What exactly is different in that quote, compared to what I wrote?”
What you wrote appears to introduce just the inconsistency of definitions that we are trying to avoid.
“As far as I know, “inertial frame” was not part of the vocabulary at that time, and it is besides the point. A group of free falling bodies towards a planet could according to Newton’s mechanics be used for local calculations as if they are in straight uniform motion, discounting the acceleration from the planet’s gravitation. On that point there is no disagreement between Newton and Einstein.”The difference is if you consider there to be a fictitious force which is locally canceling the real force (Newton) or if you consider there to be no force locally, either real or fictitious (Einstein). The former represents a convenient non inertial frame and the latter represents an inertial frame.
Generally, when any non inertial frame is used, it is chosen specifically because it has this property. It produces a fictitious force that cancels out some real force.
I will have to read the rest of the reference, but at least the section 2.6 does not seem to support your usage.
“Not according to Landau, as I already cited….”What exactly is different in that quote, compared to what I wrote?
” [..] However, in Classical Mechanics the free falling frame isn’t formally inertial. It merely yields the same results as an inertial frame, because the inertial forces introduced by it cancel gravity. That is conceptually different from GR, where the free falling frame is actually considered inertial.”
Not according to Landau, as I already cited; apparently they managed to successfully avoid such inconsistencies of terms (it could depend however on which English edition one uses).
“As far as I can see, in Landau there is no conflict or disagreement between theories about such terms.”If Newton himself didn’t even define the term at his time, then there can obviously be no disagreement.
However, in Classical Mechanics the free falling frame isn’t formally inertial. It merely yields the same results as an inertial frame, because the inertial forces introduced by it cancel gravity. That is conceptually different from GR, where the free falling frame is actually considered inertial.
“Then why do you claim to provide a definition of “inertial frame”?”
Not at all… Once more:
I stressed how the usage in good textbooks such as by Landau of “local inertial frame” corresponds with the treatment for local calculations by Newton of a group of free-falling bodies.
As far as I can see, in Landau there is no conflict or disagreement between theories about such terms.
“Newton’s mechanics recognized local inertial frames as follows:…”
“As far as I know, “inertial frame” was not part of the vocabulary at that time, and it is besides the point.”Then why do you claim to provide a definition of “inertial frame”?
“According to your interpretation of this definition, which frame is inertial:
– A frame at rest to the surface of a non-rotating planet?
– A frame free falling towards that planet?
– Both?”
As far as I know, “inertial frame” was not part of the vocabulary at that time, and it is besides the point. A group of free falling bodies towards a planet could according to Newton’s mechanics be used for local calculations as if they are in straight uniform motion, discounting the acceleration from the planet’s gravitation. On that point there is no disagreement between Newton and Einstein.
PS compare with modern usage:
“in a gravitational field the particle moves so that its world point moves along an extremal or, as it is called, a geodesic [..]; however, since in the presence of the gravitational field space-time is not galilean, this line is not a “straight line”, and the real spatial motion of the particle is neither uniform nor rectilinear. [..]by a suitable choice of the coordinate system one can always [turn] an arbitrary point of pace-time [into] a locally-inertial system of reference [which] means the elimination of the gravitational field in the given infinitesimal element of space-time”
– Landau & Lifchitz (Fields)
” [..] To me this quote seems to be describing the use of non-inertial frames to eliminate real forces and simplify an analysis, although it isn’t using clear terminology so I cannot be certain. I see no mention of anything local.
What is the source for this quote? I am guessing that it is something quite old, before the terminology became clarified. I believe that my presentation accurately reflects the modern usage, and it is not intended to be an historical treatise.”
[URL]http://plato.stanford.edu/entries/spacetime-iframes/#IneFraNewSpa[/URL]
“In GR inertial frames have no proper acceleration, but in Newtonian mechanics inertial frames have a proper acceleration of -g. I tried to word it in a way that is true for both.
….. a theory-neutral explanation is difficult.”
It sure is….If that sentence I quoted isn’t in your ”insights” it should be.
Keeping track of
‘local’/ ‘distant’,
‘inertial’/ ‘accelerated’ and
‘coordinate’/’proper’
are key ideas that continue to make me think….not always correctly as you can tell!!
“Why would it be “fortunate” if there is a unique metaphysical opinion in the physics community? Physics must be based on facts of observation. Consequently the situation with QT is perhaps better – except from the esoterical part! o0)”
Exactly! If all physicists would agree on this simple definition, there’d be no (imho somewhat fruitless) debate about the “right” interpretation of QT, but one would be satisfied with just the minimal interpretation. There’s much more esoterics going on in quantum theory than in relativity, except for the crackpot community…
“Not really: even Newton’s mechanics recognized local inertial frames as follows:
“If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces.” (emphasis mine)”You are reading something into this that simply isn’t there. Neither the word “local” nor “inertial” nor “frame” even appears.
To me this quote seems to be describing the use of non-inertial frames to eliminate real forces and simplify an analysis, although it isn’t using clear terminology so I cannot be certain. I see no mention of anything local.
What is the source for this quote? I am guessing that it is something quite old, before the terminology became clarified. I believe that my presentation accurately reflects the modern usage, and it is not intended to be an historical treatise.
“Newton’s mechanics recognized local inertial frames as follows:
“If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces.” (emphasis mine)”
According to your interpretation of this definition, which frame is inertial:
– A frame at rest to the surface of a non-rotating planet?
– A frame free falling towards that planet?
– Both?
“Lorentz had also a different view concerning SR. Fortunately this is overcome in the physics community, and there is a unique view about relativity. Unfortunately, one can’t say this about QT, where in some niches of the scientific universe there coexist very different interpretations and metaphysics (reaching well into the realm of esoterics), and I’m not talking about obvious crackpots ;-)).”
Why would it be “fortunate” if there is a unique metaphysical opinion in the physics community? Physics must be based on facts of observation. Consequently the situation with QT is perhaps better – except from the esoterical part! o0)
“The problem for this description is not that GR inertial frames are local and Newtonian inertial frames are global. The problem is that even locally they disagree. So stressing “local” doesn’t avoid the reason that I chose that description. [..].”
Not really: even Newton’s mechanics recognized local inertial frames as follows:
“If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces.” (emphasis mine)
The pertinent difference for physics (that is, leaving aside philosophy and nomenclature) is that GR postulates this equivalence not only for Newton’s mechanics but for all physics.
“Sure. Consider a 1 cubic meter chunk of soil. If we draw a free-body diagram of that chunk of soil then we have real pressure forces on all 6 faces of the cube. The left and right and the front and back pressures all cancel out. However, the pressure force on the top is much less than the pressure force on the bottom, so they do not cancel out and there is a net pressure force upwards.
In the Newtonian inertial frame, that upwards pressure force is exactly balanced by the downwards gravitational force.
In the GR inertial frame, the downwards gravitational force does not exist, so the upwards pressure force is unbalanced and causes the ground to accelerate upwards.”
Ah, got it, thanks!
“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?”
You can approximate a small latitude range with a cone:
[URL]https://en.wikipedia.org/wiki/Map_projection#Conic[/URL]
If you roll out the cone flat, you end up with this local picture:
[MEDIA=youtube]DdC0QN6f3G4[/MEDIA]
“Cool post! I really liked how you explained the geodesic and ground’s upward acceleration parts. I just didn’t get the “free-body diagram of a small section of the ground” part. Can you elaborate on this a bit?”Sure. Consider a 1 cubic meter chunk of soil. If we draw a free-body diagram of that chunk of soil then we have real pressure forces on all 6 faces of the cube. The left and right and the front and back pressures all cancel out. However, the pressure force on the top is much less than the pressure force on the bottom, so they do not cancel out and there is a net pressure force upwards.
In the Newtonian inertial frame, that upwards pressure force is exactly balanced by the downwards gravitational force.
In the GR inertial frame, the downwards gravitational force does not exist, so the upwards pressure force is unbalanced and causes the ground to accelerate upwards.
“To avoid unnecessary confusion it is better to follow that example: the rest frame of the free-falling apple is a “LOCAL inertial frame” in GR, so that the apple can be considered as “inertial” locally.”The problem for this description is not that GR inertial frames are local and Newtonian inertial frames are global. The problem is that even locally they disagree. So stressing “local” doesn’t avoid the reason that I chose that description.
There are many equivalent ways of defining an inertial frame. I chose one that I thought fit best with the intention of the article.
Lorentz had also a different view concerning SR. Fortunately this is overcome in the physics community, and there is a unique view about relativity. Unfortunately, one can’t say this about QT, where in some niches of the scientific universe there coexist very different interpretations and metaphysics (reaching well into the realm of esoterics), and I’m not talking about obvious crackpots ;-)).
” [..] I did think about wording similar to [Inertial frame: a coordinate system where inertial objects have no coordinate acceleration], but the problem is that Newtonian and GR inertial frames are different. In GR inertial frames have no proper acceleration, but in Newtonian mechanics inertial frames have a proper acceleration of -g. I tried to word it in a way that is true for both.
It could probably still use some improvement, but a theory-neutral explanation is difficult.”
The good textbooks that I know clearly differentiate between “inertial motion” and “inertial frames” on the one hand, and “local inertial frames” on the other hand. Those mimic inertial frames for sufficiently local measurements. There is as a consequence a consistent use of terms throughout those textbooks, independent of theory.
To avoid unnecessary confusion it is better to follow that example: the rest frame of the free-falling apple is a “LOCAL inertial frame” in GR, so that the apple can be considered as “inertial” locally.
PS: Einstein had a subtly different view of GR than the view that you describe as “the GR view”, and surely he also taught GR. And Lorentz again had a subtly different view, and he also taught GR. In fact GR is interpretation neutral, as it is foremost mathematical, making predictions of observations. What you describe is perhaps more correctly indicated as the geometric view of GR, or the Minkowskian view of GR.
“Proper acceleration: “the acceleration measured by an ideal accelerometer” [consider adding: an acceleration an observer feels]”I like that. I will add that.
“Coordinate acceleration: “the 2nd derivative of position in some given coordinate system [add: an acceleration not felt]”You do “feel” coordinate acceleration in a GR local inertial frame (since it is equal to proper acceleration).
“Inertial frame: a coordinate system where inertial objects have no coordinate acceleration [I thought an inertial frame had no proper acceleration.] [yes, you say this later:”So inertial objects (accelerometer reads 0)…….”I did think about wording similar to that, but the problem is that Newtonian and GR inertial frames are different. In GR inertial frames have no proper acceleration, but in Newtonian mechanics inertial frames have a proper acceleration of -g. I tried to word it in a way that is true for both.
It could probably still use some improvement, but a theory-neutral explanation is difficult.
“Nice insight:
“Thanks. I appreciate the encouragement.
“Yes, very nice article. I’d only make sure to say once that gravity is not due to mass (energy) only (as in the Newtonian theory of gravity) but to all forms of energy-momentum distributions. This explains why light, which is described by massless spin-1 fields is affected by gravity (bending of light at the sun as one of the most important early tests of GR; red shift of light in gravitational field) and (in principle) is a source of gravity itself.”That is a good idea, but I am not sure it is a good idea for an “everyday gravity” explanation. I also avoided any discussion of time dilation for the same reason.
I will look back and see if there is a good place to put that in without much distraction.
“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.”
I get it now, thanks.
And now I can thank you a lot for the insight article, because this was the only thing that was keeping me from understanding this issue. So thanks.
“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.
“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 constant distance. This is impossible on a flat surface, but possible in a curved surface.”
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?
Yes, very nice article. I’d only make sure to say once that gravity is not due to mass (energy) only (as in the Newtonian theory of gravity) but to all forms of energy-momentum distributions. This explains why light, which is described by massless spin-1 fields is affected by gravity (bending of light at the sun as one of the most important early tests of GR; red shift of light in gravitational field) and (in principle) is a source of gravity itself.
Great post, DaleSpam!
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 doesnt 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.
Cool post! I really liked how you explained the geodesic and ground's upward acceleration parts. I just didn't get the "free-body diagram of a small section of the ground" part. Can you elaborate on this a bit?
[I could not find the two comments already posted]good write up….unsure of background education experience you are aiming at….accelerometer: maybe an explanation??….eg, it measures proper acceleration relative to free fall…Proper acceleration: "the acceleration measured by an ideal accelerometer" [consider adding: an acceleration an observer feels]Coordinate acceleration: "the 2nd derivative of position in some given coordinate system [add: an acceleration not felt]Inertial frame: a coordinate system where inertial objects have no coordinate acceleration [I thought an inertial frame had no proper acceleration.] [yes, you say this later:"So inertial objects (accelerometer reads 0)…….How about equivalence principle…That helped me at first….Nice insight:"In Newtonian mechanics gravity is considered to be a real force, despite the fact that 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 in order to correct your accelerometer readings for the presence of gravity.