Insights Understanding the General Relativity view of gravity on Earth - Comments

[Dale]

DaleSpam submitted a new PF Insights post

Understanding the General Relativity View of Gravity on Earth

Continue reading the Original PF Insights Post.

 

[PeterDonis]

Great post, DaleSpam!

 

[vanhees71]

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.

 

[Finny]

[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.

 

[ShayanJ]

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?

 

[Dale]

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.

 

[ShayanJ]

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.

 

[Dale]

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.

 

[Dale]

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.

 

[PWiz]

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?

 

[harrylin]

[..] 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.

 

[vanhees71]

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 ;-)).

 

[Dale]

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.

 

[Dale]

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.

 

[A.T.]

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:

https://en.wikipedia.org/wiki/Map_projection#Conic

If you roll out the cone flat, you end up with this local picture:

 

[PWiz]

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!

 

[harrylin]

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.

 

[harrylin]

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!

 

[A.T.]

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?

 

[Dale]

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.

 

[vanhees71]

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!

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...

 

[Finny]

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!

 

[harrylin]

[..] 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.

http://plato.stanford.edu/entries/spacetime-iframes/#IneFraNewSpa

 

[harrylin]

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)

 

[A.T.]

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"?

 

[harrylin]

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.

 

[A.T.]

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.

 

[harrylin]

[..] 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).

 

[A.T.]

Not according to Landau, as I already cited...

What exactly is different in that quote, compared to what I wrote?

 

[Dale]

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.

 

[harrylin]

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.

 

[harrylin]

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.

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 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)"

 

[vanhees71]

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).

 

[A.T.]

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.

 

[harrylin]

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".

 

[harrylin]

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.

 

[A.T.]

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.

 

[PAllen]

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.

 

[harrylin]

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".

 

[A.T.]

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?

 

[PAllen]

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.

 

[Dale]

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?

 

[PAllen]

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).

 

[Dale]

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.

 

[PAllen]

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.

 

[PAllen]

...

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.

 

[PeterDonis]

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?

 

[vanhees71]

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?

 

[PAllen]

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)², 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_j(t)x^j, 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'].

 

[Dale]

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.