To clarify, Ibix is referring to a 2D ##(x, y)## Cartesian coordinate system covering all of the Earth's surface, not a 3D ##(x, y, z)## Cartesian coordinate system covering all of the Earth including its interior.Ibix said:For the same reason you can't set up a Cartesian coordinate system covering all of Earth - it won't work on a curved surface because initially parallel lines will not generally remain so.
"Stationary" has no physical meaning. In general, stationary means with constant spatial coordinates.Kashmir said:1) with respect to whom are A and B stationary?
Yes, in the sense that we expect that metric to represent the spacetime outside the Earth. You can't allocate ##x, y, z## coordinates arbitraily.Kashmir said:2) in this particular example do coordinates (x, y, z) have any physical significance?
Careful. There is a physical meaning to the term for a certain class of curved spacetimes. See my further comments below in response to the OP.PeroK said:"Stationary" has no physical meaning.
See the bolded addition I made above. It's important. See below.Kashmir said:So we can't have a 'Three perpendicular rods and a clock' as a coordinate system throughout a curved spacetime with the same properties as a global inertial frame in flat spacetime , although we may use it in a small region.
With respect to the gravitating body. Note, though, that the term "stationary" has two different possible meanings:Kashmir said:1) with respect to whome are A and B stationary?
In this particular example, yes, because the coordinates have been carefully chosen so that both meanings of "stationary" given above match: objects such as A and B, whose spatial coordinates are constant (note that this requires that the coordinate chart we chose has one timelike coordinate, ##t##, and three spacelike coordinates, ##x##, ##y##, and ##z##--but you can't just tell this from the names of the coordinates, you have to look at the metric), also are "hovering" at a constant distance from a gravitating body whose gravitational field is not changing with time. That is only possible in special scenarios like this one, where there is a gravitating body whose gravitational field is not changing with time.Kashmir said:2) in this particular example do coordinates (x, y, z) have any physical significance?
Expanding further on my previous post, with regard to this point:Kashmir said:So we can't have a 'Three perpendicular rods and a clock' as a coordinate system throughout a curved spacetime
You mean Spacelike coordinates ?PeterDonis said:spatial coordinates.
What distance are we talking about here? ##ds^2##?PeterDonis said:constant distance
Which time? coordinate time?PeterDonis said:gravitational field does not change with time
I think a lot of your questions are answered in Chapter 7. With something like GR perhaps you should try to press on instead of stopping in your tracks. Park these questions for now. No book can explain everything at once.Kashmir said:You mean Spacelike coordinates ?
What distance are we talking about here? ##ds^2##?
Which time? coordinate time?
Any coordinate that is "spatial" will have to be spacelike, yes.Kashmir said:You mean Spacelike coordinates ?
The integral of ##ds^2## along a spacelike curve of constant coordinate time.Kashmir said:What distance are we talking about here? ##ds^2##?
If the time coordinate is chosen properly, which it is in the example you give, yes. The more technical definition is that the spacetime is stationary if it has a timelike Killing vector field. One can always choose a time coordinate ##t## such that the metric is independent of ##t## in a spacetime that has a timelike Killing vector field (proving this is often an exercise in GR textbooks).Kashmir said:Which time? coordinate time?
Hartle introduces Killing vectors in Chapter 8 (Geodesics), which is why I think the OP needs to press on and see whether subsequent chapters answer these questions.PeterDonis said:If the time coordinate is chosen properly, which it is in the example you give, yes. The more technical definition is that the spacetime is stationary if it has a timelike Killing vector field.
Omitting the motivations and historical connections, and also the
detailed calculations, I state succinctly the principles that determine
the relativistic idealization of a GPS system. These determine the
results that Ashby presents in his tutorial.
A method for making sure that the relativity effects are specified correctly
(according to Einstein’s General Relativity) can be described rather briefly.
It agrees with Ashby’s approach but omits all discussion of how, historically
or logically, this viewpoint was developed. It also omits all the detailed
calculations. It is merely a statement of principles.
One first banishes the idea of an “observer”. This idea aided Einstein
in building special relativity but it is confusing and ambiguous in general
relativity. Instead one divides the theoretical landscape into two categories.
One category is the mathematical/conceptual model of whatever is happen-
ing that merits our attention. The other category is measuring instruments
and the data tables they provide.
The conceptual model for a relativistic system is a spacetime map or
diagram plus some rules for its interpretation. For GPS the attached Figure
is a simplified version of the map. The real spacetime map is a computer
program that assigns map locations xyzt to a variety of events.
The constant Φ0 is chosen so that a standard SI clock “on the geoid” (e.g.,
USNO were it at sea level) would give, inserting its world line x(t), y(t), z(t)
into equation (1), just dτ = dt where dτ is the physical proper time reading
of the clock.
wiki said:In the 1970s, it became clear that the clocks participating in TAI were ticking at different rates due to gravitational time dilation, and the combined TAI scale, therefore, corresponded to an average of the altitudes of the various clocks. Starting from the Julian Date 2443144.5 (1 January 1977 00:00:00), corrections were applied to the output of all participating clocks, so that TAI would correspond to proper time at the geoid (mean sea level).