About spacetime coordinate systems

[cianfa72]

Hi,

There is a point that, in my opinion, is not quite emphasized in the context of general relativity. It is the notion of spacetime coordinate systems that from the very foundation of general relativity are assumed to be all on the same footing. Nevertheless I believe each of them has to be specified by an "operational procedure" at least from a conceptual point of view. Otherwise what could be the meaning of coordinate values for a given event ?

Consider for instance the Schwarzschild metric where the metric tensor is defined in an (implied) coordinate system for the (curved) spacetime. In that case which is (at least from a conceptual point of view) the operational procedure to follow to assign Schwarzschild coordinates to a generic event ?

Thanks

 

[Ibix]

You define a set of non-rotating concentric spheres centred on the black hole. The r coordinate of a sphere of radius area A is $\sqrt{A/4\pi}$. Then you just mark latitude and longitude lines on the spheres and line all the spheres' markings up. Clocks tick slow relative to coordinate time, so you need clocks that can be deliberately set to count $\sqrt{1-2GM/c^2r}$ coordinate seconds per second. You pick one to be the master clock and set all clocks to zero when it would show zero - the correction to subtract out light travel time is simply determined by radar.

That's the formal way. Practically, you observe bodies in free fall to measure $r$. Then you just deploy a buoy and declare it to be $\theta,\phi=0,0$ and its clock to be the master and use slightly fancified versions of standard surveying techniques to locate anything else, I should think.

 

[Dale]

Otherwise what could be the meaning of coordinate values for a given event ?

Just a labeling of events in the manifold

 

[cianfa72]

You define a set of non-rotating concentric spheres centred on the black hole. The r coordinate of a sphere of radius A is $\sqrt{A/4\pi}$.

maybe A here is the concentric sphere surface area...not the radius

Clocks tick slow relative to coordinate time, so you need clocks that can be deliberately set to count $\sqrt{1-2GM/c^2r}$ coordinate seconds per second.

Thus, as far as I can understand, to "label" the coordinate time $t$ the physical clock located on the sphere with coordinate radius $r$ has to be deliberately "adjusted" to tick $\sqrt{1-2GM/c^2r}$ coordinate seconds per (let me say) "proper time" second ?

Just a labeling of events in the manifold

Sure, but without clear operational procedure rules how can we actually assign labels to events in the manifold ?

 

[Ibix]

maybe A here is the concentric sphere surface area...not the radius

Yes - now corrected above. Sorry.

Thus, as far as I can understand, to "label" the coordinate time $t$ the physical clock located on the sphere with coordinate radius $r$ has to be deliberately "adjusted" to tick $\sqrt{1-2GM/c^2r}$ coordinate seconds per (let me say) "proper time" second ?

Yes. Coordinate time isn't necessarily the time shown on anyone's clock. It'll usually have a simple relationship to someone's clock, however.

 

[Nugatory]

Thus, as far as I can understand, to "label" the coordinate time $t$ the physical clock located on the sphere with coordinate radius $r$ has to be deliberately "adjusted" to tick $\sqrt{1-2GM/c^2r}$ coordinate seconds per (let me say) "proper time" second ?

That's right. One second of proper time at a given point on the surface of the sphere is not equal to one second of Schwarzschild coordinate time. So if we want a properly constructed clock to read Schwarzschild coordinate time instead of proper time, we have to deliberately maladjust it - which makes it less useful for an observer standing near the clock, but perhaps more useful to a far distant observer.

Sure, but without clear operational procedure rules how can we actually assign labels to events in the manifold ?

The manifold is a mathematical abstraction, a set of points that meet certain additional requirements, and the coordinate assignment is a function mapping four-tuples of real numbers onto that set. I don't need to operationalize anything to write down such a function, I just need paper and a sharp pencil.

The more interesting question is whether the function I choose - which IS the coordinate system! - is useful. If it can be easily "operationalized" then it is likely useful; everyone uses Cartesian coordinates operationalized with tape measures and framing squares when building a house. However, that's not a necessary condition; for example Kruskal coordinates are invaluable for understanding a thought experiment in which we drop a probe into a black hole, and no one bothers with an explanation of how they might be operationalized; we just write down the transformations and proceed.

 

[Dale]

Sure, but without clear operational procedure rules how can we actually assign labels to events in the manifold ?

There is no one fixed operational procedure, and I am not convinced that such a procedure is necessary. For specific coordinate charts you could define a tailored operational procedure for that chart. For other charts you could either generate such a procedure directly or you could mathematically transform from one of the established coordinate systems.

The operational procedures are more closely related to the metric than to the coordinate chart.

 

[cianfa72]

for example Kruskal coordinates are invaluable for understanding a thought experiment in which we drop a probe into a black hole, and no one bothers with an explanation of how they might be operationalized; we just write down the transformations and proceed.

ok let me say from a mathematical point of view (just transform from Schwarzschild coordinates to get Kruskal ones) but how we should "interpreter" the metric tensor in the given (Kruskal) coordinates ?

The operational procedures are more closely related to the metric than to the coordinate chart.

Maybe this is the point not entirely clear to me: could you better elaborate that ?

 

[jbriggs444]

The operational procedures are more closely related to the metric than to the coordinate chart.

Maybe this is the point not entirely clear to me: could you better elaborate that ?

Let us drop down to two dimensions and consider a map of the state of Illinois laid out by latitude in degrees and longitude in degrees. On this map we draw in points for Chicago and Peoria. Note that the lines of longitude are straight while the lines of latitude are curved. But we have a metric that can tell us how close nearby points are to one another based on their latitude and longitude coordinates. If we want to use the map to figure out how far it is from Chicago to Peoria on a particular path, we can integrate the metric along that path.

We could have used different coordinates. We could have used a grid laid out in north-south miles and east-west degrees. We could have laid out the grid on some diagonal pattern. Or even patterns far more baroque, like gerrymandered congressional district lines.

The coordinates we use are irrelevant. It is the metric integrated along the path that tells us the distance from Chicago to Peoria.

 

[Nugatory]

ok let me say from a mathematical point of view (just transform from Schwarzschild coordinates to get Kruskal ones) but how we should "interpreter" the metric tensor in the given (Kruskal) coordinates ?

The interpretation of the metric tensor is that $ds^2=g_{\mu\nu}dx^\mu dx^\nu$; that works in all coordinate systems.

 

[cianfa72]

We could have used different coordinates. We could have used a grid laid out in north-south miles and east-west degrees. We could have laid out the grid on some diagonal pattern. Or even patterns far more baroque, like gerrymandered congressional district lines.
The coordinates we use are irrelevant. It is the metric integrated along the path that tells us the distance from Chicago to Peoria.

Sure, but I believe we need a rule to assign latitude and longitude values to a place on Earth in order to "identify" Earth locations on it otherwise I've no interpretation for a geodesic path between 2 points labeled with (latitude, longitude) pair of values...

 

[jbriggs444]

Sure, but I believe we need a rule to assign latitude and longitude values to a place on Earth in order to "identify" Earth locations on it otherwise I've no interpretation for a geodesic path between 2 points labeled with (latitude, longitude) pair of values...

I am not sure what you are driving at.

The end points are labelled "Peoria" and "Chicago". I can calculate the distance from Peoria to Chicago using the coordinates of the endpoints determined using sextants and clocks and a metric derived from spherical trigonometry. Or I can measure it with a pedometer along the path. If my model is good, the two will match.

 

[Dale]

Maybe this is the point not entirely clear to me: could you better elaborate that ?

The coordinate charts are just labels. They are mappings from events in the manifold to points in R4. The only requirements are that they must be 1-to-1 and smooth. There is no requirement that the charts have any operational meaning, only that they be mathematically “nice”.

The metric then is what gives the link to the physics/geometry. The metric is what allows you to take two nearby labels and compute the time or distance or angle or speed that some set of measuring devices would operationally record between the labels. Similarly for a worldline, the metric is what allows you to take the labels along the worldline and determine the proper time or proper acceleration. The metric gives the operational meaning.

 

[Ibix]

Sure, but I believe we need a rule to assign latitude and longitude values to a place on Earth in order to "identify" Earth locations on it otherwise I've no interpretation for a geodesic path between 2 points labeled with (latitude, longitude) pair of values...

Any rule will do, as long as it uniquely assigns a set of coordinates to a point. The standard latitude and longitude work by picking a point (the north pole) and measuring the distance from it to assign latitude, and picking a direction to call zero longitude and measuring angles from there. But any rule will do. You identify your location by recognising a couple of features you know and triangulating. The metric tells you how to interpret angles and distances, since over large distances, Euclidean geometry does not apply.

The same applies for coordinates on spacetime. Any rule will do, although there are commonly used standards. Again, you identify a point you recognise and triangulate to locate yourself. Again, the metric tells you how to interpret angles and distances in a non-Euclidean space.

 

[cianfa72]

The coordinate charts are just labels. They are mappings from events in the manifold to points in R4. The only requirements are that they must be 1-to-1 and smooth. There is no requirement that the charts have any operational meaning, only that they be mathematically “nice”.

The metric then is what gives the link to the physics/geometry. The metric is what allows you to take two nearby labels and compute the time or distance or angle or speed that some set of measuring devices would operationally record between the labels. Similarly for a worldline, the metric is what allows you to take the labels along the worldline and determine the proper time or proper acceleration. The metric gives the operational meaning.

It is fine but just to clarify my point consider the following:

Suppose you land on a unknown to you planet and someone has given you a map for it including metric tensor. All you know currently is there exist a mapping (a chart) from the actual planet locations to the map delivered to you. However is up to you to "interpreter" that map that is recognize which coordinate labels represent physical places on the planet surface (or in other words find out a rule/procedure in order to assign those coordinate labels to physical places on the planet).

With that in mind, I believe the same applies for spacetime: suppose you know a solution of Einstein Field Equation giving you a metric tensor in a given coordinate system. Now if you do not have or develop a rule/operational procedure to assign those coordinate labels to physical events in spacetime (basically "linking" them) how can you usefully employ the metric tensor to compute physical quantities (as you said for instance the proper time elapsed between two events) ?

 

[Dale]

All you know currently is there exist a mapping (a chart) from the actual planet locations to the map delivered to you. However is up to you to "interpreter" that map that is recognize which coordinate labels represent physical places on the planet surface (or in other words find out a rule/procedure in order to assign those coordinate labels to physical places on the planet).

I think that you may misunderstand the meaning of “map” in this context. A map is a function. For every element of one set it identifies a corresponding element of the other set, and in this case since the mapping is one-to-one it means that you can go either way. So having a coordinate chart means that for every event in spacetime you know the corresponding coordinates and for every coordinates you know the corresponding event in spacetime. There is no need for additional interpretation, the map includes the interpretation.

There is nothing operational at all, it is simply a labeling. Again, the operational part comes with the metric. With the map you know the coordinates of London and Paris, but with the metric you know the distance. The distance is what you would measure operationally, the coordinates are just labels.

Now if you do not have or develop a rule/operational procedure to assign those coordinate labels to physical events in spacetime (basically "linking" them)

Again, what you are calling a “linking” is already what is known as a mapping in this context. It is purely mathematical, not operational.

 

[stevendaryl]

Suppose you land on a unknown to you planet and someone has given you a map for it including metric tensor. All you know currently is there exist a mapping (a chart) from the actual planet locations to the map delivered to you. However is up to you to "interpreter" that map that is recognize which coordinate labels represent physical places on the planet surface (or in other words find out a rule/procedure in order to assign those coordinate labels to physical places on the planet).

I guess I see what you mean. A description of the universe in terms of coordinates won't do us any good if we don't know what those coordinates mean. You could identify certain features of the universe such as black holes, but you wouldn't know whether to be worried about falling into them if you didn't know where you were relative to them.

But the point of General Relativity being independent of coordinate systems is that it gives YOU the freedom to describe the universe however you want. You pick a description, and GR can tell you, in terms of that description, how spacetime will curve.

 

[cianfa72]

But the point of General Relativity being independent of coordinate systems is that it gives YOU the freedom to describe the universe however you want. You pick a description, and GR can tell you, in terms of that description, how spacetime will curve.

Yes, as you highlighted a whatsoever mapping of the spacetime events is good (provided it is 1-1 and smooth) but the key point is: you need to define it !

Consider again the derivation of Schwarzschild solution https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

As far as I can understand, we start from a a continuous set of concentric spherical shells each labeled with $r$ coordinate (the Schwarzschild radius). Then using $(\theta,\phi)$ polar and azimuthal angle spherical coordinates and searching for a spherically symmetric, static and vacuum solution we end up with the metric in the form:

$ds^2= A(r)~dr^2 + r^2 d\theta^2 + r^2\sin^2{\theta}~d\phi^2 + B(r)~dt^2$

Actually we know exactly the meaning of $(\theta,\phi)$ coordinates while $r$ and $t$ up to now are just two 1-1 smooth coordinates. Going further using vacuum field equations we're able to work out $A(r)$ and $B(r)$. Thus at this last stage, knowing the complete metric, we are able to "interpret" the coordinate $r$ as the "reduced radius" (linked to shell area $A$ via $\sqrt{A/4\pi}$) and the coordinate time $t$ as the physical local proper time scaled by $\sqrt{1-2GM/c^2r}$

Now I believe we can ask : what is the coordinate time $t$ ? We begun with a smooth function and now from the equation $ds^2=0$ for null geodesics we can develop a procedure to synchronize "coordinate clocks" sitting on spherical shells: basically we can start with a coordinate clock far away from center of gravity (here coordinate time $t$ is the same as proper time) sending a light ray at time $T$ (as recorded by the local far away clock) and calculating $t$ from $ds^2=0$ for light rays reaching remote coordinate clocks on the shells having given $(r,\theta,\phi)$ coordinates.

Doe it make sense ?

 

[stevendaryl]

Now I believe we can ask : what is the coordinate time $t$ ? We begun with a smooth function and now from the equation $ds^2=0$ for null geodesics we can develop a procedure to synchronize "coordinate clocks" sitting on spherical shells: basically we can start with a coordinate clock far away from center of gravity (here coordinate time $t$ is the same as proper time) sending a light ray at time $T$ (as recorded by the local far away clock) and calculating $t$ from $ds^2=0$ for light rays reaching remote coordinate clocks on the shells having $(r,\theta,\phi)$ coordinates.

Doe it make sense ?

Yes, I think that makes sense. The Schwarzschild coordinates are characterized by:

1. One time coordinate and three spatial coordinates.
2. The metric tensor components expressed in those coordinates are independent of the time coordinate.
3. The proper acceleration of an observer at "rest" (constant values of the spatial coordinates) depends only on $r$.
4. The collection of all points at a given value of $r$ is a sphere with area $4\pi r^2$
5. $\theta$ and $\phi$ are the usual angular coordinates to describe points on a sphere.
6. When exchanging light signals between observers at "rest" at different values of $r$, it takes the same amount of time to travel in each direction.
7. For any observer at rest, the value of the time coordinate advances linearly with the advance of elapsed time on a standard clock at rest at that location.
8. The constant of proportionality goes to 1 as $r \rightarrow \infty$

 

[cianfa72]

Yes, I think that makes sense. The Schwarzschild coordinates are characterized by:
6. When exchanging light signals between observers at "rest" at different values of $r$, it takes the same amount of time to travel in each direction

Here with "same amount of time to travel..." you mean the difference in coordinate time $t$ between the coordinate time $t$ recorded by a coordinate clock at the event of light signal starting point and that at the event of the arrival one, right ?

 

[Dale]

Coordinate time doesn’t need to be associated with specific clocks at particular locations.

 

[cianfa72]

Coordinate time doesn’t need to be associated with specific clocks at particular locations.

Consider a free stone falling towards the center of gravity starting at a given location with $(r,\theta,\phi)$ coordinates . We can follow its path trough spacetime in the map (working out for instance the proper time $\tau$ through the path) but to locate it in the map we also need to know the coordinate time $t$ of the starting event. How can we address this point ?

 

[Dale]

but to locate it in the map we also need to know the coordinate time ttt of the starting event. How can we address this point ?

I already addressed this point above in post 16. You appear to misunderstand the mathematical meaning of the term “map”. Please read my previous comments and respond to any particulars you don’t follow.

 

[cianfa72]

I already addressed this point above in post 16. You appear to misunderstand the mathematical meaning of the term “map”. Please read my previous comments and respond to any particulars you don’t follow.

I believe I got it: a map is basically a function mapping spacetime events to the corresponding coordinates (quoting what you said having a coordinate chart means that for every event in spacetime you know the corresponding coordinates and for every coordinates). So my "headache" here is this: which is actually the implied map about coordinate time $t$ by the Schwarzschild metric (in other words how is defined the map (mapping function) for the coordinate time $t$ with respect to the spacetime metric takes actually the Schwarzschild form ?)

Possibly could you kindly give me an actual example in using that metric for my better understanding ?Thank you

 

[stevendaryl]

Here with "same amount of time to travel..." you mean the difference in coordinate time $t$ between the coordinate time $t$ recorded by a coordinate clock at the event of light signal starting point and that at the event of the arrival one, right ?

Yes. You send a light signal radially at coordinate time $t_1$, it reaches a second observer at time $t_2$, and he immediately sends a return signal, which reaches you at time $t_3$. Then the time coordinates are such that $t_2 = \frac{t_1 + t_3}{2}$.

 

[cianfa72]

You define a set of non-rotating concentric spheres centred on the black hole. The r coordinate of a sphere of radius area A is $\sqrt{A/4\pi}$. Then you just mark latitude and longitude lines on the spheres and line all the spheres' markings up. Clocks tick slow relative to coordinate time, so you need clocks that can be deliberately set to count $\sqrt{1-2GM/c^2r}$ coordinate seconds per second. You pick one to be the master clock and set all clocks to zero when it would show zero - the correction to subtract out light travel time is simply determined by radar.

Another doubt about the following: respect to what is assumed the black hole non-rotating ?

 

[Nugatory]

Another doubt about the following: respect to what is assumed the black hole non-rotating ?

Speed is always relative so when we say that something is at rest or moving we have to say what that's with respect to. But proper acceleration is not relative in that sense; we can use an accelerometer to measure acceleration without reference to any external object. Different points on the surface of a rotating object will have different accelerations, and that is sufficient to determine that it is rotating.
For a real-life example of how we can measure rotation, google for "Foucault's Pendulum ".

 

[Ibix]

Another doubt about the following: respect to what is assumed the black hole non-rotating ?

In addition to Nugatory's comments, you can duck the whole issue by simply requiring spherical symmetry. If I launch a pair of projectiles with the same component of velocity towards the black hole and opposite tangential velocities, their orbits will always be mirror images for a Schwarzschild black hole - there is no frame-dragging effect.

 

[Dale]

which is actually the implied map about coordinate time t by the Schwarzschild metric (in other words how is defined the map (mapping function) for the coordinate time t with respect to the spacetime metric takes actually the Schwarzschild form ?

I am sorry, I don’t understand the question here. Are you asking if a real object can have exactly a schwarzschild metric?

 

[cianfa72]

I am sorry, I don’t understand the question here. Are you asking if a real object can have exactly a schwarzschild metric?

No, maybe I'm not able to explain my doubt. Trying again :-)
Start with definition of map as function as highlighted in your post 16

For every element of one set it identifies a corresponding element of the other set, and in this case since the mapping is one-to-one it means that you can go either way. So having a coordinate chart means that for every event in spacetime you know the corresponding coordinates and for every coordinates you know the corresponding event in spacetime. There is no need for additional interpretation, the map includes the interpretation.

Search for a solution of Einstein Field Equation (EFE) with given symmetry. For instance in Schwarzschild case start with spatial polar coordinates $(r,\theta,\phi)$ and a coordinate time variable $t$. Which conditions those functions (map) have to comply with ? Polar spatial coordinates basically "implement" a smooth map from spherical shells' locations to $(r,\theta,\phi)$ values. What about coordinate time map ? Surely there exist the requirement to map "close" events in time to "close" coordinate time $t$ values however up to now we have not a procedure to assign a coordinate time value $t$ to an event occurring at given $(r,\theta,\phi)$ spatial coordinates.

Going further proceed working out the complete EFE Schwarzschild solution. At that point we can go back finding out a "procedure interpretation" for coordinate time $t$ (see for instance stevendaryl post 19).

Hope this clarify the point

 

[Dale]

Search for a solution of Einstein Field Equation (EFE) with given symmetry. For instance in Schwarzschild case start with spatial polar coordinates (r,θ,ϕ) and a coordinate time variable t. Which conditions those functions (map) have to comply with ?

The only conditions that the function must comply with is that it is one-to-one and smooth. The functions do not need to respect the underlying symmetry of the manifold.

Symmetries of the manifold are called Killing vectors. Spherical symmetry means that there are three Killing vectors corresponding to rotations about three orthogonal axes. In addition, a static spacetime means that there is one additional Killing vector corresponding to the symmetry under translations in time.

Now, coordinate systems on this spacetime are not required to respect these symmetries. However, the symmetries also imply the existence of coordinate systems that do respect those underlying symmetries. In such coordinates the metric simplifies in specific ways which makes the calculation of the metric easier. This is shown in detail in Sean Carroll’s Lecture Notes On General Relativity. Using the simplifications available for symmetry-respecting coordinates results in a particularly nice form for the metric.

Surely there exist the requirement to map "close" events in time to "close" coordinate time t values

Yes, that is the smoothness criterion.

 

[sweet springs]

For any coordinate you choose, there is a general procedures to measure proper time and space length and thus find all the metric tensor components.
See section 84 of [link removed].

[Mentors' note: a link to what appears to be a pirated version of " The classical theory of fields" by Landau and Lifshitz has been removed. If anyone is aware of a authorized online reproduction, please PM any mentor so that we can post it here]

 

[cianfa72]

Speed is always relative so when we say that something is at rest or moving we have to say what that's with respect to. But proper acceleration is not relative in that sense; we can use an accelerometer to measure acceleration without reference to any external object.

I'm aware of marvelous Mach point of view about inertia as basically due to the entire universe mass distribution. In that context we can conceive proper acceleration as relative, actually, to the overall mass distribution. Which is the current point of view in the context of general relativity ?

 

[Nugatory]

Which is the current point of view in the context of general relativity ?

General relativity is not inherently Machian: its predictions for the proper acceleration measured at the surface of a rotating body are independent of the hypothetical distribution of hypothetical distant masses in the rest of the universe.

Whether GR is actively hostile to Mach's Principle is a different question. With neither a precise statement of exactly what is meant by "Mach's Principle" nor a testable theory based on that precise definition and that makes different predictions than GR, it's easy to start an argument by asking this question, not so easy to end it.
My advice is to learn GR first; after you understand it, you can decide for yourself whether Mach's principle further advances your understanding. However, if you can't wait... googling for "Mach's Principle and GR" will find many links, and even some to older threads here. Just be aware that the Google search rankings do not reliably distinguish between fringe and real science.

 

[PeterDonis]

its predictions for the proper acceleration measured at the surface of a rotating body are independent of the hypothetical distribution of hypothetical distant masses in the rest of the universe.

Not really. The prediction you refer to assumes an asymptotically flat spacetime, which is equivalent to assuming that whatever other stress-energy exists in the universe is distributed spherically symmetrically outside some large distance from the rotating body, with vacuum inside that distance. The prediction will change if that assumption is violated.

The "non-Machian" nature of GR in this case is better described, I think, by the observation that the asymptotically flat assumption is also satisfied by a spacetime in which the rotating body is absolutely alone in the universe, meaning the spacetime becomes Minkowski at very large distances from the body. But the asymptotic flatness in this version is not due to any stress-energy elsewhere in the universe; it's just there. (Another way of putting it would be to observe that there are multiple vacuum solutions to the Einstein Field equation in GR; but since they are all vacuum solutions, their stress-energy content is all the same--i.e., none--so whatever the difference is between them, it can't be due to a difference in stress-energy, which is the only kind of difference that a purely "Machian" theory would allow.)