On the Nature of ds[in GR]

  • Thread starter Anamitra
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All this is over-complicated and missing key points. Coordinates have *no* meaning by themselves. They can have meaning only in conjunction with a metric expressed in them (note that the metric itself can be defined without reference to coordinates). In particular, there is no conceivable meaning to talking about different metrics on the same coordinates. If the actual physical situation is unchanged, what you mean is you've changed coordinates producing a changed expression of the metric. If the physical situation is different, then you have different coordinates *and* different metric - you just can't attach meaning to the coordinates in the abstract, separate from the metric.
We can always set up a coordinate system in an arbitrary manner[for example we may think in terms of spherical or rectangular systems as three dimensional time-slices]. Then we can find out metrics that match against the physical aspects of the problem[this should include gravity and perhaps other factors according to the nature of the problem].

The coordinate system is of course arbitrary----it does not have to have a definite physical meaning.But once we use the physical aspects of the problem to impose the metric coefficients on them,the whole thing becomes meaningful.

This in no way serves as any impediment to my suggestions/thought experiments.
 
Preparing a coordinate system is like putting /attaching labels.You are not allowed to take off these labels at future points of time.Then you attach metric coefficients according to the physical nature of the problem. If the nature of the problem changes you simply change the metric coefficients[in a consistent way] without disturbing the labels.
 
The essential point is to have a meaningful system composed of metrics and coordinates corresponding to some physical system/situation[which includes gravity]. If the physical nature of the problem changes, you simply change the metrics without disturbing the coordinate labels.
 
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Preparing a coordinate system is like putting /attaching labels.You are not allowed to take off these labels at future points of time.
Yes you are. That is called a coordinate transformation and you are allowed to do it as often as you like.
 

JesseM

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Preparing a coordinate system is like putting /attaching labels.You are not allowed to take off these labels at future points of time.
And how do you "prepare a coordinate system" or "attach labels" to a region of spacetime that's so completely unknown to you that you can't even predict the metric there? What physical features of this unknown region are you attaching the labels to, so that later when you learn about what actual physical events occurred there you have a unique way of determining the coordinates of these events?

As I said before, one option would be to just assume you have an array of clocks which you use to define coordinate times and worldlines of constant coordinate position, and you know the clocks will still be in the unknown region (because they were in its past light cone) though you don't know how they'll behave. But if this is your method, you need to specify that it is since it will have some implications for your later argument...if you can think of some other method, you need to specify that. Right now it seems like you just haven't really given any thought to the problem though.
 
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And how do you "prepare a coordinate system" or "attach labels" to a region of spacetime that's so completely unknown to you that you can't even predict the metric there? What physical features of this unknown region are you attaching the labels to, so that later when you learn about what actual physical events occurred there you have a unique way of determining the coordinates of these events?

As I said before, one option would be to just assume you have an array of clocks which you use to define coordinate times and worldlines of constant coordinate position, and you know the clocks will still be in the unknown region (because they were in its past light cone) though you don't know how they'll behave. But if this is your method, you need to specify that it is since it will have some implications for your later argument...if you can think of some other method, you need to specify that. Right now it seems like you just haven't really given any thought to the problem though.
The process is simple:

Metric coefficients+coordinate system----> Meaningful idea[It corresponds to some physical problem]


You choose any particular problem problem--let us call it "The Initial Problem"
Find out the metric: metric coefficients+coordinates
The above metric should correspond to the physical nature of the problem
If the physical situation changes [ex: a high density erratic mass distribution approaches the system], you simply change the metric coefficients without disturbing the coordinate labels

[Incidentally one could use "flying labels" as coordinates for a stationary system. No harm, so long as the metric coeff+coordinates combination[which we call the metric ] gives us a correct depiction of the physical situation. But calculations might become tedious requiring too much of diligence]
 

JesseM

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Metric coefficients+coordinate system----> Meaningful idea[It corresponds to some physical problem]
But you were talking about placing coordinates in a region of spacetime where you don't yet know the metric coefficients, right? If so, nothing in your post explains how we are supposed to do that.
 
But you were talking about placing coordinates in a region of spacetime where you don't yet know the metric coefficients, right? If so, nothing in your post explains how we are supposed to do that.
You may go through post #121
You may also go through post #116

We may not always be aware of the metric ahead of us[I mean to say,the future] but we may always speculate that changes might occur----that we might get a time like,space like or null connection in future--there is a glorious uncertainty in the whole aspect of the problem
 

PAllen

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We can always set up a coordinate system in an arbitrary manner[for example we may think in terms of spherical or rectangular systems as three dimensional time-slices]. Then we can find out metrics that match against the physical aspects of the problem[this should include gravity and perhaps other factors according to the nature of the problem].

The coordinate system is of course arbitrary----it does not have to have a definite physical meaning.But once we use the physical aspects of the problem to impose the metric coefficients on them,the whole thing becomes meaningful.

This in no way serves as any impediment to my suggestions/thought experiments.
JesseM has suggested several times you think about what it means to set up a coordinate system. I doubt I can do better, but I'll try again.

Suppose you want to label an event B 3 units in x direction from event A (events are points in space time; they have no history - they are specific events somewhere, sometime, in the history of the universe). This labeling has no meaning at all until you know the metric and can express it in terms of x and other labels. Depending on how you do this, 3 in x direction can mean 3 hours later on a clock, 3 kilometers east, 3 degrees counterclockwise, whatever. It is only the metric that gives x any meaning at all. If the metric says x direction is timelike, than x has the character of time for some clock; if the metric says it is spacelike, then it is distance for some path of simultaneity.

More naturally, you can set up coordinates by (perhaps idealized) measurements. Then the measurements determing the nature of of the coordinates. Measurements, of course, take full account of the metric. If you define x by a mechanism for measuring distance, it will represent distance no matter where or when in the universe you do it, no matter what the gravitational field.

If you are 'thinking' about the the interval from (t,x)=(5,5) to (5,7), where 5 is in the future, and you have don't know the metric for this region of space *time*, and don't define any measurement you will do,, then you cannot have any expectation of what they mean. I cannot fathom what you mean by 'expecting' a meaning for this separately from a measurement procedure or defining the metric.

Note, if you define this, for example, by saying that when my watch says 3, I will define my spacetime position to be (3,5), then I will send out a rader signal and if I get it back when my watch says 7, (still calling my postion x=5), then the event of its bouncing off something I will label (5,7). Using such a procedure you would know, at all times, and all gravity situations, that you would be defining a spacelike interval between (5,5) and (5,7), and you could call them 2 lightseconds apart (thus calling your watch units seconds, and x unit lightseconds). An observer elsewhere in the universe might disagree radically on how far apart these events were, but they would certainly agree the separation between them was spacelike.
 

JesseM

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You may go through post #121
You may also go through post #116
Neither post contains any information about what physical procedure we are supposed to use to attach coordinate labels to a region of spacetime where we don't know the metric. Can you please just give a specific answer to this question?
 

PAllen

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If the physical situation changes [ex: a high density erratic mass distribution approaches the system], you simply change the metric coefficients without disturbing the coordinate labels
Maybe this is the core confusion. What are these labels attached to while you are changing the metric? If they are attached to physical events and measurements, the metric is discovered by these measurements, not the other way around. If they are not, then the only meaning they have is determined by the way the metric is expressed in terms of these labels. Depending on how you change the metric, you may redefine x as time on a clock, distance along a ruler, or angle.
 

PAllen

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Maybe I see a way to define what Anamitra is getting at in a sensible way. Suppose you define 'reasanoble coordinate' for observing what you can see in the universe. Specifically, suppose you choose to extend Fermi Normal cordinates as far in distance and time as you reasonably can. You find that out to 10 lightyears, there are no signficant deviations from Lorentz geometry except near your sun and planets. You think you can extend them well into the future because you haven't detected anything that would make this invalid. Then an isolated black hole passes a few light years away. Now you find that doing your best to extend these coordinates (still keeping them Fermi Normal based on your world line as the time axis), you can't avoid varying coordinate speed of light near the black hole; necessarily, this also means that a coordinate path defined e.g. by: ( x - x0) = .5 (t - t0) that is is timelike for t0 before the arrival of the black hole, and for any x0 far from the black hole, now describes a spacelike curve for some x0 and t0.

Of course the only sense in which you can say this is unexpected is that you previously lacked knowledge of the approaching black hole (or you didn't know about GR). So it is strange to call this unexpected. Further, there is, of course, no spactime path that changed nature because of the approaching black hole. The best you could say is that for t0 in your future you made an erroneous guess about the corresponding metric. It was your guess that got corrected, not any spacetime path changing nature.
 
When we frame the metrics for a stationary field [eg:Schwarzschild Geometry] we extend the time axis into the distant future expecting nature to be kind towards us[maintaining the stationary nature].Then we have the metrics for the stationary field.In case something happens[some gravitational upheaval] we can maintain our coordinate grid and change the nature of the metrics.We may also think of changed metrics for the future speculating different types of gravitational changes.
This was of course implied in the previous posts
 
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JesseM

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When we frame the metrics for a stationary field [eg:Schwarzschild Geometry] we extend the time axis into the distant future expecting nature to be kind towards us.
You can only extend it under the assumption that the metric will be stationary (or some other assumption about the metric). If you make this assumption and then say "I wonder whether some point at a future time T and position X will lie on a timelike path from my current position", but then the metric changes contrary to your assumption, how are you supposed to decide what physical event in spacetime actually has coordinates T and X? The labels T and X simply become meaningless if your basis for them was the assumption that the metric would stay stationary when in fact it didn't, you have to construct a new coordinate system if you want to label events in the region of spacetime that didn't match your expectations.
 
You can only extend it under the assumption that the metric will be stationary (or some other assumption about the metric). If you make this assumption and then say "I wonder whether some point at a future time T and position X will lie on a timelike path from my current position", but then the metric changes contrary to your assumption, how are you supposed to decide what physical event in spacetime actually has coordinates T and X? The labels T and X simply become meaningless if your basis for them was the assumption that the metric would stay stationary when in fact it didn't, you have to construct a new coordinate system if you want to label events in the region of spacetime that didn't match your expectations.

Actually I am constructing a new metric with the old coordinate grid[t,x,y,z] and new metric coefficients.
 

JesseM

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Actually I am constructing a new metric with the old coordinate grid[t,x,y,z] and new metric coefficients.
But the "old coordinate grid" doesn't refer to any unique coordinate system any more, since the physical meaning of the coordinates was dependent on the old metric. There are an infinite number of different ways you could extend the coordinate system in the region with the known (stationary) metric into the new region with a different metric, and depending on how you do it the metric coefficients at each coordinate would be different.

You understand that on the same physical spacetime there can be many different coordinate systems, and the equations expressing the metric coefficients in terms of that coordinate system will be different in each one, right? For example, Schwarzschild coordinates and Kruskal coordinates both cover the same physical spacetime, the nonrotating uncharged Schwarzschild black hole spacetime.
 
But the "old coordinate grid" doesn't refer to any unique coordinate system any more, since the physical meaning of the coordinates was dependent on the old metric. There are an infinite number of different ways you could extend the coordinate system in the region with the known (stationary) metric into the new region with a different metric, and depending on how you do it the metric coefficients at each coordinate would be different.
The metric coefficients together with the coordinates corresponded to the description/attributes of the existing physical system[let us consider a stationary one like the Schwarschild geometry].We may extend the coordinate system into the future in a manner as if the same stationary description continued into the future. But in effect the physical conditions may change due to gravitational changes.In such a case we maintain the previous coordinate grid and change/adjust the metric coefficients to make the new metric match against the new physical conditions.

[We may think of different coordinate systems with different coefficients describing the same physical conditions at the initial state.We may extrapolate each such system[coordinate grid] into the future--in a manner as if the same physical conditions persisted up to distant future.If there is a gravitational change we simply change the metric coefficients ,keeping the coordinate grid intact.

If some dense body visits the earth we can always maintain our old t,r,theta,phi system and adjust the metric coefficients to get a metric that best describes the new physical conditions.]

One can ,of course, extend the coordinate system into the future in infinitely possible ways.
But a transformation can always be worked out between your system and the one I would be using according to my procedure.
[The same physical point gets different coordinate descriptions in different frames
Sets of physical points[curves] are described by different coordinates in different coordinate systems.We should keep in our mind the transformation laws--they mean exactly what I have said in the previous line. ]
 
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PAllen

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In such a case we maintain the previous coordinate grid and change/adjust the metric coefficients to make the new metric match against the new physical conditions.
JeseM and I have told you several times that this operation is undefinable. I think this is a core misunderstanding in this discussion. Try defining *precisely* what you mean by maintaining a coordinate system divorced from a metric, and see where that get's us.
 

JesseM

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We may extend the coordinate system into the future in a manner as if the same stationary description continued into the future.
Are you claiming that this description defines a unique extension of the coordinate system?
Anamitra said:
One can ,of course, extend the coordinate system into the future in infinitely possible ways.
But a transformation can always be worked out between your system and the one I would be using according to my procedure.
Again, are you claiming that your "procedure" defines a unique extension, such that if we follow it than we don't have to choose between "infinitely possible ways" of extending the coordinate system into the region with the new metric? If so, do you have any clear idea of how to define the procedure for creating such a unique extension in mathematical terms (or give a reference to the literature which shows how to do this), or have you just managed to convince yourself that this is possible using verbal arguments but have not actually worked out the mathematical details?
 
Let us take the metric:

[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx}^{2}{-}{g}_{22}{dy}^{2}{-}{g}_{33}{dz}^{2}[/tex]

Now the physical situation changes[due to gravitational effects] leading to a new metric. We choose the representation:

[tex]{{ds}^{'}}^{2}{=}{{g}_{00}}^{'}{{dt}^{'}}^{2}{-}{{g}_{11}}^{'}{{dx}^{'}}^{2}{-}{{g}_{22}^{'}}{{dy}^{'}}^{2}{-}{{g}_{33}}^{'}{{dz}^{'}}^{2}[/tex]

If we choose the transformations:
[tex]{t}^{'}{=}{f0{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{x}^{'}{=}{f1{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{y}^{'}{=}{f2{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{z}^{'}{=}{f3{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]

Such that,
[tex]{{g}_{00}}^{'}{{dt}^{'}}^{2}{=}{g}_{00}{f0}{dt}^{2}[/tex]
[tex]{{g}_{11}}^{'}{{dx}^{'}}^{2}{=}{g}_{11}{f1}{dx}^{2}[/tex]
[tex]{{g}_{22}}^{'}{{dy}^{'}}^{2}{=}{g}_{22}{f2}{dy}^{2}[/tex]
[tex]{{g}_{33}}^{'}{{dz}^{'}}^{2}{=}{g}_{33}{f3}{dz}^{2}[/tex]

Then we may write:
[tex]{{ds}^{'}}^{2}{=}{g}_{00}{f0}{dt}^{2}{-}{g}_{11}{f1}{dx}^{2}{-}{g}_{22}{f2}{dx}^{2}{-}{g}_{33}{f3}{dz}^{2}[/tex]

[(x,y,z) may not be rectangular Cartesian coordinates]
 
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We take the metric:

[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx}^{2}{-}{g}_{22}{dy}^{2}{-}{g}_{33}{dz}^{2}[/tex]

Now the physical situation changes[due to gravitational effects] leading to a new metric. We choose the representation:

[tex]{{ds}^{'}}^{2}{=}{{g}_{00}}^{'}{dP}^{2}{-}{{g}_{11}}^{'}{dQ}^{2}{-}{{g}_{22}^{'}}{dR}^{2}{-}{{g}_{33}}^{'}{dS}^{2}[/tex]
P,Q,R,S are the new suitable coordinates
If we choose the transformations:
[tex]{P}{=}{f0{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{Q}{=}{f1{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{R}{=}{f2{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{S}{=}{f3{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]

Such that,


[tex]{{ds}^{'}}^{2}{=}{g}_{00}{f0}{dt}^{2}{-}{g}_{11}{f1}{dx}^{2}{-}{g}_{22}{f2}{dx}^{2}{-}{g}_{33}{f3}{dz}^{2}[/tex]

We have the old coordinate system in operation.This seems to provide a greater amount of flexibility.

[(x,y,z) may not be rectangular Cartesian coordinates]
 

PAllen

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We take the metric:

[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx}^{2}{-}{g}_{22}{dy}^{2}{-}{g}_{33}{dz}^{2}[/tex]

Now the physical situation changes[due to gravitational effects] leading to a new metric. We choose the representation:

[tex]{{ds}^{'}}^{2}{=}{{g}_{00}}^{'}{dP}^{2}{-}{{g}_{11}}^{'}{dQ}^{2}{-}{{g}_{22}^{'}}{dR}^{2}{-}{{g}_{33}}^{'}{dS}^{2}[/tex]
P,Q,R,S are the new suitable coordinates
If we choose the transformations:
[tex]{P}{=}{f0{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{Q}{=}{f1{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{R}{=}{f2{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]
[tex]{S}{=}{f3{(}{t}{,}{x}{,}{y}{,}{z}{)}[/tex]

Such that,


[tex]{{ds}^{'}}^{2}{=}{g}_{00}{f0}{dt}^{2}{-}{g}_{11}{f1}{dx}^{2}{-}{g}_{22}{f2}{dx}^{2}{-}{g}_{33}{f3}{dz}^{2}[/tex]

We have the old coordinate system in operation.This seems to provide a greater amount of flexibility.

[(x,y,z) may not be rectangular Cartesian coordinates]
For any given physical situation, there are an infinite number of choices for f0,...f4 that will work (giving different meaning to t,x,y,z).[Imagine there is one; do a coordinate transform; now you have another]. How do you pick which to use? This gets right at why the operation 'preserving coordinates as you change the metric' has no possible meaning.
 
We can always use the boundary conditions to sieve out the appropriate solutions.One may assume continuous transformation of the physical situation to make things convenient.
 

PAllen

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We can always use the boundary conditions to sieve out the appropriate solutions.One may assume continuous transformation of the physical situation to make things convenient.
This would not remove an uncountably infinite set of choices. I believe, instead of the fiction of 'maintaining a coordinate grid' you need to talk about the 'maintaining some operational definition of coordinates'. Please carefully review my post #137. It describes the closest you can get to what you are trying to say.
 
A set of differential equations should have a unique solution set corresponding to a given set of boundary conditions. We may try out different techniques--but the aim is to find a solution set that fits into the boundary conditions.If we can do this--the job is done.We can get the correct solution from a set of infinite solutions.
 

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