Can different paths in spacetime have the same separation?

[JesseM]

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.

 

[Anamitra]

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 Schwarzschild 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. ]

 

[PAllen]

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]

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?

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?

 

[Anamitra]

Let us take the metric:

$${ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx}^{2}{-}{g}_{22}{dy}^{2}{-}{g}_{33}{dz}^{2}$$

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

$${{ds}^{'}}^{2}{=}{{g}_{00}}^{'}{{dt}^{'}}^{2}{-}{{g}_{11}}^{'}{{dx}^{'}}^{2}{-}{{g}_{22}^{'}}{{dy}^{'}}^{2}{-}{{g}_{33}}^{'}{{dz}^{'}}^{2}$$

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

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

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

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

 

[Anamitra]

We take the metric:

$${ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx}^{2}{-}{g}_{22}{dy}^{2}{-}{g}_{33}{dz}^{2}$$

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

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

Such that,


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

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]

We take the metric:

$${ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx}^{2}{-}{g}_{22}{dy}^{2}{-}{g}_{33}{dz}^{2}$$

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

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

Such that,$${{ds}^{'}}^{2}{=}{g}_{00}{f0}{dt}^{2}{-}{g}_{11}{f1}{dx}^{2}{-}{g}_{22}{f2}{dx}^{2}{-}{g}_{33}{f3}{dz}^{2}$$

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.

 

[Anamitra]

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]

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.

 

[Anamitra]

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.

 

[PAllen]

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.

It is easey to see that the boundary conditions do nothing for you. Suppose one solution consistent with them. Do any of uncountably infinite coordinate transformations, you are still consistent with them *and* with your set up.

You really need to let go of the idea of coordinate grid having any meaning (separate from a metric; or unless defined with a fixed operational definition). If you consult books on GR, you will find 100% unanimity that coordinates by themselves are meaningless. More, that points in spacetime have no meaning; only material objects and measurements have meaning.

 

[kg4pae]

Consider the family of helical paths:
$$\left(ct,R \; cos(\omega t)+R, R \; sin(\omega t), 0\right)$$
Where $$\omega=2\pi/T$$

This helix connects the events (0,0,0,0) and (cT,0,0,0) with a smooth path. Those events are also connected by a straight timelike path.

The only objection I have with this is that the "T" in $$\omega=2\pi/T$$ is the period around the helix, whereas the "T" in (cT,0,0,0) is the time to decay for the muon. Otherwise what you've said is right on.

 

[Dale]

Yes, they are the same. Why do you object to that?

 

[kg4pae]

Yes, they are the same. Why do you object to that?

Actually I don't object. It's just that it wasn't specified in the original post. Your restriction that the muon exists such that it moves exactly one circle of the helix in its lifetime took some thinking for me to accept, considering that individually muons decay randomly. - But this is just an example of a curved path in spactime, not really about muons, right?

 

[PAllen]

Actually I don't object. It's just that it wasn't specified in the original post. Your restriction that the muon exists such that it moves exactly one circle of the helix in its lifetime took some thinking for me to accept, considering that individually muons decay randomly. - But this is just an example of a curved path in spactime, not really about muons, right?

Please, the muon was introduced to possibly clarify issues around what types of paths can exist between two events. Nothing else besides one post in this whole thread (before yours) deals with muons. The issue Dalespam was clarifying is long since settled, and even for that, the muon was really irrelevant.