Curved coordinates and the frame of reference that applies to them

[geordief]

In the ongoing Gravity and Space thread,Swansont wrote

"Photons have no mass, and even if they did, the effect would not be due to velocity, since all photons travel at c. They have energy and momentum, owing to their frequency (or wavelength). 

The effects on our observations is real, but spacetime is a coordinate system. Like latitude and longitude on a globe,

the coordinate system is not flat, it is curved,

because that's the proper coordinate system to use to describe the very real effects. When you look at them on a flat map (with a projection that lacks distortion), the lines are not straight. But latitude and longitude are not physical objects."

 

 

In any particular scenario ,does this  curved coordinate system have a  natural point of "origin" or can one ,in theory set the origin  at any point  one wants?

 

Can one have different (all curved) coordinate systems that allow one to describe any particular gravitational scenario?

 

Say there is a body of mass  and one wants to show the gravitational field in its vicinity,must we set the origin of the coordinate system at the centre of gravity of the massive object?

[studiot]

12 minutes ago, geordief said:

1)

Curved coordinates and the frame of reference that applies to them

 

2)

In any particular scenario ,does this  curved coordinate system have a  natural point of "origin" or can one ,in theory set the origin  at any point  one wants?

 

3)

Can one have different (all curved) coordinate systems that allow one to describe any particular gravitational scenario?

 

4)

Say there is a body of mass  and one wants to show the gravitational field in its vicinity,must we set the origin of the coordinate system at the centre of gravity of the massive object?

1)

Really it should be the other way round. The coordinate sytem applies to (or in) a particular frame of reference.
This question is really about the difference ebtween a frame and a coordinate system.
Possibly the best distinction is that frame can travel. We talk of a travelling frame and a fixed (or laboratory) frame.
The coordinate system (curvilinear or linear) is the measurement scheme in use. But the coordinate system does not travel.

2)

You should forget points of origin in relativity.

Not all coordinate systems have an origin. For instance cylindrical coordinates have an axis of orign.

3)

One can use different coordiante systems (schemes of measurement) in any situation. Some are usually more convenient than others.

4)

No, why should we ?

[joigus]

15 minutes ago, geordief said:

In any particular scenario ,does this  curved coordinate system have a  natural point of "origin" or can one ,in theory set the origin  at any point  one wants?

The answer is no, there is no particular point of origin. But that doesn't mean we can always map every point from within our chart. In general we need local charts, which are restricted patches of coordinates. Take the sphere as an example: You cannot map the whole sphere with one smooth coordinate system, though there is no particular point in the sphere that requires you to leave it out of the coordinate patch because of any geometric distinction.

20 minutes ago, geordief said:

Can one have different (all curved) coordinate systems that allow one to describe any particular gravitational scenario?

Yes AFAIK, as long as there is nothing physical that requires a singularity (e.g., static, non-rotating black holes.)

24 minutes ago, geordief said:

Say there is a body of mass  and one wants to show the gravitational field in its vicinity,must we set the origin of the coordinate system at the centre of gravity of the massive object?

Not necessarily, although arguments of symmetry force you to consider coordinates that take into account the centre of the gravitating object if you want to describe a wide-distance range of phenomena. But in fact, when you want to describe test particles crossing the horizon, more convenient coordinate patches are advisable than those distinguishing the centre of attraction. Coordinates around the centre of crossing.

[swansont]

38 minutes ago, geordief said:

Say there is a body of mass  and one wants to show the gravitational field in its vicinity,must we set the origin of the coordinate system at the centre of gravity of the massive object?

Must we? No. But in many cases this will be the most convenient one to use.

[geordief]

22 minutes ago, studiot said:

1)

Really it should be the other way round. The coordinate sytem applies to (or in) a particular frame of reference.
This question is really about the difference ebtween a frame and a coordinate system.
Possibly the best distinction is that frame can travel. We talk of a travelling frame and a fixed (or laboratory) frame.
The coordinate system (curvilinear or linear) is the measurement scheme in use. But the coordinate system does not travel.

 

 

 

 

 

1)Yes I felt the title was a bit mangled but found it had to articulate my thought

 If  the frame "travels" and the coordinate system is unchanged  the resulting diagram of the field will look different ,won't it?Will the curves be different in different areas?

4) so,say for a BH we can have a frame at any  distance  and watch the (local?) gravitational field change in real time as the frame moves wrt it?

 

 

 

 

Edited November 4, 2020 by geordief

[geordief]

1 hour ago, geordief said:

 

 

1)Yes I felt the title was a bit mangled but found it had to articulate my thought

 If  the frame "travels" and the coordinate system is unchanged  the resulting diagram of the field will look different ,won't it?Will the curves be different in different areas?

4) so,say for a BH we can have a frame at any  distance  and watch the (local?) gravitational field change in real time as the frame moves wrt it?

 

 

EDIT: either "travels"  because the physical  system evolves or "travels" because we decide  to place the frame elsewhere (hope that makes sense)

[swansont]

51 minutes ago, geordief said:

 

 

EDIT: either "travels"  because the physical  system evolves or "travels" because we decide  to place the frame elsewhere (hope that makes sense)

You have to be careful. You don't get to arbitrarily "place the frame elsewhere" because mixing frames of reference causes troubles. Many values are only valid in the frame in which they are calculated. Invariant ones are an exception.

studiot mentioned moving frames which are common in SR, but I don't know what the applicability is to GR. In a curved geometry, things are different.

As Markus said in the other thread "If the geometry is flat, then that means the relationship between any pair of neighbouring events will be the same, regardless of where/when in spacetime you are (like on a flat sheet of paper). If spacetime is curved, then this is no longer true - the relationship between a given pair of neighbouring events depends on where that pair of events is located in space and time."

 There are situations where you can assume the geometry is flat, and in that case, things are as they are in SR. Once you can't make that assumption, things are different. As was discussed, energy is no longer conserved, because conservation assumes you've stayed in one frame. It's not an invariant quantity, so it's not preserved when you change frames.

 

[geordief]

20 minutes ago, swansont said:

You have to be careful. You don't get to arbitrarily "place the frame elsewhere" because mixing frames of reference causes troubles. Many values are only valid in the frame in which they are calculated. Invariant ones are an exception.

studiot mentioned moving frames which are common in SR, but I don't know what the applicability is to GR. In a curved geometry, things are different.

As Markus said in the other thread "If the geometry is flat, then that means the relationship between any pair of neighbouring events will be the same, regardless of where/when in spacetime you are (like on a flat sheet of paper). If spacetime is curved, then this is no longer true - the relationship between a given pair of neighbouring events depends on where that pair of events is located in space and time."

 There are situations where you can assume the geometry is flat, and in that case, things are as they are in SR. Once you can't make that assumption, things are different. As was discussed, energy is no longer conserved, because conservation assumes you've stayed in one frame. It's not an invariant quantity, so it's not preserved when you change frames.

 

When I wrote the "physical system evolves or travels"  I see that that can be interpreted as mixing frames but I had it in mind that there could be  a series of measurements in different frames that could be compared even if this was not strictly accurate.

 

I was imagining the picture of the gravitational field  from one frame morphing into the next,subsequent  one albeit  with errors creeping in at each step  but still the "adjacent " pictures taken from 2 very close frames  could still be run,visually  like a movie ,even if with mistakes.   

 

I see there is probably  no practical benefit to this; but that is what I had in mind.

[swansont]

2 minutes ago, geordief said:

 I see there is probably  no practical benefit to this; but that is what I had in mind.

"Why would you do this?" certainly comes to mind. 

[geordief]

4 minutes ago, swansont said:

"Why would you do this?" certainly comes to mind. 

I think it was to reassure myself that something like this could be done  and so that I had acquired something of an understanding of the subject (even flawed understandings can be helpful if corrected)😉

[studiot]

I think you will find that the idea of a frame came along well before relativity in the early days of classical differential geometry.
I don't know if it was introduced by Frenet or someone else but consider two space curves, one a dead straight line and the other a coil spring or helix.

In classical geometry we have the fixed frame in which these curves reside and we can obtain equations of the line or helix in terms of the fixed frame.

However we can also consider these curves as 'generated' by a moving set of axes that also twists and turns as it moves from one point to the next.

The differential geometry amounts to calculatiing the change in orientation of this 'moving frame'.

I don't have the time at the moment to draw helpful pictures but google 'frenet frames' and choose images should produce plenty.

[Markus Hanke]

20 hours ago, geordief said:

Say there is a body of mass  and one wants to show the gravitational field in its vicinity,must we set the origin of the coordinate system at the centre of gravity of the massive object?

Just a quick to add to the other (excellent) answers here - the "gravitational field" is described by tensor quantities, so you are free to choose whichever coordinate system is most convenient, since such quantities do not depend on your choice of coordinates. To put it differently - events and their relationships are physically real and everyone agrees on them, but what you call those events is largely arbitrary. 

In the same manner you could replace all the street names in your town or city, without affecting the physical layout of that town in any way. People might get confused, but life would go on as usual  

[SergUpstart]

21 hours ago, geordief said:

Say there is a body of mass  and one wants to show the gravitational field in its vicinity,must we set the origin of the coordinate system at the centre of gravity of the massive object?

You don't have to. But setting the origin of the coordinate system to the center of mass dramatically simplifies the mathematical description of the gravitational field, since the problem becomes centrally symmetric

[joigus]

1 hour ago, SergUpstart said:

You don't have to. But setting the origin of the coordinate system to the center of mass dramatically simplifies the mathematical description of the gravitational field, since the problem becomes centrally symmetric

Only if the energy-momentum source is spherically symmetric.

[studiot]

Since this topic is about coordinates, it is a good idea to understand what they are and mean.

I think the place to start is with familiar simple rectangular coordinates.

Bear with me and check these over first because we are all so used to these that important differences come as a surprise when we move to oblique and curvilinear coordinates.
And many curvilinear systems are oblique.

The surprises come that oblique axes contain a little bit of each axis in components and resolutes and curved axes add the additional complication of the curve and deciding what parallel means.
This mixing has implications for transformation between coordinate systems and derivatives in differential geometry.

Edited November 5, 2020 by studiot

[geordief]

8 hours ago, studiot said:

Since this topic is about coordinates, it is a good idea to understand what they are and mean.

 

Thanks.Yes I have fairly recently (past 12 months) had a go at  non -orthogonal coordinate systems ,going through part of an  online  explanatory series.**

I think I even came across the idea of curved coordinate systems in the "Relativity,the Special and General Theory" by Einstein which I think it may have been you who recommended that I buy.

It was perhaps not greatly fleshed out in that slimmish book but was very interesting then  even if hard to understand

He referred to the idea as a kind of Gaussian  random coordinate system  which ,for some reason he likened to a "mollusk"

also online   https://www.marxists.org/reference/archive/einstein/works/1910s/relative/ch29.htm

 

** although I have only a fairly rough idea in my head now of what it was about.

Edited November 6, 2020 by geordief

[swansont]

9 hours ago, geordief said:

Thanks.Yes I have fairly recently (past 12 months) had a go at  non -orthogonal coordinate systems ,going through part of an  online  explanatory series.**

I think I even came across the idea of curved coordinate systems in the "Relativity,the Special and General Theory" by Einstein which I think it may have been you who recommended that I buy

I’m not sure if your two statements here are connected, but curved coordinate systems are (or at least can be) orthogonal. Physicists tend to use orthogonal coordinate systems.

[geordief]

33 minutes ago, swansont said:

I’m not sure if your two statements here are connected, but curved coordinate systems are (or at least can be) orthogonal. Physicists tend to use orthogonal coordinate systems.

No ,I didn't have it in my mind that curved co-ordinate systems  would be an example of non-orthogonal co-ordinate systems .

The progression ** that was in my mind was simply from simple systems  ( Cartesian is the term ,I think) to non -orthogonal  to (fairly amazingly) curved co-ordinate systems (which ,in  Einstein's the book was even referred to as "random")

 

**as in the progression in my learning or acquaintance with the concepts.

Edited November 6, 2020 by geordief

[studiot]

6 hours ago, swansont said:

I’m not sure if your two statements here are connected, but curved coordinate systems are (or at least can be) orthogonal. Physicists tend to use orthogonal coordinate systems.

 

6 hours ago, geordief said:

No ,I didn't have it in my mind that curved co-ordinate systems  would be an example of non-orthogonal co-ordinate systems .

The progression ** that was in my mind was simply from simple systems  ( Cartesian is the term ,I think) to non -orthogonal  to (fairly amazingly) curved co-ordinate systems (which ,in  Einstein's the book was even referred to as "random")

 

**as in the progression in my learning or acquaintance with the concepts.

 

I believe it was Marcus Hanke, (though it may have been Mordred) who recently advised against messing with curved or skew coordinate systems for the reasons I was going to publish, largely along the lines of geordie's post.
Yes they are more difficult, that is why they are avoided when practicable.

However they have had much use, I was going to say that Radio Navigation systems have been in use since the 1920s, but Wiki says that the earliest recorded one was 1907.

Here is a hyperbolic system due to  Decca.