How to call an invariant quantity?

[kent davidge]

When referring to an invariant quantity, which one is better "coordinate independent" or "frame independent"?

 

[Ibix]

I can't see that it matters particularly. If you are talking about frames, say frame independent. If you are talking about coordinates say coordinate independent. Unless anyone else knows a fine distinction that I don't?

 

[A.T.]

When referring to an invariant quantity, which one is better "coordinate independent" or "frame independent"?

Is it frame invariant or coordinate invariant?

 

[Vanadium 50]

Is it frame invariant or coordinate invariant?

I don't know. Is it?

The distance between New York and Miami is coordinate invariant. But it is not frame invariant.

 

[PeterDonis]

The distance between New York and Miami is coordinate invariant. But it is not frame invariant.

By not frame invariant, I assume you mean the distance would appear length contracted to someone flying past the Earth at relativistic speed.

However, this also makes the distance coordinate dependent. So I don't think it's correct to say it's coordinate invariant.

 

[Vanadium 50]

Mr. Davidge is trying to get us into quibbling mode, and we have falled into his trap.

 

[etotheipi]

The distance between New York and Miami is coordinate invariant.

Couldn't you establish one coordinate system that measures distance along the Earth's surface from an origin to a point on the surface, and another coordinate system that measures straight line distance from an origin to a point on the surface. Then the distance between New York and Miami would not be coordinate independent?

 

[Ibix]

Couldn't you establish one coordinate system that measures distance along the Earth's surface from an origin to a point on the surface, and another coordinate system that measures straight line distance from an origin to a point on the surface

It depends what you mean. In purely spatial terms, no. Those distance measures exist whatever your coordinate system. I can calculate the distance from here to the shops in a Sun centered Cartesian coordinate system if I am so inclined and it will be the exact same ~1 mile it is if I measure on a paper map.

Things get a bit more complex if we allow for relativity - as noted distance is not invariant. So then your choice of coordinate system can matter, but only because you are measuring something that isn't invariant.

 

[etotheipi]

It depends what you mean. In purely spatial terms, no. Those distance measures exist whatever your coordinate system. I can calculate the distance from here to the shops in a Sun centered Cartesian coordinate system if I am so inclined and it will be the exact same ~1 mile it is if I measure on a paper map.

I was thinking that (just in the classical case, here) the non-Cartesian coordinate system overlaid on the surface of the sphere could be defined to measure the great-circle distance whilst a standard Cartesian coordinate system (with origin anywhere, it won't make a difference) would measure the straight line distance. Perhaps that's not how it works, in which case I do apologise to $^{50}\text{V}$

 

[Nugatory]

When referring to an invariant quantity, which one is better "coordinate independent" or "frame independent"?

It’s easy to create ambiguities with natural language, which is why the truth is always in the math.

Here we can define the distance between two points (necessarily at rest relative to one another) as the difference between their spatial coordinates. Or we can define it as the proper time between emission and reception of a round-trip light signal, divided by $2c$. One way the distance is frame and coordinate-dependent, the other way it is frame-dependent.

Any discussion of which term better describes “the distance” is misdirected and just shows that we haven’t been clear about what we mean by those words. Sure, we can add adjectives and say things like “coordinate distance”, but unless we provide the math we’re still depending on our audience to guess what we mean

 

[PAllen]

It’s easy to create ambiguities with natural language, which is why the truth is always in the math.

Here we can define the distance between two points (necessarily at rest relative to one another) as the difference between their spatial coordinates. Or we can define it as the proper time between emission and reception of a round-trip light signal, divided by $2c$. One way the distance is frame and coordinate-dependent, the other way it is frame-dependent.

Actually, if you define "distance between two objects mutually at rest" by proper time either one measures for a light signal round trip to the other (divided by 2), you have an invariant: proper distance between them. It could be computed in any coordinates or frame, and will always be the same.

 

[PAllen]

I was thinking that (just in the classical case, here) the non-Cartesian coordinate system overlaid on the surface of the sphere could be defined to measure the great-circle distance whilst a standard Cartesian coordinate system (with origin anywhere, it won't make a difference) would measure the straight line distance. Perhaps that's not how it works, in which case I do apologise to $^{50}\text{V}$

A cartesian coordinate system with Euclidean metric cannot be constructed on a sphere. If you instead set up a "cartesian like" coordinate system of a region of a sphere, it would necessarily have a non-Euclidean metric such that all distances come out the same as spherical coordinates with the metric expressed in those coordinates. In Riemannian geometry, fully generally (as opposed to pseudo-Riemannian geometry of relativity), all distances along any path are invariant with respect to arbitrary coordinates.

 

[Ibix]

I was thinking that (just in the classical case, here) the non-Cartesian coordinate system overlaid on the surface of the sphere could be defined to measure the great-circle distance whilst a standard Cartesian coordinate system (with origin anywhere, it won't make a difference) would measure the straight line distance. Perhaps that's not how it works, in which case I do apologise to $^{50}\text{V}$

I think you might be confusing coordinate differences with distances. With Cartesian coordinates in Euclidean space you can just take the difference between the coordinates of a pair of points and get a displacement vector joining the two points, and calculate the length to get the straight line distance. That doesn't work with any other coordinate system. Converting infinitesimal coordinate differences into distances is what the metric does, and you then need to specify a path and integrate along it to get the total distance.

Certainly a smart choice of coordinates helps with making that integral simple. If you do want to calculate the distance along the surface of the Earth then spherical polars with their equator passing through the points of interest let's you calculate the great circle distance easily. But that distance isn't intrinsic to polar coordinates, and is the same if you carry out the more painful integration in Cartesian coordinates.

 

[etotheipi]

Oh yeah, okay I'm being slightly stupid. `Distance along a certain trajectory is evaluated as an integral of the $\sqrt{g_{ij} dx^{i} dx^{j}}$ along that trajectory. Along the surface of the sphere and through the Earth are two different trajectories, so will naturally have different distances. You could use either coordinates to measure the distances along either of the trajectories. Thanks for being patient

 

[Nugatory]

Actually, if you define "distance between two objects mutually at rest" by proper time either one measures for a light signal round trip to the other (divided by 2), you have an invariant: proper distance between them. It could be computed in any coordinates or frame, and will always be the same.

Yes, of course you’re right, thanks,

Now I’m trying to remember which examp,e I was trying to use...

 

[Wei Guo]

In special relativity, invariant quantity doesn't change with frame system or coordinate system. They are both dependent on the human factor. Frame system and coordinate system are two different concepts. Different observers adopt the different frame systems, and for a single observer(a frame system), one can adopt different coordinate systems, e.g. Cartesian and polar coordinate systems, to make the problem easier to handle.

By the way, in special relativity, one should distinguish conserved quantity and invariant quantity. Conserved quantity means some quantities don't change with time, it emphasizes the physics process, while invariant quantity means some quantities don't change with the coordinate(frame) system, it emphasizes the coordinate transformation. For example, energy is a conserved quantity, not an invariant quantity, (static) mass is an invariant quantity, not a conserved quantity, the charge of a charged particle is neither an invariant quantity nor a conserved quantity.

 

[vanhees71]

A (local) frame is a coordinate-independent concept. It's defined by a tetrad field, which is an invariant mathematical object in both special and general relativity. It must be a coordinate-independent concept because it defines a real-world setup for all measurements.

All physics is formulated such that it is completely invariant under the choice of coordinates. That's so in Newtonian, special and general relativistic physics. E.g., it doesn't matter whether you discribe the Kepler problem in Newtonian mechanics in spherical or Cartesian coordinates.