I Reference frame vs coordinate chart

[vanhees71]

I hope this will not end again in a battle about words.

From a formal point of view, I'd say a family of local reference frames in GR is defined by a coordinate chart together with a tetrad field. Local inertial frames across this chart are then further defined by a tetrad field parallel transported from an arbitrary point within the chart along arbitrary time-like geodesics (a time-like congruence) covering (at least a part of) the chart.

A nice book elucidating this from both the theoretical and observational point of view is

https://www.amazon.de/dp/B08BWQGXH6/

 

[cianfa72]

Any other point of view about the claims in post #51 ? Thanks all.

 

[ergospherical]

don’t poke this sleeping bear again babes

 

[Dale]

The bear is too busy this week.

 

[cianfa72]

The thing is that you can analyze a set of clocks and rulers using any coordinate system. So there is not a unique link between the two.

Sorry to go back to this topic: suppose you have 'labeled' each clock in the set with different spatial coordinate values and employed as the coordinate time the proper time measured by each of them (it is in fact a timelike coordinate). So far so good.

Then, as you said, suppose we want to analyze the same set of physical clocks using another coordinate chart. As an example of it we could re-label the spatial coordinate values assigned to each of them and attach to each of them a wristwatch with an arbitrary time rate.

This way we have defined a new coordinate chart in which physical clocks are at rest as well.

Does it make sense ? Thank you.

 

[Dale]

Sorry to go back to this topic: suppose you have 'labeled' each clock in the set with different spatial coordinate values and employed as the coordinate time the proper time measured by each of them (it is in fact a timelike coordinate). So far so good.

Then, as you said, suppose we want to analyze the same set of physical clocks using another coordinate chart. As an example of it we could re-label the spatial coordinates assigned to each of them and attach to each of them a wristwatch with an arbitrary time rate.

This way we have defined a new coordinate chart in which physical cloks are at rest as well.

Does it make sense ? Thank you.

Yes, that makes sense to me.

 

[cianfa72]

So the claim "take the rest frame of such and such body" is actually not uniquely defined. Namely there exist infinite rest reference frames defined up to re-labeling of their spatial coordinates and re-definition of timelike coordinate.

 

[Dale]

So the claim "take the rest frame of such and such body" is actually not uniquely defined. Namely there exist infinite rest reference frames defined up to re-labeling of their spatial coordinates and re-definition of timelike coordinate.

That is true. However, if such and such body is inertial then there is a standard convention that is implied and understood.

 

[cianfa72]

That is true. However, if such and such body is inertial then there is a standard convention that is implied and understood.

Which is that implied standard convention ?

 

[Dale]

Which is that implied standard convention ?

The standard inertial frame with Einstein synchronization convention where such and such body is at rest.

 

[cianfa72]

The standard inertial frame with Einstein synchronization convention where such and such body is at rest.

So that does mean: take a family of free bodies (zero proper acceleration) at rest w.r.t. the given (free/inertial) such and such body in a region surrounding it (just to fix ideas we can imagine a wristwatch attached to each of them).

Note that 'at rest' does actually mean that the round-trip time of 2-way light signals exchanged between those bodies does not change. Then, as pointed out in a recent PF thread, we can consistenly apply the Einstein synchronization convention to synchronize such wristwatches (the resulting one-way speed of light in the frame being defined is the universal constant value c).

Label every wristwatch (or body) with fixed different spatial coordinate values and take the proper time of each of them as the coordinate time of the frame being defined.

The map defined as above is the standard inertial frame you were talking about. Btw we're aware of we can do that just in a limited spacetime region in the context of GR. In flat spacetime instead (SR) there is no limit in principle to extend such standard inertial frame to the entire spacetime.

Make sense ? Thank you.

 

[Dale]

Make sense ? Thank you

Yes, that makes sense.