I Requesting clarification about metric tensor

[pervect]

I am not sure I agree with this. You can have a coordinate system on a manifold that is not equipped with a metric.

Well, for instance, when the IAU specifies the Barycentric Celestial Reference System (BCRS), they do it by giving the line element for the metric tensor. I'd have to agree that if one is dealing with a general manifold without a metric, that that technique wouldn't work. But in GR we do have a pseudo-Riemannian metric, and it's common physics practice to define the coordinates by specifying the metric.

The point Misner is making (at least in my reading of him) is that if you know the metric, you can calculate whatever proper intervals you like. And Misner regards proper intervals as represents the reading of any physical measuring instrument.

One first banishes the idea of an “observer”. This idea aided Einstein
in building special relativity but it is confusing and ambiguous in general
relativity. Instead one divides the theoretical landscape into two categories.

One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments
and the data tables they provide.

For GPS the measuring instruments can be taken to be either ideal SI atomic clocks in trajectories determined by known forces, or else electromag-netic signals describing the state of the clock that radiates the signal.

My own paraphrase of what Misner is saying here. Physical readings are defined as being taken according to the SI measurement system. The fundamental SI units are the meter (for distance), the second (for time), and the kg (for mass), but if one know the value of the fundamental constants G and c, there is only one real fundamental unit required, the second. The other SI units can also be derived from the second with the correct information - the value of Boltzman's constant for temperature, the radiation weighting curve for the candela, the elementary charge of the electron for the ampere, and the number of particles in the mole.

Furthermore, we only need (and want) to measure proper time intervals, to avoid introducing human conventions into our definition of what "physical insturements" read. Introducing a synchronization convention takes us into the realm of "conceptual model of what is happning", rather than being what Misner is trying to define (in the imprecise English language) as a 'physical measuring instruement'.

 

[Nugatory]

Basically, specifying the metric is equivalent to specifying the coordinate system.

I am not sure I agree with this. You can have a coordinate system on a manifold that is not equipped with a metric

Does it suffice to replace "specifying the metric is equivalent to..." with "specifying the metric implies..."? We can have a coordinate system without a metric, and we can view the metric tensor as an abstract geometric object that exists independent of any coordinate system, but putting something to the right of the $=$ sign in $ds^2=...$ does pretty much specify a coordinate system.

 

[Orodruin]

Does it suffice to replace "specifying the metric is equivalent to..." with "specifying the metric implies..."? We can have a coordinate system without a metric, and we can view the metric tensor as an abstract geometric object that exists independent of any coordinate system, but putting something to the right of the $=$ sign in $ds^2=...$ does pretty much specify a coordinate system.

In general no. If the spacetime has symmetries, then the metric will take the same form in many different coordinate systems. This is quite evident in the case of Minkowski space and our usual treatment of it. In fact, we define Lorentz transformations as exactly those coordinate transformations that preserve the form of the metric.

 

[Nugatory]

In general no. If the spacetime has symmetries, then the metric will take the same form in many different coordinate systems. This is quite evident in the case of Minkowski space and our usual treatment of it. In fact, we define Lorentz transformations as exactly those coordinate transformations that preserve the form of the metric.

D'oh. Got it.