The relationship between coordinate systems and reference frame

[paultsui]

Hi there,

I am confused about the relationship between coordinate systems and reference frame in GR.
I understand the coordinate systems can be used to describe reference frames, for example, Local inertial frames in GR can be defined by Riemann Normal Coordinates.

However, take the Schwarzschild Geometry for example, we have
$$ds^{2} = - (1 - \frac{2GM}{c^{2}r})(cdt)^{2} + (1 - \frac{2GM}{c^{2}r})dr^{2} + r^{2}(d\theta^{2} + sin^{2}\theta d\phi^{2})$$

Obviously, when we write the Schwarzschild Geometry, we have already assume a coordinate in place, with coordinates (t, r, $\theta, \phi$)

My question is, does the coordinate system (t, r, $\theta, \phi$) used in defining the line element happens to define a reference frame?

 

[Jonathan Scott]

A reference frame or "frame of reference" is a general term for the scheme used to describe locations and times of events. In GR, there are two main types of reference frame:

The first is a coordinate system such as Schwarzschild coordinates, which is effectively an unlimited large-scale map of what is happening, but the relationship between the map and local measurements varies with location according to the metric.

The other is a local observer reference frame, which typically describes events from the point of view of an observer over a region of space-time which is small enough to appear approximately flat so that Special Relativity mechanics applies locally and a corresponding SR coordinate system can be used to describe events within that small region. If the observer is in free fall, the local frame is inertial, but if the observer is being held in place against a gravitational field, the local frame is typically assumed to be subject to a constant acceleration.

See the Wikipedia article Frame of reference for more information.

 

[Mentz114]

In GR we can define a frame of reference for an observer characterized by a 4-velocity relative to the source of the field.

For example the Schwarzschild vacuum there are at least 3 interesting observers.

1. static observer who experiences a proper acceleration to remain stationary. The velocity is U^μ = ( 1,0,0,0)

2. radially free falling ( Gullstrand-Painleve chart), velocity U^μ = ( 1,-sqrt(2m/r),0,0)

3. orbiting in the azimithumal plane ( Hagihara frame ).
$$U^0=-\frac{\sqrt{r}\,\sqrt{r-3\,M}}{3\,M-r},\ \ U^3=\frac{\sqrt{r-3\,M}\,\sqrt{M}}{3\,r\,M-{r}^{2}}$$

The last two are geodesic. These observers each carry a local Minkowski metric on their worldline.

A good article can be found here
http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

 

[paultsui]

Thank you for your replies.

But I am still not entirely understand the concept.

So when we write down the line element (or a metric), we assumed a coordinate system.
Do you guys mean the coordinate system is not related to a physical reference frame directly (i.e. no physical reference frame uses that particular coordinate system to describe the events observed)?

You guys seem to suggest that the coordinate system in the line element is actually used to describe to the physical frame of reference being used. But, obviously, the coordinates used by any physical frame of reference is different.

Another thing I don't understand is the the four velocity - it seems that the components of four velocities are w.r.t. coordinates basis. While physical measurements, such as the momentum of a particle w.r.t. to a particular frame, correspond to the 4-momentum w.r.t. orthonormal basis defined in that particular frame. What causes this distinction?

 

[Jonathan Scott]

Thank you for your replies.

But I am still not entirely understand the concept.

So when we write down the line element (or a metric), we assumed a coordinate system.
Do you guys mean the coordinate system is not related to a physical reference frame directly (i.e. no physical reference frame uses that particular coordinate system to describe the events observed)?

You guys seem to suggest that the coordinate system in the line element is actually used to describe to the physical frame of reference being used. But, obviously, the coordinates used by any physical frame of reference is different.

Space-time in GR is curved. If you want to describe where and when events happen, you can either describe them locally using an approximately flat view over a region of space which is sufficiently small that the curvature can be ignored, or you can map them using a coordinate system where there is a varying relationship (described by the metric) between the map and each little local region of space-time. If you're trying to describe an orbit, you need to use the map approach, as it is the deviation from flatness (mainly the variation of the time component of the metric) which causes the test object to follow the orbit.

Obviously what actually happens doesn't depend on the choice of the type of map, but there are multiple equivalent ways of mapping the same space-time and the same events.

It is of course possible to create a coordinate system which locally matches the observer's frame of reference but which then continues to map a larger region using a metric. This can be done for example by rescaling an isotropic coordinate system to match a local observer.

 

[Mentz114]

So when we write down the line element (or a metric), we assumed a coordinate system.
Do you guys mean the coordinate system is not related to a physical reference frame directly (i.e. no physical reference frame uses that particular coordinate system to describe the events observed)?

You guys seem to suggest that the coordinate system in the line element is actually used to describe to the physical frame of reference being used. But, obviously, the coordinates used by any physical frame of reference is different.

It depends. The stationary observer in the examples I gave uses the Schwarzschild coordinates. The G-P observer uses the Schwarzschild r but has a transformed time coordinate. Another wrinkle is that the metric may be rewritten in transformed coordinates, and look different from the original.

Another thing I don't understand is the the four velocity - it seems that the components of four velocities are w.r.t. coordinates basis. While physical measurements, such as the momentum of a particle w.r.t. to a particular frame, correspond to the 4-momentum w.r.t. orthonormal basis defined in that particular frame. What causes this distinction?

I don't understand the question. The 4-velocities of the observer frames are in the holonomic (coordinate) basis, but there is always a tetrad that takes them to (1,0,0,0).
Actually the 4-velocities are interpreted as vector fields.