Distinction between coordinate and reference frame

[Andre' Quanta]

How is it possible to distinguish a change of coordinate from a change o reference frame? I had this problem while i was studying Rindler' s coordinates: is it only another way to describe a Minkowsky space-time region or does it rappresent a region of the space time as described by an accelerated observer?
I am sure that i am doing i logical mistake, but i can ' t see it

 

[Orodruin]

How is it possible to distinguish a change of coordinate from a change o reference frame?

A change of reference frames is a change of coordinates. The physics are always independent of the coordinates used. You can compute how things appear for an accelerating observer in any set of coordinates.

The thing that changes the most when allowing general curvilinear coordinate systems on Minkowski space is the notion of surfaces of simultaneity (which probably is good riddance anyway, I think it causes more problems than it solves when people try to think of it conceptually).

 

[Nugatory]

How is it possible to distinguish a change of coordinate from a change o reference frame? I had this problem while i was studying Rindler' s coordinates: is it only another way to describe a Minkowsky space-time region or does it rappresent a region of the space time as described by an accelerated observer?
I am sure that i am doing i logical mistake, but i can ' t see it

Rindler coordinates are indeed just another way of describing the flat Minkowski spacetime, just as we can choose to use polar or Cartesian coordinates to describe that spacetime. Rindler coordinates have the property that a particular accelerating observer is not changing position in those coordinates so they're convenient to use when analyzing things from the point of view of that observer, just as Cartesian coordinates are convenient to use when analyzing things from the point of view of observers moving at constant velocities.

For just about all informal purposes, you can consider choosing a reference frame to be equivalent to choosing a coordinate system.

 

[WannabeNewton]

How is it possible to distinguish a change of coordinate from a change o reference frame?

Yes there is a distinction and knowing the distinction will save you a lot of conceptual hurdles in the future, trust me.

To start with, an observer is a timelike worldline with 4-velocity $u^{\mu}$ and an orthonormal basis, also known as a local Lorentz frame or tetrad $e_{\hat{\alpha}}$ with $e_{\hat{0}} = u$ (the hat just indicates a basis) such that $e_{\hat{\alpha}}$ is transported along the worldline under some transport law e.g. Lie transport, Fermi transport, or parallel transport. Physically the Lorentz frame represents a local set of three orthogonal meter sticks or gyroscopes and an ideal clock carried by the observer.

An observer can use $e_{\hat{\alpha}}$ to define a comoving local coordinate system $x^{\mu}$ (e.g. a Fermi-normal coordinate system) with clocks that are e.g. Einstein synchronized but the coordinate system isn't necessary to define the observer. Operationally, the observer takes the meter sticks and clock they carry and places an identical set at each point in some neighborhood of space (relative to the observer) using say parallel transport.

Now a coordinate transformation is of course just a specification $x'^{\mu} = x'^{\mu}(x^{\nu})$ and associated quantities will transform as usual e.g. $V'^{\mu} = \frac{\partial x'^{\mu}}{\partial x^{\nu}}V^{\nu}$. On the other hand a change of reference frame is always a Lorentz boost $V'^{\hat{\alpha}} = \Lambda^{\hat{\alpha}}{}{}_{\hat{\beta}}V^{\hat{\beta}}$ where I have again used hats to indicate that we are now evaluating the components of the 4-vector $\vec{V}$ in the local Lorentz frame $e_{\hat{\alpha}}$ and then boosting to a new local Lorentz frame $e_{\hat{\alpha'}}$ at some event at which the two frames are co-located.

The difference between the two is the coordinate transformation acts on the coordinate (unhatted) indices whereas the Lorentz transformation acts on the tetrad (hatted) indices. The importance of this distinction will only become apparent the more you go along in SR/GR.

For example, in Minkowski space-time I can of course transform from Minkowski coordinates of an inertial frame to rotating coordinates but this is not the reference frame of the associated congruence of rotating observers. On the other hand I can boost from an inertial frame to the frame of a rotating observer using a Lorentz boost which is a change of reference frame.

A more complicated example would be the coordinate transformation from Schwarzschild to Eddington-Finkelstein coordinates, the latter of which is certainly not a reference frame. This would be in contrast to boosting from the local Lorentz frame of a static observer to that of an observer in circular orbit right when the orbiting observer passes by the static observer.

C.f. chapter 6 of MTW, section 13.6 of MTW, section 2.1 of Sachs and Wu "General Relativity for Mathematicians", chapter 3 of Eric Gourgoulhon "Special Relativity in General Frames", and http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

 

[Mentz114]

Yes there is a distinction and knowing the distinction will save you a lot of conceptual hurdles in the future, trust me.

To start with, an observer is a timelike worldline with 4-velocity $u^{\mu}$ and an orthonormal basis, also known as a local Lorentz frame or tetrad $e_{\hat{\alpha}}$ with $e_{\hat{0}} = u$ (the hat just indicates a basis) such that $e_{\hat{\alpha}}$ is transported along the worldline under some transport law e.g. Lie transport, Fermi transport, or parallel transport. Physically the Lorentz frame represents a local set of three orthogonal meter sticks or gyroscopes and an ideal clock carried by the observer.
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Good answer. Also, boosting a tetrad cannot change the metric but a coordinate transformation may.

I can't remember the details but are the holonomic basis vectors and frame basis vectors distinguished by different derivative ( I forget which derivative )

 

[WannabeNewton]

I can't remember the details but are the holonomic basis vectors and frame basis vectors distinguished by different derivative ( I forget which derivative )

I'm not sure what you mean. Are you asking if the connection coefficients differ? If so then yes. The connection coefficients for an arbitrary tetrad are $\Gamma^{\hat{\alpha}}_{\hat{\beta}\hat{\gamma}} = \frac{1}{2}g^{\hat{\alpha}\hat{\mu}}(g_{\hat{\mu}\hat{\beta}, \hat{\gamma}} + g_{\hat{\mu}\hat{\gamma},\hat{\beta}} - g_{\hat{\beta}\hat{\gamma},\hat{\mu}}+c_{\hat{\mu}\hat{\beta} \hat{\gamma}} + c_{\hat{\mu}\hat{\gamma}\hat{\beta}} - c_{\hat{\beta}\hat{\gamma}\hat{\mu}} )$ where $c_{\hat{\alpha}\hat{\beta}}{}{}^{\hat{\gamma}}$ are the structure constants defined by the Lie bracket of the basis vectors. For an orthonormal basis $g_{\hat{\alpha}\hat{\beta},\hat{\gamma}} = 0$ whereas for a coordinate basis $c_{\hat{\alpha}\hat{\beta}}{}{}^{\hat{\gamma}} = 0$.

 

[stevendaryl]

A change of reference frames is a change of coordinates. The physics are always independent of the coordinates used. You can compute how things appear for an accelerating observer in any set of coordinates.

Actually, specifying a frame of reference only provides a standard for rest, and a standard for simultaneity. Those two standards do not uniquely specify a coordinate system. For example, using rectangular coordinates versus spherical coordinates doesn't mean a change of reference frame. Maybe mathematically, a frame of reference is an equivalence class of coordinate systems.

 

[PeterDonis]

specifying a frame of reference only provides a standard for rest, and a standard for simultaneity.

I think this is a matter of terminology; the term "frame of reference" has multiple possible meanings, one of which is what you say, but it's not the only one. An equivalent way of expressing this definition would be to say that a "frame of reference" specifies a congruence of timelike worldlines and a set of spacelike hypersurfaces on an open region of spacetime, such that each worldline intersects each hypersurface exactly once.

Another possible meaning is "coordinate chart"; often the two terms are used interchangeably. Note that a coordinate chart may not specify a frame of reference in the above sense, since there is no requirement that a coordinate chart consist of one timelike and three spacelike coordinates.

Yet another possible meaning is "frame field", which is basically what WannabeNewton described. A frame field specifies a set of 4 orthonormal vectors at each event in an open region of spacetime (and also requires that certain properties are satisfied, such as continuity). This meaning has the most direct physical interpretation, since the 4 vectors at each event can be directly interpreted as the 4-velocity of an observer at that event and three orthonormal spatial axes that he uses to make measurements at that event.