Is Rotation in x and y Invariant at the Origin in a Rotating Coordinate System?

[pervect]

If we look at the rotational transformation (specifically in the x-y plane) we get

x' = cos(wt)*x - sin(wt)*y
y' = cos(wt)*y + sin(wt)*x
z' = z
t' = t

and we can write

dx' = cos(wt)*dx - sin(wt)*dy -w*(y*cos(wt)+x*sin(wt))*dt
dy' = cos(wt)*dy + sin(wt)*dx + w*(x*cos(wt)-y*sin(wt))*dt
dz' = dz
dt' = dt

We can intepret dx,dy,dz, and dt as one-forms, the basis of the cotangent space.

When we substitute x=y=z=t=0, we find that

dx' = dx
dy' = dy
and of course
dz' = dz
dt' = dt

This means that rotation in x and y leaves the basis one-forms invariant at the origin (the origin of the rotation).

Letting u_1 = dx, u_2=dy, u_3 = dz, and u_4 = dt, we can ask what the basis vectors of the tangent space are.

These will be just uⁱ = g^iju_j

Because the basis one-forms are not changed at the origin by rotation, and the above equation converts the basis one-forms into basis vectors, the basis vectors are also not changed at the origin by rotation.

The argument appears to me to be completely general - it says that tranforming a tensor quantity into a spatially rotating coordinate system does not have and can never have any effect on a tensor quantity at the origin of the rotation.

Does anyone see any flaws to this argument?

 

[Aether]

The argument appears to me to be completely general - it says that tranforming a tensor quantity into a spatially rotating coordinate system does not have and can never have any effect on a tensor quantity at the origin of the rotation.

Does anyone see any flaws to this argument?

The argument is based on a transform which assumes the Einstein clock synchronization convention, and it is therefore not completely general. A truly general linear transformation is given on page 508 of Mansouri-Sexl I.

 

[pervect]

The argument is based on a transform which assumes the Einstein clock synchronization convention, and it is therefore not completely general. A truly general linear transformation is given on page 508 of Mansouri-Sexl I.

Your objection isn't terribly clear, is your objection related to the question of whether or not

x' = cos(wt)*x - sin(wt)*y
y' = cos(wt)*y + sin(wt)*x

represents a general spatial rotation?

[Insert disclaimer that specific choice of coordinates x & y for coordinates is not general, but that this is apparently not the issue being raised.]

I don't own and am not familiar with the reference you cite (and the citation is not explicit enough for me to track it down).

I'm guessing from your cryptic comments about clock synchronization that perhaps you are defining a spatial rotation as a group with more than
the three parameters of the Euler angles?

 

[Aether]

Your objection isn't terribly clear, is your objection related to the question of whether or not

x' = cos(wt)*x - sin(wt)*y
y' = cos(wt)*y + sin(wt)*x

represents a general spatial rotation?

[Insert disclaimer that specific choice of coordinates x & y for coordinates is not general, but that this is apparently not the issue being raised.]

I don't own and am not familiar with the reference you cite (and the citation is not explicit enough for me to track it down).

R. Mansouri, and R.U. Sexl, A Test Theory of Special Relativity: I. Simultaneity and Clock Synchronization, General Relativity and Gravitation, Vol. 8, No. 7 (1977), p. 508. If you don't have access to this journal, let me know and I'll scan-in the Mansouri-Sexl papers (there are three papers in the series, all in the same volume but beginning on pages 497, 515, & 809 respectively) and send them to you.

Most, if not all, of the subsequently published tests of local Lorentz invariance reference this series of papers.

I'm guessing from your cryptic comments about clock synchronization that perhaps you are defining a spatial rotation as a group with more than
the three parameters of the Euler angles?

The spatial coordinates per se aren't at issue, but the temporal coordinate is a function of the spatial coordinates in the general linear transform.

 

[George Jones]

I don't see any problem with rotations not affecting quantities at the origin.

I learned interesting stuff by thinking about your example. For example,
when wr > 1, t' is a spacelike coordinate, and the basis vectors associated with the primed coordinate system form a set of 4 spacelike linearly independent vectors!

I convinced myself that this is not crazy in a couple of different ways. One way was to think of simpler coordinates for Minkowski spacetime for which this is true. Define coordinates T and X in terms of inertial t and x by

T = t
X = -2t + x.

This is perfectly good coodinate system for Minkowski spacetime, yet the coordinate vectors are both spacelike, in spite of the fact that t = T!

Regards,
George

 

[pervect]

I didn't think about the sign reversal of g_tt in my example that much other than a sense of caution where g_tt became zero, I'm used to coordinate systems breaking down if they get too far away from the origin.

But the coordinate system (T,X) in youre example qualifies at least as weird in my book, even if it's not quite certifiably crazy :-).

[add]
It is weird to have a coordiante system defined by four space-like vectors, but if we take three space-like vectors and one time-like vector, we can "tilt" the time-like vector by forming a linear combination of it with one of the space-like vectors, so that it becomes space-like, while still keeping it linearly independent from the other vectors (though not of course orthogonal to them). I.e. we take a*time + b*space, and make a small (but non-zero), and b large. The result is a coordinate system with three space-like vectors.

 

[George Jones]

It is weird to have a coordiante system defined by four space-like vectors, but if we take three space-like vectors and one time-like vector, we can "tilt" the time-like vector by forming a linear combination of it with one of the space-like vectors, so that it becomes space-like, while still keeping it linearly independent from the other vectors (though not of course orthogonal to them). I.e. we take a*time + b*space, and make a small (but non-zero), and b large. The result is a coordinate system with three space-like vectors.

This type of reasoning led me to my coordinate system - I tend to think with vectors more than coordinates. It also leads to bases made up of 4 timelike vectors.

I also was motivated by some other stuff that I have read recently. Lately, I have been trying to understand Roger Penrose's plausibility argument for his crazy idea that gravity causes collapse of the wavefunction. He makes the point that having t' = t hold says little about the relationship between the tangent vectors $\partial / \partial t'$ and $\partial / \partial t$. The former is calculated holding the primed spatial coordinates constant, while the latter is calculated with the unprimed spatial coordinates held constant, so differences in spatial coordiantes cause differences in time vectors even when time coordinates are equal.

The more I think about Penrose's ideas, the less crazy they seem, and I might start a thread in the Quantum Physics forum about them. I still thing that it's a real longshot that his ideas are true, but he, at least, has suggested experiments that are currently being pursued. These experiments should either result in verification of his ideas, or in the quantum superposition states of the heaviest object ever (by far!) superposed. Either result would be very exciting.

Thinking about your rotating coordinate system tacked to a rigidly rotating disk resulted in a google search that produced an interesting hit on John Baez's webpages. Stresses that hold the disk together seem to indicate that general relativity comes into play, unlike the case for linear acceleration.

Regards,
George