What Are Rosen Coordinates in Gravitational Wave Analysis?

[CoordinatesPLZ]

I see a number of gravitational wave analytic solutions with the metric given in terms of Rosen coordinates. I have no idea what these coordinates are. How do I perform a coordinate transformation from Rosen coordinates to traditional (t,x,y,z) Euclidean\Cartesian coordinates? Also, is there a difference between Rosen coordinates and Eisenstein-Rosen coordinates?

Thanks
CoordinatesPLZ

 

[Bill_K]

A coordinate system for the description of gravitational plane waves:

ds² = 2 du dv + g_ij(u) dyⁱ dy^j.

As opposed to Brinkmann coordinates for the same wave:

ds² = H(u) du² + 2 du dv + dx² + dy².

Here's a reference that discusses both, and how to convert one to the other.

 

[CoordinatesPLZ]

That doesn't directly answer my question. Also, I am not familiar with Brinkmann coordinates. If we suppose the line element in Rosen coordinates in flat space-time is:

ds²=2 du dv+dy₁²+dy₂²

Then we can deduce the possible coordinate mapping to Cartesian coordinates of

t = (u-v)/√2
x = y₁
y = y₂
z = (u+v)/√2,

Assuming a <-1,1,1,1> metric signature

Is this the correct mapping from Rosen coordinates to Cartesian coordinates?

 

[Bill_K]

That doesn't directly answer my question.

Sorry. What IS your question? You say you want to know what Rosen coordinates "are", but that question has no answer. Coordinates in a curved spacetime do not always have a simple interpretation.

Also, I am not familiar with Brinkmann coordinates.

If you're interested in gravitational waves, you should learn about Brinkmann coordinates. Rosen coordinates have several drawbacks. One: they are not unique. Two: they can develop coordinate singularities (caustics). That's why they are not generally used.

If we suppose the line element in Rosen coordinates in flat space-time is:

ds²=2 du dv+dy₁²+dy₂²

That's one possibility. Here's another instance of a plane wave in Rosen coordinates that is also flat:

ds² = 2 du dv + u²(dy₁² + dy₂²)

Then we can deduce the possible coordinate mapping to Cartesian coordinates of

t = (u-v)/√2
x = y₁
y = y₂
z = (u+v)/√2,

Assuming a <-1,1,1,1> metric signature

Is this the correct mapping from Rosen coordinates to Cartesian coordinates?

In a flat spacetime, (u, v, y₁, y₂) are called light-cone coordinates, and (t, x, y, z) are called Minkowski coordinates. The terms Euclidean and Cartesian do not apply!

 

[CoordinatesPLZ]

My problem is I have a formula for the metric tensor in Rosen coordinates and I would like the metric tensor in Minkowski spacetime. To go from one to the other I need the coordinate mapping, thus, how do I transform the coordinates from Rosen coordinates to Minkowski Coordinates?

 

[Bill_K]

My problem is I have a formula for the metric tensor in Rosen coordinates and I would like the metric tensor in Minkowski spacetime. To go from one to the other I need the coordinate mapping, thus, how do I transform the coordinates from Rosen coordinates to Minkowski Coordinates?

Can't be done. A gravitational wave is a curved spacetime, and Minkowski coordinates exist only in flat space. You can't turn a curved spacetime into a flat spacetime just by changing the coordinates. A gravitational wave is not just Minkowski space written in some weird set of coordinates.

You can certainly do the z = (u + v)/√2, t = (u - v)/√2 thing if you like, and that will make at least part of the metric look more familiar, but there will be terms left over.