Null Dust Solution - Einstein Tensor

Einstein Tensor

The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer.

In the case of a null dust solution, an adapted frame

(a timelike unit vector field and three spacelike unit vector fields, respectively) can always be found in which the Einstein tensor has a particularly simple appearance:

Here, is everywhere tangent to the world lines of our adapted observers, and these observers measure the energy density of the incoherent radiation to be .

From the form of the general coordinate basis expression given above, it is apparent that the stress-energy tensor has precisely the same isotropy group as the null vector field . It is generated by two parabolic Lorentz transformations (pointing in the direction) and one rotation (about the axis), and it is isometric to the three dimensional Lie group, the isometry group of the euclidean plane.