Hi, MeJennifer,
MeJennifer said:
Am I correct in understanding that this not an exact metric but a weak field metric?
No, that is exact! See for example
http://www.arxiv.org/abs/gr-qc/0001069 or the well if microscopically illustrated discussion in Black Hole Physics by Frolov and Novikov.
To see this, you need only know a coordinate transformation to some chart with which you are already familiar. Almost certainly the Schwarzschild chart is the obvious choice. So let me write these two line elements, using \tau for Painleve time coordinate and t for Schwarzschild time coordinate:
Schwarzschild chart:
ds^2 = -(1-2m/r) \, dt^2 + 1/(1-2m/r) \, dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < t < \infty, \; 2m < r < \infty, 0 < \theta < \pi, \; -\pi < \phi < \pi
Painleve chart:
ds^2 = -(1-2m/r) \, d\tau^2 + 2 \, \sqrt{2m/r} \, d\tau \, dr + \, dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
-\infty < \tau < \infty, \; 0 < r < \infty, 0 < \theta < \pi, \; -\pi < \phi < \pi
The coordinate transformation from the Schwarzschild chart is given by
\tau = t + \int \frac{\sqrt{2m/r}}{\sqrt{1-2m/r}} \, dr = t + \sqrt{8mr} - 4 m \, \operatorname{arctanh}(\sqrt{r/2m})
The frame field defining the Lemaitre observers (with nonspinning local Lorentz frames attached to each observer) is given in the Painleve chart by:
\vec{e}_0 = \partial_\tau - \sqrt{2m/r} \, \partial_r = \vec{X}
\vec{e}_1 = \partial_r
\vec{e}_2 = \frac{1}{r} \, \partial_\theta
\vec{e}_3 = \frac{1}{r \, \sin(\theta)} \, \partial_\phi
(The coefficients of the coordinate derivatives are the components of the four vector fields.) In terms of this frame, the acceleration vector of our observers is \nabla_{\vec{X}} \vec{X} = 0, confirming that these observers are inertial (free-falling), and from the form of \vec{e}_0 you can see that they are radially infalling, and also that as r goes to infinity, \vec{e}_0 becomes more and more "upright", i.e. the Lemaitre observers "start falling from rest at infinity".
Since these are simply four vector fields, and since the notion of a pair of orthonormal vector fields is a geometric (or "coordinate-free") notion, such frame fields are geometric structures which do not depend upon using any coordinate chart, although to write them down explicitly we must adopt some coordinate chart. But once we have a frame in hand we are free to change to any other chart should that be convenient. For example, you can easily represent the above frame field in the original Schwarzschild coordinates.
The so-called expansion tensor (see MTW, or Hawking and Ellis, or the newer book by Eric Poisson, A Relativist's Toolkit) is
H[\vec{X}]_{\hat{a} \hat{b}} = \sqrt{m/2/r^3} \, \operatorname{diag} \, (1,-2,-2)
The first component is positive, which means that each Lemaitre observer is pulling away from his neighoring Lemaitre observers radially. (You can verify this visually by plotting the integral curves of \vec{e}_0 in the \tau, r plane.) The second two components are negative, which means that each Lemaitre observer is approaching his neighbors orthogonally to his direction of infall toward the hole.
The expansion tensor is a three dimensional tensor because it has been projected into a hyperplane element orthogonal to \vec{e}_0, which in this situation happens to correspond to being projected into the hyperslice \tau=\tau_0. In the expression for the expansion tensor, the hats on the indices are often used to emphasize that the components are taken with respect to a frame field (orthonormal basis, anholonomic basis) rather than a coordinate basis. The vector field in brackets emphasizes that the expansion tensor is evaluated with respect to a particular timelike congruence, in this case the one given by the timelike unit vector field \vec{e}_0 from our frame field.
Also, the Riemann tensor can be decomposed into three pieces (actually only two are independent, since this is a vacuum solution), the electrogravitic and magnetogravitic tensors:
E[\vec{X}]_{\hat{m}\hat{n}} = \frac{m}{r^3} \; \operatorname{diag} (-2, \, 1, \, 1)
B[\vec{X}]_{\hat{m}\hat{n}} = 0
Here, the electrogravitic tensor is exactly the same as the tidal tensor for the analogous static spherically symmetric gravitational field in Newtonian gravitation! The first component is negative, which means that our observers experience a radial tidal
tension. The second two components are positive, which means that our observers experience tidal
compression orthogonally. As this shows, the famous "sphaghettification" is nothing unknown to Newtonian physics, just more intense inside a black hole than what we experience at the surface of the Earth. The magnetogravitic vanishes, which means that our observers measure no "gravitomagnetic effects". However, other observers, such as Hagihara observers, who are in stable circular orbits in the exterior region, and are represented by a different frame field, would observe such effects.
BTW, if we compute these tensors with respect to the timelike unit vector of the frame modeling the physical experience of
static observers (this frame is of course only defined outside r=2m), we obtain the same "Coulomb form" for the electrogravitic tensor, and the magnetogravitic tensor again vanishes, so the static observers do not measure magnetogravitic effects.
We can also compute the three-dimensional Riemann tensor giving the "intrinsic curvature" of the hyperslices (see the books cited above), which turns out to be
r_{\hat{m}\hat{n}\hat{p}\hat{q}} = 0
as we should expect from the spatial line element
d\sigma^2 = dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),
0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
which is of course simply the usual spatial line element for a flat euclidean three-space (in a polar spherical chart).
By the way, the normals to one of our slices \tau=\tau_0 are given by our timelike unit vector field (tangents to the world lines of the Lemaitre observers where they intersect \tau=\tau_0), and the expansion tensor wrt \vec{X} = \vec{e}_0 then gives (the negative of) the extrinsic curvature tensor; see again the books cited above.
(Personal note referring to my post on how I learned gtr: the Painleve chart is the chart which I rediscovered when I was first reading MTW, using only high school "circular" trig -> "hyperbolic" trig. Basically, I drew curves orthgonal to the world lines of the infalling Lemaitre observers in the Schwarzschild chart, and figured out what integral I needed to evaluate to pull down these curves to form straight lines, i.e. to pull the hyperslices into the shape of coordinate planes. Finding the line element in the new chart doesn't actually require evaluating the integral I wrote out above. Then of course I noticed something unexpected: in the new chart, the spatial line element is just the usual polar spherical line element!)
To repeat:
all of these expressions are
exact; no approximations are involved.
What is the difference between a holonomic (e.g. a coordinate basis) and an anholonomic basis (frame field)? Simply that these are made up of vector fields, so we can compute the commutators (Lie brackets) of pairs of these vector fields. In a coordinate basis such as
\partial_\tau, \; \partial_r, \; \partial_\theta, \; \partial_\phi
(the coordinate basis of the Painleve chart), these commutators all vanish. But the commutators of our frame field do not all vanish.
