A plane symmetric static vacuum solution?
Hi again, Michel,
lalbatros said:
I would like to know about a possible plane Schwarzschild geometry.
This would be the analog of a uniform and infinite gravitational field, or a uniform acceleration.
I would like to compare it with the solution of the uniformly accelerated motion in SR.
This is suprisingly tricky!
The quick answer is that the so-called Levi-Civita A3 vacuum (c. 1917), aka the Taub plane symmetric vacuum, aka the plane symmetric case of the Kasner vacuum (1921), aka the Taub plane symmetric vacuum (1953), is a static vacuum solution whose hyperslices (orthogonal to static observers) have planar symmetry. It often turns up in various coordinate charts, such as the Levi-Civita chart
ds^2 = -\frac{dt^2}{z} + \left( dx^2 + dy^2 \right) + z \, dz^2, <br />
-\infty < t, x, y < \infty, 0 < z < \infty
the isotropic chart
ds^2 = -dT^2/dZ^2 + Z^4 \; \left( dX^2 + dY^2 + dZ^2 \right)
the Kasner chart
ds^2 = -\zeta^{-2/3} \, d\tau^2 + \zeta^{4/3} \, (dx^2 + dy^2) + d\zeta^2
and the Weyl canonical chart (I'm running out of letters)
ds^2 = -\frac{dt^2}{z+\sqrt{z^2+r^2}} <br />
+ \frac{ 2 z^2+r^2 + 2 z\, \sqrt{z^2+r^2}}{2 \sqrt{z^2 +r^2} \; \left(dz^2+dr^2) + (z + \sqrt{z^2+r^2} ) \, r^2 \, d\phi^2
A longer and possibly more interesting answer might begin by noting that the Weyl family of all static axisymmetric vacuum solutions shows that they all arise from axisymmetric harmonic functions:
ds^2 = -\exp(2 u) \, dt^2 + \exp(-2u) \left( \exp(2 v) ( dz^2+dr^2) + r^2 \, d\phi^2,
u_{zz} + u_{rr} + \frac{u_r}{r} = 0
v_z = 2 \, u_z \, u_r, \; \; v_r = r ( u_z^2 - u_r^2)
Here, the two first order equations for v(z,r) in terms of u(z,r) are consistent exactly in case the second order equation for u(z,r) alone is satisfied. This happens to be the Laplace equation (for axisymmetric functions u(z,r,\phi).) So given any axisymmetric harmonic equation u, we can compute v by quadratures (integrals) and obtain the metric of the corresponding Weyl vacuum solution. But of course the Laplace equation is Newton's vacuum field equation, and expanding the line element to first order in u,v you can see that the in the weak-field approximation, the gravitational field is completely determined by u, so we can think of u as being analogous to the Newtonian potential.
For an excellent discussion of the Kasner vacuums, see Hawking and Ellis, The Large Scale Structure of Space-Time. The plane symmetric Kasner vacuum also arises in the study of colliding plane wave (CPW) solutions of the Einstein equation; see Griffiths, Colliding Plane Waves in General Relativity. And it arises when we use the ideas of inverse scattering (as in the solution of the KdV equation) to obtain (vacuum) solutions of the Einstein equation; see Belinsky and Verdaguer, Gravitational Solitons.
Now the surprise: the correspondence between Newtonian potentials and Weyl vacuum solution is not at all straightforward in the strong field situation. For example, a constant potential u gives the cartesian cart for the Minkowski vacuum, but the potential of an infinite uniform density ray (with linear density 1/2) gives the Rindler chart! The potential u(z,r)=m/r does not give the Schwarzschild vacuum, but a distinct solution known as the Chazy-Curzon vacuum, which is not even spherically symmetric! The Schwarzschild vacuum is in fact obtained from the potential of a uniform density thin rod of length 2m and density 1/2. And to come to the point, the potential of a uniform density thin plate, u(z,r) = mz, does not really correspond (except in the weak-field limit) to the solution you probably want! Rather, the A3 vacuum arises (see the Weyl canonical chart given above) from a uniform density ray with another (constant) density.
Pervect already noted that the "Rindler metric" is probably what you want for your second request.
Chris Hillman
