I am also interested in *General Relativity* and in particular in vacuum solutions of Einstein's Field Equations. On March 16, 2013 I detected the following metrics

ds^{2}= (dt^{2}- dx^{2}- dy^{2}- dz^{2}) - (w dt + w dz)^{2}with w = w(t,x,y,z),

which are Ricci flat, if the following conditions hold

d^{2}w^{2}/dt^{2}+ d^{2}w^{2}/dz^{2}- 2 d^{2}w^{2}/dt dz = 0,

d^{2}w^{2}/dx^{2}+ d^{2}w^{2}/dy^{2}= 0,

d^{2}w^{2}/dx dz + d^{2}w^{2}/dx dt = 0,

d^{2}w^{2}/dy dz + d^{2}w^{2}/dy dt = 0.

It is easy to find functions, which fulfill these conditions, but the corresponding metrics are often merely parametrizations of the flat Minkowski space-time. However, the function

w(x,y) = sqrt(a ln(b x^{2}+ b y^{2}) + c),

which solves the Laplace equation for w^{2} (second condition), provides a non-trivial stationary and axially symmetric solution of the field equations. This solution looks to me like the vacuum analogue of the Gödel metric. For more information read the following paper:

It is also interesting to investigate already known solutions of the field equations. In the following paper I present the metric tensor of the Kerr-Taub-NUT metric and instanton in four different coordinate systems and describe the relevant coordinate transformations in detail:

Last update: November 29, 2013 | Back to the Homepage |