# Einstein's Field Equations

Ulric Kintzel

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

ds2 = (dt2 - dx2 - dy2 - dz2) - (w dt + w dz)2   with   w = w(t,x,y,z),

which are Ricci flat, if the following conditions hold

d2w2/dt2 + d2w2/dz2 - 2 d2w2/dt dz = 0,
d2w2/dx2 + d2w2/dy2 = 0,
d2w2/dx dz + d2w2/dx dt = 0,
d2w2/dy dz + d2w2/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 x2 + b y2) + c),

which solves the Laplace equation for w2 (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:

Although the Kerr metric is well known, there do not seem to be any representations describing the metric in a general coordinate system in which the axis of rotation does not coincide with the z-axis. This case is described in the following work:

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