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I have posted this question in the PF relativity forum because I am trying to understand the derivation of the Lense-Thirring (L-T) metric. Various sources suggest that L-T produced a solution of Einstein’s field equations of general relativity in 1918, just a few years after Schwarzschild in 1916 and Einstein GR publication in 1915. However, papers describing the derivation of this metric seem thin on the ground. I have only found 2 that appear to directly discuss the L-T metric, e.g.

L-T Frame Dragging

This paper cites the Schwarzschild metric in equation (1) and the L-T metric in (2) stating that it consists of the Schwarzschild metric plus only one additional cross term, which is said to accommodate a small rotational velocity:

[tex]ds^2 = c^2 \left( 1-Rs/r \right) dt^2 - \left( 1-Rs/r \right) ^{-1} - r^2 \left( d \theta^2 + sin^2 \theta d \phi^2 \right) + 2 \frac {2GMa}{c^2r} sin^2 \theta dt d \phi [/tex][tex]where....Rs = \frac {2GM}{c^2} [/tex]

However, this paper doesn’t explain the derivation and it is not obvious to me how this cross-term is produced. One idea was that you might apply a rotation in the form of a substitution, e.g.

[tex]d \phi^2 = \left( d \phi - \omega dt \right)^2 = d \phi^2 - 2 \omega d \phi dt+ \omega^2 dt^2 [/tex]

This method is adopted in the link below, but leads to an additional term in [itex]dt^2[/itex] and doesn’t seem to explain the L-T cross term above. However, the derivation of the L-T metric in section IV (p.7) proceeds from an isotropic form of the Schwarzschild metric developed in section III (p.6). Although, the author claims that the result in section IV ‘constitutes’ the L-T metric, it is not clear how this results aligns to the cross term above, which appears to use normal [r] coordinates.

Simple Derivation of L-T

Would really appreciate any insights or pointers to freely available documents that might help explain this metric. Thanks

Footnote:

Not wishing to overload this initial post with too many questions, there seems to be an debate as to how the Schwarzschild, L-T and/or Kerr metric might be applied to black holes and still be consistent with Mach’s principle, as possibly illustrated in the reference below. Again, would appreciate any insights on this issue, although I recognise that I need to better understand the maths behind some of these metrics first.

The Lense–Thirring Effect and Mach’s Principle

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# Lense-Thirring Metric

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