Deriving L-T Metric: Understanding Schwarzschild & Einstein's GR

[mysearch]

Hi,
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:
$$ds^2 = c^2 \left( 1-Rs/r \right) dt^2 - \left( 1-Rs/r \right) ^{-1} - r^2 \left( d \theta^2 + sin^2 \theta d \phi^2 \right) + 2 \frac {2GMa}{c^2r} sin^2 \theta dt d \phi$$$$where...Rs = \frac {2GM}{c^2}$$
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.
$$d \phi^2 = \left( d \phi - \omega dt \right)^2 = d \phi^2 - 2 \omega d \phi dt+ \omega^2 dt^2$$
This method is adopted in the link below, but leads to an additional term in $dt^2$ 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

 

[Bill_K]

Insight: This is the field of a rotating mass. And the key word is "small", a small rotational velocity. In other words, linearized gravity.

In linearized gravity, the space-time cross term g_0i is analogous to the magnetic vector potential in electromagnetism. And in electromagnetism, the magnetic vector potential of a magnetic dipole is A = (m x r)/r³. The cross term in the metric is therefore (A·dr) dt = (m x r · dr) dt/r³ = (m · r x dr) dt/r³.

The z component of r x dr is x dy - y dx = r² sin²θ dφ, giving us for the cross term

Ma/r sin²θ dφ dt

 

[mysearch]

Bill,

Thanks for the insight and the reference to linearized gravity, which I will follow up. To be honest, I hadn’t seen the relevance of electromagnetic considerations in terms of Schwarzschild, Lense-Thirring or Kerr metrics, as all these metrics are based on uncharged assumptions. While I believe that such issues are taken into consideration in the Reissner-Nordström and Kerr-Newman metrics, I have not really looked at them as yet. As such, I assumed that the low-speed rotation of (L-T) metric might be derived from modifying the Schwarzschild metric based purely on gravitational considerations and the possible addition of a $\phi - \omega t$ rotation. In part, this seems to be the approach taken by Berman, as per reference 2 in post #1, although he uses isotropic coordinates, although I could still not reconcile the L-T cross term with this approach either. For example, if I assume a circular equatorial solution, i.e. where $\theta=90, sin^2 \theta=1$ and angular momentum $L=mvr=m \omega r^2$, might the cross term in the L-T metric be interpreted as:
$$2 \frac {2GMa}{c^2r} sin^2 \theta dt d \phi = 2 \frac {Rs}{r} \frac {L}{M} dt d \phi$$
However, as indicated, I could not resolve this cross term, based on any low speed gravitational assumptions linked to a modified Schwarzschild metric, which appear to suggest 2 additional terms, again assuming circular equatorial simplifications, i.e.
$$\omega \frac {L}{M}dt^2$$ and
$$2 \frac {L}{M} dt d \phi$$
While I will continue to research into these issues, my main interest was really concerned with the use of various metrics in a number of ‘novel’ galactic models now being proposed. However, I guess it might be best to raise any questions in this context in the astrophysics forum. Thanks

 

[Bill_K]

mysearch, I only invoked electromagnetism as an analogy. The equations are the same, making it easy to write down the solution for the cross term. No implication that the solution is charged.

PS - You can't get the field of a rotating mass by writing Schwarzschild in a rotating coordinate system. Totally different conceptually! And as you've seen, it leads to a cross term which is totally different.

 

[mysearch]

mysearch, I only invoked electromagnetism as an analogy. The equations are the same, making it easy to write down the solution for the cross term. No implication that the solution is charged.

Thanks for the clarification. I will review the idea of linearized gravity and the ‘analogy’ in more detail.

PS - You can't get the field of a rotating mass by writing Schwarzschild in a rotating coordinate system. Totally different conceptually! And as you've seen, it leads to a cross term which is totally different.

While you are probably right, I was led towards this line of thinking by the approach of Berman, who appears to adopt this method on the isotropic version of the Schwarzschild metric.

Again, more by way of a footnote, my interest in these metrics was in connection with galactic frame dragging. While I think the Kerr metric might be more appropriate, its derivation is beyond my mathematical pay-grade and is said to be non-Machian. This was why I was initially looking at more ‘simple’ metrics