Abstract
The standard Lense–Thirring metric is a century-old slow-rotation large-distance approximation to the gravitational field outside a rotating massive body, depending only on the total mass and angular momentum of the source. Although it is not an exact solution to the vacuum Einstein equations, asymptotically the Lense–Thirring metric approaches the Kerr metric at large distances. Herein we shall discuss a specific variant of the standard Lense–Thirring metric, carefully chosen for simplicity, clarity, and various forms of improved mathematical and physical behaviour, (to be more carefully defined in the body of the article). We shall see that this Lense–Thirring variant can be viewed as arising from the linearization of a suitably chosen tetrad representing the Kerr spacetime. In particular, we shall construct an explicit unit-lapse Painlevé–Gullstrand variant of the Lense–Thirring spacetime, one that has flat spatial slices, a very simple and physically intuitive tetrad, and extremely simple curvature tensors. We shall verify that this variant of the Lense–Thirring spacetime is Petrov type I, (so it is not algebraically special), but nevertheless possesses some very straightforward timelike geodesics, (the “rain” geodesics). We shall also discuss on-axis and equatorial geodesics, ISCOs (innermost stable circular orbits) and circular photon orbits. Finally, we wrap up by discussing some astrophysically relevant estimates, and analyze what happens if we extrapolate down to small values of r; verifying that for sufficiently slow rotation we explicitly recover slowly rotating Schwarzschild geometry. This Lense–Thirring variant can be viewed, in its own right, as a “black hole mimic”, of direct interest to the observational astronomy community.
Highlights
Two years after the discovery of the original Schwarzschild solution in 1916 [1], in 1918 Lense and Thirring found an approximate solution to the vacuum Einstein equations at large distances from a stationary isolated body of mass m and angular momentum J [2]
The only region where one should trust the Kerr solution is in the asymptotic regime, where in any case it reduces to the Lense–Thirring metric
+r dθ + sin θ dφ − 3 + O 4 dt r r. This modified metric again asymptotically approaches standard Lense–Thirring (1) at large distances, but has the three very strong advantages that (i) for J = 0 it is an exact solution of the vacuum Einstein equations, (ii) that the azimuthal dependence is in partial Painlevé–Gullstrand form, with gφφ2 = gφφ2, and (iii) that all the spatial dependence is in exact Painlevé–Gullstrand type form, in the sense that the constant-t spatial 3-slices are flat, and so very easy to interpret
Summary
Two years after the discovery of the original Schwarzschild solution in 1916 [1], in 1918 Lense and Thirring found an approximate solution to the vacuum Einstein equations at large distances from a stationary isolated body of mass m and angular momentum J [2]. (trying to explicitly calculate approximate curvature components for this approximate metric is quite slow, and the results are quite horrid). It took another 45 years before Roy. Kerr found the corresponding exact solution in 1963 [11,12]. The only region where one should trust the Kerr solution (as applied to a real rotating star or planet) is in the asymptotic regime, where in any case it reduces to the Lense–Thirring metric Two of the virtues of putting the metric into Painlevé–Gullstrand form is that mathematically it is easy to work with, and physically it is very easy to interpret—in particular, the analogue spacetimes built from excitations in moving fluids are typically (conformally) of Painlevé–Gullstrand form [31,32,33,34,35,36,37,38,39,40,41,42], and so give a very concrete and physically intuitive visualization of such spacetimes
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