Painleve-Gullstrand form of the Lense-Thirring spacetime

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 and clarity. In particular we shall construct a unit-lapse Painleve-Gullstrand version of the Lense-Thirring spacetime that has flat spatial slices, some straightforward timelike geodesics, (the"rain"geodesics), and simple curvature tensors.


Introduction
Only 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]. In suitable coordinates [2][3][4][5][6][7][8][9][10]: Here the sign conventions are compatible with MTW [5] (33.6), and Hartle [7] (14.22). It took another 45 years before Kerr found the corresponding exact solution in 1963 [11,12]. Nevertheless the Lense-Thirring metric continues to be of interest for two main reasons: (1) Lense-Thirring is much easier to work with than the full Kerr solution; and (2) For a real rotating planet or star, generically possessing non-trivial mass multipole moments, the vacuum solution outside the surface is not exactly Kerr; it is only asymptotically Kerr [9]. (There is no Birkhoff theorem for rotating bodies in 3+1 dimensions [13][14][15][16][17].) Consequently, 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.
Below we shall recast the standard Lense-Thirring metric of equation (1.1) into Painlevé-Gullstrand form -in this form of the metric (up to coordinate transformations) one has so that the constant-t spatial 3-slices of the metric are all flat, and the lapse function is unity (g tt = −1). See the early references [18][19][20], and more recently [21][22][23][24][25]. (Note that the vector v i , representing the "flow" of space, is minus the shift vector in the ADM formalism.) One of the virtues of putting the metric into Painlevé-Gullstrand form is that it is particularly easy to work with and to interpret -in particular, the analogue spacetimes built from excitations in moving fluids are typically (conformally) of Painlevé-Gullstrand form [26][27][28][29][30][31][32][33][34][35][36][37], and so give a very concrete visualization of such spacetimes.

Variants on the theme of the Lense-Thirring metric
Let us now take the Lense-Thirring metric and seek to simplify it in various ways, while retaining the good features of the asymptotic large-distance behaviour.
First, we note that at J = 0, for a non-rotating source we do have the Birkhoff theorem so it makes sense to consider the metric This modified metric asymptotically approaches standard Lense-Thirring, but has the strong advantage that for J = 0 it is an exact solution of the vacuum Einstein equations.
Second, "complete the square". Consider the metric This modified metric again asymptotically approaches standard Lense-Thirring, but now has the two advantages that for J = 0 it is an exact solution of the vacuum Einstein equations and that the angular dependence is now in partial Painlevé-Gullstrand form: g φφ (dφ−v φ dt) 2 . See the early references [18][19][20], and more recently references [21][22][23][24][25].
Third, put the r-t plane into standard Painlevé-Gullstrand form [18][19][20][21][22][23][24][25]. (Note that v r = − 2m/r for a Schwarzschild black hole.) We then have This modified metric again asymptotically approaches standard Lense-Thirring, but has the two advantages that for J = 0 it is an exact solution of the vacuum Einstein equations, and that all the spatial dependence is in Painlevé-Gullstrand type form, in the sense that the constant-t spatial 3-slices are now flat.
Fourth, drop the O(1/r 4 ) terms in the φ dependence. That is, consider the specific and explicit metric: By construction for J = 0 this is the Painlevé-Gullstrand version of the Schwarzschild metric [18][19][20][21][22][23]. By construction at large distances this asymptotically approaches the "standard" form of Lense-Thirring as given in equation (1.1), and so it also asymptotically approaches Kerr. By construction even for J = 0 this metric is in Painlevé-Gullstrand form. (In particular, with flat spatial 3-slices, and as we shall soon see, unit lapse, and easily constructed timelike geodesics.) These observations make this specific form (2.4) of the Lense-Thirring spacetime interesting and worth investigation.
From (2.4) it is easy to read off the metric components and thence to verify that (2.6) Note particularly that g tt = −1, so that the lapse function is unity; this fact will be particularly useful when we come to analyzing the geodesics.

Spacetime curvature
While the specific Lense-Thirring spacetime variant we are interested in, that of equation (2.4), is not (exactly) Ricci-flat, it is easy to calculate the Ricci scalar and Ricci invariant and verify that they are "suitably small". We have and Note that all the right things happen as J → 0. Note that all the right things happen as r → ∞. Ultimately, it is the observation that these quantities fall-off very rapidly with distance that justifies the assertion that this is an "approximate" solution to the vacuum Einstein equations.
A more subtle calculation is to evaluate the Weyl invariant: Note that this is what you would expect for Schwarzschild, 48m 2 /r 6 , plus a rapid falloff angular-momentum-dependent term, O(J 2 /r 8 ). Similarly for the Kretschmann scalar we have It is a little bit trickier to calculate the Ricci and Einstein tensors.
• The "simplest" form we have found for the Ricci tensor is this -after raising one index: Notice the perhaps somewhat unexpected pattern of zeros and minus signs. Notice the rapid fall-off at large distances.
• The "simplest" form we have found for the Einstein tensor is this -after raising one index: Notice the perhaps somewhat unexpected pattern of zeros and minus signs. Notice the rapid O(J 2 /r 6 ) fall-off at large distances.
• Because (considered as matrices) these tensors are upper triangular it is easy to extract the Lorentz-invariant eigenvalues, defined by det(X a b − λ δ a b ) = 0, or equivalently, det(X ab − λ g ab ) = 0.
Notice the rapid O(J 2 /r 6 ) fall-off at large distances.
• Algebraically, this implies that the Ricci and Einstein tensors are type I in the Hawking-Ellis classification [38,39].

"Rain" geodesics
At least some of the timelike geodesics, the "rain" geodesics corresponding to a test object being dropped from spatial infinity with zero initial velocity and zero angular momentum, are particularly easy to analyze. (These are someimes called ZAMOszero angular momentum observers.) Consider the vector field so V a is a future-pointing timelike vector field with unit norm, a 4-velocity. But then this vector field has zero 4-acceleration: Thus the integral curves of V a are timelike geodesics.
Specifically, the integral curves represented by are timelike geodesics. Integrating two of these equations is trivial so that the time coordinate t can be identified with the proper time of these particular geodesics, and θ ∞ is the original (and permanent) value of the θ coordinate for these particular geodesics.
Furthermore, algebraically one has so these particular geodesics mimic Newtonian infall from spatial infinity with initial velocity zero.
Finally note that which is easily integrated to yield Here φ ∞ is the initial value of the φ coordinate (at r = ∞) for these particular geodesics. Note the particularly clean and simple way in which rotation of the source causes these "rain" geodesics to be deflected.

On-axis geodesics
Working on-axis we have either θ = 0 or θ = π, andθ = 0. Working on-axis we can, without loss of generality, also chooseφ = 0. Then we need only consider the t-r plane, and the specific variant of the Lense-Thirring metric that we are interested in effectively reduces to That is, we effectively have This observation is enough to guarantee that on-axis the geodesics of this Lense-Thirring variant are identical to those for the Painlevé-Gullstrand version of the Schwarzschild spacetime. (For a related discussion, see for instance the discussion by Martel and Poisson in reference [24].) For the on-axis null curves x a (t) = (t, r(t)) we have g ab (dx a /dt) (dx b /dt) = 0 implying That is, for on-axis null curves (as expected for a black hole) we have For on-axis timelike geodesics we parameterize by proper time x a (τ ) = (t(τ ), r(τ )).
As r → ∞ one has which provides a physical interpretation for the parameter k. Indeed is the asymptotic "gamma factor" of the on-axis geodesic (which may be less than unity, and dr dt ∞ might formally be imaginary, if the geodesic is bound). As k → 1 the negative root corresponds to the "rain" geodesic falling in from spatial infinity with zero initial velocity, so that dr/dt = − 2m/r, while the positive root yields This represents an outgoing timelike geodesic with dr dt asymptoting to zero at large distances. Overall, the on-axis geodesics of our variant Lense-Thirring spacetime are quite simple to deal with.

Equatorial geodesics
For equatorial geodesics we set θ = π/2, and consequentlyθ = 0. Working on the equator we need only consider the t-r-φ hypersurface, and our variant of the Lense-Thirring metric effectively reduces to That is, we effectively have and thence

Equatorial null geodesics
For equatorial null geodesics let us parameterize the curve x a (λ) = (t(λ); x i (λ)) by some arbitrary affine parameter λ. Then the null condition g ab (dx a /dt) (dx b /dt) = 0 implies From the time translation and azimuthal Killing vectors, K a = (1; 0, 0, 0) a → (1; 0, 0) a andK a = (0; 0, 0, 1) a → (0, 0, 1) a , we construct the two conserved quantities: Explicitly these yield and Eliminating dt/dλ between these two equations we seẽ This can be solved, either for dφ/dt or for dr/dt, and then substituted back into the null condition (6.4) to yield a quadratic, either for dr/dt or for dφ/dt. These quadratics can be solved, exactly, for dr/dt or for dφ/dt, but the explicit results are messy. Recalling that the Lense-Thirring spacetime is at heart a large-distance approximation, it makes sense to peel off the leading terms in an expansion in terms of inverse powers of r.
For dr/dt one then finds where P (r) and Q(r) are rational polynomials in r that asymptotically satisfy Fully explicit formulae for P (r) and Q(r) can easily be found but are quite messy to write down.
Similarly for dφ/dt one finds dφ dt = 2J r 3 −k kr 2 P (r) ± 2m rk kr 2 Q (r) (6.11) whereP (r) andQ(r) are rational polynomials in r that asymptotically satisfỹ (6.12) Fully explicit formulae forP (r) andQ(r) can easily be found but are quite messy to write down. Overall, while equatorial null geodesics are in principle integrable, they are in practice not entirely tractable.

Equatorial timelike geodesics
For equatorial timelike geodesics the basic principles are quite similar. First let us parameterize the curve x a (τ ) using the proper time parameter. Then the timelike normalization condition From the time translation and azimuthal Killing vectors, K a = (1; 0, 0, 0) a → (1; 0, 0) a andK a = (0; 0, 0, 1) a → (0, 0, 1) a , we construct the two conserved quantities: K a dx a dτ = k; andK a dx a dτ =k. (6.14) Explicitly these yield Eliminating dt/dτ between these two equations we seẽ Eliminating dt/dτ between (6.16) and (6.13) we seẽ Equation (6.17) can be solved, either for dφ/dt or for dr/dt, and then substituted back into the modified timelike normalization condition (6.18) to yield a quadratic, either for dr/dt or for dφ/dt. As for the null geodesics, it is useful to work perturbatively at large r.
Fot dr/dt one then finds where P (r) and Q(r) are rational polynomials in r that asymptotically satisfy Fully explicit formulae for P (r) and Q(r) can easily be found but are quite messy to write down.
Similarly for dφ/dt one finds dφ dt = 2J r 3 −k kr 2 P (r) ± 2m rk kr 2 Q (r) (6.21) whereP (r) andQ(r) are rational polynomials in r that asymptotically satisfỹ Fully explicit formulae forP (r) andQ(r) can easily be found but are quite messy to write down. Overall, while equatorial timelike geodesics are in principle integrable, they are in practice not entirely tractable.

Physically relevant parameters
Note that in SI units For uncollapsed objects (stars, planets) we may proceed by approximating the source as a constant-density rigidly rotating sphere. In the Newtonian approximation Thence in geometrodynamic units we have the approximations Furthermore, (defining r Schwarzschild = 2m in geometrodynamic units), Another useful dimensionless parameter is Using this discussion it is possible to estimate the parameters m, J, a = J/m, a/m = J/m 2 and J/R 2 source for the Earth, Jupiter, Sun, Sagittarius A * , the black hole in M87, and our own Milky Way galaxy. See table 1. Secondly, observe that the Lense-Thirring metric should really only be applied in the region r > R source , and for uncollapsed sources we certainly have J/R 2 source ≪ 1, while even for collapsed sources we still see J/R 2 source 1. The fact that the dimensionless number J/R 2 source ≪ 1 for the Earth, Jupiter, Sun, (and even the Milky Way galaxy), is an indication that Lense-Thirring spacetime is a perfectly good approximation for the gravitational field generated by these sources once one gets beyond the surface of these objects.
These observations are potentially of interest when studying various black hole mimickers [40][41][42][43][44]. (To include a spherically symmetric halo of dark matter in galactic sources, simply replace m → m(r). The existence of the gravitationally dominant dark matter halo is really the only good reason for treating spiral galaxies as approximately spherically symmetric.) 8 Singularity, horizon, ergo-surface, and the like Now recall that the Lense-Thirring metric really only makes sense for r > R source . In fact the Lense-Thirring metric is likely to be a good approximation to the exterior spacetime geometry only for J/r 2 ≪ 1, that is r ≫ √ J. But one can nevertheless pretend to believe the Lense-Thirring metric for r → 0, and investigate the horizon and ergo-surface.
Extrapolating our variant of the Lense-Thirring spacetime down to r = 0 one sees that there is a point curvature singularity at r = 0.
Extrapolating our variant of the Lense-Thirring spacetime down to r = 0 note that ∇ a r becomes timelike for r < 2m. That is, g ab ∇ a r∇ b r = g rr = 1 − 2m r , and this changes sign at r = 2m. Thence for r < 2m any future-pointing timelike vector must satisfy V a ∇ a r < 0. That is, there is a single horizon at the Schwarzschild radius r = 2m, an outer horizon with no accompanying inner horizon.
Extrapolating our variant of the Lense-Thirring spacetime down to r = 0 note that one cannot "stand still" once g tt < 0. That is, the time-translation Killing vector becomes spacelike once g ab K a K b = g tt > 0 corresponding to That is, there is an ergo-surface located at That is, On axis we have r E (θ = 0) = r E (θ = π) = 2m, so that on axis the ergo-surface touches the horizon at r H = 2m. Near the axis, (more precisely for J sin 2 θ/m 2 ≪ 1), the formula for r E (θ) can be perturbatively solved to yield r E (θ) = 2m 1 + J 2 sin 2 θ 4m 4 − and depending on whether J ≪ m 2 or J ≫ m 2 .
Generally we have a quartic to deal with, while there is an exact solution it is so complicated as to be effectively unusable, and the best we can analytically say is to place the simple and tractable lower bounds and r E (θ) > 4 (2m) 4 + 4J 2 sin 2 θ. (8.8) For a tractable upper bound we note r E (θ) = 2m + 4J 2 sin 2 θ r E (θ) 3 < 2m + 4J 2 sin 2 θ (2m) 3 , (8.9) whence r E (θ) < 2m 1 + J 2 sin 2 θ 4m 4 < 2m 1 + J 2 4m 4 . (8.10) Overall, if one does extrapolate our variant of the Lense-Thirring spacetime down to r = 0, one finds a point singularity at r = 0, a horizon at the Schwarzschild radius, and an ergo-surface at r E < 2m 1 + J 2 4m 4 . While such extrapolation is astrophysically inappropriate, it may prove interesting for pedagogical reasons.

Conclusions
What have we learned form this discussion?
First, the specific variant of the Lense-Thirring spacetime given by the metric ds 2 = −dt 2 + dr + 2m/r dt 2 + r 2 dθ 2 + sin 2 θ dφ − 2J r 3 dt 2 (9.1) is a very tractable and quite reasonable model for the spacetime region exterior to rotating stars and planets. Because this metric is in Painlevé-Gullstrand form, the physical interpretation is particularly transparent. Furthermore, with the slight generalization m → m(r), that is, ds 2 = −dt 2 + dr + 2m(r)/r dt 2 + r 2 dθ 2 + sin 2 θ dφ − 2J r 3 dt 2 (9.2) one can accommodate spherically symmetric dark matter halos, so one has a plausible approximation to the gravitational fields of spiral galaxies. Best of all, this specific variant of the Lense-Thirring spacetime is rather easy to work with.