Measuring the Whirling of Spacetime

Transcription

1 Measuring the Whirling of Spacetime Lecture series on Experimental Gravity (revised version) Kostas Glampedakis

2 Prologue: does spin gravitate? M 1 M 2 System I: F = GM 1M 2 r 2 J 1 J 2 System II: M 1? M 2 In Newtonian gravity these systems share the same gravitational force (ignoring the rotational deformation). This is not true anymore in General Relativity

3 Newtonian vs General Relativistic gravity Newtonian field equations GR field equations 2 =4 G G ab = 8 G c 4 T ab Source: mass density Gravitational field: scalar Source: energy-momentum tensor (includes mass densities/currents) Gravitational field: metric tensor g ab

4 Frame dragging The freely falling test body (could be a photon) has zero angular momentum. The gravitating source has spin J (pointing towards you) and the spacetime is described by the line element: ds 2 = g tt dt 2 + g rr dr 2 +2g 't d'dt + g d 2 + g '' d' 2 The test body acquires an angular velocity: = d dt = g t g 2GJ c 2 r 3 The small body is dragged along the direction of the source s rotation: also known as the Lense-Thirring effect. Frame dragging is closely related related to the mixed metric component. g t

5 So... spin does gravitate In the previous slide we explained frame-dragging using a geometrical picture, i.e. in terms of the metric and the spacetime line element. ds 2 = g tt dt 2 + g rr dr 2 +2g 't d'dt + g d 2 + g '' d' 2 An alternative (and equivalent) approach is to view frame-dragging as a contribution to the total gravitational force. This is a good choice when the gravitational field is weak and one can add post-newtonian corrections to the familiar Newtonian force. This approach also shares some similarities with electromagnetic theory, where currents and dipole electric/magnetic moments are interactive. Based on this formal similarity we will next talk about gravitomagnetism.

6 Gravitomagnetism Consider again the big body of mass M and spin J and in the limit of weak gravity: GM/rc 2 1 At leading order we have the usual Newtonian potential: = GM r The leading order potential due to the spin of the source is g it = 2 c Ai, The gravitomagnetic field is: A = G cr 3 ( J r ) B = A = GJ cr 3 3(Ĵ ˆr)ˆr Ĵ Gravitomagnetic force on a particle of mass m: F gm = 2m c v B

7 Gyroscopes and gravity The gravitomagnetic character of GR also means that the spin S of a test-body (the gyroscope ) will itself couple to a given background gravitational field. M,J S Spin is forced to precess: t S = S p Precession frequency [Schiff (1960)]: (Figure credit: L. Barack) p = 3GM GJ 2c 2 ( r v )+ r3 c 2 r 3 3(Ĵ ˆr) Ĵ Mass-spin coupling. This is the geodetic effect. Spin-spin coupling. This is frame-dragging.

8 Gyroscopes and gravity The gravitomagnetic character of GR also means that the spin S of a test-body (the gyroscope ) will itself couple to a given background gravitational field. M,J S Spin is forced to precess: t S = S p Precession frequency [Schiff (1960)]: (Figure credit: L. Barack) p = 3GM GJ 2c 2 ( r v )+ r3 c 2 r 3 3(Ĵ ˆr) Ĵ Mass-spin coupling. This is the geodetic effect. Spin-spin coupling. This is frame-dragging.

9 Measuring the whirling of spacetime As any other theoretical prediction, the notion of a gravitating spin must be tested experimentally. Mercury s orbital motion provided the first evidence of another GR effect (perihelion precession). Unfortunately, the expected frame-dragging on Mercury is too small. A more viable method is high-precision tracking of objects orbiting Earth. The geodetic effect has been measured at the level of ~ 1% accuracy, using lunar laser ranging data (and navigational data from the Cassini probe). Frame-dragging has been measured, to a ~ % accuracy, using the ultra-precise laser tracking system of the LAGEOS satellite constellation.

10 The Gravity Probe B mission The world s most advanced drag-free gyroscopes in polar orbit (~650 km), completing over 5000 revolutions per year. Gyro spin-axis precession measured with respect to some fixed star. GP-B launched in 2004, at a cost ~ $ 700 million (and 90 related PhDs completed!). Data collected between 28/8/ /8/2005. Mission objective: to measure the gyro spin-axis precession induced by the coupling with Earth s mass (geodetic effect) and spin (frame-dragging) to unprecedented accuracy. The effect is tiny: p A NASA mission

11 Gravity Probe B: basic theory GP-B s gyros are sensitive to both contributions to the precession frequency. p The geodetic effect : geod = 3GM 2c 2 ( r v ) r3 The frame dragging : fd = GJ c 2 r 3 3(Ĵ ˆr)ˆr Ĵ For a gyro in a polar orbit: ( note that. Why? ) fd geod fd geod

12 The Gravity Probe B in action!

13 GP-B: a partial success The GP-B mission was designed to measure the geodetic & frame-dragging effect with < 1% accuracy. Many things can go wrong in such delicate experiments, especially when there is no possibility for post-launch repairs! Unfortunately, in the case of GP-B the gyroscopes proved to be more noisy than expected.

14 GP-B: gyro precession history

15 GP-B: final results Geodetic effect s precision: 0.28% Frame-dragging s precision: 19%

16 GP-B: another verification of GR!

17 Whirling to the extreme: Kerr black holes Black holes have become part of everyday astrophysics. Supermassive BHs ( M M ) reside in most galactic nuclei (including our own Galaxy). Lighter BHs ( M 10M ) can be found in Galactic accreting systems. The famous Kerr solution of the GR equations provides a precise description of astrophysical spinning BHs. It depends on just two (!) parameters: the mass M and spin J. Existing astrophysical data suggest rapidly spinning BHs. The spin might even power enormous jets. The BH singularity is cloaked by an event horizon provided the spin is below a maximum: a J/M M

18 Welcome to the ergoregion! Matter and radiation in the vicinity of a Kerr hole experience strong frame dragging. Remaining at a fixed position is not possible within the radius (given in G=c=1 units) r erg ( )=M + M 2 a 2 cos 2 The event horizon is at: r H = M + p M 2 a 2 r H <r<r erg In the ergoregion everything has to rotate with the black hole; frame-dragging rules!

19 The future: gravitational waves Observation of gravitational waves from small BHs orbiting around supermassive ones should provide high-precision evidence for frame dragging and the existence of ergoregions. (Figure credit: L. Barack) (Figure credit: S.A. Hughes)

20 Epilogue: analogies...and I cherish more than anything else the Analogies, my most trustworthy masters. - Johannes Kepler It wouldn t be too surprising if your mental picture of frame-dragging around BHs looks like this picture. How far can we push this analogy?

21 Acoustic BHs: frame-dragging in your bathtub Remarkably, a fluid flow like the one in a draining bathtub can be mathematically described in terms of an effective acoustic metric (Unruh 1981). This looks like: ds 2 = c 2 sdt 2 + dr 2 A r dt + rd' 2 B r dt (The sound speed plays the role of c) Sound waves propagating in such flow can be trapped behind an acoustic horizon and/or frame-dragged (for this to happen ) v flow >c s r H = A c s, r erg = p A2 + B 2 c s acoustic BH (or dumb hole )

22 Summer assignments

23 Assignment I: motion of spinning particles Search the literature for the equations of motion of a spinning particle in GR. Elementary particles (electrons etc.) have intrinsic (quantum) spin. Does this spin couple to gravity? Assuming a uniform gravitational field, does a spinning particle fall at the same rate as a non-spinning one? What if the field is non-uniform? g

24 Assignment II: ergoregions and superradiance Do rotating compact stars (for example neutron stars) have ergoregions? The Kerr black hole ergoregion is the arena where a phenomenon called superradiance takes place. Superradiance is a process by which the BH s rotational energy can be tapped. It is also closely related to the more familiar Penrose process. Try to learn more about it!