ap-3-10.dvi Acta Polytechnica Vol. 50 No. 3/2010 Gravitomagnetism J. Bičák In the Introduction to our talk,we explaineda sim- ple thought experiment to indicate that gravitomag- netism has to arise if some basic relations, such as the dependence of inertial mass on velocity given by the standard special-relativistic formula, the dependence of active gravity on the inertial mass, and the Lorentz invariance, take place: the effect on a test mass be- tween two uniformly moving parallel streams of linear distributions ofmasses was considered from the frame inwhich they bothmovewith the samevelocity but in opposite directions, and from the frame in which one of the streams is at rest (see, e.g., [1] for details). We then summarized the basic ideas and re- sults of the recent experiment Gravity Probe B, in which four gyroscopes placed in a satellite orbiting the Earth measure the gravitomagnetic field caused by Earth’s rotation. After a long period of prepa- ration, observation and data analysis it has now been concluded that “the combined four-gyro re- sult gives a statistical uncertainty of 14% (∼ 5 marcsec/yr) for frame-dragging” (in other words, for the presence of gravitomagnetic effects) – see http://einstein.stanford.edu/highlights/status1.html. In the main part of the talk, we summarized our recent works with Joseph Katz from the Hebrew Uni- versity in JerusalemandDonaldLynden-Bell from the Institute ofAstronomyof theUniversity ofCambridge onMach’sprinciple andgravitomagneticeffects ingen- eral relativity and cosmology. Einstein was strongly influenced by Mach’s idea that the inertia of a particle here and now arises as a consequence of its interaction with other particles in the universe. What do we understand by “Mach’s principle” today? In our comprehensivework [2] sum- marizing a number of our preceding contributions to gravitomagnetic effects andMach’s principle, we start out from the general formulation of Mach’s principle by Hermann Bondi in his Cosmology [3]: Local iner- tial frames are determined by the distribution of en- ergy and momentum in the Universe by suitable aver- ages. Inmathematical terms, we investigate the valid- ity of such a formulation for the case of general linear perturbations of standard “background” models, i.e., of isotropic and homogeneous cosmologicalmodels de- scribedby theFriedmann-Lemâıtre-Robertson-Walker solutions of Einstein’s field equations. In particular, we focus on thoseofEinstein’s equations for linearper- turbations which represent constraints on initial data. In suitable coordinates (“gauges”), these constraints are represented by elliptic equations which connect the distribution of matter and energy described by an energy-momentum tensor of physical matter and fields with the geometry described by the metric ten- sor and its derivatives. In these gauges, the local in- ertial frames are determined instantaneously by the distribution of matter and energy. One has to realize that the physical effects associatedwith Mach’s ideas, e.g. the “dragging of inertial frames” (“gravitomag- netic effects”) have a global character and require spe- cial coordinate systems, special gauges. As was noted by Dieter Brill in the discussion during the conference on various aspects of Mach’s principle (on the basis of which a very valuable book [4] arose), “Mach’s prin- ciple can show the way to give physical meaning to quantities which are usually considered as coordinate dependent.” In more recent papers [5, 6], we show that within General Relativity, any general statement of Mach’s principle that attributes all dragging of inertial frames solely to the distribution of energy and momentum of matter as the originof inertia is false: Gravitomagnetic effects are also caused by gravitational waves. To show this, we investigatewaveswhich do not depend on one spatial coordinate (say the “z−coordinate”). We find that there is an almost flat cylindrical region near the z-axis of a revolving gravitational wave pulse (which inevitably has no “physical” energy-momentum) and demonstrate that the inertial frame in the cylindrical interior rotates relative to the inertial frame at great distances. Our aim was to produce a nice clean ex- ample of the rotation of the inertial frame in an al- most flat region surrounded by rotating gravitational waves. An extreme example of inertia due to gravita- tionalwaves alone is providedbyGowdy’s universe [7], a closed world that contains nothing except gravita- tionalwaves. Oneofourultimate aims is todiscuss the meaning of Mach’s principle in fully nonlinear general relativity, and particularly its application to such sys- tems as Gowdy’s universe. Another question of great interest is what bearing Mach’s principle has on the existence of dark energy. References [1] Schutz, B.: Gravity from the ground up (an in- This lecture, here briefly summarized, was dedicated to Jiří Niederle, whose broad interests and contributions in mathematical physics also include gravity theories. 69 Acta Polytechnica Vol. 50 No. 3/2010 troductory guide to gravity and general relativ- ity), CambridgeUniversityPress,Cambridge2003, XXVI+462 pages. [2] Bičák, J., Katz, J., Lynden-Bell, D.: Cosmological perturbation theory, instantaneous gauges and lo- cal inertial frames, Phys. Rev. D76 (2007) 063501, 31 pages. [3] Bondi, H.: Cosmology, Cambridge University Press, Cambridge 1961, VII+182 pages. [4] Barbour, J., Pfister, H. (editors): Mach’s Prin- ciple: From Newton’s Bucket to Quantum Gravity, Birkhauser, Boston–Basel–Berlin 1995, VII+536 pages. [5] Bičák, J., Katz, J., Lynden-Bell, D.: Gravitational waves and dragging effects, Class. Quantum Grav. 25 (2008) 165018, 19 pages. [6] Lynden-Bell,D., Bičák, J., Katz, J.: Inertial frame rotation induced by rotating gravitational waves, Class. QuantumGrav.25 (2008) 165017, 13 pages. [7] Gowdy, R. H.: Gravitational waves in closed uni- verses, Phys. Rev. Lett. 27 (1971), p. 826–829. Jiří Bičák Institute of Theoretical Physics Charles University V Holešovičkách 2, 180 00 Prague 8 70