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several galactic low - mass x - ray binaries ( lmxbs ) exhibit quasi - periodic variability ( qpos ) of their x - ray fluxes , with pairs of @xmath0khz frequencies typical in the neutron - star sources @xcite .
kluniak and abramowicz @xcite suggested that these twin khz qpos are a manifestation of non - linear resonance that can occur between modes of oscillation in an accretion disk in strong field einstein s gravity , but not in newton s @xmath1 potential , and pointed out that on this hypothesis the same resonances should be present in black hole systems .
pairs of high frequency qpos should have been present where only single hhz qpos had been reported in microquasars .
such pairs of several hhz frequencies have now been reported in four or five black - hole systems , all in a 3:2 ratio @xcite , substantially strengthening the case for resonance .
our originally suggested @xcite explanation for khz qpos in neutron stars was based on these general properties of non - linear resonance , which seemed to us to correspond to the essential features of the observed twin frequency peaks :    1 .
the frequencies of non - linear oscillations @xmath2 depend on amplitude , and for this reason they may be time dependent and may differ from the fixed eigenvalue frequencies @xmath3 of the system , @xmath4 .
2 .   non - linear resonance may occur over a wide frequency range @xmath5 .
3 .   both ( resonant ) frequencies increase or decrease `` in step '' with each other .
the eigenfrequencies of resonant modes are approximately in the ratio of small integers , e.g. , 2:1 .
these ideas initially received a cool reception ( see author s note in @xcite ) , because it was not generally appreciated that frequency ratios close to 3:2 actually occur for khz qpos in neutron star sources , and no evidence for twin peaks in black hole sources was known at the time .
in addition to the high frequency qpos , features in the power density spectra can be identified at lower frequencies , and at least one frequency @xmath6 was long known to be correlated with @xmath7 , one of the khz / hhz frequencies in neutron - star / black - hole systems @xcite .
quasi - periodic modulations of the flux ( dwarf nova oscillations , dnos ) were first discovered in cataclysmic variables ( white dwarfs ) and these are analogous in many respects to the qpos in lmxbs @xcite .
however , unlike the khz / hhz qpos , the highest frequency dnos do not come in pairs .
this is consistent with the idea that the high frequency pairs arise in accretion disks only in strong gravity .
the relativistic resonance model of black hole qpos is based on fundamental features of strong gravity .
today , it is motivated by observations that sharply illuminate the physical nature of qpos :    1 .
the correlation @xmath8 between low and high frequency qpos in black hole , neutron stars , and white dwarf sources extending over six orders of magnitude @xcite , proves that in general the qpos are a hydrodynamic phenomenon , and can not be attributed to mere kinematic effects , such as doppler modulation of emission from isolated bright spots .
( @xmath6 may be the `` ninth wave '' @xcite . ) 2 .
the frequencies of twin peak hhz qpos in microquasars seem to scale with mass @xcite , @xmath9 ( figure 1 ) .
if true , this would prove their relativistic origin .
3 .   in all four microquasars with twin peak hhz qpo pairs ,
@xmath10 ( table 1 ) .
although suggestive of a resonance , the ratio 3/2 could also be a signature of overtones ( flute modes ) @xcite , or of higher modes of an mhd instability @xcite at a ` transition radius ' @xmath11 in the innermost part of the disk , which excites quasi periodic oscillations with mode frequencies @xmath12 ( in contrast with the resonance model , neither of these two models predicts @xmath13 scaling , without making some _ ad hoc _ assumptions ) .
however , there are additional properties of non - linear resonances which may help in their identification . in a non - linear resonance combination frequencies , e.g. ,
@xmath14 , and subharmonic frequencies may be present @xcite , e.g. , @xmath15 .
our resonance model may also be applied to twin peak qpo sources in neutron stars @xcite . in refs .
@xcite , and in these proceedings @xcite , we discuss a resonance in an accretion disk or torus excited by an external forcing by a millisecond pulsar . a similar
forcing is crucially important in a different , non - relativistic resonance model suggested by titarchuk @xcite .
consider a black hole , angular momentum with @xmath16 .
we use boyer - lindquist coordinates , @xmath17 , and rescale the radius with @xmath18 . ] with the mass @xmath19 and angular momentum @xmath20 . inside thin ,
almost keplerian accretion disks , matter spirals down the central black hole along stream lines that are located almost on the equatorial plane @xmath21 , and that locally differ only slightly from a family of concentric circles @xmath22 . the small deviations , @xmath23 , @xmath24 are governed , with accuracy to linear terms , by    @xmath25    here , the dot denotes a time derivative . for purely keplerian ( free )
motion @xmath26 , @xmath27 and the above equations describe two uncoupled harmonic oscillators with the eigenfrequencies @xmath28 , @xmath29 shown together with the keplerian orbital frequency , @xmath30 , in figure 2 for a non - rotating black hole , and in figure 3 for a moderately rotating one .    [ !
.the four microquasars in which two hhz qpos are observed .
they all have 3:2 ratio of frequencies .
source of data : @xcite [ cols= " < , < , < " , ]
the @xmath13 scaling of the twin peak qpos frequencies with the 3:2 ratio , was proposed by abramowicz , kluniak , mcclintock & remillard @xcite as a method for estimating black hole masses in agns and the recently discovered ultraluminous x - ray sources ( ulxs ) , based on mirabel s analogy between microquasars in our galaxy and distant quasars @xcite .
indeed , if the analogy is also valid for accretion disk oscillations , then discovering in ulxs the twin peak qpos frequencies with the 3:2 ratio , would resolve the controversy about their mass : if ulxs black holes have the same masses as microquasars , the frequencies will be @xmath31hz ; if ulxs black holes are @xmath32 times more massive , the frequencies will be @xmath33hz instead .
we thank gabriel trk for preparing all the figures and other technical help .
all figures , except figure 15 , are taken from abramowicz , kluniak , stuchlik , & trk @xcite .
figure 15 is taken from our work with r.  remillard & j.  mcclintock @xcite .
most of the work reported here was done at the silesian university of opava , in the czech republic , and at the astrophysical fluids facility in leicester university , england .
it was supported by the european commission grant _ access to research infrastructure action of the improving human potential program _ and by the polish kbn grant 2p03d01424 .
we thank j.  almergren , m.  bursa , j.  horak , v.  karas , f.  lamb , j .-
lasota , w.  lee , c.  mauche , j.  mcclintock , r.  remillard , l.  rezzolla , j.  schnittman , e.  spiegel , p.  rebusco , m.  van  der  klis , and r.  wagoner , for their sugestions and comments on this presentation .
kluniak w. , abramowicz m.a . ,
2000 , phys .
( submitted ) , astro - ph/0105057 kluniak w. , abramowicz m.a . , 2001 , acta phys .
polon . , b32 , 3605 , available at + http://th-www.if.uj.edu.pl/acta/vol32/abs/v32p3605.htm                            kluniak w. , abramowicz m.a . ,
lee w.h . , 2003 ,
_ high - frequency qpos as a problem in physics : non - linear resonance _ , these proceedings : x - ray timing 2003 : rossi and beyond , ed .
p. kaaret , f. k. lamb , & j. h. swank ( melville , ny : american institute of physics ) ,
in all the microquasars with two hhz qpos , the ratio of the frequencies is 3:2 , supporting our suggestion that a non - linear resonance between two modes of oscillation in the accretion disk plays a role in exciting the observed modulations of the x - ray flux . 
 we discuss the evidence in favor of this interpretation , and we relate the black hole spin to the frequencies expected for various types of resonances that may occur in nearly keplerian disks in strong gravity . for those microquasars where the mass of the central x - ray source is known , the black hole spin can be deduced from a comparison of the observed and expected frequencies .     
 address = department of astrophysics , chalmers university , gteborg , sweden     address = copernicus astronomical center , ul . 
 bartycka 18 , 00 - 716 warszawa , poland     address = institute of astronomy , zielona gra university , ul . 
 lubuska 2 , 65 - 265 zielona gra , poland