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the gravitomagnetic clock effect ( synthetically gclock effect ) is in a sense a phenomenon announced since the early days of relativity@xcite .
it is due to the fact that the proper time measured by a clock is indeed the four dimensional  length  of the world line of the clock between two given events : different allowed four dimensional paths between the same pair of events correspond to different proper time intervals and produce desynchronization of initially synchronous clocks .    when treating rotating systems ( and clocks )
an additional feature one finds is that a global splitting between space and time is no more possible : time on a rotating platform becomes polytropic@xcite in the sense that the space hypersurface of a rotating observer is a helicoid and it intercepts the world line of the observer in an infinite number of events : these are all nominally  simultaneous  though corresponding to different proper times of the observer .
this fact does not really affect the nature and geometric interpretation of the clock effect .
the revolving  clocks  used to measure proper times may in fact be various types of particles , such as electrons@xcite , neutrons@xcite , cooper pairs@xcite , calcium atoms@xcite , which have been used to test the sagnac effect for quantum objects .
of course one can use light as a  clock  finding an asymmetry between pro- and retrograde rotation manifested in a phase difference rather than a difference in proper times , now identically null .
this is the very sagnac effect , known since 1913 or even before and named after g. sagnac@xcite , who interpreted it with an antirelativistic and  etherial 
( in the sense of the ether theory ) attitude .
the sagnac effect is nothing else than a specialization of what we are speaking about since the beginning of this section and earned a whole literature for itself during many decades@xcite ; the simplest and most complete description of it is indeed the geometrical one@xcite@xcite .
an effect with actual clocks , and in particular the synchrony defect due to the time polytropy , has been evidenced for the first time with satellites in 1985@xcite .
the introduction of nonspinning or spinning masses in the empty space time does not in principle modify the situation , but for computational complications . the latter case ( spinning sources of gravity )
is however the one to which the expression 
gravitomagnetic clock effect  has been reserved .
the name comes from the fact that , in weak field approximation , the gravitational field may be decomposed into a gravitoelectric ( radial ) field and a gravitomagnetic ( solenoidal ) field in analogy with electromagnetism : the gclock effect is determined by the gravitomagnetic component .
this description and treatment has been particularly emphasized by mashhoon and others@xcite @xcite , but a general treatment is in principle rather simple and can be given before any approximation@xcite .
the importance of the gclock effect is in the hope that it can be used to test the influence of the angular momentum of the source on the gravitational field .
in fact the revolution time of a freely orbiting clock on a circular trajectory around a spinning mass is different according to whether the orbit is prograde or retrograde .
the difference between the two periods expressed as the times needed to recover a fixed azimuth in the inertial frame of a distant observer is@xcite : @xmath0 where @xmath1 is a length given by the ratio between the angular momentum of the source @xmath2 and its mass @xmath3 multiplied by @xmath4 : @xmath5 .
the result ( [ assoluto ] ) is true for any axisymmetric metric with an @xmath1 value independent from time@xcite .
the independence of ( [ assoluto ] ) from the radius of the orbit is a remarkable feature ; furthermore the numerical value of @xmath6 for the earth is rather promising being in the order of @xmath7 s.    when adopting the viewpoint of an observer in the vicinity of the central mass the situation worsens ; then one should use proper times of the clocks and refer for instance to specific events , such as the conjunctions of counterorbiting objects . in that case the time difference on the clocks around the earth , @xmath8 , is in the order of @xmath9 s @xcite@xcite .
besides the actual numbers that should be measured ( which , at least in the case of @xmath6 , are indeed big ) , a real experiment around the earth encounters a series of difficulties not easy to be surmounted@xcite @xcite , one of them is the stability and coincidence between opposite orbits . considering this specific problem
electromagnetic waves come again on stage as probes more interesting than actual clocks .
in fact a single ring of orbiting mirrors could be used to reflect at the same instant both prograde and retrograde  light 
beams in a sort of sagnac experiment in space . the influence of gravity on the issue of such an experiment can be worked out as a general relativistic correction of the classical effect@xcite , but the problem can be better solved in principle and in general using the method outlined in ref .
this is precisely the aim of this paper ; in sect . ii the essence of the method is recalled ; in sect .
iii the specific case of the sagnac effect is treated ; sect .
iv contains the approximated results valid for the earth environment ; finally in sect .
v some conclusions will be drawn .
the method proposed is based on the fact that in any axially symmetric and stationary space time any constant @xmath10 and @xmath11 world line is drawn on the surface of a cylinder which is indeed flat , no matter what the global curvature is : on that surface simple minkowskian geometry works . in particular when the coordinate angular velocity @xmath12 is fixed the world line on the cylinder s surface is a helix , which becomes a straight line in the development of the cylinder in a plane .
the situation is most simply described graphically : see fig .
1 , which shows the opened cylindrical world tube .
the world lines of steadily rotating objects are helices which in turn become straight lines when opening the cylinder on a plane . when a line reaches the boundary of the cut at @xmath13 or @xmath14 it bounces back or forth by @xmath15 continuing on the other side .
considering a pair of objects revolving in opposite directions with the same coordinate speed , their world lines are symmetric with respect to a static observer at the origin ( vertical straight line there ) and cross each other again at the same event for such observer : the revolution times both in the objects proper times and in the observer s time are exactly the same for both .
if however the observer is in turn steadily rotating his world line is an oblique straight line and the intersecting events with the world lines of the objects do not coincide any more : the revolution times measured by the observer differ from each other and so do the proper times of the two objects .
the interval between _ a _ and _  b , _ in fig . 1 _ ,
_ is then proportional to the proper time lapse ( for the observer ) between the completions of one revolution started simultaneously by the two probes .
this description is valid for light beams too : in this case the @xmath16 interval is an actual measure of the sagnac effect .
when the central body possesses an angular momentum , the equivalent in its vicinity of the static inertial observers of the previous case are the so called locally non rotating observers ( lnro)@xcite , i. e. the observers to whom radially infalling matter appears locally as non rotating .
the lnro s themselves are indeed rotating as seen by a distant inertial observer ; their coordinate angular speed is : @xmath17 ( the @xmath18 s are elements of the metric and the use of polar coordinates is understood ) . in practice
the considerations about the flatness of the cylindrical surface containing the world lines of constant @xmath10 and @xmath11 remain unchanged , but now the world lines of co - rotating and counter - rotating light rays have a different inclination to cope with the role of an lnro and his skew world line ; in other words after leaving the lnro at @xmath19 the two beams must cross each other again on the ( oblique ) world line of the lnro , in order it to be equivalent to the static observer of rotation free space times .
graphically the situation is the one shown in fig .
the entity of the sagnac effect is measured by the interval between @xmath20 and @xmath21 which is clearly different from the one between @xmath22 and @xmath23 in fig . 1 , depending on the value of the parameter @xmath1 which is hidden in the @xmath24 element of the metric tensor .
what is visually determined by the simple graphic treatment of the previous section can be put in analytical form by the use of minkowskian geometry in two dimensions .
let us restrict our study to light rays and the sagnac effect in axisymmetric metrics .
events @xmath22 and @xmath23 in fig . 1 ( @xmath25 and @xmath21 in fig .
2 ) are localized as the intersections of two straight lines : the world line of the observer and the world line of a light beam . when an angular momentum is absent
the former s equation is @xmath26 , where @xmath27 is the observer s angular coordinate speed .
the equation of the world line of a light ray constrained to move along a circular trajectory is in turn @xmath28 for the counter - rotating beam and @xmath29 for the co - rotating one ; the equations reproduce the lines respectively ascending to the left through @xmath1 and ascending to the right through @xmath30 in fig .
1 . the coordinate angular speed of light for a circular path in a plane orthogonal to the symmetry axis ( let us say it is the equatorial plane of the system )
is @xmath31 ; this comes from a null interval at constant @xmath10 and @xmath11 , i.e. from @xmath32 , posing @xmath33 .
the intersections between the straight world lines of the observer and the beams identify points @xmath22 ( counter - rotating beam ) and @xmath23 ( corotating beam ) with their @xmath34 and @xmath35 coordinates : @xmath36 the @xmath37 ( @xmath38 ) sign corresponds to event @xmath22 ( @xmath23 ) . the proper time corresponding to the interval @xmath39 ( @xmath40 ) in fig .
1 is in practice the length ( divided by @xmath4 ) of the hypotenuse of a right - angled triangle in a two dimensional flat minkowski space time , whose other sides are @xmath41 and @xmath42 .
the explicit result is : @xmath43 and the sagnac delay turns out to be @xmath44 the @xmath45 term has been eliminated thanks to the explicit expression of @xmath46 .
when @xmath47 the coordinate angular speed of light on a circular path is obtained from @xmath48 .
there are two solutions : @xmath49 ( @xmath37 is for prograde motion , @xmath38 is for retrograde motion ) .
now the intersection method leads to : @xmath50    the proper observer s intervals of time are obtained as before , but for the fact that now we have to use the equivalent of carnot s theorem of euclidean geometry rather than the equivalent of pythagora s : @xmath51    finally the sagnac delay is : @xmath52 the parameter @xmath53 has been eliminated using the explicit expression for the coordinate angular speed of an lnro .
if now we pose @xmath54 , we consistently verify that it is : @xmath55 a truly inertial observer ( @xmath56 ) would find : @xmath57    formula ( [ deltatau ] ) treats @xmath27 and @xmath10 as independent parameters ; the situation changes when the observer is freely orbiting . for a circular geodesic trajectory in the equatorial plane
there are two solutions for @xmath27 @xcite ( again @xmath37 is pro- and @xmath38 is retrograde motion of the observer ) : @xmath58 commas denote ordinary partial differentiation .. correspondingly ( [ deltatau ] ) becomes : @xmath59
considering the situation in the surroundings of our planet the appropriate metric is the one corresponding to a weak axisymmetric field .
actually this means that : @xmath60    consequently it is : @xmath61    now using ( [ deltatau ] ) we obtain :    @xmath62    for not too high observer s velocities and radii of the order of the size of the earth ( [ detauearth ] ) becomes : @xmath63    one easily recognizes , in the order , the traditional sagnac term , then a purely special relativistic correction , a 
correction and finally an angular momentum correction which is independent from the velocity of the observer .
when the observer is freely orbiting on a circular trajectory the starting point is formula ( [ completo ] ) ; developing and keeping only the lowest order terms up to the one containing the angular momentum , the result is @xmath64 now summing the time differences for the two rotation directions we obtain @xmath65    the order of magnitude of this quantity is @xmath9 s.
we have given a general geometric treatment of the problem of the phase shift of electromagnetic waves moving in a closed circular path about a massive spinning body .
applying the general methodology to the terrestrial environment reproduces approximate results obtainable by various alternative techniques . in terms of time differences accumulated during round trips around the earth
the numbers remain extremely small and partly smaller than some values typical of the gclock effect .
the idea of using electromagnetic waves in some orbital sagnac type experiment deserves however some attention because of a few advantages light has .    to begin with ,
if a chain of orbiting mirrors or transponders are considered it is possible to use them simultaneously for prograde and retrograde rays , thus ensuring the coincidence of the two trajectories and the significance of the time of flight difference .
next we can notice that , even for an  experimental apparatus  as big as the orbit of a satellite , the time of flight of light for a round trip is @xmath66 @xmath67 s. in that time the observer on a spacecraft could move for a few hundred meters at most and an observer on the surface of the earth would move much less than that ; this fact in principle allows for experiments at different values of the equivalent angular velocity @xmath27 , to be maintained for a few seconds at most .
in fact in the time delay ( [ approssi ] ) three terms depend on @xmath68 and one is independent from it : in principle determining the values for positive or negative @xmath27 close to zero could allow for the detection of the invariant @xmath1 term .
the scheme one can imagine is an observer on the earth moving westward at variable linear speeds of the order of @xmath69 m / s and sending and receiving signals to and from a ring of orbiting mirrors .
finally , varying the wavelength of the waves could in principle allow to tune the phase difference originated by the @xmath1 term , making it correspond to a predefined number or fraction of interference fringes and evidencing its presence with respect to the special relativistic or purely gravitational shifts .
new techniques to evaluate the clock effect using light are described . 
 these are based on the flatness of the cylindrical surface containing the world lines of the rays constrained to move on circular trajectories about a spinning mass . 
 the effect of the angular momentum of the source is manifested in the fact that inertial observers must be replaced by local non rotating observers . starting from this an exact formula for circular trajectories 
 is found . 
 numerical estimates for the earth environment show that light would be a better probe than actual clocks to evidence the angular momentum influence . 
 the advantages of light in connection with some principle experiments are shortly reviewed .