NIKHEF Gravity Geometry and Physics JW van Holten NIKHEF PO Box DB Amsterdam NL c Lectures presented at the Summerschool of Theoretical Physics Saalburg Germany Sept Abstract Geometry is present in physics at many levels most prominently in the theory of gravity In these introductory lectures geometrical concepts like manifolds geo desics curvature and top ology are introduced as a to ol to describ e and interpret physical phenomena in spacetime including the gravitational eld itself The fo cus is not only on p ossible static structures eg black holes but also on dynamical eects like waves and traveling domain walls Although quantum gravity itself is outside the scop e of these lectures it is briey discussed how geometry can b e used even in that context to characterise congurations dominating the pathintegral in various circumstances including typical quantum pro cesses like vacuum tunneling Many of these geometrical metho ds can also b e used in other branches of physics by linking the dynamics of a system with the geometry of its conguration space some examples are mentioned Contents Gravity and Geometry The gravitational force Fields Geometrical interpretation of gravity Curvature The Einstein equations The action principle Geo desics Curves and geo desics Canonical formulation Action principles Symmetries and Killing vectors Phasespace symmetries and conservation laws Example the rigid rotor Dynamics of spacetime Classical solutions of the gravitational eld equations Plane fronted waves Nature of the spacetime Scattering of test particles Symmetry breaking as a source of gravitational waves Coupling to the electromagnetic eld Black holes Horizons The Schwarzschild solution Discussion The interior of the Schwarzschild sphere Geo desics Extended Schwarzschild geometry Charged black holes Spinning black holes i The Kerr singularity Blackholes and thermo dynamics Topological invariants and selfduality Topology and top ological invariants Topological invariants in gravity Selfdual solutions of the Einstein equations RungeLenz vector and Yano tensors ii Chapter Gravity and Geometry The gravitational force Gravity is the most universal force in nature As far as we can tell from ob servations and exp eriments every ob ject every particle in the universe attracts any other one by a force prop ortional to its mass For slow moving b o dies at large distances this is a central force inversely prop ortional to the square of the distance As the action is recipro cal and since according to Newton action and reaction forces are equal in magnitude the expression for the gravitational force b etween two ob jects of mass M and M at a distance R is then determined to have the unique form M M F G R The constant of prop ortionality Newtons constant of gravity has dimensions of acceleration p er unit of mass times an area Therefore its numerical value obviously dep ends on the choice of units In the MKS system this is G m kg s It is also p ossible and sometimes convenient to x the unit of mass in such a way that Newtons constant has the numerical value G In the natural system of units in which also the velocity of light and Plancks constant are unity c h this unit of mass is the Planck mass m P q m h cG kg P GeVc Newtons law of gravity is valid for any two massive b o dies as long as they are far apart and do not move to o fast with resp ect to one another In particular it describ es the motions of celestial b o dies like the mo on circling the earth or the planets orbiting the sun as well as those of terrestrial ob jects like apples falling from a tree or canon balls in free ight Ever since Newton this unication of celestial and terrestial mechanics has continued to impress p eople and has had a tremendous impact on our view of the universe It is the origin and basis for the b elief in the general validity of physical laws indep endent of time and place Fields Although as a force gravity is universal Newtons law itself has only limited validity Like Coulombs law for the electrostatic force b etween two xed charges Newtons law holds strictly sp eaking only for static forces b etween b o dies at rest Moreover and unlike the electric forces there are mo dications at smaller nite distances which can b e observed exp erimentally For example if the gravitational force would have a pure R dep endence the orbits of particles around a very heavy central b o dy would b e conic sections ellipses parab olas or hyperb olas dep ending on the energy and angular mo mentum in accordance with Keplers laws The observation of an excess in the precession of the p erihelion of the orbit of Mercury around the sun by LeVerrier in and improved by Newcomb in was one of the rst clear indica tions that this is actually not the case and that the gravitational force is more complicated The exact form of the gravitational forces exerted by moving b o dies is a problem with many similarities to the analogous problem in electro dynamics The understanding of electro dynamical phenomena greatly improved with the introduction of the concept of lo cal eld of force This concept refers to the following characteristics of electro dynamical forces the inuence of electric charges and currents and of magnetic p oles extends throughout empty space the force on a standard test charge or test magnet at any given time dep ends on its lo cation with resp ect to the source of the eld and the relative state of motion changes in the sources of the elds give rise to changes in the force on test ob jects at a later time dep ending on the distance the sp eed of propagation of disturbances in empty space is nite In the case of electro dynamics this sp eed turned out to b e a universal constant the sp eed of light c One of the most striking consequences of these prop erties which follow directly from the mathematical description of the elds as expressed by Maxwells equations is the existence of electromagnetic waves lo cal variations in the elds which transp ort energy momentum and angular momentum over large distances through empty space Maxwells predictions were magnicently veried in the exp eriments of Hertz conrming the reality and characteristics of electromagnetic waves From these results it b ecame clear for example that light is just a sp ecial variety of such waves Therefore optical and electromagnetic phenomena have a common origin and can b e describ ed in a single theoretical framework The concept of eld has found other applications in physics for example in the phenomenological description of uids and gases Treating them as continua one can describ e the lo cal pressure and temp erature as elds with lo cal variations in these quantities propagating through the medium as sound waves or heat waves As in electro dynamics also in gravity the concept of eld has taken a central place What Maxwell achieved for electromagnetism was accomplished by Ein stein in the case of gravity to obtain a complete description of the forces in terms of space and timedep endent elds This eld theory is known as the general theory of relativity One of the most remarkable asp ects of this theory is that it provides an interpretation of gravitational phenomena in terms of the geometry of spacetime Many detailed presentations of the geometrical descrip tion of spacetime can b e found in the literature including a number of b o oks included in the references at the end of these lecture notes A general outline of the structure of the theory is presented in the following sections Geometrical interpretation of gravity The rst step in describing a lo cal eld b oth in electro dynamics and gravity is to sp ecify the p otentials The force on a test particle at a certain lo cation is then computed from the gradients of these p otentials In the case of gravity these quantities have not only a dynamical interpretation but also a geometrical one Whereas the electromagnetic eld is describ ed in terms of one scalar and one vector p otential together forming the comp onents of a four vector A x the gravitational eld is describ ed by p otentials which can b e assembled in a symmetric four tensor g x This tensor has a geometrical interpretation as the metric of spacetime determining invariant spacetime intervals ds in terms of lo cal co ordinates x ds g x dx dx In the absence of gravity spacetime ob eys the rules of Minkowski geometry well known from sp ecial relativity In cartesian co ordinates ct x y z the metric of Minkowski space is diag 1 We use greek letters taking values to denote the spacetime comp onents of four vectors and tensors Then the invariant ds takes the form ds dx dx c dt dx dy dz For a test particle there is a simple physical interpretation of invariant space