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General Relativity Cosmology GENERAL RELATIVITY AND COSMOLOGY Lecture notes Poul Olesen The Niels Bohr Institute Blegdamsvej DK Copenhagen Denmark Autumn Preface The following lecture notes on general relativity and cosmology grew out of a one semester course on these topics and classical gauge theory by Jan Amb jrn and the present author Subsequently semesters were abandoned and replaced by Blo cks which have an extension of approximately only two months Therefore the classical eld theory part which anyhow strongly needed a revision was dropp ed and the general relativity and cosmology chapters were revised These lecture notes are intro ductory and do not in anyway pretend to b e comprehen sive Several imp ortant topics have b een left out For example gravitational radiation discussed at all There are two reasons for the brevity of the notes the alloted is not time was short a couple of months four hours a week and it was hop ed that by making the notes equally short there is a bigger chance of getting through that general relativity and cosmology are exciting sub jects Sometimes the trouble with exp osing the b eauty of physics is that one has to walk a very long way so people start to feel that they rather walk in a desert than in a b eautiful garden Students hungry for more comprehensive studies are referred to the enormous literature Poul Olesen Contents General relativity The principle of equivalence Gravitation and geometry Motion in an arbitrary gravitational eld The Newton limit The principle of general covariance Contravariantand covariant tensors Dierentiation A prop erty of the determinant of g Some sp ecial derivatives Some applications to physics Curvature Parallel transp ort and curvature tensor Prop erties of the curvature The energymomentum tensor Einsteins eld equations for gravitation The timedep endent spherically symmetric metric A digression A simpler metho d for computing The Christoel symb ols for the timedep endent spherically symmetric metric The Ricci tensor The Schwarzschild solution Birkho s theorem The general relativistic Kepler problem Deection of lightby a massive body Black holes Kruskal co ordinates Painleves version of the Schwarzschild metric Tidal forces and the Riemann tensor The Tidal force from the Schwarzschild solution The energymomentum tensor for electromagnetism The ReissnerNordstrom solution The spherically symmetric solution in dimensions Cosmic strings Cosmology The cosmological problem The cosmological standard mo del CONTENTS A geometric interpretation of the Rob ertsonWalker metric Hubbles law Higher order correction to Hubbles law Einsteins equations and the Rob ertsonWalker metric The Big Bang Existence of the big bang the initial singularity The age of the Universe according to Big Bang Discussion of the fate of the Universe Fitting parameters to observations The cosmic microwave radiation background The matter dominated era The closed Universe The at Universe The op en universe Inclusion of the cosmological constant Discussion of the life time of the Universe Causality structure of the big bang The horizon problem Ination Observational evidence for the cosmological constant The end of cosmology An inhomogeneous universe without Problems Some constants Some literature Chapter General relativity The principle of equivalence Einsteins general theory of relativity is a b eautiful piece of art which connects gravita tional elds with geometry of space and time and thus provides a scheme in which our universe can be discussed Einsteins starting p ointwas the principle of equivalence which can b e understo o d in the context of Newtons mechanics We have the general equation of motion F m x i where x is the acceleration and m is the inertial mass for a given force the acceleration i is smaller the larger the mass is ie the body is more inert the larger m is i In a constant gravitational eld the force is given by F m g g g where g and m are constants It is clear that a priori the parameter m is not related to g g the inertial mass Newton made exp eriments where the p erio d of oscillation of a p endulum made up from dierent materials were studied and he found no variation with m m i g Later on manyvery precise exp eriments were made which showed that m m to a high i g accuracy and this was accepted to such an extent that most text b o oks to day and at Einsteins time do not b other to put any indices on the masses Let us consider a constantgravitational eld With m m m one has the equation i g of motion x g Thus if we intro duce the co ordinate y x gt we get y Therefore we conclude that an observer living in the ysystem sees no eect of the gravi tational eld b ecause eq shows that particles move in straight lines as if there was gt is just no force On the other hand eq shows that the observer is freely falling GENERAL RELATIVITY the displacement p ertinent toafreefall All this is true irresp ectiveofany mass b ecause m m If m m the co ordinate transformation would have to b e replaced by i g i g m g g t y x m i and hence the y system would dep end on which material we consider through the ratio m m g i When m m the transformation is universal and is easily seen to eliminate g i the gravitational eld also if other forces eg electrostatic forces are at work If the gravitational eld varies in space we can apply the transformation in a suciently small domain In Newtonian mechanics we therefore know that an observer in a suciently small freely falling elevator is unable to detect a gravitational eld Einsteins principle of equivalence generalizes this to any physical phenomena In any arbitrary gravitational eld it is p ossible at each spacetime p oint to select lo cally inertial systems freely falling small elevators such that the laws of physics in these are the same as in sp ecial relativity One can use this statement to obtain some insight into the way in which gravity inuences other physical phenomena by writing down in each of the small elevators some law of physics and then transform it to a general co ordinate system In the next section we shall consider the simplest example namely a particle which is freely falling in an arbitrary gravitational eld Some remarks on the history of the Einsteinian version of the equivalence principle After having nished the sp ecial theory of relativity Einstein thought ab out the prob lem of how Newton gravity should b e mo died in order to t in with sp ecial relativity At this p oint Einstein exp erienced what he called the happiest thoughtofmy life namely that an observer falling from the ro of of a house exp eriences no gravitational eld Gravitation and geometry Let us consider a particle which moves under the inuence of a gravitational eld only h of the innitely many freely falling systems of inertia we can apply sp ecial Thus in eac relativity with no forces acting on the particle In sp ecial relativityanevent is describ ed byafourvector y y y where y is the time Since the elevators are a priori small we need however to consider an innitesimal four vector dy dy dy The prop er time d dy dy is an invariant ie if we make a Lorentz transformation from y to y then d dy dy dy dy Most of the historical remarks in these notes are taken from MacTutor History of Mathematics httpwwwhistorymcsst andrewsacukHistTopicsGeneral relativityhtml where much more infor mation can b e found Here d means d This convention of leaving out the bracket in the square of innitesimal quantities will b e used in the following unless it leads to confusion GRAVITATION AND GEOMETRY For light d and eq says that the sp eed of light jdydtj is equal to one in all systems MichelsonMorleys exp eriment The prop er time has the following physical interpretation Let us consider a clo ckoranyphysical system which sp ecies a time eg a particle whichdecays with a certain life time whichby denition marks time by small intervals dt when the clo ck is at rest In the rest system the velo city v dy dy vanishes Thus d dy v dy in the rest system Thus d dy the interval b etween twoticks on the clo ck at rest In amoving system d dy v p v which leads to the formula
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