ICTP Lecture Notes Series Page 1 of 1 ICTP - The x^bdus Sal am International C entre for Theoretical Physics. Trieste, Italy ICTP Lecture Notes Series Volume IV (ISBN 92-93003-02-0) - Nov 200i Astroparticle Physics and Cosmology Jtll (La)7eX& PS sources Volume IV: Ills 0Q4.tar. SZ (S 3 90 2 75 byte s) (1TE: Ulistai-campresssdfAi contains, LiTeX,postscript(.ps) and/or encapsulated postscript (jeps)figure s and also the postscriptversionof each kctiure ruate J. Astroparticle Physics and Cosmology Editors: G. Senjanovic, A. Yu. Smirnov and G. Thompson (ICTP) Intro du cti on : [ Sources: PS , PDF ] Contents • Relativistic Gravity and Some A strop hy si cal Applications [ Sources-. PS . PDF ] Robert V. Wagoner • Lectures on Dark Matter [Sources-. PS .PDF ] U. Seljak • Dark Matter and Particle Physics [ÄuttttK, PDF ] A. Masiero and S. Pascoli • Finite T emp erature Field Theory and Phase Transitions [ Sources. PS , PDF ] M.-P. Lombardo • Lectures on Cosmic Topological Defects [ Sources-. PS . PDF ] Tamnay Vackaspaû • NeutrinoAstrophysics [Sources-. PDF ] Esteban Roidet • Particle and Astrophysics Aspects of Ultrahigh Energy Cosmic Rays [ Sources: PS . PDF ] G. Slgl • HighEnergyParticlesfrom $\gam m a$-r ay Bur sts \ Sources : PS , PDF ] Eli Waxman F or queries or suggestions please contact vi a E-mail: pub_off@i etp .trie ste .it • ICTP Lecture Notes Series http://users.i ctp.it/~pub_off/l ectur esZvol4.html 2007-01-09 Relativistic Gravity and Some Astrophysieal Applications Robert V. Wagoner* Department of Physics, Stanford University, Stanford, USA Lectures given at the Summer School on Astroparticle Physics and Cosmology Trieste, 12-30 June 2000 LNS014001 * [email protected] Abstract These lectures are intended to provide an overview of gravitation theory and a framework for various applications. We will first discuss the foundations and structure of metric theories of gravity, developed from the consequences of various extremal action principles. Then we restrict our consideration to scalar-tensor theories of gravity, and in- troduce various astrophysical applications. The weak-field applications are the index of refraction of gravity and gravitational waves. The strong-field applications are compact objects, their surrounding parti- cle orbits, and cosmology. Except for waves and cosmology, effects of the scalar field are assumed negligible, reducing the theory to general relativity. Contents 1 Key concepts and principles 5 2 Tensor algebra in metric spacetimes 7 2.1 Vector 8 2.2 Tangent space 8 2.3 Basis 8 2.4 Tensor 9 2.5 Metric tensor ........................... 9 2.6 Change of basis 10 2.7 Four-volume element 10 2.8 Gradient of a function 10 3 Extremal action principles for matter and their consequences 11 3.1 Stress-energy tensor 11 3.2 Energy-momentum conservation 11 3.3 Test-particle equation of motion ................ 12 3.4 Gradient and covariant derivative of a tensor 13 3.5 Geodesic deviation Riemann tensor . 14 4 Gravitational field equations 15 4.1 Scalar-tensor theories 15 4.2 Weak-field equations 17 5 Weak-field applications 19 5.1 Index of refraction of gravity .................. 19 5.2 Gravitational waves 20 6 Strong-field applications 21 6.1 Compact objects ......................... 21 6.2 Orbits 23 6.3 Cosmology . 24 6.3.1 Kinematical analysis 24 6.3.2 Dynamical analysis 26 References 28 Relativistic, Gravity 5 1 Key concepts and principles Gravitation is the most fundamental interaction, affecting all forms of mass- energy. This allows its geometrical description, at least within the classical (nonquantum) regime that we shall consider. The scope of this regime is indicated in figure 1. Useful results rather than detailed derivations will be emphasized. The approach that we will take and some applications that we will briefly consider are outlined in figure 2. Size (centimeters) Figure 1: The domains of our knowledge of the physical world, with the mass and size of representative objects indicated. A theory of quantum gravity is needed at distances less than the Planck length Lp = TiGjc3. The textbooks that are closest to our viewpoint are by Schutzfl] at the introductory level and by Misner, Thorne, and Wheeler[2] at the compre- hensive level. Except for only setting the speed of light c = 1, but not the 6 R. V. Wagoner gravitational constant G, we adopt their notation and conventions. Universality of Free-Fall , ,t • IInertial frames T I Einstein Equivalence Principle t general covariance j Metric Theories of Gravity| ^elisor algebra Classical matter action (coupled to metric) Sx" ! Stress-energy tensor) [Conservation lawsj [Particle motion. | t IGeodesic deviation Gradient of a tensor! Riemann tensor Gravitational field equations I •j Weak field equations I X. f ilndex of refraction Gravitational vavesl [Compact objects I {Cosmology| Figure 2: The structure of these lectures. We shall first consider the broad class of metric theories of gravitation. This class is defined by the specification of how gravity affects matter. The manner in which matter generates gravity is separately specified by each theory within this class. Metric theories are based upon a few key concepts and principles: 1) Universality of Free-Fall (UFF) This principle states that if a test particle is placed at an initial event and is given an initial velocity there, its subsequent worldline through spacetime will be independent of its structure (i.e., all forms of energy 'fall' at the same rate). A test particle is conveniently defined as one whose charge, mass, and size are reduced until experimental results are unchanged. Relativistic, Gravity 8 fri '2.j Coördiüstß tué A coordinate (or reference) frame can be visualized as a continuous set of spatially labeled 'clocks' filling spacetime. Such generalized clocks merely provide the time label of events. Infinitesimal distances are best measured by the radar method. The proper distance is c/2 times the round trip travel time, as measured by an ideal clock carried by the fiducial observer. The time associated with the measurement is the average of the photon s mi s s iQ ri and reception times. 3) Inertial reference frame This is a local coordinate frame in which any free test particle is unaccel- era.ted (to a specified accuracy) within a small specified region of spacetime. It can always be constructed at any point (event) in spacetime (if UFF is valid). A physical realization is a nonrotating lab m free-fii.ll, small enough that the effects of tidal gravitational forces are negligible. 4) Einstein Equivalence Principle (EEP) This principle states that in all inertial frames, the nongravitational laws of physics are those formulated within special relativity (when local Lorente coordinates are employed). Employing the EEP, we will see how gravity emerges from a local anal- ysis. Demanding that the laws of physics retain their form under general coordinate transformations xß = xß (icQ) (general covariance) will then al- low us to determine how matter couples to gravity: via a metric tensor. This Q metric tensor has components gßV (a: ) which can be put ir. t tltß Minkowski a form with dgßt,jdx = 0, in every inertia! frame (which is then called a local Lurentz frame). Note that all tensor equations (T = 0) are generally covariant. Such an equation will thus be true in all coordinate systems if it is known to be true in any one (such as a local Lorentz frame). 2 Tensor algebra in metric spacetimes Tensor algebra is the study of geometrical objects (scalars, vectors, tensors, ...) at any fixed point Pa, These objects exist independent of any coordi- nates, so form the proper description of physical reality. 8 R. V. Wagoner 2.1 Vector In curved spacetime, we require a local definition of a vector. A familiar one is the tangent vector to some curve V(X) at the point Vq where the vector exists, where DV is the infinitesimal displacement vector. If the path parameter A is chosen as the proper time r (for a nonzero mass particle), the tangent vector is the four-velocity U = dVjdr. 2.2 Tangent space The vectors at any point Vq form this abstract four dimensional vector space. All geometrical objects at this point reside in this tangent space (not in spacetime). 2.3 Basis A basis is a set of four linearly independent vectors ea (ct — 0,1,2,3) at a point VO- Any vector v at Vq can be represented by its components VA : a v = v e.•a (summation convention) . Consider some coordinate system: four functions xa(V). A (global) coordi- nate basis is then a ea - dV/dx , indicated in figure 3. We shall only employ such bases. Figure 3: Coordinate basis vectors in the tangent space at Vo, tangent to the coordinate curves there. Relativistic, Gravity 9 Most other aspects of tensor algebra are direct generalizations from spe- cial relativity in each tangent space, as follows. 2.4 Tensor A tensor can be thought of in at least two ways: • As a direct product: T = T" '"'ea Ü&ßQS'" • • As a linear operator on vectors, giving a scalar (number): a ß Q ß Tin. v,...) = Tie.,. eß).. .)u v •••= Taß...u v ••• . The second equality defines the components of a tensor. 2.5 Metric tensor This generalization of the Minkowski metric of special relativity, with com- ponents Tfov = diag.(—1,1,1,1), produces the scalar product of vectors: u • v = g(u, v). Its components then also represent scalar products: g(eQ, e;3) = eQ * = gaß = g(ef9, eQ) = gßa . Other aspects are: 2 x tT • The interval is = g(d'P, dV) 5= gnßdxf d^ (dV - dx elT). • Its inverse, given by = fj.-- • 'Raising and lowering indices': 1 % = v • ef], = glV'e,.. e/t) = g^v ' , 1 t/ T vfJ. _ gi -®g(Tl/v = g''' V(r .