http://sites.science.oregonstate.edu/coursewikis/GGR/ 2020-01-26T14:21:56-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/ http://sites.science.oregonstate.edu/coursewikis/GGR/lib/images/favicon.ico text/html 2011-05-29T09:41:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/acknowledge?rev=1306687260 First and foremost, I thank my wife and colleague, Corinne Manogue, for discussions and encouragement over many years. Her struggles with the traditional language of differential geometry, combined with her insight into how undergraduate physics majors learn --- or don't learn --- vector calculus have been a major influence on my increased use of differential forms and orthonormal bases in the classroom. text/html 2014-04-23T20:49:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/bang?rev=1398311340 Slightly rewriting Einstein's equation for the Robertson-Walker metric, as given in~(ss)Robertson-Walker Metrics, we obtain first \begin{equation} \frac{\dot{a}^2+k}{a^2} = \frac{8\pi\rho+\Lambda}{3} \end{equation} and then \begin{equation} \frac{2\ddot{a}}{a} = \Lambda - 8\pi p - \frac{\dot{a}^2+k}{a^2} = \frac23 \Lambda - \frac{8\pi}{3} \left(\rho+3p\right) \end{equation} A physically realistic model that is not empty will have strictly positive energy density $(\rho>0)$ and nonnegative pr… text/html 2014-04-23T21:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/bending?rev=1398313080 Recall from previous sections that the null geodesics of the Schwarzschild geometry satisfy \begin{align} \dot\phi &= \frac{\ell}{r^2} \\ \dot{t} &= \frac{e}{\left(1-\frac{2m}{r}\right)} \\ \dot{r}^2 &= e^2 - \left( 1-\frac{2m}{r} \right) \frac{\ell^2}{r^2} \label{ngeo} \end{align} Setting \begin{equation} u = \frac{1}{r} \end{equation} we have \begin{equation} \dot{r} = -\frac{1}{u^2} \dot{u} = -r^2 \dot{u} = -\ell \> \frac{\dot{u}}{\dot\phi} = -\ell \> \frac{du}{d\phi} \end{equation} so … text/html 2013-12-18T09:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/birkhoff?rev=1387387020 Birkhoff's Theorem states that the Schwarzschild geometry is the only spherically symmetric solution of Einstein's equation. This result is remarkable, in that the Schwarzschild geometry has a timelike symmetry (Killing vector), even though this was not assumed; spherically symmetric vacuum solutions of Einstein's equation are automatically time independent! We outline the proof of Birkhoff's Theorem below. text/html 2013-12-19T11:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/books?rev=1387480920 \begin{itemize}\item Edwin F. Taylor and John Archibald Wheeler, Exploring Black Holes, Addison Wesley Longman, 2000. An elementary introduction to the relativity of black holes, using line elements. Not easy to read, but worth it. \item James B. Hartle, Gravity, Addison Wesley, 2003. An ``examples first'' introduction to general relativity, discussing applications of Einstein's equations before presenting the mathematics behind the equations. text/html 2013-12-18T09:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/charge?rev=1387387080 text/html 2013-12-17T21:54:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/circle?rev=1387346040 Figure 1: Defining the (circular) trigonometric functions via a circle. What is the fundamental idea in trigonometry? Although often introduced as the study of triangles, trigonometry is really the study of circles. One way to construct the trigonometric functions is as follows: \begin{itemize}\item Draw a circle of radius $r$, that is, the set of points at constant distance $r$ from the origin. \item Measure arclength $s$ along the circle by integrating the (square root of the) line eleme… text/html 2013-12-18T08:04:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/conserve?rev=1387382640 We have seen that the energy-momentum of a cloud of dust, according to an observer with 4-velocity $\vv$, is given by \begin{equation} \Tvec_{\vv} = \rho\,\uu \,(-\uu\cdot\vv) \end{equation} If the dust is moving at speed $v=\tanh\beta$ with respect to the observer $\vv$, then the 4-velocity $\uu$ of the dust takes the form \begin{equation} \uu = \begin{pmatrix} 1\\ \us\\ \end{pmatrix} \cosh\beta \label{ucomp} \end{equation} where we have introduced the notation $\us$ for the 3-velocity of t… text/html 2013-12-18T08:11:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/constcurv?rev=1387383060 In any dimension and signature, it turns out there is a unique (local) geometry of constant (sectional) curvature, which can be classified by the value of the scalar curvature $R$. These three possibilities are illustrated for two dimensions with Euclidean signature in Figure~1, showing a plane (zero curvature), sphere (constant positive curvature), and Lorentzian hyperboloid (constant negative curvature), respectively. In each case, the surface is two-dimensional, even though the representa… text/html 2013-12-18T08:11:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/cosconst?rev=1387383060 Einstein was not happy with some of the predictions of his theory, notably, as we will see in the next chapter, that the universe is expanding. He therefore looked for a way to modify his theory to permit a static universe. There is in fact just one way to do so; the only other divergence-free geometric object (of the correct size, that is, with the correct number of components) is the line element itself! Using the fact that \begin{equation} {*}\sigma^i = \pm \sigma^j\wedge\sigma^k\wedge\sig… text/html 2013-11-06T09:38:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/cosmo?rev=1383759480 Cosmology is, simply put, the study of the universe. Relativistic cosmology is the study of solutions of Einstein's equation which may represent the broad features of the universe as a whole. One constraint on such models is the seemingly innocuous statement that the sky is dark at night. Why is this surprising? In a Euclidean universe with a uniform distribution of stars, both the density of matter and the apparent luminosity fall off as $1/r^2$. Thus, the perceived brightness of every thi… text/html 2014-04-23T20:29:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/cosprin?rev=1398310140 Figure 1:A foliation of spacetime by hypersurfaces of constant cosmic time. Figure 2:A foliation of spacetime by a family of observers. Figure 3:In a spacetime that is both isotropic and homogeneous, the cosmic observers are orthogonal to the cosmic surfaces. The simplest cosmological models are based on the principle that the universe is the same everywhere. This principle is sometimes referred to as the cosmological principle, and can be thought of as an extreme generalization of the Cop… text/html 2013-12-18T09:14:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/divein?rev=1387386840 We verify here that the divergence of the Einstein vector-valued 1-form $\GG$ vanishes, that is, that \begin{equation} d\gamma^i + \omega^i{}_j\wedge\gamma^j = 0 \label{diveineq} \end{equation} where \begin{equation} \gamma^i = \frac12 \epsilon_\ell{}^{ijk} \,\Omega_{jk} \wedge \sigma^\ell \end{equation} as shown in~(ss)Components of the Einstein Tensor. Using the Bianchi identities and the structure equations to evaluate the first term, we have \begin{align} 2 &(d\gamma^i + \omega^i{}_j\wed… text/html 2013-12-18T09:15:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/divmet2d?rev=1387386900 text/html 2013-08-31T11:38:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/divmetric?rev=1377974280 We now consider the general case, for which \begin{equation} d\rr = \sigma^i\,\ee_i \end{equation} Recalling that the Hodge dual of $\sigma^i$ satisfies \begin{equation} \sigma^i\wedge{*}\sigma^i = g(\sigma^i,\sigma^i)\,\omega \end{equation} we can write \begin{equation} {*}\sigma^i = \frac{1}{(n-1)!}\,\epsilon^i{}_{j...k}\,\sigma^j\wedge...\wedge\sigma^k \end{equation} where the $n$-index object $\epsilon_{ij...k}$ is the alternating symbol in $n$ dimensions, sometimes called the Levi-Civita s… text/html 2013-11-05T17:55:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/dust?rev=1383702900 Let's summarize the discussion in the preceding section, but without the restriction to Minkowski space. Matter is described by the energy-momentum 1-forms \begin{equation} T^i = T^i{}_j\,\sigma^j \end{equation} which can be combined into a single vector-valued 1-form \begin{equation} \Tvec = T^i \ee_i \end{equation} Consider an observer with velocity vector $\vv$ satisfying \begin{equation} \vv \,d\tau = d\rr \end{equation} and drop the arrow to write as usual \begin{equation} v=\vv\cdot d\rr … text/html 2014-04-30T21:24:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/einstein?rev=1398918240 As discussed in the previous sections, matter in general relativity is described by a vector-valued 1-form \begin{equation} \Tvec = T^i\,\ee_i \end{equation} where \begin{equation} T^i = T^i{}_j\,\sigma^j \end{equation} are the energy-momentum 1-forms. The components $T^i{}_j$ are (also) components of the energy-momentum tensor. We assume that the energy-momentum tensor is symmetric, that is, that \begin{equation} T_{ji} = T_{ij} \end{equation} which can also be expressed as the requirement th… text/html 2013-12-18T09:14:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/einsteinc?rev=1387386840 text/html 2013-11-05T20:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/einsteineq?rev=1383711720 As discussed in the previous sections, matter is described by the energy-momentum tensor, which we now write as \begin{equation} \Tvec = T^i{}_j \sigma^j \ee_i = {*}\tau^i \ee_i \end{equation} and conservation of matter requires that $\Tvec$ be divergence-free, that is, that \begin{equation} d{*}\Tvec = \zero \end{equation} Meanwhile, it turns out that there is a unique divergence-free vector-valued 1-form which can be constructed from the curvature, namely the Einstein tensor \begin{equation} … text/html 2014-04-30T21:29:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/einsteing?rev=1398918540 In Newtonian theory, gravity is described by the gravitational potential $\Phi$. It can be shown that tidal acceleration in the $\uu$ direction is just the difference in the acceleration due to gravity at nearby points, which can be expressed as a directional derivative of the gravitational field, namely \[ -(\uu\cdot\grad)(\grad\Phi) \] More precisely, this expression describes the vector change in the displacement between two freely falling objects, which are (originally) separated in the $\u… text/html 2014-04-23T20:51:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/frw?rev=1398311460 Current observations show that \begin{equation} \rho \gg p \end{equation} (in appropriate units!); the universe is presently ``matter dominated''. We therefore consider models with \begin{equation} p = 0 \end{equation} which are called Friedmann-Robertson-Walker models, or just Friedmann models. text/html 2013-12-21T11:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/fvac?rev=1387652460 We first consider Robertson-Walker cosmologies that are also vacuum solutions, that is, for which \begin{equation} \rho = 0 = p \end{equation} Friedmann's equation reduces to \begin{equation} \dot{a}^2 = \frac{\Lambda a^2}{3} - k \label{frwvac} \end{equation} text/html 2014-04-30T21:46:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/geodesics?rev=1398919560 When is a curve ``straight''? When its tangent vector is constant. We have \begin{equation} \vv = \frac{d\rr}{d\lambda} = \dot\rr \end{equation} or equivalently \begin{equation} \vv\,d\lambda = d\rr = \sigma^i\ee_i \end{equation} so that the components of $\vv$ are given by \begin{equation} v^i d\lambda = \sigma^i \label{vcomp} \end{equation} Differentiating, we obtain \begin{align} d\vv = d(v^i \ee_i) &= dv^i \ee_i + v^i d\ee^i \nonumber\\ &= (dv^j + \omega^j{}_i v^i) \ee_j \nonumber\\ … text/html 2013-12-18T08:00:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/geodev?rev=1387382400 Consider a family of geodesics, with velocity vectors of the form \begin{equation} \vv = \dot{\rr} = \Partial{\rr}{\tau} \end{equation} Figure 1:A family of geodesics. as shown in Figure~1. Since these curves are geodesics, their velocity vectors satisfy \begin{equation} \dot{\vv} = 0 \end{equation} Label the geodesics by some coordinate(s) $s$, and consider the separation vector \begin{equation} \uu = \rr' = \Partial{\rr}{s} \end{equation} between nearby geodesics. By construction, we have… text/html 2015-04-18T10:53:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/geodev2?rev=1429379580 As before, let $\vv=\dot{\rr}$ denote the velocity vectors of a family of geodesics, and $\uu=\rr'$ the separation vector between them. Then along the geodesics we have \begin{align} \vv\,d\tau &= d\rr \\ v^k\,d\tau &= \sigma^k \end{align} and similar expressions hold along curves with tangent vector $\uu$, namely \begin{align} \uu\,ds &= d\rr \\ u^k\,ds &= \sigma^k \end{align} text/html 2013-12-18T08:19:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/gps?rev=1387383540 text/html 2013-10-29T18:16:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/grphysics?rev=1383095760 There are three key ideas underlying Einstein's theory of general relativity: \begin{itemize}\item Principle of Relativity: The principle of relativity originates with Galileo, and says that the results of experiments do not depend on relative (uniform) motion of observers. Einstein's contribution was to apply this principle to electromagnetism. Since the speed of light can be computed from the constants in Maxwell's equations, the speed of light must be independent of the observer. This pos… text/html 2013-09-29T09:41:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/gsr?rev=1380472860 A spacetime diagram in special relativity is just a diagram drawn using hyperbola geometry. Vertical lines represent the worldline of an observer standing still (in the given reference frame). Horizontal lines represent a ``snapshot'' of time, according to that observer. text/html 2014-04-23T21:04:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/hubble?rev=1398312240 In special relativity, light emitted by a moving observer at one frequency is received by a stationary observer at another frequency. This is the Doppler effect, which can be computed using Figure~1. In this case, the two heavy lines correspond to observers moving at different velocities, and the two lighter lines correspond to pulses of light, emitted by the moving observer (on the right), and received by the stationary observer (on the left). Simple (hyperbolic) triangle trigonometry can b… text/html 2013-12-12T10:02:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/hyperbola?rev=1386871320 Figure 1: Defining the hyperbolic trigonometric functions via a (Lorentzian) hyperbola. We now apply the same procedure to Lorentzian hyperbolas rather than Euclidean circles, as illustrated in Figure 1. \begin{itemize}\item Draw a hyperbola of ``radius'' $\rho$, that is, the set of points at constant (squared) distance $r^2=x^2-t^2$ from the origin. \item Measure arclength $\tau$ along the hyperbola by integrating the (absolute value of the square root of the) line element. That is, inte… text/html 2014-04-23T15:44:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/kerr?rev=1398293040 The Schwarzschild geometry can be regarded as a real slice of a complex geometry, where one can perform a complex rotation and consider the resulting real slice. The resulting line element, known as the (Boyer-Lindquist form of the) Kerr metric, describes a rotating black hole, and is given by \begin{align} ds^2 &= - \frac{\Delta}{\rho^2} \left(dt-a\sin^2\theta\,d\phi\right)^2 + \frac{\sin^2\theta}{\rho^2} \left( (r^2+a^2)\,d\phi-a\,dt \right)^2 \nonumber\\ &\qquad + \frac{\rho^2}{\Delta} d… text/html 2014-04-24T16:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/kruskal?rev=1398381480 Figure 1: Kruskal geometry. Since Kruskal-Szekeres coordinates $U$ and $V$ are well-defined at $r=2m$, we can use them to extend Schwarzschild geometry to $r<2m$. The maximally extended Schwarzschild geometry, also called the Kruskal geometry is obtained by extending the domains of $U$, $V$ as much as possible, and is shown in Figure~1. text/html 2014-04-30T21:45:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/line?rev=1398919500 The fundamental notion in geometry is distance. One can study shapes without worrying about size or scale, but that is topology; in geometry, size matters. So how do you measure distance? With a ruler. But how do you calibrate the ruler? Figure 1:The infinitesimal version of the Pythagorean Theorem in rectangular coordinates. text/html 2014-04-23T20:07:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/matter?rev=1398308820 Figure 1:Equally spaced particles, moving uniformly to the right. Figure 2:A spacetime diagram of equally spaced particles, moving uniformly to the right. Figure 3:Determining the spacing between moving particles, as seen by an observer at rest. The dot denotes a right angle. The 2-momentum of an object in (2-dimensional) Minkowski space moving at speed $v=\tanh\beta$ is given by \begin{equation} \bp = m\,\bu = \begin{pmatrix} E \\ p \\ \end{pmatrix} = \begin{pmatrix} m\,\cosh\beta \\ m\,… text/html 2014-04-23T21:28:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/mercury?rev=1398313680 Recall from previous sections that the timelike geodesics of the Schwarzschild geometry satisfy \begin{align} \dot\phi &= \frac{\ell}{r^2} \\ \dot{t} &= \frac{e}{\left(1-\frac{2m}{r}\right)} \\ \dot{r}^2 &= e^2 - \left( 1-\frac{2m}{r} \right) \left( 1+ \frac{\ell^2}{r^2} \right) \label{tgeo} \end{align} Setting \begin{equation} u = \frac{1}{r} \end{equation} we have \begin{equation} \dot{r} = -\frac{1}{u^2} \dot{u} = -r^2 \dot{u} = -\ell \> \frac{\dot{u}}{\dot\phi} = -\ell \> \frac{du}{d\p… text/html 2014-04-23T20:54:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/missing?rev=1398311640 Non-vacuum Friedmann solutions can be classified using similar methods to those used in the vacuum case. We restrict the discussion here to some special cases. In particular, we assume throughout this section that $\Lambda=0$, which brings Friedmann's equation to the form \begin{equation} \dot{a}^2 = \frac{8\pi\rho a^2}{3} - k \end{equation} text/html 2014-04-30T21:54:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/nutshell?rev=1398920040 We summarize here the basic properties of differential forms. See Chapter~12 for further details. Wedge Products Differential forms are integrands, the things one integrates. So $dx$ is a differential form (a 1-form), and so is $dx\,dy$ (a 2-form). However, orientation matters; think about change of variables, where for instance \begin{equation} du\,dv = \Jacobian{u}{v}{x}{y} \> dx\,dy \end{equation} where $\Jacobian{u}{v}{x}{y}$ denotes the determinant of the Jacobian matrix of $(u,v)$ w… text/html 2013-09-29T10:45:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/ocircle?rev=1380476700 Further information about the possible orbits in the Schwarzschild geometry can be obtained by analyzing the potential more carefully. Using the same conventions as before, we have \begin{equation} \frac12 \dot{r}^2 = \frac{e^2-1}{2} - V \label{rdot} \end{equation} with \begin{equation} V = \frac12\left(1-\frac{2m}{r}\right) \left(1+\frac{\ell^2}{r^2}\right) - \frac12 \end{equation} Differentiating $V$ with respect to $r$ yields \begin{equation} V' = \frac{dV}{dr} = \frac{mr^2-\ell^2r+3m\ell^2… text/html 2013-12-17T22:03:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/onewton?rev=1387346580 Figure 1:Newtonian particle trajectories in the gravitational field of a massive object (heavy dot) can be circular, elliptical, or hyperbolic, all with the (center of the) gravitational source at one focus. Radial trajectories are also possible; these are the only trajectories that hit the source. How do objects move in Newtonian gravity? We all know that objects in orbit move on ellipses, and that ``slingshot'' hyperbolic trajectories also exist. The various possibilities, also including… text/html 2013-12-20T18:15:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/onull?rev=1387592100 We repeat the computations in the previous section for lightlike orbits. We still have \begin{align} \ell &= \Pvec\cdot\vv = r^2 \dot\phi \\ e &= -\Tvec\cdot\vv = \left(1-\frac{2m}{r}\right)\dot{t} \end{align} but now \begin{equation} 0 = \left(1-\frac{2m}{r}\right)\,\dot{t}^2 - \frac{\dot{r}^2}{1-\frac{2m}{r}} - r^2\,\dot\phi^2 \end{equation} which leads to \begin{equation} \dot{r}^2 = e^2 - \left(1-\frac{2m}{r}\right)\frac{\ell^2}{r^2} \label{nullgeo} \end{equation} text/html 2013-08-22T17:41:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/oradial?rev=1377218460 Consider now radial geodesics, for which $\ell=0$, so that \begin{equation} \dot{r}^2 = e^2 - \left(1-\frac{2m}{r}\right) \end{equation} A radial geodesic represents a freely falling object with no angular momentum. The energy $e$ determines the radius $r_0$ at which the object is at rest, since if $\dot{r}=0$ at $r=r_0$ then \begin{equation} r_0 = \frac{2m}{1-e^2} \end{equation} Since $r$ must be nonnegative, we must have \begin{equation} e^2 \le 1 \end{equation} and we will assume that $e$ is … text/html 2013-12-12T10:08:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/orbits?rev=1386871680 In (ss)~Newtonian Motion and (ss)~Schwarzschild Geodesics, we have seen that the trajectory of an object moving in the gravitational field of a point mass is controlled by the object's energy per unit mass ($e=E/M$), and its angular momentum per unit mass ($\ell=L/M$). In the Newtonian case, we have \begin{equation} \frac12 \dot{r}^2 = e - \left(-\frac{m}{r} + \frac{\ell^2}{2r^2}\right) \label{rdotn} \end{equation} from Equation~(11) of (ss)~Newtonian Motion (where we have set $G=1$), and in… text/html 2014-04-23T15:42:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/penrose?rev=1398292920 A Penrose diagram is a spacetime diagram in which points at infinity are included. This is accomplished by rescaling the metric, that is, replacing $ds^2$ by $\Omega^2\,ds^2$, where the conformal factor $\Omega$ typically behaves like \begin{equation} \Omega \sim \frac{1}{r} \end{equation} Points with $\Omega=0$ therefore correspond to points ``at infinity''; more formally, this construction adds a conformal boundary to the original spacetime. Since the metric has merely been rescaled by the c… text/html 2013-05-11T21:16:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/preface?rev=1368332160 This book is an introduction to general relativity, intended for advanced undergraduates or beginning graduate students in either mathematics or physics. The goal is to describe some of the surprising implications of relativity without introducing more formalism than necessary. ``Necessary'' is of course in the eye of the beholder, and this book takes a nonstandard path, using differential forms rather than tensor calculus, and trying to minimize the use of ``index gymnastics'' as much as poss… text/html 2013-08-02T21:25:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/prereqs?rev=1375503900 text/html 2015-04-16T11:39:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rain?rev=1429209540 Figure 7:A hyperbolic triangle showing the hyperbolic angle between shell observers and rain observers. We digress briefly to introduce ``rain'' coordinates adapted to freely falling observers who start from rest at $r=\infty$, each of whom moves along a geodesic as described in the preceding section. As shown there, a shell observer sees such observers fall past them with speed \begin{equation} \tanh\beta = -\sqrt{\frac{2m}{r}} \end{equation} Drawing a right triangle to these proportions, as … text/html 2013-10-31T18:25:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/raincon?rev=1383269100 Recall from (ss)~Rain Coordinates that \begin{align} \sigma^T &= dt + \frac{\sqrt{\frac{2m}{r}}}{1-\frac{2m}{r}} \>dr \\ \sigma^R &= \frac{dr}{1-\frac{2m}{r}} + \sqrt{\frac{2m}{r}} \>dt \end{align} which allows us to introduce rain coordinates ($T$,$R$) defined by \begin{align} dT &= dt + \sqrt{\frac{2m}{r}}\frac{dr}{1-2m/r} \\ dR &= \sqrt{\frac{r}{2m}}\frac{dr}{1-2m/r} + dt \end{align} The Schwarzschild line element in rain coordinates then takes the form \begin{equation} ds^2 = -dT^2 + \frac{… text/html 2014-04-24T10:09:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/raincurv?rev=1398359340 text/html 2013-12-12T10:05:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rindext?rev=1386871500 A useful technique in exploring an unknown geometry is to follow light beams, which leads to the use of null coordinates. Equivalently, factor the line element. In Minkowski space, we have \begin{equation} ds^2 = dx^2 - dt^2 = (dx+dt)(dx-dt) \end{equation} Thus, introduce coordinates \begin{align} u &= t-x \\ v &= t+x \end{align} which brings the line element to the form \begin{equation} ds^2 = -du\,dv \end{equation} Why are null coordinates useful? Because the surfaces (curves) along which $… text/html 2014-04-23T15:43:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rindgeo?rev=1398292980 Taking the ``square root'' of the line element, we have \begin{equation} d\rr = \rho\,d\alpha\,\Ahat + d\rho\,\Rhat \end{equation} where \begin{align} \Ahat\cdot\Ahat &= -1 \\ \Rhat\cdot\Rhat &= 1 \\ \Ahat\cdot\Rhat &= 0 \end{align} Since the line element depends only on $\rho$, but not on $\alpha$, there is a Killing vector \begin{equation} \Avec = \rho\,\Ahat \end{equation} since \begin{equation} \Avec\cdot\grad f = \Partial{f}{\alpha} \end{equation} (and of course $\grad f\cdot d\rr=df$). … text/html 2013-12-12T09:23:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rindler?rev=1386868980 We digress briefly from our discussion of the Schwarzschild geometry in order to consider a much simpler geometry with many of the same properties. Figure 1: The shaded region is the Rindler wedge in Minkowski space, which is covered by the Rindler coordinates ($\rho$,$\alpha$). text/html 2013-12-17T22:04:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rindprop?rev=1387346640 What do Rindler coordinates represent? Consider a worldline with $\rho=\hbox{constant}$, as shown in Figure~1. Since $x^2-t^2=\hbox{constant}$, this worldline is a timelike hyperbola, and is in fact one of our calibrating hyperbolas from special relativity, at constant ``distance'' $\rho$ from the origin. text/html 2013-12-19T12:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rn?rev=1387483260 Assuming a spherically symmetric line element of the form \begin{equation} ds^2 = -f\,dt^2 + \frac{dr^2}{f} + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2 \end{equation} where $f$ is an arbitrary function of $r$, Einstein's equation with an electromagnetic source, representing a point charge, can be solved for $f$, yielding \begin{equation} f = 1 - \frac{2m}{r} + \frac{q^2}{r^2} \end{equation} The resulting spacetime is known as the Reissner-Nordstr\"om geometry, and represents a black hole with… text/html 2019-05-26T16:33:37-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rw?rev=1558913617 We now study simple models of the universe, and therefore assume both homogeneity and isotropy. As discussed in~(ss)Cosmological Principle, homogeneity implies that spacetime is foliated by spacelike hypersurfaces $\Sigma_t$; isotropy implies there are preferred ``cosmic observers'' orthogonal to these hypersurfaces. Thus, each $\Sigma_t$ represents an instant of time according to these cosmic observers. We can therefore assume without loss of generality that the surfaces are labeled using … text/html 2013-12-18T09:15:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/rwcurv?rev=1387386900 The Robertson-Walker line element is \begin{align} ds^2 &= -dt^2 + a(t)^2 \Biggl( \frac{dr^2}{1-kr^2} \Biggr.\nonumber\\ &\qquad\Biggl.{} + r^2 \left( d\theta^2 + \sin^2\theta\,d\phi^2 \right) \Biggr) \end{align} It now follows immediately that \begin{align} d\rr &= dt\,\That + \frac{a(t)\,dr}{\sqrt{1-\kappa r^2}}\,\rhat + a(t)\,r\,d\theta\,\that + a(t)\,r\,\sin\theta\,d\phi\,\phat \end{align} and the basis 1-forms are \begin{align} \sigma^t &= dt \\ \sigma^r &= \frac{a(t)\,dr}{\sqrt{1-… text/html 2014-04-23T15:45:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwarz?rev=1398293100 The Schwarzschild geometry is described by the line element \begin{equation} ds^2 = -\left(1-\frac{2m}{r}\right)\,dt^2 + \frac{dr^2}{1-\frac{2m}{r}} \\ + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2 \label{schwmet} \end{equation} As we will see later, this metric is the unique spherically symmetric solution of Einstein's equation in vacuum, and describes the gravitational field of a point mass at the origin. Yes, it also describes a black hole, but this was not realized for nearl… text/html 2013-10-31T22:26:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwcon?rev=1383283560 We have so far related curvature to geodesic deviation by working in rain coordinates, a special coordinate system adapted to the radial geodesics we chose to study. But this relationship is geometric, and therefore independent of the coordinates we choose. To demonstrate this geometric invariance, we revisit the problem of geodesic deviation using our original Schwarzschild coordinates. First, we need the connection. text/html 2015-05-19T19:20:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwcurv?rev=1432088400 The Schwarzschild line element is \begin{equation} ds^2 = -f\,dt^2 + \frac{dr^2}{f} + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2 \end{equation} where we have written $f$ for the function \begin{equation} f(r) = 1-\frac{2m}{r} \end{equation} with derivatives \begin{align} f' &= \frac{df}{dr} = \frac{2m}{r^2} \\ \fpp &= -\frac{4m}{r^3} \end{align} It now follows immediately that \begin{equation} d\rr = \sqrt{f}\,dt\,\That + \frac{dr}{\sqrt{f}}\,\rhat + r\,d\theta\,\that + r\,\sin\theta\,d\phi\,\ph… text/html 2014-04-23T15:41:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwext?rev=1398292860 We now apply the same technique used in Chapter~ in (ss)~Rindler, to extend the Rindler geometry to the Schwarzschild geometry. Consider a radial light beam, so that $d\phi=0$ (and as usual $\theta=\pi/2$). Then the line element becomes \begin{equation} ds^2 = -\left(1-\frac{2m}{r}\right)\,dt^2 + \frac{dr^2}{1-\frac{2m}{r}} = -\left(1-\frac{2m}{r}\right) \left( dt^2 - \frac{dr^2}{\left(1-\frac{2m}{r}\right)^2} \right) \end{equation} which we can factor as \begin{equation} ds^2 = -\… text/html 2013-12-17T22:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwgeo?rev=1387346460 A fundamental principle of general relativity is that freely falling objects travel along timelike geodesics. So what are those geodesics? Taking the square root of the line element, the infinitesimal vector displacement in the Schwarzschild geometry is \begin{equation} d\rr = \sqrt{1-\frac{2m}{r}}\,dt\,\That + \frac{dr\,\rhat}{\sqrt{1-\frac{2m}{r}}} + r\,d\theta\,\that + r\sin\theta\,d\phi\,\phat \label{drschw} \end{equation} where \begin{align} \That\cdot\That &= -1 \\ \rhat\cdot\rhat &= 1… text/html 2014-04-23T15:42:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwgeom?rev=1398292920 The basic features of the Schwarzschild geometry are readily apparent by examining the metric. \begin{itemize}\item Asymptotic Flatness: As already discussed, the Schwarzschild geometry reduces to that of Minkowski space as $r$ goes to infinity. \itemSpherical Symmetry: The angular dependence of the Schwarzschild metric is precisely the same as that of a sphere; the Schwarzschild geometry is spherically symmetric. It is therefore almost always sufficient to consider the equatorial plane … text/html 2014-04-23T15:47:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/schwobs?rev=1398293220 An observer is really an army of observers stationed at all points in space. More formally, the worldlines of a family of observers foliate the spacetime. Each observer records what he or she sees, and the resulting logs are later compared. This process is often loosely described as ``an observer seeing''; a better description would be ``a family of observers recording''. text/html 2013-12-18T08:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/standard?rev=1387383480 Figure 1:The expansion of the Einstein-de Sitter cosmology. Figure 2:The expansion of the dust-filled Robertson-Walker cosmology. Figure 3:The hyperbolic analog of the dust-filled Robertson-Walker cosmology. Figure 4:The expansion of the radiation-filled Robertson-Walker cosmology. The standard cosmological models are models without cosmological constant ($\Lambda=0$). We first consider Friedmann models ($p=0$), for which \begin{equation} \dot{a}^2 = \frac{C}{a} - k \end{equation} If $k=… text/html 2014-04-30T21:47:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/symmetries?rev=1398919620 As discussed in the previous section, geodesics are the solutions to a system of second-order differential equations. These equations are not always easy to solve; solving differential equations is an art form. However, dramatic simplifications occur in the presence of symmetries, as we now show. text/html 2014-04-30T21:51:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/sympolar?rev=1398919860 In polar coordinates, we have \begin{equation} d\rr = dr\,\rhat + r\,d\phi\,\phat \end{equation} and the line element is \begin{equation} ds^2 = dr^2 + r^2\,d\phi^2 \end{equation} Both of these expressions depend explicitly on $r$, but not $\phi$. Thus, the line element does not change in the $\phi$ direction, and we expect ``$\Partial{}{\phi}$'' to be a Killing vector. But what vector is this? text/html 2014-04-30T21:50:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/symsphere?rev=1398919800 On the sphere, we have \begin{equation} d\rr = r\,d\theta\,\that + r\sin\theta\,d\phi\,\phat \end{equation} and the line element is \begin{equation} ds^2 = r^2\,d\theta^2 + r^2\,\sin^2\theta\,d\phi^2 \end{equation} Both of these expressions depend explicitly on $\theta$, but not $\phi$ (or $r$, which is constant). Thus, the line element does not change in the $\phi$ direction, and we again expect ``$\Partial{}{\phi}$'' to be a Killing vector. text/html 2014-04-30T21:40:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/symthm?rev=1398919200 We show here that coordinate directions in which the metric (line element) doesn't change always correspond to Killing vectors. Theorem: Suppose that $\{x=y^0,y^1,...\}$ are orthogonal coordinates, so that \begin{equation} d\rr = h\,dx\,\xhat + \sum\limits_i h_i\,dy^i\,\yhat{}^i \end{equation} and suppose further that the coefficients $h=h_0,h_1,...$ do not depend on $x$, that is, suppose that \begin{equation} \Partial{h}{x} = 0 = \Partial{h_i}{x} \end{equation} Then $\XX = h\xhat$ is a Kill… text/html 2014-04-23T18:27:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/tensors?rev=1398302820 text/html 2014-04-23T19:59:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/tidal?rev=1398308340 ~~~~~~~~~~~~~ ~ Figure 1: Nearby objects falling radially, either next to each other (at equal radius), or directly on top of each other. Consider two nearby objects falling radially. We first consider the case where both objects start at rest on the same $r=\hbox{constant}$ shell, so that they have the same energy $e$. This situation is shown in the first diagram in Figure~1. The separation between the objects is therefore \begin{equation} \Delta s = r \,\Delta\phi \end{equation} and … text/html 2014-04-23T20:02:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GGR/book/content/tidal2?rev=1398308520 The components $R^i{}_{jk\ell}$ of the curvature 2-forms form a tensor, known as the Riemann curvature tensor. Tensor components can be computed in any basis, then converted to any other basis using the appropriate change-of-basis transformations. Put differently, tensors are linear maps on vectors, and can therefore easily be evaluated on any vectors, not just on basis vectors.