http://sites.science.oregonstate.edu/coursewikis/GSR/ 2020-01-26T14:29:23-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/ http://sites.science.oregonstate.edu/coursewikis/GSR/lib/images/favicon.ico text/html 2011-11-18T10:45:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/acknowledge?rev=1321641900 This book grew out of class notes [ 3 ] for a course on Reference Frames, which in turn forms part of a major upper-division curriculum reform effort, entitled Paradigms in Physics, which was begun in the Department of Physics at Oregon State University in 1997. The class notes were subsequently published online in wiki format [ 4 ], and are now used as the primary text in that course; an abbreviated version was also published as a journal article [ 5 ]. text/html 2012-06-05T15:07:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/addition?rev=1338934020 Consider the line through the origin which makes an angle $\phi$ with the (positive) $x$-axis, as shown in the first sketch in Figure 3.7. What is its slope? The equation of the line is \begin{equation} y = x \,\tan\phi \end{equation} so that the slope is $\tan\phi$, at least in ``unprimed'' coordinates. Consider now the line through the origin shown in the second sketch, which makes an angle $\phi$ with the (positive) $x'$-axis. What is its slope? In ``primed'' coordinates, the equation of… text/html 2011-11-21T10:51:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/boost?rev=1321901460 Suppose the observer $O'$ is moving to the right with speed $v$. If $x$ denotes the distance of an object from $O$, then the distance between the object and $O'$ as measured by $O$ will be $x-vt$. But using the formula for length contraction derived above, namely $\Delta x'=\gamma\,\Delta x$, we see that \begin{equation} x' = \gamma\,(x-vt) \label{xlorentz} \end{equation} text/html 2012-06-05T15:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/conservation?rev=1338934920 Suppose that (Newtonian) momentum is conserved in a given frame, that is \begin{equation} \sum m_i v_i = \sum \mtwo_j \vtwo_j \end{equation} (Both of these would be zero in the center-of-mass frame.) Changing to another frame moving with respect to the first at speed $v$, we have \begin{eqnarray} v_i &=& v'_i + v \\ \vtwo_j &=& \vtwo'_j + v \end{eqnarray} so that \begin{equation} \sum m_i (v'_i+v) = \sum \mtwo_j (\vtwo'_j+v) \end{equation} We therefore see that \begin{equation} \sum m_i v'_i =… text/html 2012-04-07T16:05:01-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/dexp?rev=1333839901 Hyperbolic trigonometric functions are usually defined using formulas (Equations (1) and (2) of (ss)4.1) and it takes some work (and independent knowledge of the exponential function) to show directly that our definition is equivalent to this one. We turn this on its head and instead define \begin{equation} \exp(\beta) = \cosh\beta + \sinh\beta \label{expdef} \end{equation} The function $\exp(\beta)$ as defined in~($\ref{expdef}$) has an immediate geometric interpretation, as shown in Figure… text/html 2012-04-07T15:43:33-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/dhtrig?rev=1333838613 The derivation in the previous section carries over virtually unchanged to hyperbolic trigonometric functions. Recall from Section 4.2 the geometric definition of the hyperbolic trigonometric functions in (Equations (2) and (3) of (ss)4.2) as shown in Figure 4.2. text/html 2012-04-07T13:14:36-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/diagrams?rev=1333829676 What is a spacetime diagram? Simply a diagram showing both where things are, and when they were there. For example, the first diagram in Figure 5.1 represents somebody sitting still (but getting older!), and the second diagram in Figure 5.1 represents somebody moving to the right at constant speed. Such trajectories are called worldlines. One early French railroad timetable shows the worldlines of each train! In relativity, the convention is to have the $t$-axis be vertical, and the $x$… text/html 2012-06-05T15:48:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/diffgeom?rev=1338936480 In 2 dimensions, Euclidean geometry is the geometry of a flat piece of paper. But there are also of course curved 2-dimensional surfaces. The simplest of these is the sphere, which has constant positive curvature and is a model of (double) elliptic geometry. Another important example is the hyperboloid, which also has constant curvature, and is a model of hyperbolic geometry. Hyperbolic and elliptic geometry form the 2 main categories of non-Euclidean geometries. text/html 2011-11-20T09:11:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/distance?rev=1321809060 The key concept in Euclidean geometry is the distance function that measures the distance between two points. In two dimensions, the (squared!)~distance between a point $B=(x,y)$ and the origin is given by \begin{equation} r^2 = x^2 + y^2 \end{equation} which is of course just the Pythagorean Theorem; see Figure 3.1. text/html 2012-06-05T19:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/doppler?rev=1338948060 The frequency $f$ of a beam of light is related to its wavelength $\lambda$ by the formula \begin{equation} f\lambda = \cc \end{equation} How do these quantities depend on the observer? Consider an inertial observer moving to the right in the laboratory frame who is carrying a flashlight that is pointing to the left; see Figure~6.6. Then the moving observer is traveling along a path of the form $x'=x'_1=\hbox{const}$. Suppose the moving observer turns on the flashlight (at time $t'_1$) just … text/html 2012-06-05T18:59:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/dot?rev=1338947940 In Euclidean geometry, (squared!)~distances can be described by taking the dot product of a vector with itself. Denoting the unit vectors in the $x$ and $y$ directions by $\xhat$ and $\yhat$, respectively, then the vector from the origin to the point $(x,y)$ is just \begin{equation} \rr = x\,\xhat + y\,\yhat \end{equation} whose (squared) length is just \begin{equation} |\rr|^2 = \rr\cdot\rr = x^2 + y^2 \end{equation} text/html 2012-06-05T21:12:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/dtrig?rev=1338955920 (The material in this chapter is based on the presentation in [ 11 ]. Consider once again the point $P=(r\,\cos\theta,r\,\sin\theta)$ on the circle of radius~$r$, as shown in Figure~3.2, and recall the geometric definition of the basic trigonometric functions in (Equations (1) and (2) of (ss)3.2) text/html 2010-07-13T17:57:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/einstein?rev=1279069020 All of the above problems come from the fact that, even without worrying about gravity, the surface of the Earth is not an inertial frame. An inertial frame is, roughly speaking, one in which Newton's laws do hold. Playing catch on a train is little different from on the ground --- at least in principle, and so long as the train is not speeding up or slowing down. Furthermore, an observer on the ground would see nothing out of the ordinary, it merely being necessary to combine the train's veloc… text/html 2012-05-28T19:09:24-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/emfield?rev=1338257364 Why have we done all this? Well, first of all, note that, due to antisymmetry, $\bT$ has precisely 6 independent components. Next, compute $\bT'$, using matrix multiplication and the fundamental hyperbolic trig (4) of (ss)4.1. As you should check for yourself, the result is \begin{equation} T'{}^{\mu\nu} = \pmatrix{ 0& a'& b'& c'\cr -a'& 0& f'& -e'\cr -b'& -f'& 0& d'\cr -c'& e'& -d'& 0\cr } \end{equation} where \begin{eqnarray} a' &=& a \\ b' &=& \cish b-f \\ c' &=& \cish c+e \\ d' &=& d… text/html 2012-06-05T15:40:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/emlorentz?rev=1338936000 We now investigate more general transformations of electric and magnetic fields between different inertial frames. Our starting point is the electromagnetic field of an infinite flat metal sheet, which is derived in most introductory textbooks on electrodynamics, such as Griffiths [ 6 ], and which we state here without proof. text/html 2012-05-28T19:08:29-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/energy?rev=1338257309 This mystery is resolved by recalling that momentum is mass times velocity, and that there is also a ``$t$-component'' to the velocity. In analogy with the 2-velocity, we therefore define the ``2-momentum'' to be \begin{equation} \bp = m \pmatrix{\cc\frac{dt}{d\tau}\cr \frac{dx}{d\tau}} = m\cc \pmatrix{\cosh\alpha\cr \sinh\alpha\cr} \end{equation} The second term is clearly the momentum, which we denote by $p$, but what is the first term? If the object is at rest, $\alpha=0$, and the fir… text/html 2012-01-01T14:01:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/formulas?rev=1325455260 The key formulas for analyzing the collision of relativistic particles can all be derived (3) and (4) of (ss)9.4. Taking the difference of squares leads to the key (7) of (ss)9.4 relating energy, momentum, and (rest) mass, which holds also for massless particles. (3) and (4) of (ss)9.4 leads directly to \begin{equation} \gamma = \cosh\alpha = \frac{E}{mc^2} = \frac{1}{\sqrt{1-\uucsq}} \end{equation} and \begin{equation} \sinh\alpha = \frac{p}{mc} = \uc\,\gamma \end{equation} and dividing the… text/html 2012-06-05T15:16:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/gr?rev=1338934560 Just as one studies flat (2-dimensional) Euclidean geometry before studying curved surfaces, one studies special relativity before general relativity. But the analogy goes much further. The basic notion in Euclidean geometry is the distance between two points, which is given by the Pythagorean theorem. The basic notion on a curved surface is still the distance ``function'', but this is now a statement about infinitesimal distances. Euclidean geometry is characterized by the line element \begi… text/html 2010-07-14T15:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/haddition?rev=1279145880 What is the slope of the line from the origin to the point $A$ in Figure 4.2? The equation of this line, the $\yy'$-axis, is \begin{equation} x = \yy \tanh\beta \end{equation} Consider now a line with equation \begin{equation} x' = \yy' \tanh\alpha \end{equation} What is its (unprimed) slope? Again, slopes don't add, but (hyperbolic) angles do; the answer is that \begin{equation} x = \yy \tanh(\alpha+\beta) \end{equation} which can be expressed in terms of the slopes $\tanh\alpha$ and $\tanh\… text/html 2012-06-05T21:52:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hdistance?rev=1338958320 We saw in the last chapter that Euclidean trigonometry is based on the circles, sets of points which are a constant distance from the origin. Hyperbola geometry is obtained simply by using a different distance function! Measure the ``squared distance'' of a point $B=(x,\yy)$ from the origin using the definition \begin{equation} \rho^2 = x^2 - \ysq \label{squared} \end{equation} Then ``circles'' of constant distance from the origin become hyperbolas with $\rho=\hbox{constant}$, and we further r… text/html 2010-07-13T18:09:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hprojections?rev=1279069740 We can ask the same question as we did for Euclidean geometry. Consider a rectangle of width 1 whose sides are parallel to the unprimed axes. How wide is it when measured in the primed coordinates? It turns out that the width of the box in the primed coordinate system is less than 1. This is length contraction, to which we will return in the next chapter, along with time dilation. text/html 2012-06-05T21:53:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hrotations?rev=1338958380 By analogy with the Euclidean case, we define a hyperbolic rotation through the relations \begin{equation} \pmatrix{x\cr \yy} = \pmatrix{\cosh\beta& \sinh\beta\cr \sinh\beta& \cosh\beta\cr} \pmatrix{x'\cr \yy'} \label{hyperrot} \end{equation} This corresponds to ``rotating'' both the $x$ and $\yy$ axes into the first quadrant, as shown in Figure 4.2. While this may seem peculiar, it is easily verified that the ``distance'' is invariant, that is, \begin{equation} x^2 - \ysq \equiv x'^2 … text/html 2012-05-28T19:19:28-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/htrig?rev=1338257968 The hyperbolic trigonometric functions are usually defined using the formulas \begin{eqnarray} \cosh\beta &=& \frac{e^\beta + e^{-\beta}}{2} \label{coshdef}\\ \end{eqnarray}\begin{eqnarray} \sinh\beta &=& \frac{e^\beta - e^{-\beta}}{2} \label{sinhdef} \end{eqnarray} and then \begin{equation} \tanh\beta = \frac{\sinh\beta}{\cosh\beta} \end{equation} and so on. We will discuss an alternative definition below. The graphs of these functions are shown in Figure 4.1. text/html 2011-12-08T10:49:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/htritrig?rev=1323370140 ~ Figure 4.3: A hyperbolic triangle with $\tanh\beta={3\over5}$. We now recast ordinary triangle trigonometry into hyperbola geometry. Suppose you know $\tanh\beta={3\over5}$, and you wish to determine $\cosh\beta$. One can of course do this algebraically, using the identity \begin{equation} \cosh^2\beta = {1\over1-\tanh^2\beta} \end{equation} But it is easier to draw any triangle containing an angle whose hyperbolic tangent is $3\over5$. In this case, the obvious choice would be the tria… text/html 2013-05-29T16:50:43-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwangles?rev=1369871443 \begin{enumerate}\item The mast of a sailboat leans at an angle $\theta$ (measured from the deck) towards the rear of the boat. An observer on the dock sees the boat go by at speed $v\ll c$ (so that you do not need to use relativity to do this problem). What angle does the observer say the mast makes? \item A child on the boat throws a ball into the air at the same angle $\theta$. What angle does the observer on the dock say the ball makes with the deck? (Ignore the subsequent influence of g… text/html 2012-04-07T17:37:51-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwcolliding?rev=1333845471 Consider the head-on collision of 2 identical particles each of mass $m$ and energy $E$. \begin{enumerate}\item In Newtonian mechanics, what multiple of $E$ is the energy $E'$ of one particle as observed in the reference frame of the other? \item In special relativity, what is the energy $E'$ of one particle as observed in the reference frame of the other? \item Suppose we collide 2 protons ($mc^2=1 \hbox{ GeV}$) with energy $E=30 \hbox{ GeV}$. Roughly what multiple of $E$ is $E'$ in this case?… text/html 2013-04-28T11:15:57-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwcolliding2?rev=1367172957 A pion of (rest) mass $m$ and (relativistic) momentum $p={3\over4}mc$ decays into 2 (massless) photons. One photon travels in the same direction as the original pion, and the other travels in the opposite direction. Find the energy of each photon. text/html 2012-06-05T15:31:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwcolliding3?rev=1338935460 A particle of mass $m$ whose total energy is twice its rest energy collides with an identical particle at rest. If they stick together, what is the mass of the resulting composite particle? What is its speed? Figure 10.5: Spacetime diagram for the collision of identical particles. text/html 2012-04-07T17:40:56-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwcolliding4?rev=1333845656 A pion (mass $m_\pi$) at rest decays into a muon (mass $m_\mu$) and a massless neutrino ($m_\nu=0)$. Find the momentum $p$, the energy $E$, and the speed $v/c$ of the muon in terms of $m_\pi$ and $m_\mu$. Figure 10.6: Spacetime diagram for the decay of a pion. text/html 2012-06-05T21:13:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwcosmic?rev=1338955980 Consider muons produced by the collision of cosmic rays with gas nuclei in the atmosphere 60 kilometers above the surface of the earth, which then move vertically downward at nearly the speed of light. The half-life before muons decay into other particles is 1.5 microseconds ($1.5\times10^{-6}$~s). \begin{enumerate}\item Assuming it doesn't decay, how long would it take a muon to reach the surface of the earth? \item Assuming there were no time dilation, approximately what fraction of the muons… text/html 2012-06-05T19:07:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwdoppler?rev=1338948420 ~ Figure 7.12: Computing Doppler shift. \begin{enumerate}\item A rocket sends out flashes of light every 2 seconds in its own rest frame, which you receive every 4 seconds. How fast is the rocket going? \end{enumerate} 1. First solution: This situation is shown in the first drawing in Figure~7.12. In order to find the hyperbolic angle $\alpha$, draw a horizontal line as shown in the enlarged second drawing, resulting in the system of equations \begin{eqnarray} \tanh\alpha &=& \frac… text/html 2012-06-05T15:13:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwem1?rev=1338934380 Suppose you know that in a particular inertial frame neither the electric field $\EE$ nor the magnetic field $\BB$ has an $x$ component, but neither $\EE$ nor $\BB$ is zero. Consider another inertial frame moving with respect to the first one with velocity $v$ in the $x$-direction, and denote the electric and magnetic fields in this frame by $\EE{}'$ and $\BB{}'$, respectively. \begin{enumerate}\item What are the conditions on $\EE$ and $\BB$, if any, and the value(s) of $v$, if any, such that … text/html 2013-03-25T16:16:51-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwem2?rev=1364253411 Suppose that in a particular inertial frame the electric field $\EE$ and magnetic field $\BB$ are neither perpendicular nor parallel to each other. \begin{enumerate}\item Is there another inertial frame in which the fields $\EE{}''$ and $\BB{}''$ are parallel to each other? \item Is there another inertial frame in which the fields $\EE{}''$ and $\BB{}''$ are perpendicular to each other? (You may assume without loss of generality that the inertial frames are in relative motion parallel to t… text/html 2012-06-05T19:04:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwgetaway?rev=1338948240 The outlaws are escaping in their getaway car, that goes ${3\over4}c$, chased by the police, moving at only ${1\over2}c$. Realizing they can't catch up, the police attempt to shoot out the tires of the getaway car. Their guns have a muzzle velocity (speed of the bullets relative to the gun) of~${1\over3}c$. \begin{enumerate}\item Does the bullet reach its target according to Galileo? \item Does the bullet reach its target according to Einstein? \item Verify that your answer to question 2 is th… text/html 2012-06-05T15:28:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwmass?rev=1338935280 Two identical lumps of clay of (rest) mass $m$ collide head on, with each moving at $\frac35 c$. What is the mass of the resulting lump of clay? First solution: We assume this is an elastic collision, that is, we do not worry about the details of the actual collision. Conservation of momentum doesn't help here --- there is no momentum either before or afterwards. So we need to use conservation of energy. After the collision, there is no kinetic energy, so we have \begin{equation} E' = Mc^… text/html 2012-06-05T21:59:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwpractice?rev=1338958740 \begin{enumerate}\item In a laboratory experiment a muon is observed to travel 800 m before disintegrating. The lifetime of a muon is $2\times10^{-6}~\hbox{sec}$, so the speed must be \begin{equation} v = \frac{800~\hbox{m}}{2\times10^{-6}~\hbox{sec}} = 4\times10^{8} \hbox{m/sec} \end{equation} which is faster than the speed of light! Identify the error in this computation, and find the actual speed of the muon. text/html 2012-02-03T22:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwtravel?rev=1328336220 Alpha Centauri is roughly 4 lightyears from Earth. Dr.~X travels (at constant velocity) from Earth to Alpha Centauri in 3 years. Immediately upon her arrival at Alpha Centauri, she turns on a powerful laser aimed at the Earth. (Ignore the motion of the Earth!) text/html 2011-12-01T14:33:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hwtwin?rev=1322778780 \begin{enumerate}\item On their 21st birthday, one twin gets on a moving sidewalk, which carries him out to star X at a speed $\frac45c$; the other twin stays home. When the traveling twin gets to star X, he immediately jumps onto the returning moving sidewalk and comes back to earth, again at speed $\frac45c$. He arrives on his 39th birthday (as determined by his own watch). text/html 2012-06-05T15:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/hyperboloid?rev=1338934620 We move now to three-dimensional Minkowski space, with ``squared distance'' function given infinitesimally (with $c=1$) by \begin{equation} ds^2 = dx^2 + dy^2 -dt^2 \end{equation} which can of course be positive, zero, or negative. We consider the hyperboloid \begin{equation} x^2 + y^2 - t^2 = -\rho^2 \label{3dmetric} \end{equation} where the radius $\rho$ is a positive constant, and we further restrict ourselves to the branch with $t>0$. text/html 2011-11-18T17:56:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/interval?rev=1321667760 Direct computation using the Lorentz transformations shows that \begin{eqnarray} x'^2 - \csq t'^2 &=& \gamma^2 \, (x-vt)^2 - \gamma^2 \,\left( \cc t - \vc x\right)^2 \nonumber\\ &=& x^2 - \csq t^2 \end{eqnarray} so that the quantity $x^2 - \csq t^2$, known as the interval, does not depend on the observer who computes it. We will explore this further in later chapters. text/html 2012-01-31T20:29:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/klein?rev=1328070540 Figure 14.5: Constructing the Klein disk. The Klein Disk model of hyperbolic geometry is also a projection of the hyperboloid. This time, however, we project from the origin into the plane $t=\sigma$. Thus, the point $U$, with coordinates given by (Equation (4) of (ss)14.2), would be mapped to the point \begin{equation} U_K = (\sigma\,\tanh\beta\cos\phi,\sigma\,\tanh\beta\sin\phi,\sigma) \end{equation} as shown in Figure~14.5, The image of the hyperboloid is again the interior of a disk, t… text/html 2013-04-01T20:04:08-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/length?rev=1364871848 We now return to the question of how ``wide'' things are. Consider first a meter stick at rest. In spacetime, the stick ``moves'' vertically, that is, it ages. This situation is shown in the first sketch in Figure~6.2, where the horizontal lines show the meter stick at various times (according to an observer at rest). How ``wide'' is the worldsheet of the stick? The observer at rest of course measures the length of the stick by locating both ends at the same time, and measuring the distanc… text/html 2012-06-05T21:56:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/lorentz?rev=1338958560 We now relate Lorentz transformations, based on the physical postulates of special relativity, to hyperbola geometry. The Lorentz transformation between a frame $(x,t)$ at rest and a frame $(x',t')$ moving to the right at speed $v$ was derived in Chapter 2. The transformation from the moving frame to the frame at rest is given by \begin{eqnarray} x &=& \gamma\,(x'+vt') \\ t &=& \gamma \left(t'+\vcsq x'\right) \end{eqnarray} where \begin{equation} \gamma = \frac{1}{\gammainv} \end{equation} Th… text/html 2012-06-05T15:34:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/magnetism?rev=1338935640 Our starting point is the electric and magnetic fields of an infinite straight wire, which are derived in most introductory textbooks on electrodynamics, such as Griffiths [ 6 ], and which we state here without proof. The electric field of an infinite straight wire (in vacuum) with charge density $\lambda$ points away from the wire with magnitude \begin{equation} E=\frac{\lambda}{2\pi\epsilon_0 r} \label{Eline} \end{equation} where $r$ is the perpendicular distance from the wire and $\epsilon… text/html 2011-03-31T22:54:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/manhole?rev=1301637240 There are two well-known paradoxes involving manhole covers, which illustrate some unexpected implications of special relativity. In the first, a 2-foot manhole cover approaches a 2-foot manhole at relativistic velocity. Since the hole sees the cover as much smaller than two feet long, the cover must fall into the manhole. It does. text/html 2013-05-18T17:55:02-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/maxwell?rev=1368924902 Maxwell's equations in vacuum (and in MKS units) are \begin{eqnarray} \grad\cdot\EE &=& \frac{1}{\epsilon_0} \,\rho \label{Gauss}\\ \end{eqnarray}\begin{eqnarray} \grad\cdot\BB &=& 0 \\ \end{eqnarray}\begin{eqnarray} \grad\times\EE &=& -\Partial{\BB}{t} \label{Faraday}\\ \end{eqnarray}\begin{eqnarray} \grad\times\BB &=& \mu_0\,\JJ + \mu_0\epsilon_0 \Partial{\EE}{t} \label{Ampere} \end{eqnarray} where $\rho$ is the charge density, $\JJ$ is the current density, and the constants $\mu_0$ and $\eps… text/html 2011-11-21T09:22:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/measure?rev=1321896120 Special relativity involves comparing what different observers see. But we need to be careful about what these words mean. A reference frame is a way of labeling each event with its location in space and the time at which it occurs. Making a measurement corresponds to recording these labels for a particular event. When we say that an observer ``sees'' something, what we really mean is that a particular event is recorded in the reference frame associated with the observer. This has nothing t… text/html 2012-05-28T19:03:42-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/momentum?rev=1338257022 Consider the ordinary velocity of a moving object, defined by \begin{equation} u=\frac{d}{dt}\,x \end{equation} This transforms in a complicated way, since \begin{equation} \cm \frac{dx'}{dt'} = \frac{\cm\frac{dx}{dt}-\vc}{1-\vcsq\frac{dx}{dt}} \label{dadd} \end{equation} The reason for this is that both the numerator and the denominator need to be transformed. Note that~($\ref{dadd}$) is just the Einstein addition formula for velocities, which we have therefore independently derived using Lo… text/html 2011-11-01T11:02:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/newton?rev=1320170520 Our daily experience leads us to believe in Newton's laws. When you drop a ball, it falls straight down. When you throw a ball, it travels in a uniform (compass) direction --- and falls down. We appear to be in a constant gravitational field, but apart from that there are no forces acting on the ball. This isn't the full story, of course, as we are ignoring things like air resistance and the spin of the ball. Nevertheless, it seems to give a pretty good description of what we observe, and s… text/html 2011-11-18T11:10:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/noneuclid?rev=1321643400 In two dimensions, Euclidean geometry is the geometry of an infinite sheet of paper. The postulates now exist in several different forms, but all address the basic properties of lines and angles. Key among them is the parallel postulate, which says that, given any line and a point not on that line, the exists a unique line through the given point which is parallel to the given line. text/html 2011-11-21T10:58:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/paddition?rev=1321901880 Suppose that, as seen from $O$, $O'$ is moving to the right with speed $v$ and that an object is moving to the right with speed $u$. According to Galileo we would simply add velocities to determine the velocity of the object as seen from $O'$: \begin{equation} u=u'+v \end{equation} This equation can be derived by differentiating the Galilean transformation \begin{equation} x=x'+vt \end{equation} thus obtaining \begin{equation} \frac{dx}{dt} = \frac{dx'}{dt}+v \end{equation} text/html 2011-11-17T20:36:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/paradoxes?rev=1321590960 It is easy to create seemingly impossible scenarios in special relativity by playing on the counterintuitive nature of observer-dependent time. These scenarios are usually called paradoxes, because they seem to be impossible. Yet there is nothing paradoxical about them! text/html 2012-01-31T20:21:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/poincare?rev=1328070060 The Poincar\'e Disk model of hyperbolic geometry is obtained from the hyperboloid by stereographic projection. Stereographic projection of an ordinary sphere maps the sphere to the $xy$-plane by projecting from the South Pole; points in the Northern Hemisphere ($z>0$) map to the interior of a disk, points in the Southern Hemisphere ($z<0$) map to its exterior, and the Equator ($z=0$) maps to the circle bounding the disk. text/html 2012-04-07T17:52:52-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/pole?rev=1333846372 A 20 foot pole is moving towards a 10 foot barn fast enough that the pole appears to be only 10 feet long. As soon as both ends of the pole are in the barn, slam the doors. How can a 20 foot pole fit into a 10 foot barn? This is the beginning of the Pole and Barn Paradox. It's bad enough trying to imagine what happens to the pole when it suddenly stops and finds itself in a barn which is too small. But what does the pole see? text/html 2012-01-01T06:10:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/postulates?rev=1325427000 The most fundamental postulate of relativity is Postulate I: The laws of physics apply in all inertial reference frames. The first ingredient here is a class of preferred reference frames. Simply put, an inertial (reference) frame is one without external forces. More precisely, an inertial frame is one in which an object initially at rest will remain at rest. Because of gravity, inertial frames must be in free fall --- a spaceship with its drive turned off, or a falling elevator. Gra… text/html 2011-11-21T08:56:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/preface?rev=1321894560 The unification of space and time introduced by Einstein's special theory of relativity is one of the cornerstones of the modern scientific description of the universe. Yet the unification is counterintuitive, since we perceive time very differently from space. And, even in relativity, time is not just another dimension, it is one with different properties. Some authors have tried to ``unify'' the treatment of time and space, typically by replacing $t$ by $it$, thus hiding some annoying minus… text/html 2012-04-07T17:55:27-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/problems?rev=1333846527 We began in Chapter 2. by using moving trains to model inertial reference frames. But we made an implicit assumption beyond assuming an ideal train, with no friction and a perfectly straight track. We also assumed that there was no gravity. Einstein's famous thought experiment for discussing gravity is to consider a ``freely falling'' reference frame, typically a falling elevator. Objects thrown horizontally in such an elevator will not be seen to fall --- there is no gravity (for a little … text/html 2012-06-05T21:21:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/projections?rev=1338956460 Consider the rectangular object of width 1 meter shown in the first sketch in Figure 3.6, which has been rotated so as to be parallel to the primed axes. How wide is it? As worded, this question is poorly posed. If width means ``extent in the $x'$ direction'', then of course the answer is 1 meter. If, however, width means ``extent in the $x$ direction'', then the answer is obtained by measuring the horizontal distance between the sides of the rectangle, which results in a value larger than… text/html 2012-01-29T17:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/proper?rev=1327886220 Let $\tau$ denote time as measured by a clock carried by an observer moving at constant speed $u$ with respect to the given frame. We call $\tau$ ``wristwatch time''~[ 2 ], or, more formally, proper time. Such a clock simply measures the ``length'' of the observer's worldline. But this length can be measured in any reference frame. text/html 2012-06-05T15:54:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/pseudosphere?rev=1338936840 Each of the preceding models of hyperbolic geometry has its advantages and disadvantages. The two Poincar\'e models are conformal, so angles are easy to measure, but lines are not obvious. Lines in the Klein model are obvious, but angles are hard to measure. And for all its similarity with the sphere, in the hyperboloid model neither lines nor angles are easy to determine. Furthermore, none of these models exists in ordinary (three-dimensional) Euclidean space. Perhaps that is no surprise; a… text/html 2012-06-05T15:59:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/references?rev=1338937140 \begin{enumerate}\item Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, W. H. Freeman, San Francisco, 1963. \item Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, second edition, W. H. Freeman, New York, 1992. \item text/html 2012-06-05T14:20:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/rotations?rev=1338931200 Now consider a new set of coordinates, call them $(x',y')$, based on axes rotated counterclockwise through an angle $\theta$ from the original ones, as shown in Figure 3.5. For instance, the $y$-axis could point towards true north, whereas the $y'$-axis might point towards magnetic north. In the primed coordinate system, the point $B$ has coordinates $(r,0)$, and the point $A$ has coordinates $(0,r)$, while the unprimed coordinates of $B$ and $A$ are $(r\cos\theta,r\sin\theta)$ and $(-r\sin\t… text/html 2012-06-05T14:36:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/rules?rev=1338932160 We begin by summarizing the rules for drawing spacetime diagrams. \begin{itemize}\item Points in spacetime are called events. \item Lines with slope $m=\pm1$ represent beams of light. \item Vertical lines represent the worldline of an object at rest. \item Horizontal lines represent snapshots of constant time, that is, events which are simultaneous (in the given reference frame). \item Lines with slope $|m|>1$ (called timelike) represent the worldlines of observers moving at constant speed. \it… text/html 2012-06-05T18:56:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/spacetime?rev=1338947760 We now return to the peculiar fact that the ``squared distance'' between two points can be positive, negative, or zero. This sign is positive for horizontal distances and negative for vertical distances. But these directions correspond to the coordinates $x$ and $t$, and measure space and time, respectively --- as seen by the given observer. But any observer's space axis must intersect the hyperbola $x^2-\csq t^2=\rho^2$ somewhere, and hence corresponds to positive ``squared distance''. Su… text/html 2012-04-07T12:12:49-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/surveyors?rev=1333825969 A brilliant aid in understanding special relativity is the surveyors' parable introduced by Taylor and Wheeler [ 1, 2 ]. Suppose a town has daytime surveyors, who determine North and East with a compass, and nighttime surveyors, who use the North Star. These notions of course differ, since magnetic north is not the direction to the North Pole. Suppose further that both groups measure north/south distances in miles and east/west distances in meters, with both being measured from the town cente… text/html 2012-01-01T13:52:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/tensors?rev=1325454720 Roughly speaking, tensors are like vectors, but with more components, and hence more indices. We will only consider one particular case, namely rank 2 contravariant tensors, which have two ``upstairs'' indices. Such a tensor has components in a particular reference frame which make up a $4\times4$ matrix, \begin{equation} T^{\mu\nu} = \pmatrix{ T^{00}& T^{01}& T^{02}& T^{03}\cr T^{10}& T^{11}& T^{12}& T^{13}\cr T^{20}& T^{21}& T^{22}& T^{23}\cr T^{30}& T^{31}& T^{32}& T^{33}\cr } \end{equa… text/html 2011-11-20T17:38:31-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/tidal?rev=1321839511 Consider now two objects falling towards the Earth, but far from it, as shown in the first sketch in Figure~13.3. Both objects fall towards the center of the Earth --- which is not quite the same direction for each object. Assuming the objects start at the same distance from the Earth, their paths will converge. Now, if they don't realize they are falling --- by virtue of being in a large falling elevator, say --- they will nevertheless notice that they are approaching each other. This is gr… text/html 2012-04-07T18:26:30-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/time?rev=1333848390 We now investigate moving clocks. Consider first the smaller dot in Figure~6.3. This corresponds to $\cc t=1$ (and $x=0$), as evidenced by the fact that this point is on the (other) unit hyperbola, as shown. Similarly, the larger dot, lying on the same hyperbola, corresponds to $\cc t'=1$ (and $x'=0$). The horizontal line emanating from this dot gives the value of $\cc t$ there, which is clearly greater than 1, and which represents the time according to the observer at rest when the moving … text/html 2012-06-05T21:31:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/timelength?rev=1338957060 We have seen that the postulates of relativity force the surprising conclusion that time is observer-dependent. We now examine this phenomenon in more detail. Consider again a train, of height $h$, with a beam of light bouncing up and down between mirrors on the floor and ceiling, as shown in the first sketch in Figure 2.4. The time between bounces can be interpreted as the ``ticks'' of a clock, and of course this interval, as measured on the train, is independent of whether the train is mov… text/html 2012-04-07T11:35:38-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/trig?rev=1333823738 Figure 3.2: Defining the (circular) trigonometric functions via a circle. Consider a point $P$ on the circle of radius $r$, as shown in Figure 3.2. The angle $\theta$ between the line from the origin to $P$ and the (positive) $x$-axis is defined as the ratio of the length the arc of a circle between $P$ and the point $(r,0)$ to the radius $r$. Denoting the coordinates of $P$ by $(x,y)$, the basic (circular) trig functions are then defined by \begin{eqnarray} \cos\theta &=& \frac{x}{r} \… text/html 2012-06-05T14:18:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/tritrig?rev=1338931080 Figure 3.3: A triangle with $\tan\theta={3\over4}$. An important class of trigonometric problems involves determining, say, $\cos\theta$ if $\tan\theta$ is known. One can of course do this algebraically, using the identity \begin{equation} \cos^2\theta = {1\over1+\tan^2\theta} \end{equation} But it is often easier to do this geometrically, as illustrated with the following example. text/html 2012-06-05T15:09:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/twin?rev=1338934140 One twin travels 24 light-years to star X at speed ${24\over25}c$; her twin brother stays home. When the traveling twin gets to star X, she immediately turns around, and returns at the same speed. How long does each twin think the trip took? Star X is 24 light-years away, so, according to the twin at home, it takes her 25 years to get there, and 25 more to return, for a total of 50 years away from earth. But the traveling twin's clock runs slow by a factor of $$\cosh\beta = \sqrt{1\over1-\ta… text/html 2012-06-05T15:17:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/uniform?rev=1338934620 Special relativity is about inertial observers, moving at constant velocity. Consider now an observer undergoing uniform acceleration. What does this mean? We start with Newton's Second Law in the form \begin{equation} F = \frac{dp}{dt} = ma \end{equation} but we use the relativistic notion of momentum, so that \begin{equation} p = mc\,\sinh\beta \end{equation} Setting $a=\hbox{constant}$ is equivalent to assuming that $p$ is a linear function of~$t$, leading to \begin{equation} \sinh\beta = \… text/html 2012-01-01T15:06:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/unify?rev=1325459160 In our brief tour of relativistic electromagnetism, we have seen how special relativity unifies physical concepts previously regarded as separate. Space and time become spacetime. Energy and momentum become 4-momentum. Charge and current densities become the 4-current density. The scalar and vector potentials become the 4-potential. And, finally, the electric and magnetic fields become the electromagnetic field tensor. text/html 2017-07-19T10:27:55-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/vaddition?rev=1500485275 What is the rapidity $\beta$? Consider an observer moving at speed $v$ to the right. This observer's world line intersects the hyperbola \begin{equation} \csq t^2 - x^2 = \rho^2 \qquad (ct>0) \end{equation} at the point $A=(\rho\,\sinh\beta,\rho\,\cosh\beta)$; this line has ``slope'' \begin{equation} \vc = \tanh\beta \end{equation} as required. Thus, $\beta$ is nothing more than the hyperbolic angle between the $\cc t$-axis and the worldline of a moving object. As discussed in the precedi… text/html 2012-01-01T13:50:00-08:00 http://sites.science.oregonstate.edu/coursewikis/GSR/book/content/vectors?rev=1325454600 In the previous chapter, we used 2-component vectors to describe spacetime, with one component for time and the other for space. In the case of 3 spatial dimensions, we use 4-component vectors, namely \begin{equation} x^\nu = \pmatrix{x^0\cr x^1\cr x^2\cr x^3\cr} = \pmatrix{\cc t\cr x\cr y\cr z\cr} \end{equation} These are called contravariant vectors, and their indices are written ``upstairs'', that is, as superscripts.