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# Relativity and Cosmology Einstein and Beyond PHY 312

Syracuse

GPA 3.93

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This 2 page Class Notes was uploaded by Ms. Bryce Wisoky on Wednesday October 21, 2015. The Class Notes belongs to PHY 312 at Syracuse University taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/225639/phy-312-syracuse-university in Physics 2 at Syracuse University.

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Date Created: 10/21/15

Revision Sheet PHY312 You are responsible for the following including knowing any equations written here I Einstein s theory of GR rests on two principles H Principle of Equivalence there is no way to distinguish locally between an inertial frame and a free falling frame of reference 0 Principle of General Coordinate Invariance an observer in an arbitrary frame of refer ence should be able to discover the same laws of physics all frames are equivalent I What do we mean by locally 27 If we do experiments of a limited precision over a small region of spacetime we cannot distinguish a free falling frame over a truly inertial frame ie we do not know whether a gravitational eld is present However for a large enough spacetime region we will be able to infer the presence of gravity via its tidal gravitational e ects that is two initially separated particles viewed from my freely falling frame will appear to approach each other I Einstein pictured tidal gravity as arising from underlying spacetime curvature Particles falling freely in a gravitational eld are pictured as following geodesics in the curved spacetime A geodesic path between two points is the straightest possible path between the points and corresponds to the maximum possible proper time Einstein s eld equations relate the density of spacetime curvature at some point in spacetime to the density of energy and mass at that point I Curved spacetime speci ed by giving a rule to compute spacetime distances at every point a metric I Know the formula for the Schwarzschild metric Ar2 Air3A Ari i EMA 1 s a r M r lt This describes a spherically symmetric source of gravitation with Ar 1 The event horizon r 2GMCg I Know the formulae for the radial distance a shell observer one static with respect to the black hole would measure in terms of far away or r coordinate r And also the time between 2 events as measured on his watch given the time measured by a far away observer t Ar Ar 7 2 shell A Atsheu AWN 3 I The energy E for a particle moving in this geometry is conserved 2 E E E me 1 TAT 4 For a particle released from rest the expression for the speed measured by a far away observer is Aquot c EA Kw lt5 The speed measured by a shell observer is given by 739 C i T For general orbital motion we have another conserved quantity the angular momentum L A Lzmr lg Arsheu Atsheu 6 7 Be able to sketch the form of the effective potential for material particles in orbit around a black hole Understand the signi cance of drawing horizontal lines of constant energy Know which points of the picture correspond to stable circular orbits which to elliptical orbits and which yield the possibility of capture Under assumptions experimental evidence favors this of homogeneity and isotropy know what those terms mean the spacetime of the Universe looks like a set of 3d spaces evolving in a cosmic time t These spaces may be positive curvature sphere at or negative curvature hyperboloid If we look at the solutions of Einstein s equations for this type of geometry we nd solutions in which the size of the Universe is changing with time In fact it seems as if the Universe was of zero size a nite time ago the Big Bang Shortly after the Big Bang the Universe was very hot and it subsequently cooled as the Universe expanded The evidence for this comes from the cosmic 3 the elements nucleosynthesis and Hubble s law of light Hubble s law two galaxies separate with a speed proportional to their distance apart The standard Big Bang cosmology has several problems most prominent of which is the observation that the observed Universe is much more homogeneous than one would have expected These problems can be solved by assuming a period of very rapid expansion in the early Universe in ation The question of what will happen to the Universe in the future is determined by the matter density and pressure the value of the cosmological constant and the sign of the spatial curvature Modern observations favor a at spatial geometry and a non zero cosmic acceleration

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