Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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This 6 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 12 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
Lecture 9 Recap 0 Recap GR rests on two postulates Principle of Equivalence mg 2 m or the statement Freely falling frames cannot be distinguished locally from inertial frames Principle of General Relativity Any frame can be used to discover the laws of physics 0 Actually gravitation can be distinguished from pure acceleration because of tidal effects two otherwise free particles situated some distance apart in a gravitational field will follow trajectories which converge diverge even when viewed from a FFF 0 Also discussed evidence that gravity influences spacetime Specif ically can be thought of as giving rise to curved or nonEuclidean geometries Such curved spaces can be described via a mathe matical object called a metric 0 We have also seen that free particles in GR follow paths which are closest to straight lines in an underlying curved spacetime these are called geodesics They correspond to a maximum proper time between events in curved spacetime o What determines the curvature Energymomentum Sum marised heuristically in Einsteins famous eld equations curvature Rdensity of energymomentum T 0 These boil down to 10 nonlinear differential equations which determine the spacetime geometry in terms of the distribution of energy and momentum in some region 0 True equations are relations between tensor elds 0 Reduce to Newtonian gravity in the weak eld limit Differ markedly for strong gravity large curvature o If we know the latter can predict the motion of test particles or equivalently if we observe certain motions of bodies can infer the geometry 0 Very hard to solve Only very symmetric cases can be done at all solution can be found for spherical symmetry which describes exterior of stars or black holes 0 Big effort to numerically solve these equations for predicting vi olent astrophysical events like black hole collisions Still enor mously hard and technical business still at least a decade away from this goal supercomputers grand challenge etc 0 We will spend the next few weeks discussing some simple solu tions of E s equations First one for spherical symmetry For this we don t need to know the formalism of tensor calculus Schwarzschild solution 0 Imagine the spacetime around a spherically symmetric source of gravity such as a star or a black hole 0 At some distance we can imagine measuring the circumference of a circle which passes through the center of the black hole For ordinary geometry we would divide this by 27139 to get a radius call it the rccordinate Do this for two different distances from the center giving two rcoordinates T1 and r2 What is the distance between the two shells with these radii T1 and r2 In ordinary Euclidean geome try it would be just 71 73 What happens if drop a plumbline from 71 down to T2 and measure it directly We will nd that it is actually larger than this I This is our rst example of curved space For the Sun radius 695980 km Find another shell with r coordinate 1 km greater 695981 km The distance between the two is 2 mm greater than 1 km l Spacetime curvature here is rather small For a black hole of one solar mass Imagine nding a shell with radius r 4km And let the outer one have radius r 5km But here the directly measured radial distance is 172km l Space time curvature is very large here I Time is also affected in such a region of spacetime Imagine sending a light signal out radially from some rcoordinate As it climbs out of the gravitational eld it loses energy This results in its period increasing If we think of the arrival of wavecrests at some point as like the ticking of a clock we can see the arriving wavecrests are further apart in time than the sent ones Thus the time measured between clock ticks will have a different value close to the black hole than far away Thus time is also distorted in a gravitational eld the curvature is in spacetime I This increase of period in light leads to the socalled gravitational redshift characteristic emission lines from atoms get shifted to longer wavelength the red end of the spectrum Inside the event horizon of the black hole this shift is in nite photons are shifted to zero energy but this means no photons l How do we get these numbers how do we specify the spacetime geometry We use the metric In general the spacetime distance between two neighboring events in a curved spacetime is given by the formula A32 ggWAr AV In SR this is just the formula for the spacetime interval with 911 C and gm 1 for all 239 2 3 4 Notice that the metric components are all constant in space i this indicates that the metric of special relativity is very special 7 it is a metric for flat spacetime sometimes called Minkowski spacetime The prin ciple of equivalence just says that I can also find a cooordinate system locally in which the metric takes this form ie smooth curved spaces always look locally flat Polar coordinates Rewrite this SR metric formula in polar co ords that is we specify a point by its distance from some origin 7 and an angle 0 from a fixed direction The spatial separation is then Al2 2 Ar T2A gt2 The spacetime distance then looks like A32 8M Ar 73302 1 Looks different in this coordinate system but still corresponds to flat spacetime Solution of GR for spherical symmetry outside an attractive mass M is very similar Schwarzschild 1915 9 Ar 2 2 2 2 As cArAt A TAG 2 The function AV is given by AO 1 o Einstein wrote to Karl Schwarzschild I had not expected that the exact solution to the problem could be formulated Your analytic treatment seems to me splendid This is valid for timelikc separated events For spacelike events multiply by minus 1 This is nonrotating uncharged spherically symmetric structure The solution for a spinning black hole was only published in 1963 almost 50 years later Holds all info on the external spacetime Note 7 gt oo corrections go to one at spacetime Also for XVI gt O correct at spacetime again Valid strictly for vanishing increments eg Ar gt 0 Metric tells us that measure of local distance varies with position If we consider a two simultaneous events in this spacetime which are at the same angle then we can immediately read off the expression for the distance between them from the spacclz39kc form for the spacetime interval Ar AW AS Arsheu 3 So predicts the distance measured by an observer stationary at some rcoordinate would be greater than the difference in the r coordinate Ar Such an observer is often called a shell observer since he lives on a certain shell or rcoordinate 0 Similarly if consider a clock at that shell and imagine measuring the time between ticks the metric timelike form gives a new shell time which is smaller than the time measured by a faraway observer At m ArAt 4 o Applies to outside of nonrotating objects Sun spins slowly so its a very good approximation Black holes have no surface applies all the way down to r O 0 Notice that if r MiG2 A is zero time shift factor is zero In nite red shift This is the event horizon a surface through which nothing not even light can pass out o What is At It is faraway time the time between two events as recorded with a clock which is remote from r O 0 Also note that time and space interchange in the formula for the metric if r lt MiG2 ie rcoordinate becomes like time This has consequences 7 once one has passed into this region of spacetime arrival at the center 7 0 occurs as inevitably as the passing of time in the external region 0 Notice that other coordinate systems can also be used to view black holes and the like the shell frame the free fall frame and this global rt frame
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