Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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This 25 page Class Notes was uploaded by Clement Bernier on Wednesday October 21, 2015. The Class Notes belongs to PHY 312 at Syracuse University taught by Don Bunk in Fall. Since its upload, it has received 9 views. For similar materials see /class/225620/phy-312-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
PHY312 lecture 16 Simon Catterall Summary of lecturelS V 7 9 Exterior spacetime to spherically symmetric time independent nonspinning gravitational source is given by Schwarzschild metric 1 A52 A7 C2At2 A7 2 7 2A62 7 A with An 1 9 7 5 2GMc2 Schwarzschild radius Event horizon Discussed motion of photon relative to far away observer shell observer free fall observer L J PHY312 lecture 16 p 2 What about massive particle W In SR learnt that correct formula for energy is At E 2 5R mc T It is conserved n GR the correct generalization is 2GM At EGR 77102 1 327 AT It is also conserved Energyatinfinity Key to analysing radial motion J PHY312 lecture 16 p 3 Principle of maximal ageing j Consider flat spacetime first Watch free particle move from event 00 thru 23513 to final T X Think of initial and final events as fixed and 23513 as variable Call two parts of motion A and B Principle of maximal ageing says i dz where 739 TA 73 This yields 1 d 2 1 d 2 g 2 Z J 27A dt 2TB dt PHY312 lecture 16 p 4 Continued In SR we have 32731 C2232 512 327 32T t2 X x2 leading to Thus the path followed by a free particle is one in which the quantity tT is conserved ie fixed This is another way to see that Em 31 is correct definition of energy Advantage can now generalize to radial motion in Schwarzschild metric J PHY312 lecture 16 p 5 Finally Think of events T 7 Ar 25 7 and O 7 Ar 7 corresponding to the geodesic motion of a test particle in the spacetime Require again that 1 dr 1 7123 27A dt 2TB dt But now TA etc depends on metric Setr Ar TA and 7 Ar TB Find 1 T t 1403105 AU B TB J PHY312 lecture 16 p 6 Falling into 21 BH 7 7 o Consider particle released from rest at infinity Energy is mc2 and is conserved Conservation of energy 2 1 2GM A152 A72 327 Combine with metric g 2GM 1 2GM At C 327 327 L J PHY312 lecture 16 p 7 Interpretation 7 7 o Notice as for light speed goes to zero for far away observer as 7 gt 7 5 9 Thus to far away observer objject never crosses the horizon I But light is redshifted away so we don t see this o What about a shell observer Using the expressions Arshell 14 12A7 and Atshell A12At 1 2 Arshell C Atshell 027 To himher the object approaches the speed of light L J PHY312 lecture 16 p 8 Energy measured by shell observer 7 7 0 Using for v ArshellAtsheu find E 2GM 1 C27 Eshell where E mc2 is energy measured by far away observer 9 As 7 gt R5 energy available to local observer becomes infinite J PHY312 lecture 16 p 9 Conclusions 7 7 Again there are no contradictions here Neither energy nor velocity are invariant physical quantities in SR or GR L J PHY312 lecture 16 p 10 Time to crunch j Once pass event horizon object will reach 7 O in finite proper time d7 Need E clth Conservation of energy means dt 1A dT 7 So find a 2GM 12 AT C 327 Integrate to find tota proper time T0 What is T0 for solar mass black hole J PHY312 lecture 16 p 11 PHY312 lecture 11 Simon Catterall Recap j Ingredients of GR principle of equivalence principle of general relativity Importance of tidal gravity Consequences of POE slowing of clocks in gravitational field deforming of spatial geometry bending of light Important hints perhaps tidal gravitational force corresponds to free motion in curved spacetime Simplest example of curved space surface of sphere Geodesics J PHY312 lecture 11 p 2 Curved Space Simplest example of nonEuclidean geometry 7 Study it need to employ a coordinate system eg x y z with 512 342 22 R2 Or spherical polar coordinates 7 6 gb with 7 R To talk about geometry we need to talk about distances between nearby points 6 qb and 6 A6 gb Agb lfl know all these distances I can recconstruct the surface A52 R2 A62 sin2 6A 2 J PHY312 lecture 11 p 3 Metric Can rewrite in a suggestive way A52 911Ax12 922Ax22 where x1 6 x2 gb and 911 R2 922 R2 Sin2 9 911 and 922 are components of an object called the metric Knowledge of the metric at each point uniquely determines the physical surface geometry Flat space 911 922 1 For curved geometry metric will vary from point to point J PHY312 lecture 11 p 4 Lengths V 7 o Ifwe know metric can compute distances between 2 points which are not close together 0 Consider the path gb constant varying 6 L R d6 7TB 0 Example of a geodesic Shortest path between two points Look locally like straight lines Set of paths with 6 constant varying gb 27 L Rsind dgb 27tRsin6 0 L Not a geodesic J PHY312 lecture 11 p 5 Geodesics V 7 p Any truly curved surface possesses neighboring geodesics which do not remain parallel over whole surface L J PHY312 lecture 11 p 6 r J 0 Curvature j How can one tell locally that the surface is not flat Ans compute the curvature Take a small vector arrow and move it around a closed path on the surface so that it always makes a fixed angle with the tangent vector to the curve at that point this is called parallel transport Compare the final vector with the initial vector and determine the angle of rotation that has been induced by the motion Then the curvature is given by the formula R net rotation angle surface area enclosed by loop J PHY312 lecture 11 p 7 Curvature of sphere j Take path that runs from equatorto N pole Then along similar path back to equator arriving 14 way along equator from start What is curvature of sphere Thus we don t need to look at the sphere from a higher dimensional space to see it is curved We can determine that property by just moving on loops on the sphere and measuring angles For general curved space these loops must be small and the curvature varies from point to point What happens for R gt 00 and the sphere approaches a flat plane PHY312 lecture 11 p 8 General coordinate systems j One final complication Want to describe sphere using any coordinate system Not just one whose basis vectors are at 900 to each other Eg even in flat space we might want to transform from Cartesian system x y to some other system 513 y 513 ax by y ca dy Original distance l2 x 2 y 2 is now given by l2 a2 c2932 b2 d2y2 2ab cday Thus need offdiagonal components of metric 912 and 921 J PHY312 lecture 11 p 9 General formula for length 7 7 2 2 L2 2721 231 gijxixj Metric is a matrix at each point of space b 9 Similarly curvature also has several components and varies over surface tensor 0 Mathematical description gets involved tensor calculus and we won t get into to it here L J PHY312 lecture 11 p 10 Curved spacetime j Proceed by analogy as 23513 y z Spacetime coordinate Coordinate system replaced by frame of reference Distance in spacetime squared given by differences in nearby spacetime coordinates weighted by metric 4 4 ds2 Z Z guyddexV 1 V21 Question What is the metric of special relativity Locally can always use a frame of reference coordinate SyStem SUCh that 911 I 07922 I 17933 I 17944 I 1 and all other components zero FFF Principle of equivalence J PHY312 lecture 11 p 11 Curvature II V 7 Mathematical expression exists for curvature in spacetime analogous to the curvature of space 0 If RneO cannot find a FFF frame where metric is simple everywhere presence of tidal gravity L J PHY312 lecture 11 p 12 Motion W In SR we learned that free motion corresponds to straight worldlines and a maximal proper time It is hence geodesic motion in a flat spacetime In GR we postulate that free motion corresponds also to maximal proper time but now in general the geodesic worldlines are curved In this way we remove the force of gravity and replace it by geodesic motion in curved spacetime Notice this structure automatically encompasses the principle of general relativity since in such a geometric theory the curved spacetime is real any coordinate system FOR is equally good to describe it 4 PHY312 lecture 11 p 13 To do If Need to flesh out this geodesic motion 7 0 Need to give an equation which tells us how curved is spacetime This must have something to do with the distribution of matter and energy Einstein s field equa ons 9 Show that these reduce to Newton s theory for small velocities and weak gravity L J PHY312 lecture 11 p 14
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