Curved SpacetimeA2290-33 1Curved SpacetimeRelativity and AstrophysicsLecture 33Terry HerterA2290-33 Curved Spacetime 2Outline A series of video clips Curvature Mass in units of length Coordinates Schwarzschild metric Spatial part Time partCurved SpacetimeA2290-33 2A2290-33 Curved Spacetime 13Curvature Curvature The geometry of spacetime need not be “flat” This means that the “distance” between “straight lines” that are initial parallel changes with space & time (spacetime) Spacetime intervals between events evidence the geometry of spacetime – its curvature Does not depend on choice of a reference frameStart parallel at equatorPositive CurvatureNegative Curvature2D AnalogiesA2290-33 Curved Spacetime 14Spatial Curvature A flat plane has zero curvature, a sphere has positive curvature, and a hyperboloid has negative curvature.FlatC = 2rPositiveC < 2rNegativeC > 2rCurved SpacetimeA2290-33 3A2290-33 Curved Spacetime 15Reduce Circumference Spherical symmetry Many objects exhibit a symmetry such that their properties change only (or nearly so) with radius e.g. stars, planets, black holes Rotation and other factors can break exact spherical symmetry  However, a great deal of physics can be learn with a much simpler mathematical approach r-coordinate Derive from measurement of circumference Can’t always get to the center with your stick (see above examples) Also call reduced circumference or coordinate radius When space (around a object) is curved – The directly measured distance between two concentric spherical shells is different than the difference between the r-coordinates of the shells. For gravity around a mass2ncecircumfererdxr2= C2/2r1= C1/212rrdxA2290-33 Curved Spacetime 16Mass in Units of Length Natural units: In spacetime (special relativity), it is natural to deal with time in unit of meters By analogy in gravitation (general relativity), it is natural to deal with mass in units of meters This enhances the link of mass to geometry What is the conversion factor to give mass in meters? Newton’s law of gravity: So that Which allows us to convert mass from kilograms to meters2rmGMFkgkg2310skgm106726.6G221623102/sm109876.8/kg/sm106726.6cGkgm10424.728kgMMkgm10424.728kgMcGM2Curved SpacetimeA2290-33 4A2290-33 Curved Spacetime 17Masses of Astronomical Objects“Geometric mass” is the conversion of mass to units of meters (multiplying by G/c2)4109km310961039Black hole in Quasar 3.8106km2.61065.21036Black hole at Galactic Center696,0001.477 km1.01.9891030Sun3.00310-6Mass(Msun)63710.444 cm5.97421024EarthRadius(km)GeometricMassMass(kg)ObjectA2290-33 Curved Spacetime 18Coordinates Orbits are planar The motions of bodies around a spherically symmetric object are confined to a plane.  This is because of symmetry – no difference between “below” and “above” the plane Polar coordinates We need two spatial coordinates to specify the motion A “natural” then to choose is polar coordinates which specify a radial distance (r-coordinate) and azimuthal angle, .xyvmrcosrxsinry222yxr xytanConverting between Cartesian and polar coordinatesandCurved SpacetimeA2290-33 5A2290-33 Curved Spacetime 19Flat Spacetime in Polar Coordinates In Cartesian coordinates the spatial separation, ds, between two objects is222dydxds ddrrd12222rddrds In polar coordinates, the distance between points 1 and 2 isFor spherically symmetric systems it is convenient to use a different form for the spacetime interval, or metric for describing spacetime. For a timelike interval:2222rddrdtd This “metric” still represents flat space time, just in polar coordinates. The standard shorthand (but not technically correct) notation is:22222drdrdtd A2290-33 Curved Spacetime 20Schwarzschild Metric In the presence of a spherically symmetric massive body, the flat spacetime metric is modified. This metric is found by solving Einstein’s field equations for general relativity Karl Schwarzschild found the solution within a month of the publication of Einstein’s general theory of relativity This metric is the solution for curved empty spacetime on a spatial plane through the center of a spherically symmetric (nonrotating) center of gravitational attraction The timelike form of the solution is: where= angular coordinate with the same meaning as in Euclidean geometry  r = the reduced circumference.  t = the “far-away” time which is measured on clocks far away from the center of attraction.222222121drrMdrdtrMd