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General Relativity II Gravitational time dilation Curved space-time & Einstein’s theory The General Theory of Relativity Einstein’s equations Some general motivations Consequences of GR Orbit of Mercury Gravitational lensing Gravitational waves I : GRAVITATIONAL TIME DILATION Recap of waves: Waves characterized by Wavelength () = distance between crests Frequency (f or ) = number of crests passing a given point per second Speed of a crest; c= Energy of a wave is proportional to frequency . The electromagnetic spectrum Small wavelength High Frequency High energy Large wavelength Low frequency Low energy Remember the tower… Laser light must lose energy as it climbs up So…frequency must decrease i.e., light is redshifted. Gravitational redshifting Imagine a clock based on frequency of laser light… 1 “tick” = time taken for fixed number of crests to pass Gravitational redshifting slows down the clock. Clocks in gravitational fields run slowly GM t grav 1 2 tspace cr if gravitatio nal field is " weak" How to live for a 1000 years! Observer on Earth would see astronauts clock running very slowly when close to black hole – astronaut would age very slowly. Gravitational time dilation has practical importance! Global Positioning System (GPS) System of satellites that emit timing signals Detector on Earth receives signals Can figure out position on Earth’s surface by measuring time delay between signals from different satellite. Need to measuring timing signal from satellite very well! If GR effects were not included, GPS positions would drift from true position by kilometers per day! MACH PRINCIPLE Frictionless experiment II: CURVED SPACE-TIME Einstein pondered several things… Success of Special Relativity showed that space & time were closely linked The “tower thought experiment” suggested that free-fall observers are (locally) free of effects of gravity He wanted to say that gravity was an illusion caused by the fact that we live in an accelerating frame… … but there is no single accelerating frame that works! Somehow, you need to stick together frames of reference that are accelerating in different directions Einstein’s suggestion 4-dimensional space-time is curved Free-falling objects move on “geodesics” (generalizations of straight lines) through curved space-time. Matter and energy causes space-time to bend. What is a geodesic? Shortest path between two points on a surface E.g. path flown by aircraft Geodesics that start parallel can converge or diverge (or even cross). Another example – a “saddle” Geodesics diverge Curved space around the Earth looks something like this… From web site of UCSD III: THE GENERAL THEORY OF RELATIVITY Within a free-falling frame, the Special Theory of Relativity applies. Free-falling particles/observers move on geodesics through curved space-time The distribution of matter and energy determines how space-time is curved. “Space-time curvature tells matter/energy how to move. Matter/energy tells space-time how to curve.” Notes: 8G G 4 T c The Einstein curvature tensor “G” is mathematical object describing curvature of 4-D space-time. The Stress-Energy tensor “T” is mathematical object describing distribution of mass/energy. Newton’s constant of gravitation “G” and the speed of light “c” appear as fundamental constants in this equation. This is actually a horrendous set of 10 coupled non-linear partial differential equations!! For weak gravitational fields, this gives Newton’s law of gravitation. IV: GR EFFECTS IN THE SOLAR SYSTEM Have already heard about bending of star light by the Sun (detected by Eddington). Orbit of Mercury: Mercury does not move in perfect ellipse but precesses-> Vulcan? sun Effect called “precession of perihelion”. Effect small - orbit twists by 5600 arc-seconds (1.56 degrees) per century With Newtonian gravity, can explain 5557 arcseconds/century as due to Gravitational effect of other planets, deformation of the Sun, non-inertial nature of Earth’s frame But still leaves 43 arc-seconds per century unexplained… Using GR, Einstein predicted (with no fiddling!) that Mercury should precess 43 arcseconds per century! V : THE BENDING OF LIGHT (GRAVITATIONAL LENSING) “The Einstein Cross” Gravitational micro-lensing Individual stars can also make a gravitational lens… microlensing. Suppose we… Look at a distant star in our galaxy Another massive (but dark) star passes in front… From web site of Ned Wright (UCLA) Causes apparent increases in brightness of stellar image VI: GRAVITATIONAL WAVES Particular kind of phenomena (e.g. orbiting stars) produce ripples in the space-time curvature… Ripples travel at speed of light through space These are called gravitational waves. Features of gravitational waves… Usually extremely weak! Only become strong when massive objects are orbiting close to each other. Gravitational waves carry energy away from orbiting objects… lets objects spiral together. The grand challenge – to compute the spiralling together of two black holes. How do we know that these waves exist? The binary pulsar (PSR1913+16) Russell Hulse & Joseph Taylor (1974) Discovered remarkable double star system Nobel prize in 1993 From Nobel Prize website Two neutron stars orbiting each other One neutron star is a pulsar – Neutron star is spinning on its axis (period of 59ms) Emits pulse of radio towards Earth with each revolution Acts as a very accurate clock! Interesting place to study GR Orbit precesses by 4 degree per year! Orbit is shrinking due to gravitational waves Very precise test of certain aspects of GR Direct detection of gravitational waves… How do you search for gravitational waves? Look for tidal forces as gravitational wave passes Pioneered by Joseph Weber (UMd Professor) Estimated wave frequency (10000Hz) Looked for “ringing” in a metal bar caused by passage of gravitational wave. Weber claimed detection of waves in early 1970s Never verified – but Weber held out to the end… Modern experiments : LIGO Laser Interferometer Gravitational Wave Observatory Two L-shaped 4km components Hanford, Washington Livington, Louisiana Will become operational very soon! Can detect gravitational waves with frequencies of about 10-1000Hz. VERY sensitive… need to account for Earthquakes and Geological movement Traffic and people! What will it see? Stellar mass black holes spiraling together Neutron stars spiraling together LISA Laser Interferometer Space Antenna Space-based version of LIGO Sensitive to lower-frequency waves (0.0001 – 0.1Hz) Can see Normal binary stars in the Galaxy Stars spiraling into large black holes in the nearby Universe. Massive black holes spiraling together anywhere in the universe!