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How the simple measurements of distances and positions of planets and stars have led to great discoveries in science 2005 is the World Year of PHYSICS 100th anniversary of Albert Einstein’s “miraculous year” of 1905 March 1905: the quantum nature of light May 1905: Brownian motion shows the existence of atoms and molecules June 1905: Special Relativity as a theory of space, time, and motion General Relativity Developed in 1907-1915 in close collaboration with mathematicians: Grossmann, Hilbert, Levi-Civita ... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. Marcel Grossmann David Hilbert Tullio Levi-Civita Measuring distances: a surveyor’s method A surveyor’s method in space: measuring the parallax effect an apparent shift of the object on the sky when viewed from different locations One needs larger baseline than the distance between the eyes to measure the distance to cosmic objects! In 1672 Giovanni Cassini together with Jean Richter (1630-1696) made parallel observations of the Mars parallax in Paris (France) and Cayenne (French Guiana, N. coast of South America) They were also able to determine the solar parallax as ~ 9 arcseconds and find the distance to the Sun (Astronomical Unit) as 140,000,000 km. Current value is 8.8 arcseconds, or 149,597,892 km. B D B D A Parallax angle A 1 arcsecond = (1 degree)/3600 The angle between these two lines is 2.5 degrees The angle of 9 arcseconds is 1000 times smaller! One needs baselines larger than the size of the Earth! Aristarchus of Samos (~310-230 BC) Using the distance between the Earth and the Moon as a baseline Using the Earth’s orbit as a baseline Even for the nearest stars the parallax is 0.3 second of arc. This is 10,000 times smaller than 1 degree! Aristotle (384-322 BC) could not find measurable parallax, so he concluded the Earth was not moving Retrograde (westward) motion of a planet occurs when the Earth passes the planet. The Copernican Revolution Nicolaus Copernicus (1473 – 1543): Heliocentric Universe (Sun in the Center) Born: 19 Feb 1473 in Torun, Poland Died: 24 May 1543 in Frombork, Poland Church cleric, but rejected a 2000-yr old paradigm Seven axioms written in a pamphlet “Little Commentary” (1514) 1. There is no one centre in the universe. 2. The Earth's centre is not the centre of the universe. 3. The centre of the universe is near the sun. 4. The distance from the Earth to the sun is imperceptible compared with the distance to the stars. 5. The rotation of the Earth accounts for the apparent daily rotation of the stars. 6. The apparent annual cycle of movements of the sun is caused by the Earth revolving round it. 7. The apparent retrograde motion of the planets is caused by the motion of the Earth from which one observes. Johannes Kepler (1571 – 1630) Kepler’s Laws of Planetary Motion 1. The orbits of the planets are ellipses with the sun at one focus. c Eccentricity e = c/a A New Era of Science I. If F = 0, velocity v const Change of motion = acceleration a v / t II. ma F A ball experiences acceleration there must be a force! Acceleration (a) is the change of a body’s velocity (v) with time (t): a a = v/t Velocity and acceleration are directed quantities (vectors)! v Different cases of acceleration: 1. Acceleration in the conventional sense (i.e. increasing speed) 2. Deceleration (i.e. decreasing speed) 3. Change of the direction of motion (e.g., in circular motion) Curved path of the Moon there must be some force! Newton’s law of gravitation Newton’s Cannon Measurements of parallaxes of the Sun, Moon, planets and stars seemed to fit Newton’s theory amazingly well … Clockwork universe One little speck on the brilliant face of Newton’s theory: The advance of the perihelion of Mercury Mercury: the closest planet to the Sun Perihelion = position closest to the sun Mercury Sun Perihelion: 46 million km; Aphelion: 70 million km Aphelion = position furthest away from the sun In reality the orbits deviate from elliptical: Mercury's perihelion precession: 5600.73 arcseconds per century Newtonian perturbations from other planets: 5557.62 arcseconds per century Remains unexplained: 43 arcseconds/century (Le Verrier 1855) Predicted the presence and position of Neptune from irregularities in Uranus’s orbit Neptune was found in 1846 exactly at the predicted position Urbain Le Verrier 1811-1877 In the eyes of all impartial men, this discovery [Neptune] will remain one of the most magnificent triumphs of theoretical astronomy … Arago I do not know whether M. Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most detested. A contemporary In 1855 Le Verrier found that the perihelion of Mercury advanced slightly more than the Newtonian theory predicted. He and others tried to explain it with a new planet Vulcan, new asteroid belt, etc. By the beginning of the XX century, it became clear that Newtonian gravity has other problems Problem with Action at a Distance Gm1m2 F1 F2 r0 2 r m1 F1 F2 m2 If ball 1 moves, ball 2 instantaneously feels it. Direct, instantaneous connection between cause and effect! Faster than light propagation?? Einstein’s idea: Newton’s theory is a weak-gravity limit of a more general theory: General Relativity Even in the weak gravity of the Earth and the Sun, there are measurable deviations from Newtonian mechanics and gravitation law! • Precession of Mercury’s perihelion • Bending of light by the Sun’s gravity General Relativity predicts new effects, completely absent in the Newton’s theory: black holes, event horizon, gravitational waves. Gravity is a strange force. It has a unique property: All bodies in the same point in space experience the same acceleration! m mM F G 2 R R M F M a G 2 m R Acceleration of Gravity Iron ball Wood ball Acceleration of gravity is independent of the mass of the falling object! This means that in the freely-falling elevator cabin you don’t feel any effects of gravity! You and all objects around you experience the same acceleration. In outer space you can imitate the effect of gravity by acceleration. "You mighta seen a house fly, maybe even a superfly, but you ain't never seen a donkey fly!" Donkey Equivalence Principle In 1907, Einstein was preparing a review of special relativity when he suddenly wondered how Newtonian gravitation would have to be modified to fit in with special relativity. At this point there occurred to Einstein, described by him as the happiest thought of my life , namely that an observer who is falling from the roof of a house experiences no gravitational field. He proposed the Equivalence Principle as a consequence:... we shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame. Immediate consequences of the Equivalence Principle: • Bending of light in the gravitational field • Time flow and frequency of light are changed in the gravitational field The bulb emits flashes of light 2 times per second c Acceleration a = gravity g An observer on the floor receives flashes faster than 2 times per second First observed on the Earth by Pound and Rebka 1960: relative frequency shift of 10-15 over the height of 22 m. Light should be bent in the gravitational field If gravity can be eliminated by motion, no special force of gravity is needed! The force of gravity is actually the acceleration you feel when you move through space-time How to explain that in the absence of any force the trajectories are not straight lines? Because space and time are curved! All bodies experience the same acceleration, but only in a small region of space. In another region this acceleration is different. Time flows with a different rate, and paths are bent differently in these two regions. m M a1 G 2 R1 M a2 G 2 R2 R1 M R2 Main idea: Space-time gets curved by masses. Objects traveling in curved spacetime have their paths deflected, as if a force has acted on them. “Curvature” of time means that the time flows with a different rate in different points in space "Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move." John Wheeler Shortest paths are called geodesics; they are not straight lines! The shortest path between two cities is not a straight line Low density star High density star About 1912 Einstein realized that the geometry of our world should be non-Euclidean. He consulted his friend Grossmann who was able to tell Einstein of the important developments of Riemann, Ricci and Levi-Civita. G.F.B. Riemann (1826-1866) When Planck visited Einstein in 1913 and Einstein told him the present state of his theories Planck said: As an older friend I must advise you against it for in the first place you will not succeed, and even if you succeed no one will believe you. Several versions of Einstein’s GR in 1913-1914 were wrong. Only in November 1915, after correspondence with Levi-Civita and Hilbert, Einstein published a paper with correct equations. Hilbert also published correct equations, in fact 5 days earlier than Einstein. On the 18th November Einstein made a discovery about which he wrote For a few days I was beside myself with joyous excitement . He explained the advance of the perihelion of Mercury with his theory. Bending of light Two British expeditions in 1919 confirmed Einstein’s prediction. The shift was about 2 seconds of arc, as predicted Gravitational lensing Gallery of lenses (Hubble Space Telescope) Black holes A black hole is a region of space-time from which nothing can escape, even light (first suggested by Laplace in 1796). Schwarzschild radius 1 GMm 2GM 2 mc Rs 2 2 Rs c Derivation is wrong and picture is wrong, but the result is correct To make a black hole from a body of mass M, one needs to squeeze it below its Schwarzschild’s radius 2GM Rs 2 c Rs K. Schwarzschild Schwarzschild radius: event horizon for a nonrotating body No signals can reach an outside observer from inside the event horizon! This is a point-of-no-return for everything that crosses it. The curvature of a 2D slice of a spherically symmetric black hole Curvature becomes infinite as we approach the singularity r =0 Gravitational bending of light paths around a black hole Approaching a black hole Circling around a black hole Time dilatation Rs 2 ds 1 dt r 2 t o t Rs 1 r t0 t As measured by a distant observer, clocks slow down when approaching a massive object Tidal forces and contraction of space squeeze and stretch the astronaut. Lateral pressure is 100 atm at a distance of 100 Rs from the event horizon Black holes are NOT big cosmic bathtub drains! Approaching a black hole R ~ Rs (strong field): gravity pull runs away Far from a black hole R >> Rs (weak field): Newtonian gravity law holds GM ag R 2 R 1 s R GM ag 2 R If our Sun collapses into a black hole, we won’t see any difference in the gravitational pull (but it will be VERY cold) Black holes in the Universe 1. Formation of galaxies 2. Collapse of massive stars 3. Early Universe? How to find the object that does not emit any radiation? By its effect on nearby objects The Galactic Center Our view (in visible light) towards the galactic center (GC) is heavily obscured by gas and dust Extinction by 30 magnitudes Only 1 out of 1012 optical photons makes its way from the GC towards Earth! Galactic center Wide-angle optical view of the GC region If one looks at this region with big telescopes and nearinfrared cameras one can see lots of stars. If one takes pictures every year it seems that some stars are moving very fast (up to 1500 kilometers per second). The fastest stars are in the very center - the position marked by the radio nucleus Sagittarius A* (cross). Distance between stars is less that 0.01 pc A Black Hole at the Center of Our Galaxy By following the orbits of individual stars near the center of the Milky Way, the mass of the central black hole could be determined to ~ 2.6 million solar masses Black hole vicinity is probably very messy … Cores of other galaxies show an accretion disk with a possible black hole “All hope abandon, ye who enter here” Dante Second scenario of black-hole formation: Death of massive stars Gravitational collapse of the iron core Supernova explosion Mass transfer in a binary system Binary systems 3 If we can calculate the total mass and measure the mass of a normal star independently, we can find the mass of an unseen companion a M1 M 2 2 ; P a – in AU P – in years M1+M2 – in solar masses Still, this is an indirect evidence. Proving the existence of black holes remains one of the greatest problems in physics. 1912 You cannot tell the difference between acceleration and gravity!