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##### Einstein’s Universe

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**Einstein’s Universe**Tuesday, February 5**Einstein – Newton smackdown!**Two different ways of thinking about gravity and space.**The Way of Newton:**Space is static (not expanding or contracting) and flat. (“Flat” means that all Euclid’s laws of geometry hold true.)**“Objects have a natural tendency to move on straight lines**at constant speed.” However, we see planets moving on curved orbits at varying speed. How do you explain that, Mr. Newton?**“There is a force acting on the planets – the force**called GRAVITY.” The gravitational force depends on a property that we may call the “gravitational mass”, mg.**Fg = gravitational force**Mg = mass of one object mg = mass of other object r = distance between centers of objects G = “universal constant of gravitation” (G = 6.7 × 10-11 Newton meter2 / kg2)**Newton gave another law that gives the acceleration in**response to any force (not just gravity)! The acceleration depends on a property that we may call the “inertial mass”, mi.**If a gravitational force is applied to an object with**gravitational mass mg and inertial mass mi, its acceleration is**Objects falling side-by-side have the same acceleration (the**same mg/mi).**Truly astonishing and fundamental fact of physics:**mg = mi for every known object! This equality is known as the “equivalence principle”.**The equivalence principle (which Newton leaves**unexplained) led Einstein to devise his theory of General Relativity. Let’s do a “thought experiment”, of the kind beloved by Einstein.**Two ways of thinking about a bear:**1) Bear has constant velocity, box is accelerated upward. 2) Box has constant velocity, bear is accelerated downward by gravity.**Two ways of thinking about light:**1) Light has constant velocity, box is accelerated upward. 2) Box has constant velocity, light is accelerated downward by gravity.**Einstein’s insight:**Gravity affects the paths of photons, even though they have no mass! Mass and energy are interchangeable: E = mc2**Einstein**Newton Mass & energy are very different things. Mass & energy are interchangeable: E = mc2 Space & time are interchangeable: part of 4-dimensional space-time. Space & time are very different things.**Light takes the shortest distance between two points.**In flat space, the shortest distance between two points is a straight line. In the presence of gravity, light follows a curved line. In the presence of gravity, space is not flat, but curved!**A third way of thinking about a bear:**3) No forces are acting on the bear; it’s merely following the shortest distance between two points in space-time.**The Way of Newton:**Mass tells gravity how much force to exert; force tells mass how to move. The Way of Einstein: Mass-energy tells space-time how to curve; curved space-time tells mass-energy how to move.**Objects with lots of mass (& energy) curve**space (& distort time) in their vicinity.**The Big Question:**How is the universe curved on large scales (bigger than stars, bigger than galaxies, bigger than clusters of galaxies)? That depends on how mass & energy are distributed on large scales.**The cosmological principle:**On large scales (bigger than stars, galaxies, clusters of galaxies) the universe is homogeneous and isotropic.**There are three ways in which space can have homogeneous,**isotropic curvature on large scales. (Apologetic aside: describing the curvature of 3-dimensional space is difficult; I’ll show 2-d analogs.)**Measuring curvature is easy…in principle.**Flat: angles of triangle add to 180° Positive: angles add to >180° Negative: angles add to <180°**Curvature is hard to detect on scales smaller than the**radius of curvature. Flat = good approximation Flat = bad approximation**d**a d = Sun-star distance a=Sun-Earth distance (1 A.U.) Parallax (flat space) August p p February p = a/d**Parallax (positivecurvature)**August p p February p < a/d**Parallax (negativecurvature)**August p p p > a/d As d→infinity, p→a/R February**Bright idea:**The smallest parallax you measure puts a lower limit on the radius of curvature of negatively curved space. Hipparcos measured ‘p’ as small as 0.001 arcsec; radius of curvature is at least 1000 parsecs.**We need Bigger trianglesto measure the curvature accurately!**d L =L/d (flat) >L/d (positive) <L/d (negative)**The most distant objects we can see are the hot & cold spots**on the Cosmic Microwave Background.**Universe became transparent when it was 350,000 years old.**The largest hot & cold spots are about 700,000 light years across. Hot & cold spots should be 1° across …if the universe is flat.**But we’ve measured the size of the hot & cold spots, and**they are 1° across! Consistent with flat (Euclidean) space.**What Would Einstein Say?**Density > critical density: Positive curvature Density < critical: Negative curvature Since our universe is close to flat, the density must be close to the critical density.**Thursday:**Midterm Exam Bring your calculator and your favorite writing utensil.