Black Holes and Curved Space-time

1. Black Holes and Curved Space-time • When a massive star collapses at the end of its life, it can become a black hole • A black is an object that is so massive that light cannot escape from it • The theory that describes gravity is called general relativity • Put forward by Einstein in 1916 • Based in simple principles but makes interesting predictions • Light does not always travel in a straight line Lecture 21

2. The Principle of Equivalence • The principle of equivalence states that the force of gravity is equivalent to force from acceleration • You cannot tell if you are floating in space or falling freely in a gravitational field • The laws of physics are the same in every inertial reference frame Lecture 21

3. Paths of Light and Matter • Let’s try a thought experiment (Einstein’s favorite technique) • Suppose the space shuttle is in orbit and one of the astronauts shines a laser from the back of the shuttle to the front • While the light is traveling, the shuttle is “falling” out of its straight path • The light would strike B’ instead of B if light were not bent by gravity Lecture 21

4. Implications of Gravity Bending Light • The space shuttle has mass while light does not • Einstein postulated that it was actually the fabric of space-time that was distorted by gravity • The light still travels in a straight line in space-time by that space was warped by the presence of a massive object • Mass tells space-time how to curve and the curvature of space tells matter how to move • The amount of distortion depends on the mass of the object • Everyday objects do not have enough mass to distort space-time Lecture 21

5. Space-time Examples • Let’s imagine we are driving to Minneapolis • We stop for lunch on the way • We can plot the space-time for this trip using 1 dimension • Imagine an ant walks across a rubber sheet • Now put a heavy weight on the rubber sheet and let the ant walk across again • The ant always walks in a straight line but his path through space-time is curved Lecture 21

6. Facts from Relativity • One’s perception of space and time depends on one’s reference frame • For these effects to be noticeable, the relative speed must be an appreciable fraction of the speed of light • Some interesting facts • The clock in a moving frame runs slow from your point of view • A fast-moving object appears to be shortened • If you are moving fast, everything around you seems to speed up Lecture 21

7. Relativistic Effects on Mercury’s Orbit • The orbit of Mercury is very eccentric • Normally the point at which Mercury is farthest from the Sun would always be the same • However, this point moves around the Sun • Precession of the perihelion Differences exaggerated for pedagogical purposes • The motion of the other planets causes part of this precession • 0.531 seconds of arc is predicted • 0.574 seconds of arc is measured • The difference is caused by the warping of space-time by the Sun • When Mercury is close to the Sun, there is a small additional push due to the warping of space Lecture 21

8. Deflection of Light • A second prediction of general relativity was that light would be bent by gravity • Einstein suggested that starlight bent by the Sun could be observed during a solar eclipse • Indeed was observed by Eddington in 1919 but not to great accuracy (20%) • Recent measurements with radio waves confirm Einstein’s predictions to 1% Lecture 21

9. Gravitational Lensing • Massive invisible objects can create images of distant objects HST picture showing gravitational lensing • On the left is an animation of gravitational lensing by a spiral galaxy Lecture 21

10. Time in General Relativity • General relativity predicts that time runs more slowly in a strong gravitational field • Experiments with atomic clocks have shown that indeed time runs more slowly in higher gravitational fields • In 1976, Viking sent a radio pulse to Earth from the other side of the Sun • The timing of the pulse agreed with relativity to within 0.1% Lecture 21

11. Black Holes • Astronomer had long speculated that stars might exist that have escape velocities faster than the speed of light • Relativity states that gravity is a curvature in space-time • A sufficiently large mass could distort space-time so that nothing, even light, could escape • If an object gets within the event horizon of a black hole, the object will disappear from our view Lecture 21

12. Event Horizon • Karl Schwarzchild calculated mathematically using relativity that an event horizon would exist around a black hole in 1916 • The event horizon does not get smaller as a collapsing star compresses • The Schwarzchild radius depends only on the mass in the black hole • About 3 km for 1 Msun • Feed the black hole, and the radius will grow • Collapse a globular cluster to a black hole and it will have a radius of 300,000 km • The gravity from black holes is just like normal star as long as you are not too close to it Lecture 21

13. A Trip into a Black Hole • We cannot see into a black hole but that does not prevent us from trying to calculate what it must be like inside a black hole • In the center there is a singularity in space-time • The laws of physics as we know them break down • Suppose a spaceship decides to enter a black hole and send out a light pulse every second • The light pulses would get farther apart and have longer wavelength as the ship went in until no more pulse would arrive • The ship would not return • Tidal forces would destroy the ship Lecture 21

14. Approaching a Black Hole Lecture 21

15. Circling a Black Hole Lecture 21

16. Evidence for Black Holes • One way to study black holes is to look at binary star systems where one partner has become a black hole • Matter will be sucked from the normal star to the black hole giving off visible radiation • Accretion disk Lecture 21

17. Gravity of a Black Hole • As long as you are a reasonable distance from the black hole, the black hole acts as is it mass is concentrated at the center (and it really is!) • Newton’s laws are in effect • The event horizon is small so ordinary distances (like 1 AU) are safe from a black hole of 1 to a few 10s of solar masses • Only if you approach to with a few solar diameters will the tidal effects and relativistic effects become apparent • Einstein’s law come into effect Lecture 21

18. The Milky Way Galaxy • The Milky Way Galaxy can be seen in the night sky as a streak of dim white light stretching from horizon to horizon • All the stars we can see in the night sky are in the Milky Way • The Milky Way is a flat, spiral galaxy containing 200 billion stars • The picture on the right is the center of the Milky Way taken with infrared Lecture 21

19. The Architecture of the Milky Way • In 1785 William Herschel discovered that the Sun belongs to a group of stars on the shape of a disk • He observed that the Sun was near the center of this system and the disk was about 6000 LY across • We now know that dust blocked Herschel’s view and the Milky Way is much larger than Herschel thought • The full disk is about 100,00 LY across • The Sun is not at the center of the Galaxy Lecture 21

20. Globular Clusters and the Center of the Galaxy • The center of the Milky Way is shrouded with dust so that one cannot look far along the plane of the Galaxy • However, one can see much farther if you look up through the dust instead of along the disk • By looking out of the plane of the Galaxy, Shapley observed globular clusters in 1917 and their distances were measured using Cepheids and RR Lyrae stars • Shapley found that the globular clusters were distributed spherically and their center was far from the Sun • Shapley postulated that these globular clusters defined the galaxy size Lecture 21

21. Overview of the Galaxy Picture toward the center of the Galaxy Schematic Diagram of the Galaxy Picture of the Galaxy taken in near infrared • The Galaxy has been studied using infrared radiation • Most of the stars and dust of the Galaxy are found in the thin disk of the Galaxy • The Galaxy disk is embedded in a spherical halo of faint stars that extends to a distance of 50,000 LY • Close to the center, the stars are no longer confined to the disk but form a nuclear bulge consisting of old stars Lecture 21

22. Interstellar Matter in the Galaxy • Radio observations shows that the Galaxy’s cold atomic hydrogen is confined to an extremely flat layer that is about 400 LY thick • In the plane of the Galaxy, this cold hydrogen extends out 80,000 LY from the center • Dust is also confined to the plane of the Galaxy being about the same thickness as the hydrogen gas but more concentrated in the spiral arms and toward the Galactic center • Large molecular are found in the spiral arms along with many young stars Lecture 21

23. The Spiral Arms of the Galaxy • The Milky Way has four spiral arms • Sagittarius-Carina, Perseus, Cygnus arms • We are located in a sub-arm called the Orion arm • There is a fourth, unnamed arm difficult to observe • The spiral arms are thought to have arisen from differential galactic rotation • The spiral arm structure is thought to get frozen in by a spiral density wave that occurs when material encounters the material in the spiral arms • Other mechanisms such as chain reaction formation of stars may also be responsible Lecture 21

24. Two Kinds of Stars • The stars in our Galaxy can be divided into two groups • Population I stars • Bright blue stars in the spiral arms • Found only in the Galactic disk • Follow nearly circular orbits around the Galactic center • The Sun is population I • This group includes many young stars • Population II stars • Stars in the halo, nuclear bulge, and globular clusters • No correlation with the spiral arms • Found throughout the Galaxy • Can be found in elliptic orbits out of the plane of the disk • This group consists entirely of old stars 12 billion years old Lecture 21

25. The Mass of the Galaxy • The Sun orbits the Galactic center every 225 million years • We can use Kepler’s laws to calculate the mass of the Galaxy inside the Sun’s orbit • Most of the luminous matter of the Galaxy is within 30,000 LY of the center • However, there is a large amount of invisible matter which causes distant objects to orbit faster Lecture 21

26. Dark Matter • The mass of the galaxy is 10 times larger than the mass of the observable objects • This mass is not in the form of • stars (we could see them) • gas (we could observe the neutral or ionized gas) • dust (would obscure major parts of the Galaxy) • This mass cannot be in the form of black holes, white dwarfs, or neutron stars • Could be brown dwarfs or massive planets or huge black hole • Could be unknown elementary particles • WIMPs Lecture 21

27. The Nucleus of the Galaxy • The center of the Galaxy is a crowded and complicated place • The first radio source found was the Galactic center • There is strong evidence that a massive black hole is at the center of the Galaxy • The VLA has shown that the radio source at the Galactic center, Sagittarius Ao, has a radius of 10 AU • Star orbit measurements show that the Galactic center is a million times denser than any known star cluster Lecture 21

28. Finding the Source • There could have been a black hole formed at the center of the Galaxy early in the history of the Galaxy • Matter is falling into the center at a rate of 1 Msun per 1000 years • At this rate, a black hole with a mass several million times the mass of the Sun could have been formed • Our galaxy is not the only one with a black hole in the center • Supermassive black holes have been observed in the center of other galaxies Lecture 21

29. The Protogalactic Cloud • Because the oldest stars are distributed in a sphere centered around the nucleus of the Galaxy, it is logical to assume that a protogalactic cloud was roughly spherical • Like in star formation, the cloud collapsed and formed a thin, rotating disk • Gravity caused the disk to clump into star clusters • There are problems with this idea in the form of “backwards clusters” that are several billion years older than the Galaxy Lecture 21

30. Collisions of Galaxies • In 1994 astronomers discovered a satellite galaxy of the Milky Way, the Sagittarius dwarf galaxy • It is approaching the Milky Way and is being torn apart by tidal forces • The Large and Small Magellanic Clouds are satellite galaxies that are spiraling closer and closer to the Milky Way • Tidal forces have pulled out dust trails in front and behind these galaxies • There are 8 other nearby galaxies that appear to have split off from the Magellanic Clouds • The Galaxy formed in two stages • The process described from a protogalactic cloud • Other stars and clusters were captured, retrograde motion Lecture 21

31. Galaxies View of very distant galaxies by the HST • In 1925 Edwin Hubble announced that the Andromeda Galaxy was a separate galaxy from the Milky Way • Previously most astronomers believed that the Milky Way was the only galaxy • Now we know that there millions and maybe billions of other galaxies Lecture 21

32. Spiral Galaxies • Spiral galaxies (like the Milky Way and Andromeda) consist of a nucleus, a halo, and spiral arms • Interstellar matter is spread throughout the disks and bright nebula and hot young stars are present • The disks are often dusty which shows when the galaxies are viewed edge-on Lecture 21

33. Types of Spiral Galaxies • Spiral galaxies rotate such that their arms appear to trail • A third of spiral galaxies have bars running through their nuclei and another third have faint bars • There are many different shapes ranging from mostly nucleus to mostly disk • Spiral galaxies contain many young stars and lots of gas and dust Lecture 21

34. Elliptical Galaxies The giant elliptical galaxy M87 with globular clusters surrounding it • Elliptical galaxies contain mainly old stars and are shaped like spheres or ellipsoids • Their light is dominated by old reddish, population II stars • Dust and gas are not conspicuous in elliptical clusters • The stars do not all orbit the center in the same direction like spirals Lecture 21

35. Irregular Galaxies Large Magellanic Cloud Small Magellanic Cloud • Everything else besides spiral and elliptical is called irregular • The best known irregular galaxies are the Large and Small Magellanic Clouds Lecture 21

36. Masses of Galaxies • The masses of spiral galaxies can be measured as the mass of the Milky Way was measured • Kepler’s law, period of star orbits • Elliptic galaxies do not rotate so Kepler cannot help us • We can look at the broadening of the absorption lines and calculate the average speed of all the stars • Elliptic galaxies are the most massive and the least massive • Irregular galaxies have less mass than spirals Lecture 21

37. Mass-to-Light Ratio • We can characterize galaxies by their mass to light ratio with the Sun mass-to-light ratio being 1 • Because galaxies have many small stars, their mass-to-light ratio is generally greater than 1 • Young galaxies have a mass-to-light ratio of 1 to 10 • Older galaxies have a mass-to-light ratio of 10 to 20 • As much as 90% of the mass of galaxies is not visible in any electromagnetic wavelength • Dark matter may have a mass-to-light ratio as high as 100 Lecture 21

38. Distance Scale for Galaxies • The distance to stars was measured using variable stars • Locating variable stars in galaxies is difficult • There are two kinds of Cepheids and they were mixed up and the distance scale to galaxies had to be increased by a factor of 2 in the 1950s • Variable stars are only visible in nearby galaxies so we need a new method • The method is to notice that galaxies come in groups and clusters with average characteristics • This method of the “standard bulb” is always used with caution Lecture 21

39. New Techniques • These new distance measuring techniques have been calibrated with nearby galaxies • The first method is to exploit the relationship between rotational velocity and luminosity for spiral galaxies • Use 21 cm radiation to measure rotational velocity • The second method involves the measurement of the bumpiness of the apparent surface of an elliptical galaxy • The less bumpy, the farther away • The two methods are complementary Lecture 21

40. The Expanding Universe • We know that the universe is expanding • The discovery of this expansion began with the search for Martians and other solar systems • Vesto Sipher, working for Lowell, was asked to measure the spectra from spiral nebula to search for chemical compositions expected for newly forming planets • The spiral nebulae are very dim and exposures of 20 to 40 hours were required • It took 20 years to measure the spectra from 40 nebulae Lecture 21

41. Redshifts • What Sipher found was that these spiral nebulae showed astounding redshifts • The lines in the spectra were all shifted toward longer wavelengths • They were all moving away from us at high speed • Only a few spirals such as M31, which we now know to be a close neighbor, were moving towards us • These measurements were announced in 1914, years before Hubble found that these spiral nebula were galaxies and before anyone knew how far away they were Lecture 21

42. The Hubble Law • In the 1920s, Edwin Hubble found a way to estimate the distance to the galaxies • He found, along with Humason, that there was a direct relationship between the distance to the galaxy and it velocity of recession • v = Hd, Hubble Law • H is the Hubble constant Lecture 21

43. Hubble’s Law • Every galaxy measured showed the relationship of velocity and distance • Implies that the universe is expanding • Having this relationship allows astronomers to measure the distance the galaxies just by measuring their redshift • Redshifts have been measured up to 90% of the speed of light • This method allows astronomers to measure the distance of galaxies far beyond previous methods Lecture 21

44. Implications of the Hubble Constant • The best measurement of H was done in 1999 using the Hubble Space Telescope viewing the spiral galaxy NGC 4603 • The distance was measured using Cepheid variable stars • The value of H was measured to be: • 70  7 km per second per megaparsec • 20  2 km per second per million light years • 1/H is the age of the universe • 15  1.5 billion years Lecture 21

45. Models for an Expanding Universe • At first one might think that an expanding universe in all directions means we are at the center of the universe • However, in a uniformly expanding universe, we and all other observers, must see the same expansion Lecture 21