# The interpretation is that gravity bends spacetime and that light follows the curvature of space.

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1 7/8 General Theory of Relativity GR Two Postulates of the General Theory of Relativity: 1. The laws of physics are the same in all frames of reference. 2. The principle of equivalence. Three statements of the principle of equivalence: 1. The inertial mass (the one that appears in Newton s second law, F = ma) is the same as the gravitational mass (the one that appears in Newton s law of gravity). 2. No experiment can be done in a closed elevator that will distinguish between being in free fall in a gravitational field or moving with constant velocity in gravity-free space. 3. No experiment can be done in a closed elevator that will distinguish between being at rest in a gravitational field or having a uniform acceleration in gravity-free spece. Consequences of the principle of equivalence (#3) 1. Gravity is a manifestation of the curvature of space. Follow the path of a light pulse in an elevator accelerating in gravity-free space. The dashed line in the top figure shows the path of the pulse as observed by an observer who remains at rest outside the elevator. Each different color rectangle represents the position of the elevator at the given time. The x s represent the locations of the pulses at each given time. The bottom figure shows the path of the pulse as seen by an observer inside the elevator. This observer sees that the pulse falls from the upper left corner of the elevator to the lower right corner. By the principle of equivalence, the same thing must happen in a elevator at rest here on Earth. (Note that we don t see this effect because of the large value of the speed of light. It travels so fast that it doesn t fall a measurable distance in the Earth s weak gravity.) We see that gravity has an effect on light bends the path of the light. The interpretation is that gravity bends spacetime and that light follows the curvature of space. Einstein showed that it is the presence of matter and energy in space that causes spacetime to curve.

2 Einstein s equations are equations for the metric of spacetime. Einstein s Field Equations schematically: Curvature = mass and energy Consider the Pythagorean theorem: Not true in this form on a curved surface. On a curved surface, we have this: The g s are called the metric of the space. The two middle ones are always the same and one can always choose a frame in which they are zero. Einstein s field equations are equations for the metric once we have the metric, we can predict the paths of particles and light through the space. Analogies: 1. Imagine a surgical rubber sheet stretched over a wooden frame. Place a large ball bearing on the sheet, and it will stretch the sheet. A second small ball bearing placed on the sheet next to the first ball bearing will roll toward it as if there were a force gravity! between the two. 2. Consider two bodies traveling with constant speed along two lines of longitude, starting at the equator and heading toward the north pole. As they travel, we see them move closer together because the two lines of longitude intersect at the north pole. From our vantage point in three dimensions, we easily see this to be the case. But a two-dimensional observer on the surface of the sphere won t be aware of the third dimension and will interpret this moving together of the two bodies as a force gravity! between the two. GR is based on the idea that gravitational fields and accelerations are equivalent Principle of Equivalence. If we do an experiment in an accelerating spaceship we will get same results as if we did the experiment sitting at rest in a gravitational field say, at the Earth s surface.

3 Another result of GR time runs slow in a gravitational field. Put John at the bottom of the spaceship and Marsha at the top. John and Marsha both have clocks. At time zero on both clocks, two things happen the spaceship begins to accelerate and John sends a pulse of light toward Marsha. The light pulse moves toward Marsha and eventually reaches her. She sees this as the first tick on John s clock. A time 1 second on John s clock, he sends a second pulse. Since Marsha is now moving away at some speed, it will take longer for this second pulse to reach Marsha than it did for the first pulse to reach Marsha. Marsha measures the time between pulses to be greater than 1 second, that is, greater than the time that John measures. Marsha concludes that John s clock is ticking more slowly than once a second. His clock runs slow. The same thing must happen for an experiment done at rest on Earth. GR has been tested in a number of different ways and has always met predictions. 1. Bending of starlight near the edge of the Sun during and eclipse is correctly predicted by GR. 2. Due to time dilation, GR predicts that laser light should drop in frequency as it travels from the basement to the attic of a building. The experiment agreed with the predictions of GR. 3. GR correctly accounts for a 43 seconds of arc per century precession of Mercury s orbit. 4. GR correctly accounts for a degrees per year precession in a binary neutron star system. 5. There are some competing theories to GR so called scaler/tensor theories. Predictions about the characteristics of binary systems in which on object is much more massive that the other are different in GR and these theories. Recently, a binary system containing a neutron star and a white dwarf where the ratio of the masses is about 12 were measured, and the results predicted by GR agreed with these observational results. GR and Black Holes: If we viewed the contraction of a black hole, we would see the contraction come to a standstill as

4 gravity became strong enough at its surface to bring time to a stop. The point at which this occurs is called the Schwartzchild radius or event horizon. Called the event horizon because at this point the escape speed is equal to the speed of light if light cannot escape, nothing can. We can calculate the Scwartzchild radius of a black hole A black hole with the mass of the Sun would have a radius of 3 km, so we can write where M is the mass of the black hole in solar masses. GR gives us some idea about what goes on inside a black hole. Inside the black, the contraction continues. GR predicts it will shrink to a singularity mathematical point of infinite density. Quantum Mechanics will probably prevent that from happening, but we don t understand the physics yet so we can t be sure. It turns out that an isolated black hole radiates it gets smaller it evaporates Hawking Radiation If a virtual particle-antiparticle pair were to be created near the event horizon of a black hole, one could escape while the other is trapped inside the hole. The one that escapes is the radiation the one that is trapped enters a negative energy state and reduces the mass of the hole. Except for pulsars black holes don t make pulsars all the methods used for detecting neutron stars can be used to detect black holes: 1. Close binary x ray spectrum is different for a black hole because once the matter enters the event horizon, the x rays are cut off. Doesn t happen for a neutrons star. 2. If a dark companion in a binary system is found to have a mass in the black hole range, it must be a black hole. 3. A black hole produces a much larger brightening and much longer brightening in gravitational lensing than does a neutron star. Chapter 11 Galaxies

5 Our own Galaxy Milky Way Galaxy MW Structure First attempt was counting stars out from the Sun. INCORRECT conclusion Disk shape with Sun at the center. Incorrect because gas and dust blocks our view after a certain distance. Harlow Shapely first astronomer to deduce the location of the center of the Galaxy. He noted that globular star cluster orbit about a point in the constellation Sagittarius. This point, about 30,000 ly from the Sun is the center of the Galaxy. We can now map the MW: 1. IR observations tell us where the dust in the Galaxy is. 2. Radio observations tell us where the hydrogen gas (H) is in the Galaxy. Uses the 21-cm emission from hydrogen. Electron has two spin states in the magnetic field of the proton. When aligned, the electron has lower energy than when opposite. When spin flips, a 21- cm photon is emitted. Structure of MW: 1. It has a basic disk shape with a diameter of 100,000 ly. 2. Center has a bulge about 10,000 ly thick. 3. Away from the bulge, thickness is about 1000 ly. 4. Looking down on the MW we see spiral arms coming out from the center much like a pinwheel. 5. It does have a bar-like structure through its nuclear bulge. 6. Gas and dust in the disk is about 500 ly across. What we find in the spiral arms: 1. Almost all the gas and dust in the Galaxy resided in the spiral arms. 2. Sun is about in the spiral arm about 2/3 of the way from the center of the Galaxy. 3. All star formation in the spiral arms. 4. All the young stars.

6 5. All open clusters. 6. No globular clusters. 7. There are some old stars in the spiral arms. What do we find in the nuclear bulge? 1. Almost no gas and dust. 2. No star formation. 3. Only old stars. These parts are the visible parts there is a non visible part called the halo of the Galaxy. Note: The halo of the Galaxy is spherical; it is not a ring. What we find in the halo: 1. Globular star clusters. 2. We have found that, using Newton s law of gravity, the visible matter in the Galaxy does not account for the motion of the visible matter in the Galaxy. Thus, either there is matter out there that is not visible (that is, not detectable) except by its gravitational effect, or Newton s law of gravity and GR are not correct or Newton s second law is not correct. MOND Modification Of Newtonian Dynamics 3. We will take the undetectable matter perspective dark matter. 4. At this point, we don t know what dark matter is. 5. However, apparently dark matter accounts for 90% of the matter in the universe! 6. Dark matter can only be detected gravitationally. Note: the halo of the Galaxy might be 5 times the size of the visible part of the Galaxy. What is the mass of the MW? Use Kepler s third law! Apply it to the orbital parameters of the Sun Note that this will only give us the mass of that part of the MW closer to its center than the Sun is. Time for the Sun to complete one orbit about the center of the MW is 250 million years. Distance from of the Sun from the center of the MW is 30,000 ly or about 8500 pc or AU.

7 Note that value is actually more like due to the fact that there is a lot of matter farther from the center of the MW than the Sun. The Nucleus of the MW its center 1. Can t see the nucleus in visible light too much stuff in the way. 2. Can observe radio waves from the nucleus see a strong radio source there Sagittarius A* or Sgr A*. 3. IR observations show stars to be tightly packed near the center. 4. Distances between stars about 1000 AU not 100,000 AU like out here in the disk. 5. We see a ring of gas that extends from about 6 pc from the center to 25 pc from the center. 6. This gas may have been pushed away from the center by an explosion 100,000 years ago. 7. From the orbital parameters of the stars near the center and Kepler s third law, we have calculated the mass of the nucleus to be 2 million solar masses. 8. The size of the nucleus is no more than 2 AU in diameter we see fluctuations in the nucleus over a time periods of 15 to 20 minutes. The size of the object is the distance light can travel in this time about 2 AU. 9. Something that small and that massive must be a black hole! Other Evidence: 1. Sgr A* does not move stars around it do, but it doesn t. 2. Evidence of black holes at the centers of other Galaxies as well.

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