Physics 171: General Relativity. An Introduction

Transcription

1 Physics 171: General Relativity An Introduction

2 Unlike anyone in physics except perhaps Newton (or Bohr?), the figure of Einstein looms large in any discussion of physics, and especially General Relativity. Einstein

3 Many books

4 Indices: Summation Convention Introduced in 1916 (with final formulation of GR). Repeated indices summed. Einstein joked I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice x i x i = x i x i

5 Newton

6 Newton s Constant: Fundamental to Gravitation

7 Features of Newton s Laws Action at a distance -- almost ghostly. A source of skepticism, initially; Newton was defensive about it. Equivalence of gravitational mass and inertial mass (Eotvos) Galilean Invariance

8 Galileo and Galilean Invariance Galileo on Relativity: Galileo Defends Copernicus Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it nor more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully, have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.

9 Modern Description of Galilean Invariance

10 Not a feature of Maxwell s equations Speed of light just a constant Einstein s Problem With Galileo/Maxwell If I pursue a beam of light with the velocity c, I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There is no such thing.

11 The fall of absolute time Newton: Newton on Absolute Time (Principia): Absolute, true and mathematical time, of itself and from its own nature, flows equably without relation to anything external Absolute space, in its own nature, without relation to anything stable, remains always similar and immovable. But: Absolute time is not an object of perception The Deity endures forever and is everywhere present, and by existing always and everywhere, He constitutes duration and space. And Mach (one of Einstein s early heroes): absolute time is a useless metaphysical concept, and cannot be produced in experience. Newton acted contrary to his expressed intention only to investigate actual facts. And Poincare Not only do we have no direct intuition of the equality of two times, we do not even have one of the simultaneity of two events occurring in different places. Poincare had much of the mathematics of special relativity, as did Lorentz, but neither made the final leap; they were too wedded to the ether concept.

12 Special Relativity Einstein: realized that the symmetry of Maxwell s equations is not Galilean invariance, but Lorentz invariance argues this is what should be taken seriously. [Demise of action at a distance] Indices again: Upstairs/downstairs: Einstein borrowed, it seems, from mathematicians: Following Ricci and Levi-Civita, we denote the contravariant character in such a way that we place the index in the upper position But other notational improvements only came later. In this way, Maxwell s equations look simple:

13 Maxwell s Equations in Relativistic Form

14 Einstein on Maxwell and the field concept A new concept appeared in physics, the most important invention since Newton s time: the field. It needed great scientific imagination to realize that it is not the charges nor the particles but the field in the space between the charges and the particles that is essential for the description of physical phenomena. The field concept proved successful when it led to the formulation of Maxwell s equations describing the structure of the electromagnetic field.

15 Einstein: Special -> General 1905: Einstein formulates the principles of special relativity. Time a relative concept. No action at a distance. 1907: Einstein asked to write a review. Thinks about gravitation. Asks how one might write a relativistic theory of gravitation. Hardly a pressing question at that time.

16 Imagine yourself in 1907 When might relativistic gravity be important? mc 2 ¼ mgm/r Gm/r ¼ 1 Earth distance from sun: cm Solar mass: gm Gm/r ¼ 10-8

17 What features of Newton s laws should you preserve, change? For slow motion, masses not too large: Newton s laws should be valid No action at a distance (interactions propagate at speed c.) New fields? Instead, Einstein focused on equivalence principle. Inertial mass: related to how we define what we mean by time (and space).

18 Einstein and equivalence principle Happiest Thought of Eisntein s Life We ll discuss this more soon, but for now: Because of this idea, the uncommonly peculiar experimental law that in the gravitational field all bodies fall with the same acceleration attained at once a deep physical meaning. Namely, if there were to exist just one single object that falls in the gravitational field in a way different from all others, then with its help the observer could realize that he is in a gravitational field and is falling in it. If such an object does not exist

19 1915: Einstein publishes his General Relativity After a long struggle, realized equivalence principle required a reformulation of gravitation in terms of the geometry of space-time. Understanding what this means will be the focus of this course. Objects (fields): metric, connections, curvature. Beautiful, but somewhat challenging, mathematical structure. We ll take in stages: first, motion of objects in gravitational fields; then determining the fields themselves.

20 21 st Century computational tools No longer quite so frightening or tedious. Useful algebraic manipulation programs on text website, elsewhere.

21 General relativity in the real world When Einstein proposed his theory, three tests: Perhelion precession of mercury (we ll compute) Gravitational redshift Bending of light by the sun (we ll compute) When I was a grad student (much later!) still only three tests; subject somewhat disreputable.

22 In 21 st Century Black holes observed, studied Gravitational waves inferred from pulsar timing; direct searches Cosmology: cosmic microwave background, structure of the universe on large scales, composition of the universe GPS a day to day technological application

23 Gravitational wave searches: Ligo, Lisa

24 Black hole at the center of the Milky Way Suspected for a long time that a supermassive black hole at center of our galaxy (and now most galaxies). About 2000, yours truly speaks at a conference right after important NASA administrator talks about huge NASA project to image the galaxy (very glitzy presentation about expensive proposed project).

25 Macarthur winner 2008 Seeing the black hole at the center of the galaxy

26

27 Cosmic Microwave Background: Fossil from 15 billion years ago

28 WMAP Satellite: Much Greater Precision

29 More quantitative (remember Y_{lm} s?)

30 We know the energy budget of the universe, now and in the past

31 We know much about the history

32 Dark Matter: about 30% of mass of universe Bullet cluster (c. 2006)

33 Dark Energy

34 Newest Instrument: GLAST (FERMI) Already observing exploding stars at billions of light years; dark matter? [UCSC!]