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F.  Black Holes

The effects of spacetime curvature on light, on time, on distance, and on spinning objects can all be seen in the familiar gravitational fields of the earth and the solar system.  General relativity implies the existence of some even more intriguing phenomena, associated with the extremely intense gravitational fields known as black holes.

Outside a black hole, spacetime has the same kind of shape that we find around a massive object, but within, the shape of spacetime takes an interesting turn.  There, the direction of time has traded places with the inward direction of space.  An object inside the black hole that followed the straightest possible path in spacetime—the path of gravitational attraction—would find the direction of its future pointing ever inward, and never toward the outside of the hole.

In fact, if matter behaved according to the expectations of 19^th-century physicists, it would be strictly impossible for the object to ever escape.  Even if a force were applied to counter gravity and "unstraighten" the object's path, it still couldn't leave; in effect, the escape velocity from a black hole's interior is larger than the speed of light.  The region outside the black hole does have an achievable escape velocity, since the directions of spacetime there are the same as they are outside an ordinary object.

Black holes don't suck matter in; they're more like trapdoors in space than vacuum cleaners.  But the trapdoor is evidently not an absolute one, because matter has some properties that physicists only began to recognize at the close of the 19^th century.  By taking account of these properties, Steven Hawking showed in 1974 that matter should slowly leak out of black holes all the time.  It turns out that the rate of escape from a black hole is inversely proportional to the hole's surface area, which will shrink as more matter escapes.  A large black hole thus emits matter and shrinks slowly, but given enough time it will become small enough to emit matter and shrink much more rapidly.

The general theory of relativity has come a long way from its early days, when only a few people in the world understood it.  Nowadays a typical class introduced to the details of general relativity will have something like a dozen students, and many such classes are conducted each year in schools all over the world.  It took physicists a while to get used to a theory of curved spacetime, and to the mathematics needed to describe and reason about the curvature.  Over the last nine decades, advanced in telescopes and experimental devices have made it possible to see more evidence of spacetime curvature in nature, and progress with computers and software has made it feasible to calculate implications of the theory that were once impractically hard to work out.

References, Links, and Comments:

With this article we come full circle, from Einstein's special relativity theory of interrelated space and time, to his general theory of curved spacetime.  Many of the references and links at the end of the earlier article deal with both installments of the relativity theory, and are thus repeated here.

• The Universe and Dr. Einstein by Lincoln Barnett
  A journalist's account of Einstein's theories, including General Relativity.
  

• The Evolution of Physics by Albert Einstein and Leopold Infeld
  Traces the main ideas of physics from Galileo to modern quantum theory in four chapters.  Relativity theory, including its generalization to curved spacetime, is one of the main subjects of chapter three.  Remarkably, this book only has two equations in it:  2 x 2 = 4 and 3 x 3 = 9!
  

• Relativity:  The Special and the General Theory by Albert Einstein
  A brief volume in three parts—"The Special Theory of Relativity", "The General Theory of Relativity", and "Considerations on the Universe as a Whole"—plus an appendix; the appendix includes descriptions of the earliest experimental confirmations of the general relativity theory.
  

• The Prinicple of Relativity by A. Einstein, H.A. Lorentz, H. Weyl, and H. Minkowski, with notes by A. Sommerfeld
  Original articles on relativity, in English or English translation, including Einstein's 1916 report "The Foundation of the General Theory of Relativity" ("Die Grundlage der allgemeinen Relativitätstheorie").
  

• The Meaning of Relativity by Albert Einstein
  Advanced mathematical treatment of both the Special and General Theories of Relativity and Einstein's attempts to extend them.
  

• The Light Cone:  an illuminating introduction to relativity by Rob Salgado
  Sections on the general theory focus on the laws of gravitation, the equivalence of gravity and acceleration, curvature, and black holes.

Some other references on the general theory include:

• The Evolution of Scientific Thought from Newton to Einstein by Abraham D'Abro
  Mainly about the development of the theory of relativity, from its 17^th-century antecedents to the mid-20^th century.  By the same author who brought us The Rise of the New Physics, the two-volume work referred to in some of the earlier articles of this series.
  

• The New Physics, edited by Paul Davies
  Early chapters in this book relate to general relativity, including "The renaissance of general relativity" by Clifford Will, which focuses on work done between 1960 and 1980.
  

• Relativity:  Special, General, and Cosmological by Wolfgang Rindler
  Advanced undergraduate/graduate level textbook.  First chapter is a remarkably clear overview of the entire subject.
  

• "The meaning of Einstein's equation" by John C. Baez and Emory F. Bunn, in American Journal of Physics, July 2005, pp. 644-652
  According to the authors of this article:  "There are many nice popularizations that explain the philosophy behind relativity and the idea of curved space-time, but most of them don't get around to explaining Einstein's equation [G = 8πκ T/c⁴] and showing how to work out its consequences.  There are also more technical introductions which explain Einstein's equation in detail—but here the geometry is often hidden under piles of tensor calculus."  This article explains the meaning of G = 8πκ T/c⁴ in terms of how a small ball of low-mass particles is affected when spacetime is curved in various ways.  The authors assume the reader is somewhat familiar with the special theory of relativity.  The article also provides a guide to 41 relativity references that the authors have found particularly helpful for students.
  The same issue of American Journal of Physics contains two other articles on astrophysical applications of general relativity:  "Superluminal apparent motions in distant radio sources" by Michał J. Chodorowski, and "Cosmology calculations almost without general relativity" by Thomas F. Jordan.
  

• "The General Relativity Tutorial" by John Baez
  "This is bunch of interconnected web pages that serve as an informal introduction to that beautiful and amazingly accurate theory of gravity called general relativity.  The goal is to explain the basic equation in this theory - Einstein's equation - with a minimum of fuss and muss."

Additional links related to specific topics mentioned above:

"The Weight of Light”, Physical Review Focus, 12 July 2005.
The story of the tower experiments by Pound, Rebka, and Snider.

"Gravity Probe B:  Testing Einstein’s Universe"
Stanford University's official website for the satellite gyroscope experiment described above.  (Gravity Probe A was the hydrogen maser clock sent on a rocket flight in 1976 to measure the difference in time flow between the rocket's flight path and the ground.)

"An introduction to general relativity, Spaceflight Now 
Describes several features of curved spacetime, with emphasis near the end on Gravity Probe B.

What goes up need not come down, if it can rise quickly enough.  While the spacetime curvature associated with any object can extend throughout the universe, this curvature is smaller at greater distance from the object.  In other words, an object's gravitational attraction is weaker, the further one is from the object.  So while anything will be decelerated by gravity as it rises, if its initial speed is fast enough then gravity will never slow it enough to reverse its course.  The speed required to avoid falling back towards an attracting object is called that object's escape velocity.

• "What is escape velocity?" at PhysLink.com tells what escape velocity is and gives the Newtonian formula linking the escape velocity from a planet or other spherical object to the object's mass and radius.
  

• "Escape Velocity" at HyperPhysics provides, in addition to the Newtonian formula, an automatic calculator that lets you find any celestial sphere's escape velocity from the sphere's mass and radius.  You can use the calculator for planets in the solar system (for which data are provided) or for other spherical objects that you might imagine.
  

• There is also an article about "Escape Velocity" at Wikipedia , which includes a list of escape velocities for the solar system, a description of how to calculate the speed required to escape from a system of objects as well as from a single object, and details of the orbits of escaping objects.

The lunar hammer-feather drop experiment is described on several websites, including:
"The legend of the leaning tower", by Robert P. Crease, physicsweb
"The Apollo 15 Hammer-Feather Drop", author/curator Dr. David R. Williams, NASA
     8.3 megabyte QuickTime movie
"A Brief History of the Exploration of the Moon", by Mark R. Whittington

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