General_Relativity

# General_Relativity - 1/14/10 General Relativity Chapter 8...

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1/14/10 1 General Relativity Chapter 8 Introduction GR is Einstein’s theory of gravitation that builds on the geometric concept of space- time introduced in SR. Is there a more fundamental explanation of gravity than Newton’s law? GR makes specific predictions of deviations from Newtonian gravity. Curved space-time Gravitational fields alter the rules of geometry in space-time producing “curved” space For example the geometry of a simple triangle on the surface of sphere is different than on a flat plane (Euclidean) On small regions of a sphere, the geometry is close to Euclidean

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1/14/10 2 How does gravity curve space-time? •With no gravity, a ball thrown upward continues upward and the worldline is a straight line. •With gravity, the ball’s worldline is curved. •It follows this path because the spacetime surface on which it must stay is curved. •To fully represent the trajectory, need all 4 space-time dimensions curving into a 5th dimension(!) •Hard to visualize, but still possible to measure t t x x No gravity gravity Principle of Equivalence A uniform gravitational field in some direction is indistinguishable from a uniform acceleration in the opposite direction Keep in mind that an accelerating frame introduces pseudo-forces in the direction opposite to the true acceleration of the frame (e.g. inside a car when brakes are applied) Elevator experiment •First, elevator is supported and not moving, but gravity is present. Equate forces on the person to ma (=0 since a=0) •F s - mg = 0 so F s = mg •F s gives the weight of the person. •Second, no gravity, but an upward acceleration a. The only force on the person is F s and so •F s = ma or F s = mg if “a” value is the same as “g” •Person in elevator cannot tell the difference between gravitational field and accelerating frame •Third, there is gravity and the elevator is also in free-fall •Fs - mg = -mg or Fs = 0 •“Weightless” Let upward forces be positive, thus gravity is -g See also http://www.pbs.org/wgbh/nova/einstein/relativity/
1/14/10 3 Einstein was bothered by what he saw as a dichotomy in the concept of "mass." On one hand, by Newton's second law ( F=ma ), "mass" is treated as a measure of an object’s resistance to changes in movement. This is called inertial mass . On the other hand, by Newton's Law of Universal Gravitation, an object's mass measures its response to gravitational attraction. This is called gravitational mass . As we will see, Einstein resolved this dichotomy by putting gravity and acceleration on an equal footing. The principle of equivalence is really a statement that inertial and gravitational masses are the same for any object. This also explains why all objects have the same

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## This note was uploaded on 02/13/2012 for the course AST 3019 taught by Professor Sarajedini during the Spring '08 term at University of Florida.

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General_Relativity - 1/14/10 General Relativity Chapter 8...

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