General Relativity

Topics

• Principle of Equivalence • Bending of Light by Gravity • Gravity and Time: Gravitational Red Shift • Gravity and Space: Motion of Mercury • Gravity, Space, and a New Geometry • Gravitational Waves

PBS Relativity Animations

• http://www.pbs.org/wgbh/nova/einstein/r elativity/ • Special Relativity Length Contraction • Equivalence Principle • Curved Space

Equivalence Principle

• Observations made in an accelerated reference frame are indistinguishable from observations made in a Newtonian gravitational field.

Acceleration & Gravity Equivalence

• To an observer inside the accelerating ship, a lead ball and a wood ball appear to fall together when released.

Trajectories

• If a ship is accelerating, the floor overtakes the ball. • An observer outside the ship sees a straightline path.

• An observer in the accelerating ship sees the path as curved; it is a parabola.

Acceleration Deflects Light

• (b) To an inside observer, the light bends as if responding to a gravitational field.

Gravity Bends Light

• According to the

principle of equivalence

,

if light is deflected by

acceleration

, it must be deflected by

gravity

. • How can gravity bend light?

Energy of Light

• Einstein's answer was that light may be massless, but it's not “energy-less” • Gravity pulls on the energy of light because energy is equivalent to mass.

• Later Einstein added the idea of curved spacetime.

Light from Star Bent by Sun

• Einstein's calculations light from a star which just grazed seconds of arc. during the total and during most of those which since.

Bending of Light?

• Why do we not notice the bending of light in our everyday environment?

Check Yourself

• Why do we not notice the bending of light in our everyday environment? • Only because light travels so fast; just as over a short distance we do not notice the curved path of a high-speed bullet, we do not notice the curving of a light beam.

Gravity and Time

• According to Einstein's general theory of relativity, gravitation causes time to slow down.

Time

• Clocks 1 and 2 are on an accelerating disk, and clock • Clocks 1 and 3 run at the same rate, while clock 2 runs slower. • From the point of view of an observer at clock 3, clock 2 runs slow because it is moving. • From the point of view of an observer at clock 1, clock 2 runs slow because it is in a stronger “centrifugal” force field.

Clock Runs Slower at Earth’s Surface

• By applying the principle of equivalence, which says that any effect of acceleration can be duplicated by gravity, we must conclude that as we move in the direction that a gravitational force acts, time will also be slowed.

Gravitational Red Shift

• All atoms emit light at specific frequencies characteristic of the vibrational rate of electrons within the atom. • Every atom is therefore a “clock,” and a slowing down of atomic vibration indicates the slowing down of such clocks. • An atom on the sun should emit light of a lower frequency (slower vibration) than light emitted by the same element on the Earth. • Since red light is at the low-frequency end of the visible spectrum, a lowering of frequency shifts the color toward the red.

Mossbauer Effect Used

• In 1960 an entirely new technique • using gamma rays from radioactive atoms • incredibly precise and confirming measurements • of the gravitational slowing of time between the top and bottom floors of a laboratory building at Harvard University.

Precession Of Mercury

• Near the sun, where the effect of gravity on time is the greatest, the rate of precession should be the greatest • the orbit of Mercury does precess—above and beyond effects attributable to the other planets

New Geometry

not rotating, C/D = when the disk is rotating, C/D does geometry is no • measuring stick along the edge of the rotating disk appears contracted • measuring stick farther in and moving more slowly is not contracted as much. • measuring stick along a radius is not contracted at all.

Sum of Triangle Angles

The sum of the angles of a triangle depends on which kind of surface the triangle is drawn on.

(a) On a flat surface the sum is 180°. (b) On a spherical surface the sum is greater than 180°. (c) On a saddleshaped surface the sum is less than

Geodesics

The light rays joining the three planets form a triangle. Since light passing near the sun bends, the sum of the angles of the resulting triangle is greater than 180°. • These lines of shortest distance are called geodesic lines or simply geodesics.

• The path of a light beam follows a geodesic.

Curved Universe

• The whole universe may have an overall curvature. • If negatively curved, it is open-ended like the saddle and extends without limit • If positively curved, it closes in on itself. • One familiar example of a positively curved space is the surface of the Earth.

Einstein’s Universe

• “Space tells matter how to move, and matter tells space how to curve. That’s it.” QuickTime Movie

Warped Space

QuickTime Movie