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/relativity/

• 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

• (a) An outside observer sees light travel horizontally in a straight line, but it strikes the wall slightly below a point opposite the window.

• (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 indicated that the light from a star which just grazed the sun should be deflected by 1.75 seconds of arc.

• It was tested during the total eclipse of 1919 and during most of those which have ocurred 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 3 is at rest in an inertial frame.

• 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

• 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.

When the disk is not rotating, C/D = π; when the disk is rotating, C/D does not equal π and Euclidean geometry is no longer valid.

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 180°.

Geodesics

• These lines of shortest distance are called geodesic lines or simply geodesics.

• The path of a light beam follows a geodesic.

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°.

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