General Relativity and the Expanding Universe

Allan Johnston

4/4/06

Overview

• The Equivalence Principle

• The bending of light

• Gravitational red shift

• The Precession of Mercury

• Curved space

• Black Holes

• Hubble’s Law

• Open or closed universe

The Equivalence Principle

• Inertial reference frames are indistinguishable from gravitational reference frames

• Gravitation is simply a form of an inertial frame of reference

• This principle shows why Newton’s inertial mass is equivalent to the mass in his Law of Universal Gravitation

An Elevator Laboratory

Figure 18.1 p. 253

The same applies to light

An observer in a frame outside sees the light travel in a straight line

An observer inside the elevator sees the light bent towards the floor

The Bending of Light

• In the frame of the observer inside the elevator, the light is bent

• Based on the equivalence principle it is not just an inertial frame that bends light, but a gravitational one as well

A Ball on a Sheet

Einstein described gravity as a mass that bends the surface of space-time

This model is useful for describing the bending of light by a large mass

The bending of light around the sun helped to solidify Einstein’s theory of relativity

Einstein is Right Again

• E=mc2 allows Einstein to assert that the energy possessed by light gives it mass which allows it to be affected by a gravitational force

• Future experiments during an eclipse proved him right

• This opens the door for many other possibilities

Gravitational Redshift

• Light loses energy as it is affected by a gravitational field

• This change is frequency is related to the distance from and the strength of the gravitational field

• Run through calculations

--p. 258

Precession of Mercury

• The orbit of Mercury was known to change 5600” of arc per century

• 43” of this were predicted by relativistic mechanics

• The wobble in the earth’s axis accounts for most of the difference, the other planets account for all but the predicted 43”

A Picture of Precession

Curved Space

• Einstein formulated his General Theory of Relativity based on a curved space time

• Originally Einstein included a repulsion term to accommodate for the static observed position of the planets

• Friedmann shows later that if the curvature of space is constant then Einstein’s equations hold up without the cosmological constant

The Angles of Triangles

• Constant positive curvature occurs on a sphere while constant negative curvature occurs on a saddle-shaped form

This can create triangles with unusual properties

Angles continued

• A triangle on a surface of constant positive curvature will have angles that add up to more than 180°

• A triangle on a surface of constant negative curvature has angles that add up to less than 180°

• This is important mainly because it shows how space can be warped based on your concept of the dimensions of space

A World Without Pi

• Consider a rigid disc --a meter stick is placed on the outer

edge --as the disc rotates the edge

approaches the speed of light --the disc is rigid and cannot contract

but the meter stick will --the radius is not in the direction of the

acceleration and is not affected

Black Holes

• A black hole occurs when the speed of light is not sufficient to reach the escape velocity from a mass

• This allows for the existence of dark matter• Einstein originally believed that a black hole could not

exist• He believed that nature would abhor such an event• He used the critical volume of the atom to support his

claims• Eventually it was found that the atom implodes to

create the black hole

What is inside a Black Hole?

• We are not really sure• In a classical black hole all matter would

be destroyed at the singularity by the stretching and squeezing waves

• Relativistic gravitation must be taken into account because it forbids the absolute

• At this point time and space become untangled

• Time is nonexistent

What does this mean?

• As there is no time, space is random

• The probabilistic froth

• Any conceivable curvature and space is possible

• The hole might contain a rip into hyperspace or to another universe

• While scientists consider that unlikely, they cannot eliminate it as a possibility

Time Warps• Wormholes• Allows large

distances to be covered quickly

• Can only be held open by “exotic material”

• Must be lined with a negative energy density material to keep from collapsing

Time Travel?

• Imagine a wormhole in this room

• Imagine one end here and one end on the rocket ship outside

• The ship travels away at near the speed of light and returns to earth 10 years later

• To the person on the ship only 12 hours has passed

• Where does the wormhole lead now?

Hubble’s Law

• The velocity of the recession of stars from earth is proportional to their distance from earth

• This was determined by observations that almost all distant objects are red shifted

• This suggests strongly that all objects in the universe are moving away from us and each other

What Does This Mean?

• Our universe is expanding

--surface of a balloon

• We only receive light from those stars that are receding at less than the speed of light

• Is our universe infinite?

--Olbers’ paradox

• Will our universe continue to expand?

Open or Closed?

• An open universe is one that will continue expanding on forever

• A closed universe will slow expansion to a stop and then collapse back in on itself

• Hubble’s Law would seem to suggest that the speed of recession is slowing because older stars are moving faster

• Possible flaw with Olbers’ paradox in that it assumes the light from more distant stars has reached us

An Open or a Closed Universe?

• As of this time we have no real proof of the infinite expansion or inevitable collapse of the universe

• There seems to be a definite deceleration due to gravitation but is it enough to stop the expansion?

• The rate of deceleration is much greater than expected for the amount of visible mass in the universe suggesting dark matter

Conclusions

• Light can be bent by a gravitational field

• Einstein predicted the bending of light as well as other minor errors in Newtonian mechanics

• Curved space gives us an entirely new area of possibilities for the universe

• The expansion of our universe is slowing