Astronomy 1143 – Spring 2014

Lecture 19:General Relativity

Framework

Postulates

Things assumed to be true about the Universe

Example: the speed of light is the same for all observers

Consequences

What happens when moving quickly or in strong gravitational fields

Example: time is different for observers who have relative motion

Framework

Tests

Observational proof

Example: the bending of light in the gravitational field of the Sun

Relevance

When special or general relativistic equations must be used

Example: Special Relativity – fast

General Relativity – strong gravity

Key Ideas:

Postulates:• Gravitational mass=inertial mass• Laws of physics are the same for all observers

Consequences• Matter tells spacetime how to curve.• Curved spacetime tells matter how to move.• Clocks run more slowly in strong gravitational

fields• Light leaving a gravitational field is redshifted

Key Ideas:

Tests of General Relativity:• Perihelion Precession of Mercury• Bending of Starlight near the Sun/Galaxies• Gravitational Redshift• Gravity Waves

Relevance• General Relativity must be used for strong

gravitational fields or for large accelerations• General Relativity is not a quantum theory and

does not work on the smallest scales

Principle of Equivalence

This is just another way of stating the principle of equivalence.

Everything falling

Principle of Equivalance

One of the postulates of General Relativity is that inertial mass=gravitational mass

• Appears in Newton as well, but Einstein elevated it to a postulate

• Consequences include• Free fall = free float, aka zero-g or weightlessness• Accelerations can mimic gravity (roller coasters and

“pulling g’s”• Matter can be thought of as moving together

through spacetime instead of being pulled by forces

Principle of Equivalence

No experiment that you can perform (in a small enough laboratory) that will tell you whether you are in zero-gravity or falling freely under the influence of gravity

Only works for gravity (not for electric or magnetic forces for example) – thing that is responsible to creating force also thing that responds to force

Principle of Equivalence

Power of Equivalence Principle

For thought experiments: what happens in an accelerating frame must also be true for gravity

Everything falling at the same rate implies that there is one thing that is guiding the motion of all particles – curved spacetime

Equivalence Principle only holds over small amounts of spacetime – smaller than the curvature of the Earth, for example

Bending of Light

Bending of Light

Light under the influence of gravity behaves the same way as in an accelerating object

A light beam in a rocket ship would appear to bend to the people in the ship.

Therefore light under the influence of gravity also will appear to bend

Light is not unaffected by gravity

NewtonNewton EinsteinEinstein

Mass & energy are very different things.

Mass & energy are interchangeable: E = mc2

Space & time are very different things.

Space & time are interchangeable: part of 4-dimensional spacetime.

General Relativity

New way of looking at gravity•Maybe the motions of masses under gravity didn’t have anything to do with the objects themselves

Curved spacetime tells matter how to move

Consequences:•Photons affected by curved spacetime•Disagreement with Newton’s Law of Gravity

Matter tells spacetime how to curve

How do objects move in curved spacetime?Newton’s 1st Law

• Objects in motion will move in straight lines unless acted upon by an outside force

• Straight line=shortest distance between two points

No forces = flat spacetime• Objects move by the shortest distance

Forces = curved spacetime• Objects move by the shortest distance

The Shortest Path...

On a flat surface:• The shortest path between two points is a

straight line.• Parallel lines stay parallel always.

On a curved surface:• The shortest path is a curved line.• Lines that start parallel can converge or diverge

at some distance away.

Curved Spacetime

Matter curves the spacetime around it:• The least paths are curved lines.• More mass = Greater spacetime curvature.• Closer = Greater spacetime curvature.• Einstein gave the equivalent of Newton’s Law of

Gravity in the form of his field equations

A freely falling object follows a curved path.

Curved spacetimearound the Sun

Curved spacetimearound a Black Hole

A New Theory of Gravity

General Relativity may be summarized as:

• Matter tells spacetime how to curve.

• Curved spacetime tells matter how to move.

Replaces the Newtonian idea of a “force” with the curvature of spacetime as the agent of Gravity.

GR has withstood all experimental tests.

Bending of Starlight

Light travels on the shortest path through spacetime.

Prediction:• Gravity bends light passing a massive object

Confirmed:• 1919 Solar Eclipse• Gravitational Lenses (1980s)

Bending of Starlight(side view)

TruePosition

Sun

ApparentPosition

Scale is exaggerated

Earth

0 0.5 1Deflection(arcsec)

1o 2o

1o

2o

View from EarthData from 1922 Eclipse

Gravitational Lenses

Gravitational Waves

Newton thought the force of gravity was instantaneously transported through space

But, remember, in special relativity, information travels at the speed of light

Gravity waves

Travel at the speed of light

Carry energy away

Have not been directly detected (yet!)

Binary Pulsar

2 pulsars orbiting each other

Can use the arrival of the pulses to map the orbits and periods very accurately

Strong gravitational field – test of theory

See system losing energy

Pulsars are getting closer together

75 millionths sec/year different in period

Nobel Prize for Hulse & Taylor

Binary Pulsar

Direct Detection of Gravity WavesGravity waves distort spacetime

Detected by laser beams moving along 2 different arms

One length squeezed, one length stretched

No detections yet• Many things can cause the

path length to change

• Tested by using duplicate facility

Clocks in Accelerating Spaceship

Clock B emits one photon pulse every 1 second according to Person B

Acceleration in this direction

Clock A emits one photon pulse every 1 second according to Person A

Clocks in Accelerating Spaceship

Person B will see Clock A’s photons arrive closer together than 1 second, as the photons have to travel less far as spacecraft catches up to them

• Clock A is running fast to Person B

Person A will see Clock B’s photons arrive farther apart than 1 second, as they have to travel farther to catch up

• Clock B is running slow to Person A

Gravitational Redshift

Gravitational field affects time

Clocks in stronger gravitational fields run slow

Predicts a gravitational redshift

Smaller number of waves/second for observer in weaker gravity (=faster clocks)

Gravitational redshift observed:• Pound & Rebka (1960 ) -- Harvard Tower• Hafele & Keating (1971) – jetlagged clocks• Scout D rocket (1976) – clocks on rockets

Practical Relativity

Global Positioning System (GPS)• 24 satellites in high Earth orbit• 20,000km altitude, 12h period (14,000 km/h)• Carry on-board atomic clocks

Relativistic effects on these clocks:• Special Relativity: 7sec/day slower• General Relativity: 45sec/day faster

Combined correction: 45–7 = 38 sec/day

Whither Newton?

Newton’s laws are approximations of GR.

Conditions:• weak gravitational fields• speeds much slower than the speed of light.

Newton’s Laws:• Work accurately in the “everyday” world.• Are mathematically much simpler.

Status of General Relativity

Passed every test – will continue to be tested

Its effects must be included in binary pulsar calculations, in collapses of stars to black holes, and in cosmology, among other applications

Not the last word in gravity – we need a theory of quantum gravity.

Applying GR to the Universe

Einstein applied his original GR equations to the whole Universe and found it should be expanding or contracting

Added this term to keep the Universe in a delicate balance

Cosmological Constant

We refer to as the cosmological constant – constant amount of energy for each chunk of space

Turns out that• Doesn’t make the Universe safely static• Universe is actually expanding• Einstein’s greatest mistake?

Not the end of the story• Accelerating Universe indicates a new source

of energy – cosmological constant?