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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?