Outline

• Gravity and Acceleration – Force – Kepler’s 3rd Law – Energy – Escape Velocity • General Relativity Special Rel. - Big Results

• Length Contraction – Distance appears shorter in a frame moving relative to you • Time Dilation – Clocks appear slower in a frame moving relative to you • Relativity of Simultaneity – Two events that occur at the same time, but different places, in one frame occur at different times in another frame. Doppler Effect

• Moving toward you = blueshift

• Moving away = redshift

• Light from moving object focused forwards Gravity The Big Four:

• Copernicus (1473-1543) – Planets orbit Sun, not Earth – but assume circular orbits (Greek idea of perfection) • Kepler (1571-1630) – Orbits elliptical, not circular – but described by simple laws • Galileo (1564-1642) – Discovers moons of Jupiter – Orbital motion is Universal • Newton (1642-1727) – Idea of forces theory of gravity Newton’s Theory Idea of “Escape Velocity”

• Gravity is a “force” – causes acceleration or deceleration (= negative acceleration) – g at Earth’s surface = 32 ft/sec/sec = 22 mph/sec = 9.8 meters/sec/sec • Example: – throw something up at 66 mph: it stops after 3 sec, then falls back in another 3 sec • How to calculate gravitational pull of a mass:

– increases with amount of mass 2×mass ⇒ 2× g

– decreases with square of distance from center of mass 1 2×distance ⇒ × g 4

Gravity

• Force between two masses is

• Kepler’s Third Law:

Velocity in Circular orbit

• Velocity

• Kepler’s 3rd Law

• Velocity Escape Velocity

• Gravitational potential energy:

• Kinetic Energy: Escape Velocity

• Gravitational potential energy:

• Kinetic Energy:

• Escape Velocity: Waste Disposal

• It is decided that Earth will get rid of it’s nuclear waste by shooting it into space. To make sure it doesn’t come back, it has to either be shot into the Sun or out of the solar system. Which was is cheaper?

• Radius of Sun = 7 × 105 km • Orbital speed of Earth = 30 km/s

The Idea of a Black Hole

John Michell (1783) , Pierre-Simon Laplace (1796):

• Ques: What happens if the escape speed from an object is greater than the speed of light? • Ans: If light consists of particles of matter, they would not be able to escape “dark star” ?

• The catch: early 19th cent - light is a wave (a disturbance), not a particle BH idea forgotten….

… until Einstein comes along General Relativity General Relativity Principle of Equivalence: Einstein 1907 g Box stationary in Box gravity field = accelerates in empty space g

Box falling freely =

g Box moves through space at constant velocity Equivalence Principle

• Special relativity: all uniformly moving frames are equivalent, i.e., no acceleration

• Equivalence principle: Gravitational field = acceleration

freely falling frames in GR = uniformly moving frames in SR. Gravitational Doppler Shift

• Total energy is always conserved • Energy and mass are equivalent • As light falls, it must gain energy • Speed of light is constant, so frequency must increase Gravitational Time Dilation

• As light falls, frequency increases • Frequency increase means a clock farther away appears to be running faster

• Opposite happens for light going up • Time near a mass appears to run slow compared to someone far away Light rays and Gravity…

• Remember: gravity bends light…

accelerating observer = gravity