
Special Theory of Relativity
 Special Relativity  space and time are intertwined but does not deal with effects of gravity
  implied that space and time are not absolute concepts, but distances and time periods measured by different observers depend on the relative velocities of different observers
  1905  Einstein
  "special" b/c only in the special case in which we ignore the role of gravity
  distances and time appear to be absolute in our daily lives (we all agree it takes x minutes to walk y meters). Seems this way when moving at ordinary speeds, but actually the distance you walk/time it takes you will be different by a tiiiiny bit. But if your friend is moving at the speed of light, these differences will be substantial

General Theory of Relativity and the Equivalence Principle
  special relativity created problems for describing gravity. General relativity solves these problems
  1915, Einstein  helps us to understand the expansion/fate of the universe, and black holes
  general because it applies with or without gravity
 matter shapes the "fabric" of spacetime in a manner analogous to the way heavy wights distort the taught rubber sheet of a trampoline
 gravity arises from distortions of spacetime. it is not a mysterious force that acts at a distance. The presence of mass causes distortions, and the resulting distortions determine how other objects move through spacetime
 time runs slowly through gravitational fields. The stronger the gravity, the more slowly time runs
 Black holes can exist in spacetime, and falling into a black hole means leaving the observable universe
 the universe has no boundaries and no center, yet it might still have a finite volume
 large masses that undergo rapid changes in motion or structure emit gravitational waves that travel at the speed of light

the Equivalence Principle
the effects of gravity are exactly equivalent to the effects of acceleration
whenever you feel weight, you can equally attribute it to the effects of gravity or acceleration, so you (and the laws of physics) cannot tell the difference between being in a closed room on earth (feel weight from gravity), or being in a closed room accelerated through space at 1g (feel weight from acceleration)

So what does special relativity tell us about the example of someone (Karen) traveling in space vs someone (Susan) standing still on earth
 nothing can travel faster than the speed of light (in a vacuum), and nothing can ever reach the speed of light
 Karen is in a spaceship moving with velocity v, Susan is stationary, watching the ship with Karen and a lamp in it > if you observe anyone or anything moving by you at a speed close to the speed of light, you will conclude that time runs more slowly for that person (in space)  they age more slowly, clock moving by ticks more slowly than your clock.
 if you observe 2 events simultaneously (flashes of light in 2 different places at the same time), the two events will not appear to be simultaneous if one is moving close to the speed of light > light goes off in spaceship, travels in a straight line, directly across the spaceship. (Travels at a distance d, at velocity c, and takes an amount of time  time _{karen} = c/d) to get there).
 In classical physics (WRONG), the velocity of light, c, would be added with the velocity of the spaceship, to find the total velocity of light. So the principle of relativity says  the speed of light is the same for all observers (even ones moving at different velocities)  WRONG
  Implies  time flows at a different rate for observers moving at different velocities
 t_{s} = t_{k} / √(1  (v/c)2)
 thus, when susan looks at Karen, it looks like time on the spaceship is flowing more slowly  karen herself is moving slower. But to Karen, its susan who appears to be moving slower.
if you measure the size of something moving by you at close to the speed of light, you will find that its length (in the direction of its motion) is shorter than it would be if the object were not moving, mass would be greater than if it were stationary

What are the two absolutes in the theory of relativity
 1. the laws of nature are the same for everyone
 2. the speed of light is the same for everyone.

What is spacetime
 special relativity tells us: three dimensions of space, and one dimension of time, together form an inseparable 4dimensional combination called spacetime
 (space and time are "different" ways of looking at the same thing)
ex: point = zero dimensions, line = 1 dimension (X), plane = 2 dimensions (X,Y), volume (like this room) = three dimensions (X,Y, Z). Space time = 4 dimensions (X, Y, Z, t)

Distances in three spatial dimensions
 the distance between two points is the same for all observers, no matter where they are standing
  Mathematically use Pythagorean theorem, just adding another dimension
 d= √( (x_{1} −x_{2})^{2} +(y_{1} −y_{2})^{2} +(z_{1} −z_{2})^{2 })

Distances in Four Dimensional Spacetime
 Exactly like three dimensional distances, but includes a time dimension too
 two "points" are now called events  they have a location and a time as measured by different observers
 The distance between two events is the same for all observers, no matter where they are standing, or how fast they are moving
Use pythagorean theorem
d=√( (x _{1} −x 2) ^{2} +(y _{1} −y 2) ^{2} +(z _{1} −z _{2}) ^{2 }− c ^{2}(t _{1} −t _{2}) ^{2 })

Our perception of gravity is actually what?
 What we perceive as gravity arises from the curvature of spacetime
  what we perceive as gravity is actually motion along a straight line is spacetime

What does the curvature of spacetime determine about the motion of freely moving masses (ex: planets)
  gravity becomes stronger, and the curvature of spacetime becomes greater as we approach the suns surface (or any body that has gravitational attraction).
  marbles (other bodies) coming near will follow thestraightest possible paths given the curvature of the sheet (spacetime) > planets orbit because they follow the straightest possible paths allowed by the shape of spacetime around them
  bodies that come in relatively slowly and close to the center would follow circular or elliptical orbits
  bodies that come in with higher speeds or from father away could loop around the center on an unbound orbit
  other possible orbit is circular orbit
A mass like the sun causes spacetime to curve, and the curvature of spacetime determines the paths of freely moving masses like the planets

Is spacetime affected by density? Mass?
What happens if you keep compressing a mass
No  spacetime doesnt change as a function of how dense an object is. ???  only affects what is immediately around it???
 greater curvature of gravity means stronger gravity
  "larger mass causes greater curvature at any particular distance away from it"
  "increase curvature of spacetime by increasing density (make it smaller in size)
  Ex: if you compressed the sun to a white dwarf, because its total mass is the same, no affect on the curvature of spacetime far from the sun, but much more curved near the suns surface > gravity is stronger.
  If you kept compressing (mass stays the same so far from the sun has no effect b/c same total mass causing the curvature of spacetime) but could eventually compress to a bottomless pit  a black hole spacetime is so curved that nothing that falls in can ever escape. The boundary that marks the "point of no return" is called the event horizon  events that occur within this horizon have no influence on the observable universe

Is General Relativity different from Newton's Gravitational Law?
  Yes! The first example of this was the measured procession of Venus' elliptical orbit
  Newton's Law: Planets travel in ellipses around the sun
  General Relativity: the ellipses on which planets travel also rotates around the sun (VERY slowly. About 1" per century!)

General Realtivity predicts that WHAT is affected by Gravity
 what does Newton's law say?
 what does General Relativity say?
 and what can we observe
Light.
 Newton's law  the force between 2 objects depends on their masses. Photons have no mass, so they shouldnt be affected by gravity
 General Relativity: the shape of spacetime is changed by a massive object, so the direction of a photon is changed as it travels in a "straight line" on warped spacetime near a massive object.
 we can see that the direction of light is affected when it passes close to a massive object (like the sun or distant galaxies)

The Southern Cross
evidence of what?
explain structure
how do we know?
evidence that light is affected by gravity
four outside dots are actually the same object (distant, luminous quasar)
center is a massive galaxy between us and the quasar
we know because they have the exact same spectrum and when one brightens, the other three brighten too.

Black Holes
spacetime is so curved that nothing that falls in can ever escape. (einstein proposed that the speed of light, c, is the "speed limit" of the universe. So if an object has an escape speed greater than c, then nothing can escape from a black hole
 The boundary that marks the "point of no return" is called the event horizon  events that occur within this horizon have no influence on the observable universe

Size of a black hole
 size = the radius of its event horizon a.k.a. the radius it would have if its geometry were flat (circles circumfrence divided by 2π
 radius of horizon = Schwarzchild Radius. Radius depends only on its mass. > 3km per Mass_{sun}

How do black holes form? Who discovered this
 Chandrashekar  if you simply increase the mass of a neutron star above 3 masses of the sun, then the neutron star will collapse into a black hole
 (big result of 20th century physics)

What happens if you get too close to a black hole?
tidal forces (the difference between the gravitational pull by the black hole on your head and feet) will rip you apart

Black holes at the center of galaxies? ours?
  Stars near the center of our Galaxy are observed to move about a dark spot in orbits
  the orbits are elliptical, periods measured and the center localized. There is nothing optically bright at that position, but the orbits indicate the central object has a mass of 10^{6} M_{sun}
  every galaxy seems to have a black hole at its center, with a mass between 10^{6}  10^{9} M_{sun}

