Principle of Equivalence Einstein 1907 Box stationary in

Principle of Equivalence: Einstein 1907 Box stationary in gravity field Box accelerates in empty space Box falling freely 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.

Tides • Problem: r 2 r 1 moon • Gravity decreases with distance => stretch…

Tides • Tides = gravity changes from place to place not freely falling ? freely falling not freely falling ? ? ?

Light rays and Gravity… • Remember: gravity bends light… accelerating observer = gravity

Light Rays and Gravity II • In SR: light rays travel on straight lines => in freely falling fame, light travels on straight lines • BUT: to stationary observer light travels on curved paths => Maybe gravity has something to do with… curvature of space ?

Curved Spacetime • Remember: Gravity warps time fast BUT: in spacetime, time and space are not separable => Both space and time are curved (warped) slow This is a bit hard to vizualize (spacetime already 4 D…)

GR: Einstein, 1915 • Einstein: mass/energy squeeze/stretch spacetime away from being “flat” • Moving objects follow curvature (e. g. , satellites, photons) • The equivalence principle guarantees: spacetime is “locally” flat • The more mass/energy there is in a given volume, the more spacetime is distorted in and around that volume.

GR: Einstein, 1915 • Einstein’s “field equations” correct “action at a distance” problem: Gravity information propagates at the speed of light => gravitational waves r?

Curvature in 2 D… • Imagine being an ant… living in 2 D • You would understand: left, right, forward, backward, but NOT up/down… • How do you know your world is curved?

Curvature in 2 D… • In a curved space, Euclidean geometry does not apply: - circumference 2 R - triangles 180° - parallel lines don’t stay parallel 2 R R R <2 R =180

Curvature in 2 D…

Geodesics • To do geometry, we need a way to measure distances => use ant (let’s call the ant “metric”), count steps it has to take on its way from P 1 to P 2 (in spacetime, the ant-walk is a bit funny looking, but never mind that) • Geodesic: shortest line between P 1 and P 2 (the fewest possible ant steps) ant P 1 P 2

Geodesics • To the ant, the geodesic is a straight line, i. e. , the ant never has to turn • In SR and in freely falling frames, objects move in straight lines (uniform motion) • In GR, freely falling objects (freely falling: under the influence of gravity only, no rocket engines and such; objects: apples, photons, etc. ) move on geodesics in spacetime.

Experimental Evidence for GR • If mass is small / at large distances, curvature is weak => Newton’s laws are good approximation • But: Detailed observations confirm GR 1) Orbital deviations for Mercury (perihelion precession) Newton: Einstein:

Experimental Evidence for GR 2) Deflection of light

Experimental Evidence for GR

Black Holes • What happens as the star shrinks / its mass increases? How much can spacetime be distorted by a very massive object? • Remember: in a Newtonian black hole, the escape speed simply exceeds the speed of light => Can gravity warp spacetime to the point where even light cannot escape it’s grip? That, then, would be a black hole.

Black Holes

Black Holes • Time flows more slowly near a massive object, space is “stretched” out (circumference < 2 R) • Critical: the ratio of circumference/mass of the object. If this ratio is small, GR effects are large (i. e. , more mass within same region or same mass within smaller region) 1) massive 2) small ? ? ?

The Schwarzschild Radius • GR predicts: If mass is contained in a circumference smaller than a certain size gravitational constant critical circumference mass speed of light space time within and around that mass concentration qualitatively changes. A far away observer would locate this critical surface at a radius Schwarzschild radius • Gravitational time dilation becomes infinite as one approaches the critical surface.

Black Holes • To a stationary oberserver far away, time flow at the critical surface (at RS) is slowed down infinitely. • Light emitted close to the critical surface is severely red-shifted (the frequency is lower) and at the critical surface, the redshift is infinite. From inside this region no information can escape red-shifted into oblivion

Black Holes • Inside the critical surface, spacetime is so warped that objects cannot move outward at all, not even light. => Events inside the critical surface can never affect the region outside the critical surface, since no information about them can escape gravity. => We call this surface the event horizon because it shields the outside completely from any events on the inside.

Black Holes • Critical distinction to the Newtonian black hole: Newton Einstein Nothing ever leaves the horizon of a GR black hole. • Lots of questions… What happens to matter falling in? What happens at the center? Can we observe black holes anyway? And much, much more…