Space Science

Principle of Equivalence: Einstein 1907

g

Box stationary in gravity field

g

=

Box accelerates in empty space

=

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.

Aberration of Light

Moral: direction of light beam is relative

Gravitational deflection of Light Now assume boxes are accelerating

Light path is curved

Light ray curved in accelerating frame

+

Principle of Equivalence (acceleration = gravity)

Gravity attracts light!

Paradox: How can gravity attract light if light has no mass?

MASS-ENERGY EQUIVALENCE

E = mc

2

Gravity extracts energy from escaping matter Gravity extracts energy from escaping light Gravitational redshift, time dilation Other points of view

same result:

– accelerating frames of reference - apply special relativity – spacetime is curved

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 4D…)

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

?

? ?

CURVATURE OF SPACETIME • How to tell difference between accelerating frame and gravity? – tidal forces

curvature

• Eliminates Newton’s “action at a distance” • Freely moving bodies follow “shortest path” – not necessarily a straight line

GENERAL RELATIVITY: EINSTEIN 1915 • Matter + energy determine curvature of spacetime • Curvature of spacetime determines motion of matter + energy

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

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 2D… • Imagine being an ant… living in 2D

• You would understand: left, right, forward, backward, but NOT up/down… • How do you know your world is curved?

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

Curvature in 2D…

Curvature in 2D…

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 P1 to P2 (in spacetime, the ant-walk is a bit funny looking, but never mind that)

• Geodesic: shortest line between P1 and P2 (the fewest possible ant steps) ant P1

P2

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 its 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 < 2πR) • 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 critical circumference

2π G Circumference < 2 M ⋅ 2 c mass

gravitational constant 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

2GM RS = 2 = 3( M / M Sun ) kilometers c

• 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 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…