Astronomy 1142 Quiz 2 Review

Topics

Reference Frames

Special Relativity

General Relativity

Curvature

Tides

Testing GR

Black holes

Orbits Near Black Holes

The Event Horizon

More on Black Holes

Reference Frames

Reference frames have relative velocities, no “absolute” velocity

Inertial reference frames are non-accelerating, constant velocity

Spinning or rotating reference frames are accelerating reference frames

The laws of physics are invariant, they are the same in all reference frames

The speed of light is the same in all reference frames

Special Relativity

Two observers in different inertial reference frames will disagree over values of space and time

Lorentz factor γ=(1-v2/c2)-1/2≥1

γ≈7.3 in figure

Time dilation-observers will notice clocks in other reference frames appear to run more slowly

t'=γt

Length contraction-observers will notice ruler in other reference frames are shorter

L'=L/γ

The effects of length contraction are only in the direction parallel to the velocity

More Special Relativity

Adding velocities, you don't need the formula, just that objects moving slower than the speed of light in one reference frame are slower than the speed of light in all reference frames

Doppler shifts

λ'=λγ(1+vlos

/c)

γ has total velocity, not just los

Mass and energy

General Relativity

Gravity (acceleration) causes differences in reference framesTime moves slower next to the gravitational sourceLight close to the source is redshiftedThis is not symmetric like in SR, the observers know who is deeper to the gravitational field

Newton: matter tells gravity how to exert force, force tells matter how to moveGR: Matter tells spacetime how to curve, curved spaced time tells matter how to moveFreely falling objects follow geodesics (straight lines on curved surfaces)

Curvature

Flat: parallel lines stay parallel, angles of a triangle=180º, circumference of a circle=2πr

Positive(sphere): parallel lines converge, angles of a triangle>180º, circumference of a circle<2πr

Negative(hyperbola): parallel lines diverge, angles of a triangle<180º, circumference of a circle>2πr

How can you measure your local curvature?

Tides

The effect is stronger the “taller” the object falling in

Tides according to GR:

Positive curvature-converging geodesics will “squish”

Negative curvature-diverging geodesics will “stretch”

This is different from the “differential force” view of Newton

Tidal forces near the event horizon are weaker for more massive black holes

Ftides

~1/M2

Tests of GR

Michelson and Morley search for the aether and relative velocity of light

GR predicts twice the deflection of light as predicted by Newton

Measurements made during solar eclipses

Precession of Mercury's orbit predicted exactly by GR

Gravitational redshifts

Black Holes

Need GR to correctly describe black holes

The Schwarzschild radius-the event horizon

RSch

=2GM/c2 (3km for the sun)

The escape velocity of the black hole v=c at event horizon

Not even light can escape

Black holes only depend on Mass, Spin/Angular momentum, and charge (No “Hair”)

The Schwarzschild black hole has no spin or charge, just mass

Orbits Near a Black Hole

There are stable orbits near a black hole

The smallest stable circular orbit is at 3R

Sch for a

Schwarzschild black hole

Elliptical orbits must precess near the black hole

The ergosphere is the boundary of the static limit

Rotating space time causes retrograde orbits to become prograde if inside static limit

The Event Horizon

As you approach the event horizon:

Time dilation and redshifts

Extreme bending of light

Multiple images of objects

More and more of the sky becomes dark until the entire sky is in one small region directly opposite the black hole

As you pass the event horizon:

You don't notice anything abnormal (as long as you survive the tidal forces)

A distant observer never sees you pass the event horizon, time slows to zero

More Black Holes

What goes into a black hole doesn't matter, everything is radiated away except mass, spin/angular momentum, and charge

The singularity exists inside the event horizon

Thorne's hoop conjecture says that if you get all of the mass, spin, charge, singularity inside a hoop at R=R

Sch then you get a black hole

No “naked” singularities (Cosmic Censorship)

Energy can be extracted from a rotating or charged black hole, until zero charge and zero spin (Schwarzchild black hole)

As energy is extracted, the area of the event horizon must increase (Hawking's Area Theorem)

Questions?