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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?