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ASTR 200 : Lecture 30

More Gravity:

• Tides, GR, and Gravitational Waves

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Topic One : Tides

• Differential tidal forces on the Earth.

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How do tides work????Think about 3 billiard balls sitting in space

They fall towards the planet. The red ball accelerates more … GM/r2

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How do tides work????Now imagine sitting on the center ball...!

What do you see?

The two others are PUSHED AWAY from you!

Weird! Gravity looks repulsive?

How is this possible? Because you are in an accelerating reference frame.

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The Moon does the same thing

to Earth's surface

- Replace the billiard balls with locations on the Earth's surface.

- The near and far points on the Earth seem to bulge away from the center!

- Because Earth's oceans are less rigid, the water responds to this `tidal acceleration' the most.

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There are 2 high tides per day

Why? Because the Earth 'turns under' the Moon.

To Moon

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But tides are a completely general concept.It's just the gradient (rate of change) of the gravitational field

• Because gravity goes as 1/r2

• ...the rate of change of the gravitational force (which is what we call the `tidal force' Ft ) goes as 1/r3

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The standard thought experiment: falling into a black hole

• The normal gravitational acceleration you feel is g ~ 10 m/s2

• Turn the Sun into its Rsch ~ 3 km black hole, and you are 100 km from the

singularity. What is the tidal acceleration over your height of ~ 2m ?

• It's 2GM/r3 dr ~ 2(10-10m3kg-1s-2)(2x1030kg)/(105m)3(2m) ~106 m/s2 ~105 g

• So your head and feet will separate rather violently...

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Tidal disruption of unlucky passing objects is probably how supermassive black holes grow

• Tides distort star

• Some material drawn off (core of star might escape)

• Self-interacting material dissipates energy and settles into accretion disk

• Accretion disk friction results in mass being fed below event horizon; some small amount fueling a highly energetic AGN, till mass consumed

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Topic Two : More General Relativity

• A page from Einstein's`Zurich Notebooks'

• His first usage of the

curvature tensors

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The metric tensor gμν in Einstein's field equations

Describes wherethe matter is

Cosmologicalconstant

Describes curvatureof spacetime

Example: Spacetimemetric tensor for spherical polar coordinatesin a flat spacetime

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The Schwarzchild metric

• Describes the geometry of spacetime where one mass M exists

• Note interesting things happen as r → 0 and r → 2GM/c2

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The metric tensor gμν in Einstein's field equations

Describes wherethe matter is

Cosmologicalconstant

Describes curvatureof spacetime

Together these make Gμν

the Einstein tensor

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Row 1 Row 2 Row 3 Row 40

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Column 1

Column 2

Column 3

s

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Motion in curved space time

The trajectories are just 'locally straight' (geodesics)

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The Schwarzchild metric gives elliptical planetary orbits- (in the 'weak field limit' where r always >> Rsch)

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The 'non-Newtonian' effects on the motion thus appear first for objects as close as possible to the Schwarzchild radius – Mercury in our solar system

• A planetary orbit in GR does not precisely close back upon itself

• The net result is that the short-term `elliptical orbit' has its major axis slowly rotate, so the direction pointing to its perihelion advances slowly

• This was measured at the start of the 20th century and needed a resolution. GR provided the solution.

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Topic Three : Gravitational Waves

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Gravitational Waves• In newtonian physics, 'force' is some pull that happens between

massive objects, and the force happens infinitely fast

• In general relativity, 'force' is just provided by the curvature of space time, and accelerating mass distributions 'send out' their changing positions as gravitational waves

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Gravitational Waves• The waves travel as distortions in the metric, and propagate at the speed

of light even though they are NOT electromagnetic radiation

• A very important example of a system that generates graviational waves (or gravitational 'radiation') is a pair of orbiting massive objects

• Because the gravitational waves DO carry mass-energy, the orbital energy of the binary (and hence orbital semimajor axis) will decrease. Recall from an earlier lecture that this orbital period decay of the double pulsar was detected and basically confirmed GR but the gravitational waves were NOT directly detected.

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What detectable signal do gravity waves generate?

• Because of the way the distortion works, as a gravity wave passes a point in space, one spatial axis is contracted, and the perpendicular axis is stretched. This then reverses a half wave later.

• Because the metric is changing, the distance between two points will change, by a small amount.

• Thus, detection can be done by constantly measuring the distance between two points and looking for a periodic oscillation

• The challenge is that the distortion is less than 1 part in 1020...

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 This feat is accomplished using interference of lasersSend a laser beam in two 

orthogonal directions by splitting the light, and then recombine and look for fringes

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The LIGO detectors (Laser Interferometer Gravitational-wave Observatory)

• The two arms are 4 km in length. Each arm holds a 1-m diameter vacuum tube through which the light beam travels

• Even with this, the distances change by only 10-18 m (!)

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The kind of signal

• Seen in two detectors

• No doubt... super convincing

• The rising frequency is caused by the decreasing orbital period in the final merger stage

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Double black hole merger

• This signal is from two black holes that are spiraling toward each other as they emit copious amounts of gravitational waves

• When their event horizons touch they merge and become one larger BH

• The details of the signal give the BH masses before and after the merger

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The first few events: Mergers of two 'stellar remnant' black holes.Note the distances are gigaparsec scale...

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The Mass­Energy Budget of the MergerThe Mass­Energy Budget of the Merger

62 Solar masses

36 Solar masses

29 Solar masses36+29 = 65 ...

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The Mass­Energy Budget of the MergerThe Mass­Energy Budget of the Merger

62 Solar masses

36 Solar masses

29 Solar masses

 E = mc2 energy of 3 solar         masses emitted in            gravitational waves         in ~0.1 sec    =  5 x 1048 W

Sun's L =  4 x 1026 WMW L ~ 1010 stars ~ 4x1036 W              whoa... ~ a trillion galaxies

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This is a new `type' of astronomy!• Gravitational waves do NOT get absorbed, they propagate away at c

(getting weaker as time goes one) but pass through everything

• The current LIGO interferometers detect events at ~100 Hz, from inspiraling binaries of two black holes or two neutron stars

• Future space interferometers (LISA) can detect the lower-frequency mergers of super-massive black holes (expected to be much rarer)

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Even better: use the travel time to many PULSARS as a giant detector

• The NANO-Grav project would use radio telescopes to monitor many pulsars and detect the passage of very long-period gravitational waves