Introduction: Special Relativity •  Observation: The speed c (e.g., the speed of light) is the same

in all coordinate systems (i.e. an object moving with c in S will be moving with c in S’)

•  Therefore: If is valid in one coordinate system, it should be valid in all coordinate systems!

•  => Introduce 4-dimensional “space-time” coordinates:

•  => Introduce “metric” g that defines the “distance” between any 2 space-time points (using Einstein’s summation convention) as

•  Postulate that all products between 2 vectors and the metric is invariant (the same in all coordinate systems)

•  Meaning? If |Δct| > Δr, for a moving object, then there is one system So where Δr = 0 => rest frame for that object. => is the time elapsed in S0 between the 2 points (“Eigentime”)

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Δ! r = cΔt ⇒ cΔt( )2 − Δ

! r ( )2 = 0

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x0 = ct; x1,x2,x3( ) =! r

€

(Δs)2 = gµνΔxµΔxν ; g00 =1, g11 = g22 = g33 = −1, all others = 0

€

ds( )2

Examples •  Object moving (relative to S) with speed v along x. “Distance”

between point 1 = (0,0,0,0) (origin) and point 2 = (ct, vt, 0, 0): where τ is the “eigentime” (time elapsed between the two points in the frame S0 where the object is at rest - i.e. the system moving with v along x-axis) =>

•  Consequence: As seen from S, the clock in So is “going slow”! –  From point of view of So, it is the clock in S that is going slow! –  With similar arguments, one can prove “length contraction”,

“relativity of synchronicity” and all the other “relativity weirdness”

•  Argument can be extended to other quantities: All must come as 4-vectors or as invariant scalars (or tensors…), and the same metric applies to calculate the “invariant length” of each 4-vector –  Example: 4-momentum

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(Δs)2 = g00 ct( )2 + g11 vt( )2 + g2202 + g330

2 = ct( )2 − vt( )2 = cτ( )2

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τ = t ⋅ 1− v2 /c2 = γ−1t

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pµ =Ec, ! p

"

# $

%

& ' ; gµν pµ pν =

Ec

"

# $

%

& ' 2−! p 2 = m2c2

Now a bit more General… •  Equivalence Principle: Motion in a gravitational field is

(locally) indistinguishable to force-free motion in accelerated coordinate system S’: y = - 1/2gt2 –  Example: Free fall in elevator –  2nd example: Clock moving around circle with radius r,

angular velocity ω, speed v=rω => goes slow by factor where “g”=rω2 is the centripetal acceleration. If we replace this with a gravitational force, we must choose Φ = Upot/m such that dΦ/dr = -rω2 => Φ= -1/2 r2ω2 �

•  => New metric: g00 = 1+2Φ/c2 –  Example: clock at bottom of 50 m tower is slower by 5.10-15

than clock at top. Can be measured using Mößbauer effect! –  More general: Curved space-time!

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1− r2ω2 /c2 = 1− r"g"/c2

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⇒ τ clock = 1+ 2Φ(r) /c2 t

1 foot!

General Relativity •  Einstein’s idea: Space-time is curved, with a metric

determined by mass-energy density •  Force-free objects move along geodetics: paths that

maximize elapsed eigentime (as measured by metric) –  Example: twin paradox – it is really the stay-at-home twin that ages more

•  Most general equations complicated (differential geometry), but special manageable case: spherically symmetric mass M at rest => Schwarzschild metric

•  Examples: Falling, redshifting, bending, gravitational lensing, Event horizon (Schwarzschild radius)

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ds( )2 = 1− 2GMrc 2

#

$ %

&

' ( cdt( )2 − 1− 2GM

rc 2#

$ %

&

' ( −1

dr2 − r2 dθ( )2 + sin2θ dφ( )2[ ]

Calculation: Falling

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Radial motion Δr in 2 steps (each taking time Δt): Δr1, Δr2 = Δr-Δr1

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Δs ≈ 1+2Φ1

c 2%

& '

(

) * Δct( )2 − Δr1( )2 + 1+

2Φ2

c 2%

& '

(

) * Δct( )2 − Δr2( )2 = Δct 1+

2Φ1

c 2−

Δr1Δct%

& '

(

) * 2

+ 1+2Φ2

c 2−

Δr2Δct%

& '

(

) * 2,

- . .

/

0 1 1

≈ Δct 1+Φ1

c 2−12

Δr1Δct%

& '

(

) * 2

+1+Φ2

c 2−12Δr2Δct%

& '

(

) * 2,

- .

/

0 1 ≈ Δct 2 +

Φ0 + dΦdr

Δr12

c 2−12

Δr1Δct%

& '

(

) * 2

+Φ0 + dΦ

drΔr+Δr12

c 2−12Δr2Δct%

& '

(

) * 2,

- .

/

0 1

Find extremum of Δs w.r.t. Δr1 (for which Δr1 does Δs become max.?):

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dΔsdΔr1

=!0⇒ 0 =

12dΦdr

c 2 −Δr1Δct( )2 +

12dΦdr

c 2 −Δr2(−1)Δct( )2 =

1Δct

Δr2Δct

−Δr1Δct

&

' ( )

* + +

1c 2dΦdr

=1c 2

1Δt

v2 − v1( ) +dΦdr

&

' ( )

* + ⇒ a = −

dΦdr

q.e.d.

=> Black Holes •  Beyond a certain density, NOTHING can prevent gravitational collapse!

–  If there were a new source of pressure, that pressure would have energy (P = 1/3-2/3 u), which causes more gravitation => gravity wins over

–  Singularity in space-time (infinitely dense mass point, infinite curvature; no classical treatment possible)

•  For spherical mass at rest, Schwarzschild metric applies and we have an event horizon at r = RS = 2GM/c2 = 3km M/Msun (Schwarzschild radius) –  as object approaches rS from outside, clock appears to slow to a crawl and light

emitted gets redshifted to ∞ long wavelength –  along light path, ds = 0 => dr = ±(1-rS/r).dct => light becomes ∞ slow and never

can cross from inside rS to outside –  From outside, it takes exponential time for star surface to reach rS –  Rate of photon emission decreases exponentially (less than 1/s after 10 ms) –  All material that falls in over time “appears” frozen on the surface of event

horizon but doesn’t emit any photons or any other information –  Co-moving coordinate system: will cross event horizon in finite time => no return!

Sample trajectories

0.0E+00%

3.0E+04%

6.0E+04%

9.0E+04%

1.2E+05%

1.5E+05%

1.8E+05%

2.1E+05%

2.4E+05%

2.7E+05%

3.0E+05%

0% 0.0005% 0.001% 0.0015% 0.002% 0.0025% 0.003% 0.0035% 0.004%

Distan

ce)to

)cen

ter)[m])

Proper)Time)[s])

Trajectories)near)10)solar)mass)black)hole)

free%fall%from%∞%

light%from%EH%

light%from%1/2%EH%

0.0E+00%

3.0E+04%

6.0E+04%

9.0E+04%

1.2E+05%

1.5E+05%

1.8E+05%

2.1E+05%

2.4E+05%

2.7E+05%

3.0E+05%

0% 0.0005% 0.001% 0.0015% 0.002% 0.0025% 0.003% 0.0035% 0.004%

Distan

ce)to

)cen

ter)[m])

External)Time)[s])

Trajectories)near)10)solar)mass)black)hole)

free%fall%from%∞%

Δtlocal = 1+2Φgrav

c2Δt∞ ;Φgrav =

Vpotgrav

m= −

GMr

%

&'

(

)*

Rs =2GMc2

⇒ Δtlocal = 1− RsrΔt∞

dτ = 1− Rsr

"

#$%

&'dt∞

2 − 1− Rsr

"

#$%

&'

−1 dr2

c2)

*++

,

-..

1/2

Black holes in the Wild

•  Smallest black holes likely > 3 Msun (supernovae of 25 Msun star followed by complete core collapse) –  mostly detectable as invisible

partner in binary system –  some radiation from

accretion disks (esp. X-ray) smallest radius about 3 rS; about 5-10% of gravitational pot. energy gets converted into luminosity (much more than fusion in stars in case of ns/bh)

•  White dwarf: L ≅ Lsun (UV); ns/bh 1000’s times more (X-ray, gamma-ray)

•  Gigantonormous black holes in center of galaxies (see later in semester)

•  Primordial black holes?

Black holes in the Wild

Gravitational Waves