General relativity: Schwarzchild metric, event horizon and last stable orbit

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General relativitySchwarzchild metric, event
horizon and last stable orbit

• Chris Done
• University of Durham

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Gravity acceleration

• How to tell the difference between gravity and
  acceleration ?
• Look the same, behave the same
• Maybe they ARE the same - Einsteins happiest
  thought
• Principle of equivalence accelerationgravity
• Free fall in gravity floating in space
  (inertial frame!)

3
Special relativity

• Physics in inertial frames is special relativity!
  Towering achievement
• Throw away ideas about fixed space and fixed
  time!!!
• NOT that everything is relative!
• Fixed spacetime interval
• 1D time,
• ds2c2dt2c2dt2 - dx2
• Fixed speed c through spacetime. Mostly through
  time! But if put some through space then travel
  through time less fast.fast clocks run slow!!
• Also length contraction along direction of motion

P
ct
ds
x

• BUT only does inertial frames. Cant handle
  acceleration..

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Acceleration special relativity

• Circular motion easiest to think about
• Measure roundabout circumference (CL) and
  diameter (dL) by crawling around with ruler of
  length L
• Get ratio C/dp
• Now rotate
• Time dilation fast clocks run slow
• Length contracts along direction of motion so
  need more ruler lengths to go round C gt C!!

• But diameter unaffected.
• Ratio C/d gt p
• Cant happen!! in flat space

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Curved spaces

• Can happen in curved spaces!!
• eg sphere. Circle round equator. Circumference is
  2pr, diameter is pr so ratio is 2 lt p!!!
• But we wanted this number bigger than p
• Sphere is ve curvature curves away in all
  direction
• Inside a sphere also ve as curves towards in
  both directions
• Can get ratio gt p only in negatively curved space
  curves towards in one direction and away in
  another (saddle)

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• Acceleration Curvature (SR)
• Gravity Acceleration (EP)
• hence
• Gravity Curvature

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Gravity warped spacetime

• Straight paths on curved space!! Shortest
  distance, geodesics, inertial frames!
• NOT a spooky, action at a distance force
  (Newtonian)
• Space(time) warped by mass(energy)

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Gravity warped spacetime

• Matter tells space how to curve, curvature tells
  matter how to move

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Curved spacetime general relativity

• Flat spacetime ds2 c2dt2 c2dt2 - dr2 -
  r2sin2df2
• Solve Einstein in special case of single,
  spherically symmetric, static mass curving
  spacetime to compare with Newtonian
• ds2 c2dt2(1-2GM/c2r) c2dt2 - (1-2GM/c2r)-1 dr2
  - r2sin2df2
• Schwarzchild metric something very odd at
  rRs2GM/c2
• distance goes infinite, time goes to zero..
  Event horizon
• Below this time becomes spacelike, space
  timelike.

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Straight lines on curved space

• Minimum distance between two points! straight
  line in flat space
• Looks curved
• Everything that travels over spacetime is curved
  so gravity affects light Newtonian gravity
  affects MASS so doesnt affect light (?)
• Lightbending one of first tests of GR!

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Newtonian gravity orbits

• Newtonian orbits
• Gravity versus angular momentum
• Gravity attractive likes to be closer -1/r
• Angular momentum outwards L/r2
• Angular momentum barrier
• Minimum energy circular orbit
• Ellipse (bound)
• Hyperbolic (unbound)

V
r
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Compare with GR orbits

• Rest mass
• Large distances gravity 1/r
• Smaller distances ang. mom 1/r2
• Very small 1/r3
• Extra term!! adds to gravity.
• Makes gravity stronger!
• Weak field difference is only small
• - precession of Mercury perihelion. Another
  test of (weak field) GR
• Same sorts of orbits as before
• .except can have ones which get to r 0 where
  potential undefined! V2 lt 0 everywhere below
  r2GM/c2

V2
r
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Behaviour of maximum

• But not at all radii
• Cubic has max and min
• BUT THESE MERGE at r6GM/c2 3Rs

V2
r
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Behaviour of maximum

• Shape of the curve changes. To a point of
  inflection
• Needed maximum to hold orbit stable
• Just falls off!
• Last stable orbit
• At r6GM/c2
• NOT where vc r3GM/c2
• NOT the event horizon r2GM/c2
• GRAVITY is stronger in GR
• Black holes suck only at small r
• NOT cosmic vacuum cleaner for
• all space (bad Sci-Fi movies)

V2
r
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Event horizon

• What happens at rRs2GM/c2?
• Speed is distance/time ? c no matter where
  dropped from or how fast it was hurled towards
  the hole
• So must be infinite accelerations (could drop
  from rest just above horizon and would still be
  at c at Rs)
• Cant have fixed anything! So no sense to make a
  fixed radial grid..

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Event horizon

• ds2 c2dt2(1-2GM/c2r) c2dt2 - (1-2GM/c2r)-1 dr2
  - r2sin2df2
• Embedding diagram shows dR not spacetime
  curvature.
• True curvature ? ? at r0 and is finite (though
  large) at rRs

rRs
r0
r
t
r

• And principle of equivalence in free fall so
  is inertial frame and no difference between this
  and no gravity at all!
• until you hit r0 or rather when tidal forces
  rip you apart.

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Gravity warped spacetime

• 2 key predictions of strong field GR last
  stable orbit at 6GM/c2 and event horizon at
  2GM/c2
• Utterly extreme. Need mass of earth squashed down
  to 1cm! Or mass of sun squashed into size of
  London.
• Impossible!!!!!!!!?

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Black hole recipe I

• Can get such extreme compression in death of the
  highest mass stars
• Stars fuse 4H to He
• Lose mass, gain energy via Einsteins Emc2
• hydrogen bomb! in its stable life outward
  pressure of hot gas (fusion) balanced by inward
  pull of gravity
• Hydrogen fuel eventually exhausted. Gravity isnt!

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Black hole recipe II

• Supernovae if very massive star
• Core implodes as cant hold itself up against
  gravity
• Forms neutron star

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Black hole recipe III

• But core being hit by infalling layers from above
• Neutrons get squashed into smaller and smaller
  box, going faster and faster
• Hit c at 1.4-3x mass of sun (depends on rotation
  rate)
• no known stable state of matter can hold up
  against complete collapse
• Only need factor of 3 smaller than a neutron star
  to get to event horizon. Roughly size of last
  stable orbit!!

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Observing black holes?

• The thing about a black hole, its main
  distinguishing feature is its black! And the
  thing about space, your basic space colour is its
  black! So how are you supposed to see them ? Red
  Dwarf

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Disc Accretion

• Single particles simply orbit
• Gravitational orbits (as in solar system) mean
  inner edge faster than outer edge.
• Continuous ring of gas - Frictional viscosity
  dissipates energy as material can fall inwards
• BRIGHT accretion discs

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Bright accretion discs!

• Huge gravitational potential energy of infalling
  material so gas heated to X ray temperatures and
  very luminous.
• Disc down to last stable orbit different from
  Newtonian gravity where can always orbit closer
  by going faster.

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Galactic Binary systems

• Huge amounts of data
• Timescales
• ms year (observable!)
• hours 108 years in quasars
• Observational template of accretion flow as a
  function of L/LEdd onto 10 M? BH
• Look for last stable orbit
• Comparison sample of NS 1.4 M? objects. Same
  accretion flow PLUS surface. Difference shows
  evidence for event horizon

7 years
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Event horizon

• BH event horizon 2-0.5 Rs, last stable
  orbit 3-0.5 Rs
• Neutron stars. R3Rs so gravitational potential
  for accretion flow the same.
• Look at same M/MEdd for accretion flow
  difference reveals presence/absence of surface!

. .
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Spectra of accretion flow disc

• Differential Keplerian rotation
• Friction gravity ? heat
• Thermal emission L AsT4
• Temperature increases inwards
• GR last stable orbit gives minimum radius Rms
• For a0 and LLEdd
• Tmax is 1 keV (107 K) for 10 M?

Log n f(n)
Log n
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Disc spectra last stable orbit

• Pick ONLY ones that look like a disc!
• L/LEdd ?T4max (Ebisawa et al 1993 Kubota et al
  1999 2001)
• Constant radius over factor 10-50 change in
  luminosity
• Last stable orbit!!! Looks like Einstein GR
  (Gregory, Whisker, Beckwith Done 2004)
• Proportionality constant gives Rms i.e. a as
  know M
• Consistent with low to moderate spin not maximal

Gierlinski Done 2003
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Disc spectra last stable orbit

• Pick ONLY ones that look like a disc!
• L/LEdd ?T4max (Ebisawa et al 1993 Kubota et al
  1999 2001)
• Constant radius over factor 10-50 change in
  luminosity
• Last stable orbit!!! Looks like Einstein GR
  (Gregory, Whisker, Beckwith Done 2004)
• Proportionality constant gives Rms i.e. a as
  know M
• Consistent with low to moderate spin not maximal

Gierlinski Done 2003
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Disc spectra last stable orbit

• Pick ONLY ones that look like a disc!
• L/LEdd ?T4max (Ebisawa et al 1993 Kubota et al
  1999 2001)
• Constant radius over factor 10-50 change in
  luminosity
• Last stable orbit!!! Looks like Einstein GR
  (Gregory, Whisker, Beckwith Done 2004)
• Proportionality constant gives Rms i.e. a as
  know M
• Consistent with low to moderate spin not maximal

Gierlinski Done 2003
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But rest are not simple

• Bewildering variety of spectra from single object
• Underlying pattern
• High L/LEdd soft spectrum, peaks at kTmax often
  disc-like, plus tail
• Lower L/LEdd hard spectrum, peaks at high
  energies, not like a disc

Gierlinski Done 2003
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Accretion flows without discs

• Disc models assumed thermal plasma not true at
  low L/LEdd
• Instead hot, optically thin, geometrically thick
  inner flow replacing the inner disc (Shapiro et
  al. 1976 Narayan Yi 1995)
• Hot electrons Compton upscatter photons from
  outer cool disc
• Few seed photons, so spectrum is hard

Log n f(n)
Log n
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Qualitative and quantitative models geometry
Log n f(n)
Log n
Hard (low L/LEdd)
Soft (high L/LEdd)
Log n f(n)
Log n
Done Gierlinski 2004
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Observed GBH spectra

• RXTE archive of many GBH
• Same spectral evolution 10-3 lt
  L/LEdd lt 1

• Truncated disc?? Rms qualitative and quantitative

Done Gierlinski 2003
1.5
3.0
4.5
G (3-6.4)
1.5
1.5
3.0
3.0
4.5
4.5
G (6.4-16)
G (6.4-16)
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Lh/Ls
LS
Hard (low L/LEdd) Soft (high L/LEdd)
1
VHS
HS
US
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Conclusions

• Test GR - X-rays from accreting black holes
  produced in regions of strong gravity
• Last stable orbit (ONLY simple disc spectra) L
  ?T4max
• Corrections to GR from proper gravity must be
  smallish
• Accretion flow NOT always simple disc. Model this
  in black holes
• Compare to neutron stars and compatible with same
  accretion flow PLUS a surface
• ASTROPHYSICS ? PHYSICS