Transcript Gamma Ray Bursts: The biggest bang since the big one!

General relativity:
Schwarzchild metric, event
horizon and last stable orbit
Chris Done
University of Durham/ISAS
Gravity = acceleration
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How to tell the difference between gravity and acceleration ?
Look the same, behave the same…
Maybe they ARE the same - Einstein’s ‘happiest thought’
Principle of equivalence: acceleration=gravity
Free fall in gravity = floating in space (inertial frame!)
Gravity = acceleration
• Also solves deep problem
• Inertial mass – response to
accelerating force F=mia
• Response to gravitational
force governed by
‘gravitational charge’
Fg=mgGM/r2
• for matter falling under
gravity mg/mi=1
• No other force constant
behaves like this eg EM q/mi
• But obviously the same if
gravity = acceleration
Acceleration: special relativity
• Circular motion easiest
to think about
• Measure roundabout
circumference (CL) and
radius (rL) by crawling
around with ruler of
length L
• Get ratio C/r=2p
• Now rotate
• Length contracts along
direction of motion so
need more ruler lengths
to go round C’ > C!! But
radius unaffected.
• Ratio C’/r > 2p
• Can’t happen!! …in flat space
• Also time dilation: acceleration
= gravity slows down time
Curved spaces
• Can happen in curved spaces!!
• Eg sphere. Circle round equator. Circumference is 2pr, diameter
is pr so ratio is 2 < 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 > p only in negatively curved space – curves
towards in one direction and away in another (saddle)
Curved spaces
Gravity = Acceleration (EP)
Acceleration = Curvature (SR)
hence
Gravity = Curvature
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)
Event horizon
• So light is affected too!
• One of first tests of GR
• More gravity, deeper hole in
spacetime, higher velocity to
escape - more mass or
smaller size
• Black hole – escape velocity
is faster than light so can’t
get out!
• No change in curvature at
Earths orbit – black holes
don’t suck inexorably!
Unlike bad SF movies…
Curvature
• Riemann curvature tensor Rabgd
• Encodes how the separation x between two ‘natural paths’ changes
with distance s along the path
• If all components Rabgd = 0 then D2x/ds2=0 so x=As+B in ANY
coordinates! geodesics separate linearly (flat space, inertial frame
motion in straight line at constant speed).
• Rabgd  0 then space is intrinsically curved and TIDAL FORCES
Gravity: Energy density, Tab
• Stress energy tensor Tab – all
contributions to energy density
ie rest mass AND pressure.
• Thermal energy per particle 3kT,
number density n so 3nkT = 3P
• Pressure adds to gravity! – black
holes inevitable if P dominates!
• All forms of energy gravitate!!
• Makes sense of Special Relativity. Increase velocity so increase KE
so increase response to gravity. KE dominated by rest mass for
v<<c so constant mass. But v~c then KE dominates. Increasing
energy increases response to gravity ie increases inertial mass and
harder to increase speed! Stops you going faster than c
• Conservation of energy and momentum so derivative is zero.
Gravity (energy density) = Curvature
• Try kTab = Ra bgd
• Can’t work as can’t have 2nd order tensor (42=16 elements) equal to
a 4th order tensor (44=256 elements!)
• Sum over some of curvature terms and compress to 2nd order – Ricci
tensor Rbg = Ra bga
• Lose some information about curvature, but not a lot (symmetry)
• Try kTab = Rab but derivative of Rab = ½ Rgab NOT zero (R=Raa )
• Try kTab = Rab - ½ Rgab
• Both sides have zero derivative so could add constant Lgab
• Can rewrite as k(Tab - ½ Tgab ) = Rab where T=Taa
• Einstein equations! Lowest order way to write gravity=curvature
Einstein equations
k(Tab - 1/2 Tgab) = Rab
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Represents 10 independent equations
Einstein thought it wouldn’t ever be solved
Schwarszchild in trenches of Russian front in WWI
Solved by imposing geometry (not trying to solve in full generality)
Empty spacetime round spherically symmetric stationary massive
body so can compare to Newtonian results and TEST!!!
• Must look like flat space far from massive body
Curved spacetime: general relativity
• Flat spacetime, 3 spatial directions, spherical polar coordinates
ds2= c2dt2=c2dt2 - dr2 - r2dq2 - r2sin2df2
• Curved spacetime, 3 spatial directions, spherical polar coordinates
ds2= c2dt2=A (r) c2dt2 - B(r) dr2 - r2dq2 - r2sin2df2
• Solve for A and B from condition that Rab =0 (NOT Rabcd =0, the
space IS curved) for empty spacetime and its flat at r 
• ds2= c2dt2=(1-2GM/c2r) c2dt2 - (1-2GM/c2r)-1 dr2 - r2dq2 - r2sin2df2
• Schwarzchild metric – something very odd at r=Rs=2GM/c2
Embedding diagram
• dR = (1-2GM/c2r)-1/2 dr
• go dr in radius but proper
length dR so tilt
• dR > dr  as r  2GM/c2
Event horizon
• Embedding diagram has infinite throat at r=Rs=2GM/c2?Is it real?
• Look at satellite dropping (freefall). Person on satellite gets to centre
in finite time.
• Observer at infinity sees them initially accelerate towards the hole
but decelerate and stop at the horizon…
• Observer at horizon sees them come past at speed of light
irrespective of where they were dropped from! So is there infinite
acceleration here?? Plainly something funny going on!
• ds2= c2dt2=(1-2GM/c2r) c2dt2 - (1-2GM/c2r)-1 dr2 - r2dq2 - r2sin2df2
• Below Rs=2GM/c2 the metric terms swaps sign! So suppose held
stationary by rockets. Hence dr=dq=df=0. Then ds2 < 0 below
horizon! But then there are no real paths. Real paths MUST have a
spatial term (now +ve) to offset the dt (now –ve) term. So no such
thing as stationary observers below horizon.
Event horizon
• Embedding diagram shows dR not spacetime (Riemann) curvature.
True curvature   at r=0 and is finite (though large) at r=Rs
r=0
r=Rs
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 r=0 or rather when
tidal forces rip you apart.