1. Chris Done University of Durham/ISAS
General relativity: Schwarzchild metric, event horizon and last stable orbit
Chris Done
University of Durham/ISAS

2. Gravity = acceleration
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!)

3. 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

4. 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

5. 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)

6. Curved spaces

7. Gravity = Acceleration (EP) Acceleration = Curvature (SR)
hence
Gravity = Curvature

8. 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)

9. 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…

10. 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

11. 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.

12. 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

13. Einstein equations k(Tab - 1/2 Tgab) = Rab
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

14. 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

15. 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

16. 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.

17. 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 r t r t r t
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. r t