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

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!)

Gravity = acceleration

• Also solves deep problem • Inertial mass – response to accelerating force F=m i a • Response to gravitational force governed by ‘gravitational charge’ F g =m g GM/r 2 • for matter falling under gravity m g /m i =1 • No other force constant behaves like this eg EM q/m i • 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=2 p • Now rotate • Length contracts along direction of motion so need more ruler lengths to go round C’ > C!! But radius unaffected. • Ratio C’/r > 2 p • 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 2 p r, diameter is p r 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 R a bgd • Encodes how the separation x between two ‘natural paths’ changes with distance s along the path • If all components R a bgd = 0 then D 2 x/ds 2 =0 so x=As+B in ANY coordinates! geodesics separate linearly (flat space, inertial frame motion in straight line at constant speed). • R a bgd  0 then space is intrinsically curved and TIDAL FORCES

Gravity: Energy density, T

ab • Stress energy tensor T ab – 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<

Gravity (energy density) = Curvature

• Try k T ab = R a bgd • Can’t work as can’t have 2 nd a 4 th order tensor (4 order tensor (4 4 =256 elements!) 2 =16 elements) equal to • Sum over some of curvature terms and compress to 2 nd tensor R bg = R a bga order – Ricci • Lose some information about curvature, but not a lot (symmetry) • Try k T ab • Try k T ab = R = R ab ab but derivative of R ½ Rg ab ab = ½ Rg ab NOT zero (R=R a a ) • Both sides have zero derivative so could add constant L g ab • Can rewrite as k( T ab ½ Tg ab ) = R ab where T=T a a • Einstein equations! Lowest order way to write gravity=curvature

Einstein equations

k

(T

ab

- 1/2 Tg

ab

) = R

ab • 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 ds 2 = c 2 d t 2 =c 2 dt 2 - dr 2 - r 2 d q 2 - r 2 sin 2 d f 2 • Curved spacetime, 3 spatial directions, spherical polar coordinates ds 2 = c 2 d t 2 =A (r) c 2 dt 2 - B(r) dr 2 - r 2 d q 2 - r 2 sin 2 d f 2 • Solve for A and B from condition that R ab =0 (NOT R a bcd space IS curved) for empty spacetime and its flat at r  =0, the • ds 2 = c 2 d t 2 =(1-2GM/c 2 r) c 2 dt 2 - (1-2GM/c 2 r) -1 dr 2 - r 2 d q 2 - r 2 sin 2 d f 2 • Schwarzchild metric – something very odd at r=R s =2GM/c 2

Embedding diagram

• dR = (1-2GM/c2r) -1/2 dr • go dr in radius but proper length dR so tilt • dR > dr  as r  2GM/c 2

Event horizon

• Embedding diagram has infinite throat at r=R s =2GM/c 2 ?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!

• ds 2 = c 2 d t 2 =(1-2GM/c 2 r) c 2 dt 2 - (1-2GM/c 2 r) -1 dr 2 - r 2 d q 2 - r 2 sin 2 d f 2 • Below R s =2GM/c 2 the metric terms swaps sign! So suppose held stationary by rockets. Hence dr=d q= d f=0. T hen ds 2 < 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=R s 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.