Foundations of general relativity and black-holes

Neil J. Cornish

40 Scientific American, October 2013 Photograph by Tktk Tktk

sad1013Ande3p.indd 40 8/21/13 1:12 PM Outline

• Relativity and geometry • Equivalence principle • Connecting to Newton • Field equations and equations of motion • Black Hole solutions and orbits Relativity and Geometry

Special relativity is encoded in the geometry of Minkowski space Relativity and Geometry

x ct = Worldline of observer O’

ct = x Lines of simultaneity for O’

ct0 = (ct x) x0 = (x ct)

v 1 = = 1 2 c p Invariant Interval

ds2 = ⌘ dxµdx⌫ = c2dt2 + dx2 µ⌫ ct0 = (ct x) 10 ⌘µ⌫ x0 = (x ct) ! 01 ✓ ◆

2 µ ⌫ 2 2 2 2 ds0 = ⌘ dx0 dx0 = c dt0 + dx0 = ds µ⌫ Time Dilation

Proper time elapsed t0 = T0

Apparent time elapsed t = T

2 2 2 2 s = (ct0) = (ct) +x AB

But, x = vt

2 2 2 2 t t0 =t (1 )= ) 2

T = T ) 0 Length Contraction

Proper length x0 = L0

C Apparent length x = L

2 2 2 2 s =x =x0 (ct0) AB 2 2 2 2 s =x0 =(x + vt) (ct) AC

and t = t0

After a little algebra….

L L = 0 ) Equivalence Principle (Newtonian)

Inertial mass Gravitational charge

mG M F = mI a FG = G ˆr r2

mG = mI

“It was most unsatisfactory to me that, although the relation between inertia and energy is so beautifully derived [in Special Relativity], there is no relation between inertia and weight. I suspected that this relationship was inexplicable by means of Special Relativity” Albert Einstein, 1922 Pseudo forces in non-inertial frames (Newtonian)

The pseudo forces experienced in a non-inertial frame are automatically proportional to the inertial mass x¨ = a 2 ! x˙ ! (! x) NI ⇥ ⇥ ⇥

Uniform acceleration Coriolis Centripital

We readily accept that Coriolis and Centripital forces are due to us being in a non-internal frame tied to the surface of the Earth. If we also accept that inertial observers should be in free-fall, then gravity is also a pseudo force!

This is exactly what Einstein reasoned: “I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me: ‘If a person falls freely he will not feel his own weight’. I was startled. This simple thought made a deep impression upon me. It impelled me towards a theory of gravitation” Gravity as Geometry

Curved light path seen in accelerated frame

Straight light path seen in inertial frame Gravity as Geometry

2az ds2 = 1+ c2dt2 + dx2 + dz2 c2 ✓ ◆

Null geodesics x = ct 1 z = at2 2

a z = x2 ) 2c2 Riemannian Geometry and Geodesics

Geodesics in Riemannian geometry generalize the concept of straight lines. They extremism the interval between events, and keep heading in the same direction (parallel transport their tangent vectors)

B B ↵ 1/2 B ⌫ 2 dx dx µ dx S = ds = g↵ d = L x , d A A d d A d Z p Z ✓ ◆ Z ✓ ◆

d @L @L Euler-Lagrange: = d @(dx↵/d) @x↵ ✓ ◆

d2x↵ 1 dxµ dx⌫ dxµ dx⌫ = g↵ (g + g g ) = ↵ ) d2 2 µ,⌫ ⌫,µ µ⌫, d d µ⌫ d d

dx↵ or, introducing the covariant derivative u↵ = u↵ + u⌫ ↵ and the 4-velocity u↵ = r , ⌫ d

u u↵ =0 ) r Newtonian Gravity Using Riemannian Geometry

ds2 = (c2 + 2)dt2 + dx2 + dy2 + dz2 c2 ⌧

@ ,i Non-vanishing Christoffel symbols: i = = i =i = 0 tt @xi ,i ti it c2 + 2 ⇡

d2t Geodesic equation: 0 t = d2 ⇡ ) d2xi = x¨ = dt2 ,i ) r

Field equation: 2=4⇡G⇢ r

Note: This metric predicts light deflection by the Sun that is a factor of 2 smaller than GR Gravity as Geometry Revisited

Rocket in a uniform gravitational field =gz

ds2 = (c2 +2gz)dt2 + dx2 + dz2

d2x d2z Geodesics =0 = g dt2 dt2

1 g 2 t x = ct, z = gt ⌧ c ) 2 ⇣ ⌘ g z = x2 ) 2c2 Pseudo forces from geometry

Uniform acceleration and rotation

ds2 = (c + a x)2 (! x)2 dt2 +2c(! x) dxidt + dx2 + dy2 + dz2 · ⇥ ⇥ i

Christoffel symbols i g = ai +(! (! x))i tt ⇡ tt,i ⇥ ⇥ 1 i (g g )= c✏ !k tj ⇡ 2 ti,j tj,i ijk

Equations of motion x¨ = a 2 ! x˙ ! (! x) ⇥ ⇥ ⇥

Note: In GR it is always possible to find a coordinate system that is un-accelerated and non-rotating - free fall Fermi-Walker transport of a locally inertia coordinate system Gravity is a tidal force

Gravitational field is only locally equivalent to an accelerated reference frame

L z d2r GM Geodesic motion zˆ dt2 ⇡ R2 y

d2r GM L yˆ dt2 ⇡ R3

Geodesic deviation Tidal force Riemannian Geometry: Curvature

n+1 n R 2 2⇡ 2 ✏ Surface area of a small n-sphere in n-dimensional curved space An = 1 ✏ + ... 6(n + 1) ( n+1 ) ✓ ◆ 2

R = gµ⌫ R R = R R =  +↵  ↵  µ⌫ µ⌫ µ⌫ µ⌫ µ⌫, µ,⌫ µ⌫ ↵ µ ↵⌫

Ricci scalar Ricci tensor Riemann tensor Non-rotating free-fall & Geodesic Deviation

u e↵ = e↵ e u e r (k) (t) (k) r (t) Can always find coordinates in the neighborhood of some event: 1 g = ⌘ R x↵x + (x3) µ⌫ µ⌫ 3 µ↵⌫ O

Riemann normal coordinates

Fermi-Walker transport of this coordinate system along a geodesic defines a local inertial frame (no pseudo forces)

~x 0 u↵ (u ⇠µ)=Rµ u↵u⇠ r↵ r ↵ Nearby geodesics do not remain parallel: Geodesic Deviation ~x ⇠~ ¨i 2 i j Weak field, slow motion limit: ⇠ = c R tjt ⇠ Field Equations of General Relativity

Einstein sought a geometric theory that reduced to Newtonian gravity for weak fields and slow motion

Relativity principle demands the equations must be invariant under changes of coordinates - need tensor relations

1 4⇡G For weak fields and slow motion we saw @i@ g =4⇡G⇢ T 2 i 00 ' c2 00

The only second rank tensors that can be formed out of the metric and its second derivatives are Rµ⌫ gµ⌫ R gµ⌫

Field equations must have the form Rµ⌫ + ↵gµ⌫ R +⇤gµ⌫ = Tµ⌫

1 8⇡G Conservation of energy T µ ⌫ =0 demands ↵ = Recovering the Newtonian limit implies  = ;⌫ 2 c4

1 8⇡G R g R +⇤g = T µ⌫ 2 µ⌫ µ⌫ c4 µ⌫ Equations of motion for General Relativity

Only theory where the field equations determine the equations of motion - Early work by Einstein-Infeld-Hoffman and Papapetrou Derived by considering small concentration of matter in a background space-time

µ ⌫ Zeroth order (EIH): u u =0 rµ 1 Second order (Papapetrou): uµ u⌫ = R ⌫ S↵u rµ 2M ↵ 1 1 With radiation reaction (Gralla & Wald) uµ u⌫ = R ⌫ S↵u (g⌫ + u⌫ u)( htail htail)u u↵ rµ 2M ↵ r↵  2r ↵ Black Holes - Vacuum Solutions in GR

Most general form for a time independent, spherically symmetric spacetime: (G = c = 1)

ds2 = U(r) dt2 + V (r) dr2 + r2 d✓2 + r2 sin2 ✓d2

1 V 0 1 1 R g R =0 + 1 =0 (1) tt 2 tt ) rV 2 r2 V ✓ ◆

1 U 0 V 1 R g R =0 1 =0 (2) rr 2 rr ) rU r2 V ✓ ◆

1 2M rV (1) + r (2) = (ln(VU))0 =0 (1) V = 1 ) ⇥ ⇥ ) r ✓ ◆ 1 V = ) U Black Holes - Vacuum Solutions in GR

Schwazschild Solution:

1 2M 2M ds2 = 1 dt2 + 1 dr2 + r2 d✓2 + r2 sin ✓2 d2 Areal r r ✓ ◆ ✓ ◆

2 1 M M 4 ds2 = 2r dt2 + 1+ dr2 + r2 d✓2 + r2 sin ✓2 d2 Isotropic M 2 2r 1+ 2r ✓ ◆

2M 3 2/3 ds2 = dt2+ dR2+r2(d✓2+sin2 ✓d2)withr = (R T ) (2M)1/3 Lemaitre r 2 ✓ ◆

Don’t read too much into the form of the metric in a particular coordinate system. Areal and Isotropic metrics are singular at r=2M, r=M/2, while Lemaitre’s is not. Black Holes - Vacuum Solutions in GR

1 2M 2M Schwarzschild Solution: ds2 = 1 dt2 + 1 dr2 + r2 d✓2 + r2 sin ✓2 d2 r r ✓ ◆ ✓ ◆

Curvature singularity at r=0 but not r=2M (orthonormal basis) 2M R = R =2R =2R = 2R = 2R = rˆtˆrˆtˆ ✓ˆˆ✓ˆˆ ✓ˆtˆ✓ˆtˆ ˆtˆˆtˆ rˆˆrˆˆ rˆ✓ˆrˆ✓ˆ r3

Redshift of light sent to a distant Force required to hold station at observer diverges as the source fixed radius diverges at r=2M approaches r=2M

1 rˆ mM 1 z(r)= 1 F = 1 2M 2 1 r 1 2M 2 r r q Schwarzschild Geodesics

1 2M 2M Schwarzschild Line Element: ds2 = 1 dt2 + 1 dr2 + r2 d✓2 + r2 sin ✓2 d2 r r ✓ ◆ ✓ ◆

Symmetries and conserved quantities - 4 Killing vector fields

⇠~ = @~ ⇠~ = @~ ⇠~ =sin @~ + cot ✓ cos @~ ⇠~ = cos @~ cot ✓ sin @~ 1 t 2 3 ✓ 4 ✓

⇡ Mass, energy and angular momentum are all conserved. Can used rotational symmetry to set ✓ = 2 2M ~ t 2 2 E = ⇠1 p~ = m 1 u L = ⇠~ p~ = mr u m = p~ p~ · r 2 · ✓ ◆ ·

E˜2 1 1 dr 2 M L˜2 ML˜2 = + ) 2 2 d⌧ r 2r2 r3 ✓ ◆ Schwarzschild Geodesics Particles Ve↵ (r)

E˜2 1 1 dr 2 M L˜2 ML˜2 = + 2 2 d⌧ r 2r2 r3 ✓ ◆

Same as Kepler problem

Photons

Ve↵ (r)

dr 2 L2 2M E2 = + 1 d r2 r ✓ ◆ ✓ ◆ Schwarzschild Geodesics

! = ! = ! ✓ 6 r

Innermost stable Unstable circular photon L˜2 M 2 r = 1 1 12 circular orbit (ISCO) orbit (Light Ring) ± 2M 0 ± s L˜ 1 ✓ ◆ rISCO =6M rLR =3M @ A

Rotating Black Holes - Kerr Solution Boyer-Lindquist coordinates: 2 2 sin ✓ 2 ⌃ ds2 = dt a sin2 ✓d + (r2 + a2) d adt + dr2 +⌃d✓2 ⌃ ⌃ J a = ⌃=r2 + a2 cos2 ✓ =r2 2Mr + a2 M

Horizons: No escape from inside r+ r = M M 2 a2 ± ± Static Limit: Must rotate with BHp inside

r = M + M 2 a2 cos2 ✓ SL p ⇡ Curvature singularity when r =0 & ✓ = 2 Geodesics in Kerr

Symmetries lead to four conserved quantities: 2 2 2 p E = pt L = p m = p~ p~ C = p + · ✓ sin2 ✓

While the orbits are fully specified by the 4 conserved quantities, the paths in space can be highly non-Keplarian ! = ! = ! 6 ✓ 6 r Geodesics in Kerr Useful (random) facts

The spacetime outside a non-rotating star is Schwazschild. The spacetime outside a rotating star is not Kerr

Particles in an accretion disk do not follow geodesics (there are inter-particle forces), but geodesics do provide insight

Frame-dragging (LT precession) drives the inner accretion disk to lie in the equatorial plane (Bardeen-Pettersen)

Co-rotating and counter-rotating orbits are very different. In the high spin limit, M=a, the ISCO is at r=M (co- rotating) and r=9M (counter-rotating) Extra Slides Fermi-Walker Transport of a Tetrad

u e↵ = e↵ e u e r (k) (t) (k) r (t)

↵ This equation determines the transport of the orthonormal basis e ( k ) along ↵ an arbitrary timelike curve, such that e ( t ) is always tangent to the curve