4. General Relativity and Gravitation 4.1. The Principle of Equivalence 4.2. Gravitational Forces 4.3. The Field Equations of General Relativity 4.4. The Gravitational Field of a Spherical Body 4.5. Black and White Holes

4.1. The Principle of Equivalence Under a coordinate transformation x μ → x μ, In general, Every real symmetric matrix can be diagonalized by an orthogonal transformation: g j are the (real) eigenvalues of g. Consider Λ = O D, where D = diag(D 1, …, D d ). Λ  1 exists & real → D j  0 & real  j. → Choosing →  canonical form of the metric tensor Spacetime is locally flat (Minkowskian): at 1 point.

4.2. Gravitational Forces Lagrangian: Special relativity: Principle of covariance → all EOMs covariant under Λ that leaves η unchanged ( Poincare transformations ) General relativity: Principle of covariance → all EOMs covariant under all Λ → L is a scalar ( contraction of tensors ) Principle of equivalence → L is Minkowskian in any local inertial frame. → L contains only contractions involving g μν and g μν, σ.

Free Particles Minkowski → General:  → g : → This is also the only choice that is both covariant and linear in g. Euler-Lagrange equation:

Geodesic equation

Gravity Let( h μν small ) Non-relativistic motion: →→

→ → The only non-vanishing components of g μν,σ are g 00, j = h 00, j. → Setting gives Newtonian gravity

4.3. The Field Equations of General Relativity Electrodynamics: Gravitational field : L Particles + L Interaction Task is to find L Fields → invariant infinitesimal spacetime volume dV For a Minkowski spacetime → = Jacobian  d 4 x is a scalar density of weight  1 Only choice is ( g = det | g | is a scalar density of weight +2 ) Check: In a Minkowski spacetime

By definition f = scalar function →is a scalar is a scalar density of weight +1 is a scalar.

Lagrangian Densities = Ricci curvature κ = coupling constantΛ = cosmological constant Einstein introduced Λ to allow for a static solution, even though the vacuum solution would no longer be Minkowskian. At present: Λ = 0 within experimental precision. Recent theories: Λ  0 immediately after the Big Bang.

For N = 1:

Field Equations Euler-Lagrange equations for the metric tensor field degrees of freedom g μν are called Einstein’s field equations: stress tensor Einstein curvature tensor → Ex.4.2

Another Form of the Field Equation g νμ field eqs → Field eqs:

Newtonian Limit Newtonian theory : κ is determined by the principle of correspondence. → non-vanishing components of R must have at least two “0” indices. → 

ρ is stationary → To lowest order in h, →

4.4. The Gravitational Field of a Spherical Body The Schwarzschild Solution (1916 ): 1.ρ is spherically symmetric; so is g. 2.ρ is bounded so that g ~ η at large distances. 3. g is static (t-independent) in any coordinate system in which ρ is stationary. 2. → Note: ( r, θ, φ) are spatial coordinates only when r → . An extra C(r) factor in the “angular” term can be absorbed by →

Exterior Solutions → (2 nd order partial differential equations for g μν ) → Schwarzschild solution [see Chapter 14, D’Inverno ]: = Schwarzschild radius Singularity at r = r s will be related to the possibility of black holes.

Time Near a Massive Body Coordinate t = time measured by a stationary Minkowskian (r→  ) observer. To this observer, two events at (ct 1, x 1 ) and (ct 2, x 2 ) are simultaneous if t 1 = t 2. For another stationary observer at finite r > r S, time duration experienced = proper time interval d  with dx = 0 → two events simultaneous to one stationary observer (Δτ 1 = 0 ) are simultaneous to all stationary observers (Δt = Δτ 2 = 0 ). The finite duration Δτ of the same events (fixed dt  0) differs for stationary observers at different r.

If something happens at spatial point (r 1,θ 1,φ 1 ) for duration another stationary observer at (r obs,θ obs,φ obs ) will find For the observation of emision of light Verified to an accuracy of 10  3 by Pound and Rebka in 1960 for the emission of  rays at a height of 22m above ground using the Mossbauer effect.

For measurements done on the sun and star light, Earth’s gravity can be ignored. For starlights observed on earth, → gravitational red shift Originally, observed red shifts ~ validation of the theory of general relativity. Now: ~ validation of the principle of equivalence. → Allows for other gravitational theories, such as the Brans-Dicke theory.

Distances Near a Massive Body → Radial distance between 2 points with the same  and  coordinates is defined as where Only exterior solution known → radial distance of a point from the origin is not defined.

Consider circular path described by the equations r = a and θ = π/2. Its length, or circumference, is ( same as E 3 ) Its radius is not defined. Closest distance between 2 concentric circles r = a 1 and r = a 2 is not A “circle” of a well defined radius a about a point would appear lopsided when plotted using the spherical coordinates. Since for

The lowest order of corrections valid forare

Particle Trajectories Near a Massive Body Einstein field equations are non-linear → principle of superposition is invalid → perturbation theory inapplicable → even the 2-body problem is in general intractable One tractable class of problems: Motion of a “test” particle ( geodesics of g ) For time-like geodesics in the Schwarzschild spacetime,

Setting m = 0 makes S = 0. Hence, for massless particles, we switch to another affine parameter so that Null geodesic eqs are obtained from the geodesics by replacing τ with λ. Notable phenomena: Bending of light by the sun. Precession of Mercury. See Chap 15, D’Inverno. In practice, the r eq is usually replaced by

4.5. Black and White Holes R = radius of the mass distribution. If R > r S then singularity at r = r S is fictitious. Problem of interest: R < r S and R < r < r S

Radial Motion: Solution for r Free particle with purely radial motion ( dθ = dφ = 0 ): → → EOM for r: → → → Newton’s law

For we have→ → Outgoing Incoming Singularity at r = r S not felt

Radial Motion: Solution for t Putting into gives →→ → → Outgoing Incoming for r > r S Incoming Outgoing for r < r S

→ For an incoming particle in the region r > r S  r → r S as t →  To a Minkowskian observer, the particle takes forever to reach r = r S, the singularity in coord system ( ct, r, θ,φ) To an observer travelling with the particle, the time τ it takes to fall from r 0 to r S is finite:

Null Geodesics The null geodesics (light paths) are given by ds= 0. For radial ( d θ = d φ = 0 ) null geodesics, → Note: are not defined individually on the null geodesics. & Outgoing Incoming

For r < r S, r becomes time-like & t space-like. t = const is a time-like line → forward light cones must point towards the origin. → Increasing time:  dr > 0, increasing radial distance: c dt > 0. To a Minkowskian observer, incoming light takes forever to reach r = r S.

Eddington-Finkelstein Coordinates Eddington- Finkelstein coordinates: null radial geodesics are straight lines. Incoming null geodesics: Set for r > r S (straight line) → Line element: regular for all r  0 Region I:r S < r <  Region II:0 < r < r S Assuming the line element to be valid for all r is called an analytic extension of from region I into region II as t → 

Advanced time parameter : Line element: Incoming null geodesics: becomes For outgoing particles with time-reversed coordinate Retarded time parameter : Line element: Analytical extension from region I into region II* (0 < r < r S ). Outgoing null geodesics: becomes

Forward light cones in region II point to the right because we are dealing with a time-reversed solution.

Black Holes Eddington-Finkelstein coordinates are not time-symmetric Incoming (outgoing) particles, time is measured byor v ( t* or w). I: future light cones point upward II: future light cones point left → no light can go from II to I  II = black hole Spherical surface at r S = event horizon To a Minkowskian observer, light emitted by ingoing particles are redshifted.

Possible way to form black holes: collapse of stars or cluster of stars. All information are lost except for M, Q, and L. Rotating black hole ~ Kerr solution. Black holes can be detected by the high energy radiation ( X  and  rays) emitted by matter drawn to it from nearby stars or nabulae. E.g., gigantic black hole at the center of our galaxy. Estimated minimum mass density  of a black hole of total mass M: For M < 10 M , ρ is too large so the star collapses only into a neutron star.

Extension Regions II I I Direction of extension Direction of particle motion is denoted by For extension I → II, no light ray can stay in II (white hole). Extension II (II) → I (I  ) shows that I and I  are identical. However, I(I  ) is distinct from region I → no overlap or extension between them. The collection of these 4 regions is called the maximal extension of the Schwarzschild solution [see Chapter 17, D’Inverno].