4. General Relativity and Gravitation

1. 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

2. 4.1. The Principle of Equivalence Under a coordinate transformation xμ → xμ , In general, Every real symmetric matrix can be diagonalized by an orthogonal transformation: gj are the (real) eigenvalues of g. Consider Λ = O D, where D = diag(D1 , …, Dd ). Λ1 exists & real → Dj  0 & real  j. → Choosing → canonical form of the metric tensor Spacetime is locally flat (Minkowskian): at 1 point.

3. 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 gμν and gμν, σ.

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

5. Geodesic equation

6. 4.2.3. Gravity Let ( hμν small ) Non-relativistic motion: → →

7. → → The only non-vanishing components of g μν,σ are g00 , j = h00 , j . → Newtonian gravity Setting gives

8. 4.3. The Field Equations of General Relativity Electrodynamics: Gravitational field : LParticles + LInteraction Task is to find LFields → invariant infinitesimal spacetime volume dV For a Minkowski spacetime = Jacobian →  d 4 xis a scalar density of weight 1 Only choice is ( g = det | g | is a scalar density of weight +2 ) Check: In a Minkowski spacetime

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

10. Lagrangian Densities = Ricci curvature κ = coupling constant Λ = 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.

11. For N = 1:

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

13. Another Form of the Field Equation gνμ field eqs → Field eqs:

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

15. ρ is stationary → To lowest order in h, →

16. 4.4. The Gravitational Field of a Spherical Body • The Schwarzschild Solution (1916 ): • 1. ρ is spherically symmetric; so is g. • ρ is bounded so that g ~ η at large distances. • 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

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

18. 4.4.2. Time Near a Massive Body Coordinate t = time measured by a stationary Minkowskian (r→) observer. To this observer, two events at (ct1 , x1) and (ct2 , x2) are simultaneous if t1 = t2. For another stationary observer at finite r > rS , time duration experienced = proper time interval d with dx = 0 → two events simultaneous to one stationary observer (Δτ1 = 0 ) are simultaneous to all stationary observers (Δt = Δτ2 = 0 ) . The finite duration Δτ of the same events (fixed dt 0) differs for stationary observers at different r.

19. If something happens at spatial point (r1 ,θ1 ,φ1) for duration another stationary observer at (robs,θobs ,φobs) 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.

20. For measurements done on the sun and star light, Earth’s gravity can be ignored. gravitational red shift For starlights observed on earth, → 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.

21. 4.4.3. 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.

22. Consider circular path described by the equations r = a and θ = π/2. Its length, or circumference, is ( same as E3 ) Its radius is not defined. Closest distance between 2 concentric circles r = a1 and r = a2 is not A “circle” of a well defined radius a about a point would appear lopsided when plotted using the spherical coordinates. Since for

23. The lowest order of corrections valid for are

24. 4.4.4. 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,

25. 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. In practice, the r eq is usually replaced by • Notable phenomena: • Bending of light by the sun. • Precession of Mercury. • See Chap 15, D’Inverno.

26. 4.5. Black and White Holes R = radius of the mass distribution. If R > rS then singularity at r = rS is fictitious. Problem of interest: R < rS and R < r < rS

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

28. Outgoing Incoming For we have → Singularity at r = rS not felt →

29. Radial Motion: Solution for t Putting into gives → → → Outgoing Incoming Incoming Outgoing → for r > rS for r < rS

30. → For an incoming particle in the region r > rS To a Minkowskian observer, the particle takes forever to reach r = rS , the singularity in coord system ( ct, r, θ,φ)  r → rS as t →  To an observer travelling with the particle, the time τ it takes to fall from r0 to rS is finite:

31. 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

32. For r < rS , 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 = rS .

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

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

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

36. Black Holes Eddington-Finkelstein coordinates are not time-symmetric Incoming (outgoing) particles, time is measured by or 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 rS = event horizon To a Minkowskian observer , light emitted by ingoing particles are redshifted.

37. 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.

38. Direction of extension Direction of particle motion is denoted by II I I II Extension Regions 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].