Theory of General Relativity

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2 Theory of General Relativity Expansion on the concept of Special relativity Special: Inertial perspectives are Equivalent (unaccelerated) General: All perspectives are equivalent

3 Let s go back to Newton F = G M 1 M 2 / R 2 F = ma When we solved this we cancelled the masses why? Why is the mass for gravity the same as the mass for acceleration?

4 Equivalence Inertial mass = gravitational mass New thought: when in free fall (an accelerating frame) you feel weightless, but this is an inertial frame or is it? Gravity = acceleration

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6 General Relativity An accelerated environment is equivalent to a gravitational environment Examples: An elevator going up space A spaceship speeding up in

7 So what? This all seems obvious but work out the implications new Gedanken experiment Shine a light across the elevator

8 But acceleration = gravity So a stationary elevator in a grav field must produce the same result GRAVITY BENDS LIGHT?!!

9 We can do this another way Light falling down a gravity field gets blue shifted (gains energy) while light going against gravity gets redshifted (loses energy)

10 Gravity Affects Light It also slows down clocks! Gravity bends light Light gains and loses energy as it moves through gravitational fields Clocks in strong gravity move slowly

11 Light curves? How can light curve from gravity? it has no mass How can it not travel in a straight line? New way of thinking gravity is not a force as Newton would describe it it is a change in geometry

12 Geometry and Mass This is a revolution the geometry of space-time is changed by the presence of mass light moves along a geodesic in this altered geometry the shortest distance between two points Mass and space-time are linked This is a field theory not a force theory

13 Einstein s Theory G = 8 G/c 4 T English translation: Local Geometry (curvature) of spacetime is defined by the distribution of mass and energy (T)

14 Easy right? This equation is so hard to solve, that a closed form solution has only been done for a handful of extremely simple cases Point mass, spherical mass, black hole uniformly distributed material in a homogeneous and isotropic condition

15 Is it true? GR makes several testable predictions bending of starlight orbit of mercury gravitational redshift binary pulsar decay gravitational radiation (gravity does not act at a distance )

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17 1919 verification of GR showing bending of starlight

18 Cosmological Principle The universe is homogeneous and isotropic on large scales True?

19 Cosmology, finally! Can solve G=8 G/c 4 T under the assumption of the cosmological principle: theoretical cosmology Observations to test the predictions of the above: observational cosmology

20 BRAIN BREAK

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22 Cassini Mission

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24 Titan True color Visible light

25 Titan Radar Map

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28 Temperature of Titan Phase Diagram for Methane (CH 4 ) Pressure on Titan

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34 Titan View from the ground

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37 Modeling the Universe

38 The real way Apply the cosmological principle to the universe: matter and energy are uniformly distributed Solve Einstein s Equations of General Relativity under these assumptions We will not be doing this

39 Our Way Apply the cosmological principle Solves Newton s equations for such a circumstance Show the real relativistic answer (which is a little bit different) and just accept it

40 Concepts needed Scale Factor: R(t) Co-moving coordinates (expand along with the universe) distances between objects remain constant (D) Distance between two objects (freely expanding) Then: R(then) D Now: R(now) D

41 Concepts (continued) Scale factor changes with time (universe is expanding/contracting) Rate of change of R = R This can be thought of as speed (there, you just did calculus!)

42 More concepts The rate of change of just the R is acceleration: R

43 Newtonian Escape Velocity Consider a small particle (mass m) at the surface of a sphere with mass M (M >> m) If the particle has kinetic of (1/2 m v 2 ) equal to it gravitational potential energy (GMm/R) then it could just escape from the gravitational field (and have zero kinetic energy when it got to R =

44 Solving for v, we find: v 2 = ( 2GM/R) = escape velocity (Newtonian) Now consider the universe, expanding at a rate R

45 Kinetic energy per unit mass (energy density = (½ mv 2 m = ½ v 2 ) define this quantity as So (twice) the energy density of the universe = 2 = R 2 = 2GM/R + 2 [ - GM/R = ] What does this mean? Expansion of the universe is related to its gravitational potential energy at it s kinetic energy at infinite size

46 Consider the possibilities R 2 = 2GM/R + 2 As R gets larger, the 2GM/R term gets smaller and smaller, R if < 0, then at some point, = 0, and the expansion stops (it never gets to R = ) If = 0, then R gets smaller and smaller, and the universe expands for ever, but just barely If > 0 it expands forever

47 Now lets get rid of M The mass of the universe is just: M = 4/3 R 3, where is the mass density of the universe R 2 = 8/3 G R2 + 2

48 Now add Einstein If we had done the full relativistic solution: R 2 = 8/3 G R2 kc 2 k = curvature +1, -1, 0 since R just a scale factor, when can choose our coordinates to make k = one of these values Friedmann Equation

49 Other changes in relativistic solution is the mass-energy density (not just mass) If = just mass this is a standard model

50 Standard Model Basics k = +1, positive curvature, closed universe (re-collapse) k = 0, flat curvature, open universe (expand forever) k = -1, negative curvature, open universe

51 The Hubble constant Remember: v = H 0 d (H 0 = v/d) v = change in distance divided by change in time = R(then) D R(now) D / (t(then) t(now)) v = ΔR D/ Δt

52 continuing v = ΔR D/ Δt H 0 = v/d (d = R D) H 0 = [ΔR D/ Δt] / R D = [ΔR/ Δt] / R = R /R = H 0 If we use R(today) and R (today)

53 The Hubble constant The Hubble constant varies with time! It is the constant of proportionality between speed and distance The ratio of R /R = H at any time, = H 0 at the present time

54 Why I want to know H Rearrange the equation: R 2 = 8/3 G R2 kc 2 H 2 = 8/3 G - kc 2 /R 2 If k=0, then = crit, and crit = 3H 2 /(8 G) ~ 10 H atoms/cubic meter = / crit

55 Friedmann equation R 2 = 8/3 G R 2 kc 2 The rate of expansion (contraction) of the universe is a function of its gravity and its curvature k= curvature term, = 0, +1, or -1 (4/3 R 2 = M/R) where M is mass of the universe If is the mass density

56 = / crit where crit is the density necessary to just close the universe When = 1 (by definition) k = 0 = 1 + kc 2 / (H 2 R 2 )

57 and curvature = 1 + kc 2 / (H 2 R 2 ) Clearly: if k = 0, = 1 for all time if k = +1, > 1 for all time, but changes if k = -1,,< 1 for all time, but changes

58 New information R = -4/3 G R + R/3 acceleration = -GM/R 2 + mysterious new term Is the cosmological constant, appears from Einstein s equations Note that looks like in its application- a mass/energy density

59 Friedmann equation + R 2 = 8/3 G R 2 kc 2 + R 2 /3

60 Two basic equations 2 R = 8/3 G R2 kc 2+ R 2 /3 Kinetic energy = gravitational energy curvature of space + energy of vacuum Or Kinetic energy gravitational energy vacuum energy = - curvature R = -4/3 G R + R/3 Acceleration = gravitational force (attractive) + vacuum force

61 Two basic equations Rሶ R 2 = 8/3 G kc2 /R 2+ /3 Kinetic energy = gravitational energy curvature of space + energy of vacuum Or Kinetic energy gravitational energy vacuum energy = - curvature Rሷ R = -4/3 G + /3 Acceleration = gravitational force (attractive) + vacuum force

62 Rules for Standard Cosmology Standard cosmology means mass dominates over energy, and the cosmological constant = 0 Three cases: k = -1, neg curve, < 1, expand forever, infinite k = 0, flat curve, = 1, expand forever, infinite k =+1, pos curve, > 1, collapses, finite

63 What is the curvature of space positive, negative or flat? How fast is the universe expanding? Is the rate of expansion changing, and if so, by how much? Inferred by solving the equations, I have no idea what this actually is. How much mass is there in the universe? Observational cosmology is the attempt to measure these quantities, now, and over cosmic time.

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65 The Einstein Static universe R 2 = 8/3 G R 2 kc 2 + R 2 /3 This is what Einstein got, and wasn t very happy about it R He wanted = 0, and = 0 (static universe) This meant = 4 G and k = 4 G R 2 /c 2 R

66 The Einstein Static universe No expansion, positive curvature, finite size infinite in time requires a specific value of the cosmological constant But it is unstable look what happens if you change R

67 What is? Looks like an energy density but it is constant doesn t go down as does A positive value for implies a force that wants to push the universe outward an negative value wants to collapse the universe The R/3 term increases as R, gets more important as the universe expands less important in a small universe

68 Any physics to all of this? can be interpreted as an energy density associated with the vacuum: As space grows, there is more vacuum, therefore more total energy associated with vacuum This is pretty hard to believe certainly seems to violate conservation of energy

69 New possibilities for a universe Unlike the standard model, the addition of the cosmological constant alters the relationship between k,, and the fate of the universe: De Sitter universe: no mass, positive cosmological constant: exponential, scale free-expansion, no big bang

70 Steady State model - De Sitter Universe + matter Requires the constant creation of mass to keep the product of R a constant No deceleration: H is unchanging throughout time no beginning or end Perfect cosmological principle Disproved if the universe is accelerating or decelerating

71 A negative Causes all models to collapse, regardless of other factors: as R increases towards infinity, always wins

72 LeMaitre Model Choose close to, but just off from crit = 4 G This causes the universe to hover at a near constant size for a long time, before expanding again Popular when the value of H 0 was inconsistent with the age of the earth

73 So what happens in the far future? Second Law of thermodynamics: Entropy always increases Entropy = disorder Defines positive time Heat Death Big Crunch